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2310.07875 | TabLib: A Dataset of 627M Tables with Context | It is well-established that large, diverse datasets play a pivotal role in
the performance of modern AI systems for text and image modalities. However,
there are no datasets for tabular data of comparable size and diversity to
those available for text and images. Thus we present "TabLib'', a compilation
of 627 million tables totaling 69 TiB, along with 867B tokens of context.
TabLib was extracted from numerous file formats, including CSV, HTML, SQLite,
PDF, Excel, and others, sourced from GitHub and Common Crawl. The size and
diversity of TabLib offer considerable promise in the table modality,
reminiscent of the original promise of foundational datasets for text and
images, such as The Pile and LAION. | Gus Eggert, Kevin Huo, Mike Biven, Justin Waugh | 2023-10-11T20:34:42 | http://arxiv.org/abs/2310.07875v1 | # TabLib: A Dataset of 627M Tables with Context
###### Abstract
It is well-established that large, diverse datasets play a pivotal role in the performance of modern AI systems for text and image modalities. However, there are no datasets for tabular data of comparable size and diversity to those available for text and images. Thus we present "TabLib", a compilation of 627 million tables totaling 69 TiB, along with 867B tokens of context. TabLib was extracted from numerous file formats, including CSV, HTML, SQLite, PDF, Excel, and others, sourced from GitHub and Common Crawl. The size and diversity of TabLib offer considerable promise in the table modality, reminiscent of the original promise of foundational datasets for text and images, such as The Pile and LAION.
## 1 Introduction
The importance of data in model training has continued to grow (Hoffmann et al., 2022). Training data volume is now considered to be roughly as important to model performance as model size (Zha et al., 2023a). This implies that large datasets are promising assets for improving the performance of AI models.
For example, in 2021 OpenAI released both CLIP and DALL-E (Radford et al., 2021; Ramesh et al., 2021), which were considered state-of-the-art for image tasks. A large part of their success was due to their training data scale of 400M image-text pairs, whereas previously the largest open dataset for image-text pairs was around 10M (Schuhmann et al., 2021). Even larger training datasets such as LAION-5B (Schuhmann et al., 2022) have fueled subsequent image models like Stable Diffusion (Rombach et al., 2022).
Given the volume and significance of information captured in tabular data, research on applying AI models to tabular data is an area of active research (Badaro et al., 2023) (Jin et al., 2022) (Dong et al., 2022). Despite this, there are not many large-scale, diverse, and accessible datasets for tabular data. We are aware of only one large scale crawl that exceeds 10M tables (WebTables (Lehmberg et al., 2016)), and only a few additional datasets have more than one million tables (WikiTables (Bhagavatula et al., 2015), GitTables (Hulsebos et al., 2023), VizNet (Hu et al., 2019)). Furthermore, the largest of these datasets (WebTables) is composed solely of HTML tables, which differ meaningfully from other common table types such as database tables, suggesting that WebTables may be insufficient for training models for diverse tasks. We believe that a larger and more diverse dataset will accelerate the advancement of tabular AI systems.
Thus, we present "TabLib", whose notable characteristics include:
* **Scale**: Over 627 million individual tables totaling 69 TiB
* **Table metadata**: 867B tokens of contextual information, such as filenames, URLs, text before and after the table in the source document, and OpenGraph metadata.
* **Diversity**: Across language, category, size, source (Common Crawl2 and GitHub3), and format (CSV, HTML, PDF, Excel, SQLite, etc.)
* **Provenance**: Table source and transformation data to enable attribution and validation
These characteristics suggest TabLib could be a useful research asset for many fields, which we discuss later in 1.2Impact. We hope that TabLib will help advance tabular data understanding and catalyze the development of AI models focused on this modality, which we refer to as _large data models_.
### Related Work
Numerous open datasets exist for the purpose of training machine learning models to understand and interpret tabular data. Some of the most significant of these datasets are detailed in Table 2 in [Badaro et al., 2023]. While high quality, existing datasets such as Spider, WikiDB, and VizNet [Vogel and Binnig, 2023, Yu et al., 2019, Hu et al., 2019] lack the size and/or diversity necessary to pre-train large data models with broad applicability.
Two data sets have noteworthy volume: WebTables [Cafarella et al., 2008] and GitTables [Hulsebos et al., 2023].
The latest WebTables corpus contains 233 million tables extracted from HTML pages from Common Crawl 4. WebTables contains a large volume of tables, but has limited diversity due to only including HTML tables from web pages.
Footnote 4: [https://webdatacommons.org/webtables/#results-2015](https://webdatacommons.org/webtables/#results-2015)
GitTables is a continuously updated library of tables extracted from "comma-separated value" files (CSVs) hosted on GitHub, containing 1 million tables. These tables tend to be structurally different from the HTML-centric WebTables [Hulsebos et al., 2023], thus an important table corpus. Compared to WebTables, GitTables is relatively small, and still only supports a single file type (CSV).
### Impact
Applying AI to tabular data is an active field of study, and there are many applications and research areas that could significantly benefit from a large, diverse dataset such as TabLib. These include:
* **Dataset Search:** Identifying corresponding tables using a set of keywords that describe the required information [Benjelloun et al., 2020, Chapman et al., 2020, Zhang and Balog, 2018]
* **Semantic Understanding:** Using data tables to create or augment general-purpose knowledge bases, and vice versa. [Dong et al., 2014, Liu et al., 2023, Jimenez-Ruiz et al., 2020, Efthymiou et al., 2017, Bonfitto, 2021, Hulsebos et al., 2019]
* **Data Integration:** Identifying tables that can be joined or unioned within a large corpus of tables. Includes schema mapping. [Dong et al., 2021, Zhang and Balog, 2019, Zhu et al., 2019, Nargesian et al., 2018, Santos et al., 2021, Srinivas et al., 2023, Zhu et al., 2017, Cong et al., 2023a,b]
* **Knowledge Extraction:** Interacting with data through natural language, via tasks like question answering and semantic parsing. [Zha et al., 2023b, Cheng et al., 2023, Zhang et al., 2023, Li et al., 2023, Pourreza and Rafiei, 2023, Talmor et al., 2021, Lin et al., 2020]
* **Table Metadata Prediction:** Predicting metadata such as column types, inclusion of personally identifiable information (PII), and data cleanliness. [Zhang, 2017, Parikh et al., 2020, Korini and Bizer, 2023]
* **Table Representation Learning:** Representing tables as a distinct modality of information for training machine learning models [Yin et al., 2020, Deng et al., 2020, Tang et al., 2021, Herzig et al., 2020, Iida et al., 2021]
## 2 Methods
### System Architecture
We built a processing pipeline that consumes raw data from data sources, extracts tables into Pandas dataframes [McKinney, 2010], serializes those dataframes into Arrow tables 5, stores each in blob storage and metadata in a SQL database, and then aggregates into Parquet files 6. To orchestrate this process, we used the Ray distributed processing framework [Moritz et al., 2018].
Footnote 5: [https://arrow.apache.org/](https://arrow.apache.org/)
Because parsing tabular data is relatively complex compared to text due to its additional structure and data types (see Formats, Parsing, and Metadata), we encountered some failure scenarios which were difficult to recover gracefully from,
such as out-of-memory errors and catastrophic regular expression backtracking. As such, we isolated each "source" as its own task instead of batching them together.
This granular task scheduling necessitated scheduling hundreds of millions of tasks. We found Ray's scheduler problematic for this, so we scheduled these tasks using a PostgreSQL database, and used Ray to maintain long-running tasks which pulled work from the database, extracted the tables and metadata, stored the tables in blob storage, and wrote the metadata back to the DB. A separate Ray actor tracked the progress of these tasks, handled timeouts and retries, occasionally aggregated batches of metadata into Parquet files, and wrote those into blob storage.
### Sources
For data about the number of tables extracted for each data source and file types, see Summary Statistics. For samples of extracted tables and metadata, see Sample Data in the appendix.
#### 2.2.1 GitHub
To reduce the amount of noise, we skipped all files under node_modules directories, and all JSON and YAML files which are generally configuration files in GitHub. Since files in GitHub often contain extensions like.csv that provide hints for the content type, we used Python's minetypes.guess_type() function to see if the file was a supported type; if not then we inspected the file's bytes using libmagic 7, and if it was still unsupported then the file was skipped. Files larger than 1 GB were also skipped.
Footnote 7: [https://www.darwinsys.com/file/](https://www.darwinsys.com/file/)
Tables extracted from GitHub repos result in the following fields in each table's **context_metadata**:
* **github_repo**: the repo name
* **github_ref**: the ref used, such as "refs/heads/master"
* **github_hash**: the shortened Git commit hash
* **github_repo_path**: the path of the file in the repo where the table was found
#### 2.2.2 Common Crawl
We used the latest crawl at the time, which was CC-MAIN-2023-23. Common Crawl results are serialized using the WARC format, which includes "request" and "response" records. We only considered response records. We discarded
Figure 1: Architecture of table extraction pipeline.
"truncated" responses which had response lengths that exceed Common Crawl's limit. If a WARC-Identified-Payload-Type record header was included in the record, then we used its minetype as a hint for detecting the content type, otherwise we used the Content-Type header in the HTTP response, and followed a similar approach as GitHub (use the minetype if possible, otherwise use libmagic). About 20% of WARC files were dropped due to issues parsing certain HTML elements with Pandas.
Tables extracted from Common Crawl WARC records result in the following fields in each table's **context_metadata**:
* **warc_path**: the path of the WARC file in Common Crawl
* **warc_record_id**: the record ID in the WARC file as specified by WARC-Record-ID
* **warc_target_uri**: the target URI of the HTTP request as specified by WARC-Target-URI
* **warc_date**: the date of the request as specified by WARC-Date
### Storage Data Model
In order to efficiently store and manage the large volume of tabular data in TabLib, we implemented a data storage model that consists of two main components: blob storage and manifests. Using this storage model, we can efficiently manage and retrieve the tables based on their metadata and content hash. This allows for easy deduplication, querying, and analysis of the dataset. A final post-processing step was performed which added the serialized tables as a column in the manifests, which is ultimately the TabLib schema, but this paper will focus on the intermediate representation because it is what the analyses are based on.
#### 2.3.1 Manifest Schema
The manifests contain metadata about the tables and are stored as partitioned Parquet files. The schema for the manifests includes the following fields:
* **bucket**: the blob storage bucket of the table
* **key**: the blob storage key of the table
* **ref**: a human-readable string describing how the table was extracted
* **ref_id**: a base64-encoded sha256 hash of the ref
* **exec_id**: a UUIDv7 generated at the time of table extraction
* **run_metadata**: serialized JSON object containing metadata about the run, including start and end times
* **context_metadata**: serialized JSON object containing metadata about the table, including:
* **extractor**: the extractor used for this table (e.g. "html", "csv", "pdf", etc.)
* **mine_type**: the detected mine type of the bytes that the table was extracted from, e.g. "text/html" for an HTML page, "text/csv" for a CSV file, etc.
* **<source-specific>**: additional fields depending on the source, see Sources
* **<datatype-specific>**: additional fields depending on the data type, see Formats, Parsing, and Metadata
#### 2.3.2 Blob Storage Key Schema
Each table in its intermediate form, before the final post-processing step, is stored as a separate blob object. The blob's content is computed by serializing the Arrow table to bytes, and compressing these bytes with gzip. Each table is assigned a unique key based on the arrow table bytes content hash. The blob storage follows the following key schema:
* /manifests/{batch}/manifest.parquet
* /tables/{batch}/{base64_sha256_of_arrow_table}
### Formats, Parsing, and Metadata
Parsing tabular data presents unique challenges that are not present when parsing text. Tasks such as inferring column data types and row delimiters are complex and error-prone. Because of this, we reused existing open-source parsers as much as possible, such as those in Pandas and pdfplumber. For most file types, we drop parsed tables with only one column, one row, all empty column names, or only numeric column names.
Below we detail each data type and a summary of the parsing logic:
## 3 Analysis and Results
### Keys and Metadata
We begin by examining the cardinalities of different keys: exec_id, ref_id, key, and content_hash, as shown in Table 2. Definitions of these values are in Manifest Schema and Blob Storage Key Schema.
The exec_id is unique across the dataset, generated upon line-item creation in the manifest. Any duplication indicates a serialization error.
The ref_id represents a unique source for a table. This should be unique across TabLib, but the current version of TabLib has some repetitions due to a bug in deduping items in the work queue. Future versions will allow tracking external data changes over time via ref_id.
The number of unique key values is substantial but not as large as unique ref_id values. This discrepancy arises because the same content table can appear multiple times within a batch (e.g., a CSV file stored multiple times in a GitHub repository with different filenames). However, key is not a global content-collision key as it includes the batch.
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Data Type** & **Method** & **Context Metadata** \\ \hline \multirow{6}{*}{HTML} & Parse with BeautifulSoup using lxml and html5lib parsers. & \multirow{6}{*}{html\_title} \\ & Then extract all \textless{table\textgreater{} elements with pandas.read\_html().} & \\ & Extract HTML metadata with metadata\_parser library. Extract “before” and “after” context with BeautifulSoup. Drop & \multirow{6}{*}{html\_metadata} \\ & \textless{table\textgreater{} elements with colspan values \textgreater{} 1000 to avoid causing} & \\ & out-of-memory errors. & & \\ \hline \multirow{6}{*}{PDF} & Use pdfplumber to extract tables. Only supports text-based tables and not image-based tables, and multi-page tables appear as a separate table per page. & \multirow{6}{*}{pdf\_broad} \\ & Use a custom SQLite VFS implementation to load the in-memory bytes with apsw, list the tables, and then parse each table with pandas.read\_sql(). & \multirow{6}{*}{\(\bullet\) pdf\_page} \\ & There are many Excel formats, minetypes, and extensions, so ignore specifics and always try parsing as XLSX using openpyxl, and then fall back to XLS using xlrd, using pandas.read\_excel(). Parse each sheet as its own table. & \\ \hline Parquet & pandas.read\_parquet() & n/a \\ \hline JSON & pandas.read\_json(orient="records") & n/a \\ \hline YAML & Use yaml.safe\_load() to convert to JSON, then pandas.read\_json(orient="records"). & n/a \\ \hline CSV & pandas.read\_csv(engine="python") & n/a \\ \hline TSV & pandas.read\_csv(engine="python") & n/a \\ \hline \hline \end{tabular}
\end{table}
Table 1: **Summary of supported data types, and how each was parsed.**
\begin{table}
\begin{tabular}{l l l} \hline \hline Key Type & Number of Tables \\ \hline exec\_id & 660840556 \\
**ref\_id** & 627208299 \\ key & 459022959 \\ content\_hash & 201376283 \\ \hline \hline \end{tabular}
\end{table}
Table 2: **Unique counts of key-like values, ordered by decreasing uniqueness. Each may be considered some definition of “table”. We will use ref_id as the definition of “table” for our analyses.**
Table content_hashes are 30.5% the size of exec_id values, indicating that most tables are not globally unique by content. The breakdown of these repeated tables is discussed further in Data Duplication.
For clarity, the term _table_ henceforth refers to a specific ref_id instance.
### Summary Statistics
We calculate the total number of tables, total uncompressed table bytes, and total columns, broken out by data source and file type. See Table 3 for a summary of the dataset statistics.
We also consider token counts from metadata fields. We used tiktoken 8 to tokenize the ref, space-separated column names, and context_metadata. Because context_metadata has nested JSON, we considered tokenizing the string of recursively-concatenated string values, instead of the serialized JSON itself (which includes JSON syntax such as commas, curly braces, and quotation marks). We compared this on a sample and found -10% less token counts in the JSON vs non-JSON versions. We decided that was tolerable, so we treated context_metadata as serialized JSON.
Footnote 8: [https://github.com/openai/tiktoken](https://github.com/openai/tiktoken)
### Power-Law Like Distributions
In examining TabLib, we found several metrics--including row-count, column-count, and domain-size (column-level unique-count) displaying distributions resembling power-law or Zipfian distributions, common in natural and social phenomena (Newman, 2005). Such distributions in our data suggest a few tables or columns hold most data, while the majority hold little. This pattern can significantly impact the design and evaluation of machine learning algorithms.
Power-law distributions are characterized by an exponent or Zipf's coefficient (\(\alpha\) in \(P(x)\propto x^{-\alpha}\)), guiding the distribution's decay rate. Our comparison revealed a higher exponent in column-count than in row-count, suggesting a faster decay and affirming the typical practice of constructing tables with rows for entities and columns for entity properties (dimension tables).
Using the powerlab library (Alstott et al., 2014), we observed exponents below 2 (e.g., \(\alpha_{rc}\approx 1.5\) for row count), which is crucial since distributions with exponents under 2 lack well-defined mean or variance--a hallmark of true
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline Source & File Type & Tables & Bytes & Columns & \multicolumn{3}{c}{Metadata Tokers} \\ \cline{6-8} & & & & Ref & Column Names & Context Metadata \\ \hline \multirow{8}{*}{Common Craw1} & CSV & 90,667 & 3.10 GB & 2,265,499 & 15,630,941 & 12,488,110 & 17,984,442 \\ & Excel & 143,012 & 3.26 GB & 1,836,491 & 210,638,376 & 10,579,484 & 60,755,137 \\ & HTML & 219,397,657 & 7025 GB & 1,076,171,440 & 2,602,724,931 & 3,686,722,801 & 493,085,023,697 \\ & JSON & 70,737 & 1.75 GB & 537,934 & 2,737,873 & 4,826,400 & 3,393,257 \\ & Payment & 1 & 4.63 MB & 13 & 123 & 30 & 161 \\ & PDF & 11,442,231 & 30.84 GB & 46,514,046 & 1,876,490,940 & 432,038,521 & 18,927,559,013 \\ & SQLite & 1,408 & 83.70 MB & 8.839 & 186,687 & 17,973 & 329,783 \\ & TSV & 4,374 & 419,829 MB & 75,989 & 569,475 & 210,506 & 696,084 \\ & YAML & 3,185 & 67.58 MB & 22,336 & 31,849 & 4,601 & 375,145 \\ \hline \multirow{8}{*}{GitHub} & Total & 231,153,272 & 742.10 GB & 1,127,412,487 & 3,518,991,195 & 4,146,888,426 & 512,095,779,088 \\ & CSV & 122,091,982 & 59.86 TB & 5,481,784,256 & 7,390,202,751 & 36,457,207,467 & 13,912,319,966 \\ & Excel & 15,787,659 & 3.02 TB & 243,597,019 & 95,834,206 & 2,016,629,675 & 5,77,869,104 \\ & HTML & 199,098,080 & 633.01 GB & 599,028,450 & 1,817,543,971 & 2,515,057,875 & 175,195,161,6693 \\ & PDF & 40,022,516 & 79.00 GB & 144,096,394 & 3,442,432,322 & 584,802,039 & 51,385,307,211 \\ & SQLite & 14,919,675 & 3.52 TB & 94,541,122 & 78,970,698 & 165,104,049 & 7,405,534,796 \\ & TSV & 4,714,115 & 15.4 TB & 94,845,931 & 26,363,169,169 & 739,989,036 & 94,089,504 \\ \cline{2-8} & Total & 396,055,027 & 68.62 TB & 7,007,816,674 & 24,489,631,027 & 2,478,790,602 & 252,089,037,310 \\ \hline \multirow{8}{*}{Total} & CSV & 122,182,649 & 59.86 TB & 5,484,049,755 & 7,405,833,692 & 36,469,695,577 & 13,930,304,408 \\ & Excel & 15,930,671 & 3.02 TB & 245,433,510 & 972,898,082 & 2,027,209,159 & 5,836,624,241 \\ & HTML & 418,567,370 & 1.0 TB & 203,199,890 & 4,149,823,402 & 6,201,780,696 & 66,020,040,390 \\ & JSON & 70,737 & 1.75 GB & 537,934 & 2,737,873 & 4,826,400 & 3,933,257 \\ & Parquet & 1 & 4.63 MB & 13 & 13 & 23 & 30 & 161 \\ & PDF & 51,64,747 & 109.48 GB & 190,520,952 & 5,220,734,172 & 1,286,840,560 & 70,312,866,224 \\ & SQLite & 14,921,083 & 3.52 TB & 84,562,951 & 729,157,385 & 165,122,463 & 7,405,864,579 \\ & TSV & 4,178,489 & 1.54 TB & 94,921,920 & 257,405,644 & 740,199,542 & 494,785,624 \\ & YAML & 3,185 & 67.58 MB & 2,236 & 31,849 & 4,601 & 37,514 \\ \hline \multirow{8}{*}{Total} & Total & 627,208,299 & 69.35 TB & 8,135,229,161 & 56,008,622,222 & 46,895,679,028 & 764,184,816,398 \\ \cline{2-8} & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: **Summary statistics table, showing counts of tables, bytes, columns, and tokens across GitHub and Common Crawl and the encountered file types.**
power-law distributions. Hence, the mean values of these metrics in our dataset might not accurately represent the data due to skewness from a few large tables.
Considering the distributions' long-tail nature, training models on raw data might present challenges (Johnson and Khoshgoftaar, 2019). Therefore, we propose training on aggregated tabular data instead. This approach, involving the compression of columns into concise and finite representations, could improve the robustness and generalizability of the resulting models, effectively addressing the issues posed by long-tail distributions.
An important caveat is that there may be a selection bias affecting this analysis, due to factors such as our exclusion of tables larger than 1 GB, Common Crawl's truncation of large responses, parsing bugs and limitations, etc. We leave a more detailed study of these factors for future work.
### Data Duplication
Data duplication is a common occurrence, and is important for downstream tasks. Some works have shown that deduplication of training data can enhance language model performance (Lee et al., 2022), necessitating an investigation into TabLib's duplicated tables.
As seen in Table 2 prior, there are many duplicates of the content hash values within the key field of the dataset. This is to be expected - many tables are duplicated across the web since they are used in different contexts by different groups of people. Within GitHub for example, there are many repositories that contain the same data, but with different names, or different versions of the same data. Whether it is an HTML table used in a frontend component, or a CSV file used in a popular data science project, there are many reasons for datasets with different contexts but the same content. Additionally, we believe that some part of this is due the practice of forking repositories on Github. See Appendix 7.1.5 for examples. To look at the duplication in the dataset, we use the content_hash of the table.
In Figure 3, we see that the behavior appears Zipf-ian, with roughly similar parameters in both sources. A notable divergence occurs around the rank 50-100 area, where GitHub has more "uneven bumps". We hypothesize that this is due to GitHub having mechanisms to copy data directly built into the platform, changing the nature of what data are commonly found.
We also consider duplicate data with different contexts. For a given set of tables with \(N\) duplicate content_hashes, there may be anywhere from \(0\) to \(N\) distinct context_metadata values for those tables. Using before and after fields, we compare total vs. distinct values among the duplicate content hash tables using a 2D histogram, color coded by density, shown in Figure 4. As illustrated by the color, most of the values can be seen in the bottom left corner, which are the smaller tables. The values along the line \(y=x\) have a high degree of uniqueness in context_metadata among the same duplicated content, whereas the values along the line \(y=0\) have higher degrees of duplication. There is a wide variety of data spanning those values, with high normalized counts along both \(y=x\) and \(y=0\), suggesting a diverse distribution of distinct context_metadata values among tables with duplicate content hashes. We leave further investigation of the implication of filtering values along such a distribution for downstream tasks to further works.
Figure 2: **Power law behavior of table statistics**. The (a) row-count, (b) column-count, and (c) domain-size (column-level unique-count) exhibit power-law-esque distributions, with a tail end following less close to a theoretical fit. The solid line shows the empirical distribution and the dotted line shows the theoretical fit given the relevant alpha value.
### Data Categories
There are an abundance of categories of tables in the real world, and we consider it critical to represent them in a single dataset. While TabLib includes table metadata, there are no explicit ground-truth labels for table categories such "Sports and Recreation" or "Financial and Economic". So we used the gpt-3.5-turbo model9 to categorize tables using the ref and the dataframe "head" (the column names and first few rows of the table), using 25 hand-picked categories. We randomly sampled 28,630 tables from TabLib and prompted gpt-3.5-turbo to categorize them, using enums with the OpenAI function call interface10. We discarded 2,364 responses which did not exactly match a requested enum value. The results of this categorization are shown in Figure 5.
Footnote 9: [https://platform.openai.com/docs/models/gpt-3-5](https://platform.openai.com/docs/models/gpt-3-5)
Footnote 10: [https://platform.openai.com/docs/plugins/getting-started/writing-descriptions](https://platform.openai.com/docs/plugins/getting-started/writing-descriptions)
As shown in Figure 5, the majority of GitHub tables are centered around the category "Software and Technology" which includes many examples of code and documentation. Outside of code-related content, there are a variety of content types including: science and research, financial and economic, retail and e-commerce, etc. Common Crawl is more balanced and diverse, with a majority of tables focused on retail and e-commerce, internet and web services, and calendars, etc. Most of the Common Crawl tables were HTML, whereas in GitHub most those HTML tables occurred in the "Software and Technology" category as documentation.
Figure 4: **2D histogram of content hash distinct values.** There is a wide variance of duplicate context_metadata values among tables with duplicated content_hash, for both CommonCrawl and Github. The y-axis is the log of the distinct context_metadata counts, and the x-axis is the log of the total number of duplicated values for a given content hash. Both are on log scale with log bins, and the color reflects a normalized density.
Figure 3: **Content Hash Duplication Frequencies By Source.** Duplication based on content_hash shows a Zipf-like distribution when comparing frequency versus rank for both Github and Common Crawl.
Having the categories may be useful for downstream tasks, such as training a model to classify or generate tables of a specific category [Korini and Bizer, 2023]. We chose a limited set of categories to label and example tables to process, and leave further investigation of the accuracy and effectiveness of these categories to future work.
### Language Breakdown
Another important aspect of diversity for language models is the language itself, as discussed in many papers such as LAION-5B [Schuhmann et al., 2022] and the Pile [Gao et al., 2020]. We classified the language of tables using langdetect11, fasttext12, and gpt-3.5-turbo, based on the column names and values of string-typed cells, joined by spaces and limited to 100 characters. With manual inspection on a small sample, gpt-3.5-turbo was the most accurate.
Footnote 11: [https://github.com/Mimino666/langdetect](https://github.com/Mimino666/langdetect)
Footnote 12: [https://github.com/facebookresearch/fastText](https://github.com/facebookresearch/fastText)
We sampled 10,000 random tables from TabLib and classified their languages using gpt-3.5-turbo, with results shown in 6. Since English was 69% of the data, English is excluded from the figure. A large portion of tables were
Figure 5: **Data Categories Breakdown by File Type and Data Source.** CC is Common Crawl, and GH is GitHub HTML is the majority of content across most categories, and GitHub is predominantly of the category “Software and Technology”. Note the x-axis has frequencies normalized by data source, and the y-axis of categories is sorted based on the normalized frequency values on GitHub. The x-axis is broken to prevent the high proportion of “Software and Technology” for GitHub from dominating the figure.
classified as "Unknown", which includes mostly numeric tables which include no human languages. See Unknown Language Example in the appendix for an example of a table with an "Unknown" language.
### Data Types Breakdown
In addition to language, tabular data also has a variety of column types. We look at the type breakdown of the columns in the dataset. Table 4 below shows the column type frequency based on the inferred table schema. Surprisingly, we found that very little of the data had timestamp or datetime columns. This is likely due to implementation details of Pandas' type inference, requiring a separate pass to parse dates and timestamps in their various forms. In some cases it may be difficult or impossible to correctly infer timestamps, such as integral UNIX epoch timestamps. We believe that overall, the distribution is dominated by parsing decisions since the data in many formats (HTML, CSV, TSV) are stored as strings first, and column type is then inferred. We leave more detailed data cleaning, post-processing, and type inference to future works.
### Embeddings
Word embeddings are vector representations of words that contain semantic meaning [16]. We can represent other features such as column names, table schemas, etc. using these word embeddings. We sampled 500,000 tables from TabLib and used the all-MiniLM-L6-v2 model of Sentence Transformers [14] to embed the column names and first few rows of each table into a word embedding. We then computed a UMAP embedding to project those into a 2D plot, shown in Figure 7. As we can see, there are many large and small clusters. Upon manual inspection, the large clusters tend to represent different languages, and the smaller clusters align semantically towards categories (see Data Categories). This technique focuses mainly on table metadata such as column names and schema, and does a poor job of representing the contents of the table itself, which we leave for future work.
\begin{table}
\begin{tabular}{l l} \hline \hline Data Type & \% of Tables \\ \hline String & 61.8\% \\ Float & 22.3\% \\ Integer & 11.3\% \\ Unknown & 3.8\% \\ Boolean & 0.74\% \\ Timestamp & 0.04\% \\ Datetime & 0.01\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: **Column type frequency.** The majority of column types are strings.
Figure 6: **Frequency estimate of non-English languages in TabLib.** Note that English had a frequency of 69% so was excluded from this figure. All languages shown had a non-zero frequency in the 10,000 table sample.
## 4 Discussion
### Ethics
#### 4.1.1 Personally Identifiable Information
TabLib captures personally identifiable information (PII), such as names, phone numbers, and email addresses. However, all data within TabLib are from publicly accessible sources, implying that the PII it contains is already available to the public. Furthermore, we acknowledge that the identification and protection of PII is an evolving field of study, and we believe that raw datasets like TabLib will be essential resources in this research.
#### 4.1.2 Potential Biases
Publicly available data, like the data in TabLib, often contain inherent biases which can be inadvertently propagated in trained models. This phenomena, well-documented in language and image models, might also permeate tabular data, whether within the actual tabular data or their accompanying descriptive context. Acknowledging the presence of possible biases, TabLib presents an opportunity to study and mitigate such prejudices, leading to the development of fairer AI systems.
#### 4.1.3 Legality of Content
The legal implications of training machine learning models using copyrighted data is a topic of ongoing debate within the machine learning community (Gao et al., 2020). However, there is much less discussion, and even less clarity on the processing and distribution of data for research purposes. Based on our understanding, we believe this falls under the purview of fair use. Additionally, it is noteworthy to mention that under U.S. copyright law, facts and data are not
Figure 7: **UMAP sample. A UMAP embedding plot generated from a sample of 500K tables from TabLib, using column names and the first few rows from each table.**
subject to copyright protection (see _Feist v. Rural Telephone_13). This aspect of the law, while not providing definitive legal clarity, adds an interesting dimension to the discussion surrounding the use of datasets like TabLib, which collect factual data in tables. We commit to remaining informed and making necessary adjustments as the legal implications of this work become clearer.
Footnote 13: [https://www.law.cornell.edu/supremecourt/text/499/340](https://www.law.cornell.edu/supremecourt/text/499/340)
#### 4.1.4 Data Licensing
TabLib is an aggregation of publicly available data. Each datum has its own specific license which must be respected. We have attempted to include provenance information for each table within its context_metadata to help find licensing information. We also recommend that this dataset be used primarily for research purposes.
### Limitations
#### 4.2.1 Source Limitations
TabLib's initial version does not include many public sources such as CKAN sources (e.g., data.gov and data.gov.uk), books (e.g., Project Gutenberg), and other datasets on the public Internet not indexed by Common Crawl. Additionally, we have not included source files larger than 1 GB, GitHub branches other than "main" or "master", or truncated Common Crawl responses. These limitations affect the diversity, volume, and distribution of data in TabLib.
#### 4.2.2 Parsing Limitations
Detecting and parsing table structures is difficult, and our current parsing capabilities are limited. For instance, PDF tables that span multiple pages are not recognized as a single table. Similarly, ambiguities in the meaning of "before" and "after" can result in PDF tables with missing or incomplete context. For HTML tables, the presence of JavaScript, CSS, and other elements can introduce noise into the context. Furthermore, our current version does not support the extraction of tables from images, whether they are standalone image files or mined in PDFs and HTML. Another challenge lies in the accurate inference of column types and the correct detection of column headers (e.g. nested column headers). These limitations could potentially affect the accuracy of the data extracted and its subsequent usability.
#### 4.2.3 Metadata Limitations
Metadata are often inaccurate, incomplete, or missing. This includes data we actively sought to include, such as provenance. It also includes data that are useful but were not intentionally captured, such as licensing.
## 5 Future Work
There are numerous areas for exploration and improvement to enhance the value of TabLib as a research asset.
* **Add New Data Sources:** Increase the size of TabLib by including other Common Crawl crawls, GitHub branches beyond master and main, and broader expansion beyond the limitations of Common Crawl.
* **Derive New Tables:** Programmatically transform existing data tables to create new data tables, thereby increasing the number of tables.
* **Enhanced Table Extraction:** Improve our current table extraction methods, particularly for complex formats like PDFs and images, to increase the accuracy and completeness of the data extracted.
* **Inclusion of Additional Metadata:** Include additional metadata, such as licensing, categorization, etc.
* **Creation of Cleaned Versions:** Develop cleaned versions of TabLib by removing categories of information such as noise, PII, etc., thereby increasing the usability of the dataset for various applications.
* **Development of Benchmarks:** Create benchmarks around TabLib for tasks like question answering and search, to encourage the use of this dataset and spur advancements in tabular data research.
* **Pre-training Large Data Models:** Explore the potential of pre-training large data models exclusively on TabLib's tabular data.
* **Bias Study and Mitigation:** Study social biases in tabular data and develop techniques to mitigate them.
## 6 Conclusion
### Key Outcomes
In this work, we present TabLib, a dataset of 627 million tables (69 TiB) with 867 billion tokens of context extracted from GitHub and Common Crawl. TabLib contains raw, minimally processed tabular data derived from formats like CSV, HTML, PDF, and Excel, along with rich contextual metadata.
Our analysis of TabLib shows its extensive coverage across a multitude of topics, languages, and data types. The dataset exhibits interesting long-tail behavior with important consequences for downstream training and evaluation. Furthermore, our duplication analysis confirms that a non-trivial portion of TabLib consists of unique tables, and a large majority of tables contains unique metadata, further enhancing its value as a resource for AI research and development.
### Acknowledgements
We would like to thank TPU Research Cloud14 for providing the compute resources to process the data.
We are grateful to GitHub and Common Crawl for making available the underlying data necessary for TabLib. | 大型、多様性の高いデータセットが現代のテキストと画像のAIシステムの性能に重要な役割を果たすと広く認められています。しかし、テキストと画像のデータセットと比較できる規模と多様性を持つ表のデータセットは存在しません。そこで、TabLibという、627000万のテーブルと合わせて69TBの容量、867000万のトークンのコンパイルを提示します。TabLibは、CSV、HTML、SQLite、PDF、Excelなどのさまざまなファイル形式から、GitHubとCommon Crawlなどのソースから抽出されました。TabLibのサイズは、多様性は、テキストと画像の基盤的データセットの当初の約束と同様の期待を秘めています。これには、The PileとLAIONのようなものです。 |
2302.08957 | Like a Good Nearest Neighbor: Practical Content Moderation and Text
Classification | Few-shot text classification systems have impressive capabilities but are
infeasible to deploy and use reliably due to their dependence on prompting and
billion-parameter language models. SetFit (Tunstall et al., 2022) is a recent,
practical approach that fine-tunes a Sentence Transformer under a contrastive
learning paradigm and achieves similar results to more unwieldy systems.
Inexpensive text classification is important for addressing the problem of
domain drift in all classification tasks, and especially in detecting harmful
content, which plagues social media platforms. Here, we propose Like a Good
Nearest Neighbor (LaGoNN), a modification to SetFit that introduces no
learnable parameters but alters input text with information from its nearest
neighbor, for example, the label and text, in the training data, making novel
data appear similar to an instance on which the model was optimized. LaGoNN is
effective at flagging undesirable content and text classification, and improves
the performance of SetFit. To demonstrate the value of LaGoNN, we conduct a
thorough study of text classification systems in the context of content
moderation under four label distributions, and in general and multilingual
classification settings. | Luke Bates, Iryna Gurevych | 2023-02-17T15:43:29 | http://arxiv.org/abs/2302.08957v3 | # Like a Good Nearest Neighbor:
###### Abstract
Modern text classification systems have impressive capabilities but are infeasible to deploy and use reliably due to their dependence on prompting and billion-parameter language models. SetFit (Tunstall et al., 2022) is a recent, practical approach that fine-tunes a Sentence Transformer under a contrastive learning paradigm and achieves similar results to more unwieldy systems. Text classification is important for addressing the problem of domain drift in detecting harmful content, which plagues all social media platforms. Here, we propose Like a Good Nearest Neighbor (LAGONN), an inexpensive modification to SetFit that requires no additional parameters or hyperparameters but modifies input with information about its nearest neighbor, for example, the label and text, in the training data, making novel data appear similar to an instance on which the model was optimized. LaGoNN is effective at the task of detecting harmful content and generally improves SetFit's performance. To demonstrate LaGoNN's value, we conduct a thorough study of text classification systems in the context of content moderation under four label distributions.1
Footnote 1: Code and data: [https://github.com/UKPLab/lagonn](https://github.com/UKPLab/lagonn)
## 1 Introduction
Text classification is the most important tool for NLP practitioners, and there has been substantial progress in advancing the state-of-the-art, especially with the advent of large, pretrained language models (PLM) (Devlin et al., 2019). Modern research focuses on in-context learning (Brown et al., 2020), pattern exploiting training (Schick and Schutze, 2021, 2022), or parameter efficient fine-tuning (Liu et al., 2022). State-of-the-art methods have achieved impressive results on the SuperGLUE (Wang et al., 2019) and RAFT (Alex et al., 2021) few-shot benchmarks. However, they are difficult to use because of their reliance on billion-parameter PLMs and prompt engineering. Constructing prompts is not trivial and may require domain expertise.
One exception to these cumbersome systems is SetFit. SetFit does not rely on prompting or billion-parameter PLMs, and instead fine-tunes a pretrained Sentence Transformer (ST) (Reimers and Gurevych, 2019) under a contrastive learning paradigm. SetFit has comparable performance to more unwieldy systems while being one to two orders of magnitude faster to train and run inference.
An important application of text classification is aiding or automating content moderation, which is the task of determining the appropriateness of user-generated content on the Internet (Roberts, 2017). From fake news to toxic comments to hate speech, it is difficult to browse social media without being exposed to potentially dangerous posts that may have an effect on our ability to reason (Ecker et al., 2022). Misinformation spreads at alarming
Figure 1: We embed training data, retrieve the text, gold label, and distance for each instance from its second nearest neighbor (\(k\)=2) and modify the original text with this information. Then we embed the modified training data and train a classifier. During inference, the NN from the training data is selected (\(k\)=1), the original text is modified with the text, gold label, and distance from the NN, and the classifier is called.
rates (Vosoughi et al., 2018), and an ML system should be able to quickly aid human moderators. While there is work in NLP with this goal (Markov et al., 2022; Shido et al., 2022; Ye et al., 2023), a general, practical and open-sourced method that is effective across multiple domains remains an open challenge. Novel fake news topics or racial slurs emerge and change constantly. Retraining of ML-based systems is required to adapt this concept drift, but this is expensive, not only in terms of computation, but also in terms of the human effort needed to collect and label data.
SetFit's performance, speed, and low cost would make it ideal for effective content moderation, however, this type of text classification poses a challenge for even state-of-the-art approaches. For example, detecting hate speech on Twitter (Basile et al., 2019), a subtask on the RAFT few-shot benchmark, appears to be the most difficult dataset; at time of writing, it is the only task where the human baseline has not been surpassed, yet SetFit is among the top ten most performant systems.2
Footnote 2: [https://huggingface.co/spaces/ought/](https://huggingface.co/spaces/ought/) raft-leaderboard(see “Tweet Eval Hate”).
Here, we propose a modification to SetFit, called Like a Good Nearest Neighbor (LaGoNN). LaGoNN introduces no parameters or hyperparameters and instead modifies input text by retrieving information about the nearest neighbor (NN) seen during optimization (see Figure 1). Specifically, we append the label, distance, and text of the NN in the training data to a new instance and encode this modified version with an ST. By making input data appear more similar to instances seen during training, we inexpensively exploit the ST's pretrained or fine-tuned knowledge when considering a novel example. Our method can also be applied to the linear probing of an ST, requiring no expensive fine-tuning of the large embedding model. Finally, we propose a simple alteration to the SetFit training procedure, where we fine-tune the ST on a subset of the training data. This results in a more efficient and performant text classifier that can be used with LaGoNN. We summarize our contributions as follows:
1. We propose LaGoNN, an inexpensive modification to SetFit- or ST-based text classification.
2. We suggest an alternative training procedure to the standard fine-tuning of SetFit, that can be used with or without LaGoNN, and results in a cheaper system with similar performance to the more expensive SetFit.
3. We perform an extensive study of LaGoNN, SetFit, and standard transformer fine-tuning in the context of content moderation under different label distributions.
## 2 Related Work
There is not much work on using sentence embeddings as features for classification despite the pioneering work being roughly five years old (Perone et al., 2018). STs are pretrained with the objective of maximizing the distance between semantically distinct text and minimizing the distance between text that is semantically similar in feature space. They are composed of a Siamese and triplet architecture that encodes text into dense vectors which can be used as features for ML. STs were first used to encode text for classification by Piao (2021), however, the authors relied on pretrained representations.
SetFit uses a contrastive learning paradigm from computer vision (Koch et al., 2015) to fine-tune STs. The embedding model is fine-tuned with a distance-based loss function, like cosine similarity, such that examples belonging to different labels are separated in feature space. This approach can relatively easily and quickly train a strong, few-shot text classifier, transforming the ST from a sentence encoder to a topic encoder.
Most related to LaGoNN is work done by Xu et al. (2021), who showed that retrieving and concatenating text from training data and external sources, such as ConceptNet (Speer et al., 2017) and the Wikitionary3 definition, can be viewed as a type of external attention that does not modify the architecture of the Transformer in question answering. Liu et al. (2022) used PLMs, including STs, and \(k\)-NN lookup to prepend examples that are similar to a GPT-3 query sample to aid in prompt engineering for in-context learning. Wang et al. (2022) demonstrated that prepending and appending training data can benefit PLMs in the tasks of summarization, language modelling, machine translation, and question answering, using BM25 as their retrieval model for speed (Manning et al., 2008; Robertson and Zaragoza, 2009).
Footnote 3: [https://www.wiktionary.org/](https://www.wiktionary.org/)
We alter the SetFit training procedure by using fewer examples to adapt the embedding model for
many-shot learning. LaGoNN decorates input text with its nearest neighbor's gold label, Euclidean distance, and text from the training data to exploit the ST's optimized representations. Compared to retrieval-based methods, LaGoNN uses the same model for both retrieval and encoding, which can be fine-tuned via SetFit. We only retrieve information from the training data for text classification.
## 3 Like a Good Nearest Neighbor
Xu et al. (2021) formulate a type of external attention, where textual information is retrieved from multiple sources and added to text input to give the model stronger reasoning ability without altering the internal architecture. Inspired by this approach, LaGoNN exploits pretrained and fine-tuned knowledge through external attention, but the information we retrieve comes only from data used during optimization. We consider an embedding function, \(f\), that is called on both training and test data, \(f(X_{train})\) and \(f(X_{test})\). Considering its success and speed on realistic, few-shot data and our goal of practical content moderation, we choose an ST that can be fine-tuned with SetFit as our
\begin{table}
\begin{tabular}{c c}
**Training Data** & **Test Data** \\ “I love this.” [positive 0.0] (0) & “So good!” [?] (?) \\ “This is great!” [positive 0.5] (0) & “Just terribel” [?] (?) \\ “I hate this.” [negative 0.7] (1) & “Never again.” [?] (?) \\ “This is awful!” [negative 1.2] (1) & “This rocks!” [?] (?) \\ \end{tabular}
\end{table}
Table 1: Toy training and test data and different LaGoNN configurations considering the first training example. Train and Test Modified are altered instances that are input into the final embedding model for training and inference, respectively. The input format is “original text [SEP] [NN gold label distance] NN instance text”. Input text is in quotation marks, the NN’s gold label and distance from the training data are in square brackets, and the integer label is in parenthesis. We present real examples of LaGoNN BOTH modified text in Appendix A.4.
Figure 2: LaGoNN LABEL uses an ST to encode training data, performs NN lookup, appends the second NN’s (\(k\)=2) gold label and distance, and optionally SetFit to fine-tune the embedding model. We then embed this new instance and train a classifier. During inference, we use the embedding model to modify the test data with its NN’s gold label and distance from the training data (\(k\)=1), compute the final representation, and call the classifier. Input text is in quotation marks, the NN’s gold label and distance are in brackets, and the integer label is in parenthesis.
embedding function.
**Encoding training data and nearest neighbors** LaGoNN first uses a pretrained Sentence Transformer to embed training text in feature space, \(f(X_{train})\). We perform NN lookup with scikit-learn (Buitinck et al., 2013) on the resulting embeddings and query the second closest NN (\(k\)=2). We do not use the NN because it is the example itself.
Nearest neighbor informationWe extract text from the second nearest neighbor and use it to decorate the original example. We experimented with different text that LaGoNN could use. The first configuration we consider is the gold label and Euclidean distance of the NN, which we call LA-BEL. We then considered the gold label, distance, and the text of the NN, which we refer to as TEXT. Finally, we tried the same format as TEXT but for all possible labels, which we call BOTH (see Table 1 and Figure 2).4 Information from the second NN is appended to the text following a separator token to indicate this instance is composed of multiple sequences. While the BOTH and TEXT configurations are arguably the most interesting, we find LABEL to result in the most performant version of LaGoNN, and this is the version about which we report results.
Footnote 4: LaGoNN requires a mapping from the label to the text the label represents, for example, \(0\) – positive and \(1\) – negative.
TrainingLaGoNN encodes the modified training data and optionally fine-tunes the embedding model via SetFit, \(f(X_{trainmod})\). After fine-tuning, we train a classifier \(CLF(f(X_{trainmod}))\), like logistic regression.
InferenceLaGoNN uses information from the nearest neighbor in the training data to modify input text. We compute the embeddings on the test data, \(f(X_{test})\), and query the NN lookup, selecting the NN (\(k\)=1) in the training data and extracting information from the training text. LaGoNN then decorates the input instance with information from the NN in the training data. Finally, we encode the modified data with the embedding model and call the classifier, \(CLF(f(X_{testmod}))\).
IntuitionAs \(f\) is the same function, we hypothesize that LaGoNN's modifications will make a novel instance more semantically similar to its NNs in the training data. The resulting representation should be more akin to an instance on which the embedding model and classifier were optimized. Our method also leverages both distance-based NN lookup and probabilistic algorithms (logistic regression) for its final prediction.
## 4 Experiments
### Data and label distributions
In our experiments, we study LaGoNN's performance on four binary and one ternary classification dataset related to the task of content moderation. Each dataset is composed of a training, validation, and test split.
Here, we provide a summary of the five datasets we studied. LIAR was created from Politifact5 for fake news detection and is composed of the data fields _context_, _speaker_, and _statement_, which are labeled with varying levels of truthfulness (Wang, 2017). We used a collapsed version of this dataset where a statement can only be true or false. We did not use _speaker_, but did use _context_ and _statement_, separated by a separator token. Quora Insincere Questions6 is composed of neutral and toxic questions, where the author is not asking in good faith. Hate Speech Offensive7 has three labels and is composed of tweets that can contain either neutral text, offensive language, or hate speech (Davidson et al., 2017). Amazon Counterfactual8 contains sentences from product reviews, and the labels can be "factual" or "counterfactual" (O'Neill et al., 2021). "Counterfactual" indicates that the customer said something that cannot be true. Finally, Toxic Conversations9 is a dataset of comments where the author wrote a comment with unintended bias10 (see Table 2).
Footnote 5: [https://www.politifact.com/](https://www.politifact.com/)
Footnote 6: [https://www.kaggle.com/c/quora-insincere-questions-classification](https://www.kaggle.com/c/quora-insincere-questions-classification)
Footnote 7: [https://huggingface.co/datasets/hate_speech_offensive](https://huggingface.co/datasets/hate_speech_offensive)
Footnote 8: [https://huggingface.co/datasets/SetFit/amazon_counterfactual_en](https://huggingface.co/datasets/SetFit/amazon_counterfactual_en)
Footnote 9: [https://huggingface.co/datasets/SetFit/toxic_](https://huggingface.co/datasets/SetFit/toxic_) conversations
Footnote 10: [https://huggingface.co/datasets/SetFit/toxic_](https://huggingface.co/datasets/SetFit/toxic_) conversations
We study our system by simulating growing training data over ten discrete steps sampled under four different label distributions: extreme, imbalanced, moderate, and balanced (see Table 3). On each step we add \(100\) examples (100 on the first, 200 on the second, etc.) from the training split sampled under one of the four ratios.11 On each
step, we train our method with the sampled data and evaluate on the test split. Considering growing training data has two benefits: 1) We can simulate a streaming data scenario, where new data is labeled and added for training and 2) We can investigate each method's sensitivity to the number of training examples. We sampled over five seeds, reporting the mean and standard deviation.
### Baselines
We compare LaGoNN against standard fine-tuning, linear probing of a Sentence Transformer, and two versions of SetFit, detailed below.
RoBERTaRoBERTa-base is a pretrained language model Liu et al. (2019) that we fine-tuned with the transformers library Wolf et al. (2020). We select two versions of RoBERTa-base: an expensive version, where we perform standard fine-tuning on each step (RoBERTa\({}_{full}\)) and a cheaper version, where we freeze the model body after step one and update the classification head on subsequent steps (RoBERTa\({}_{freeze}\)). We set the learning rate to \(1e^{-5}\), train for a maximum of 70 epochs, and use early stopping, selecting the best model after training. We consider RoBERTa\({}_{full}\) an upper bound as it has the most trainable parameters and requires the most time to train of all our methods.
Linear probeWe perform linear probing of a pretrained Sentence Transformer by fitting logistic regression with default hyperparameters on the training embeddings on each step. We choose this baseline because LaGoNN can be applied as a modification in this scenario. We select MPNET Song et al. (2020) as the ST, for SetFit, and for LaGoNN.12 We refer to this method as Probe.
Footnote 12: [https://huggingface.co/sentence-transformers/paraphrase-mgnet-base-v2](https://huggingface.co/sentence-transformers/paraphrase-mgnet-base-v2)
Logistic regressionHere, we perform standard fine-tuning with SetFit on the first step, and then on subsequent steps, freeze the embedding model and retrain only the classification head. We choose this baseline as LaGoNN also uses logistic regression as its final classifier and refer to this method as Log Reg.
\(k\)-nearest neighborsSimilar to the above baseline, we fine-tune the embedding model via SetFit, but swap out the classification head for a \(k\)NN classifier, where \(k=3\). We select this baseline as LaGoNN also relies on an NN lookup. \(k=3\) was chosen during our development stage as it yielded the strongest performance. We refer to this method as \(k\)NN.
SetFitFor this baseline we perform standard fine-tuning with SetFit on each step. On the first step, this method is equivalent to Log Reg.
LaGoNN cheapThis method modifies the training and test data via LaGoNN before fitting a logistic regression classifier. Even without adapting the embedding model, as the training data grow, modifications made to the test data may change. We refit the classification head on each step and refer to this method as LaGoNN\({}_{cheap}\), which is comparable to Probe.
LaGoNNOn the first step, we use LaGoNN to modify our training data and then perform standard fine-tuning with SetFit. On subsequent steps, we freeze the embedding model and use it to modify our data. We fit logistic regression on each step and refer to this method as LaGoNN. It is comparable to Log Reg.
LaGoNN expensiveThis version is identical to LaGoNN, except we fine-tune the embedding model on each step. We refer to this method as LaGoNN\({}_{exp}\) and it is comparable to SetFit. On the first step, this method is equivalent to LaGoNN.
## 5 Results
Table 4 and Figure 3 show our results. In the cases of the extreme and imbalanced regimes, Set
\begin{table}
\begin{tabular}{c|c}
**Dataset (and Detection Task)** & **Number of Labels** \\ \hline LIAR (Fake News) & 2 \\ Insincere Questions (Toxicity) & 2 \\ Hate Speech Offensive & 3 \\ Amazon Counterfactual (English) & 2 \\ Toxic Conversations & 2 \\ \end{tabular}
\end{table}
Table 2: Summary of datasets and number of labels. We provide the type of task in parenthesis in unclear cases.
\begin{table}
\begin{tabular}{c|c|c}
**Regime** & **Binary** & **Ternary** \\ \hline Extreme & 0: 98\% 1: 2\% & 0: 95\%, 1: 2\%, 2: 3\% \\ Imbalanced & 0: 90\% 1: 10\% & 0: 80\%, 1: 5\%, 2: 15\% \\ Moderate & 0: 75\% 1: 25\% & 0: 65\%, 1: 10\%, 2: 25\% \\ Balanced & 0: 50\% 1: 50\% & 0: 33\%, 1: 33\%, 2: 33\% \\ \end{tabular}
\end{table}
Table 3: Label distributions for sampling training data. 0 represents neutral while 1 and 2 represent different types of undesirable text.
Fit's performance steadily increases with the number of training examples. As the label distribution shifts to the balanced regime, however, SetFit's performance quickly saturates or even degrades as the number of training examples grows. LaGoNN, RoBERTa\({}_{full}\), and Log Reg, other fine-tuned PLM classifiers, do not exhibit this behavior. LaGoNN\({}_{exp}\), being based on SetFit, exhibits a similar trend, but the performance degradation is mitigated; on the \(10^{th}\) step of Amazon Counterfactual in Table 4 SetFit's performance decreased by 9.7, while LaGoNN\({}_{exp}\) only fell by 3.7.
LaGoNN and LaGoNN\({}_{exp}\) generally outperform Log Reg and SetFit, respectively, often resulting in a more stable model, as reflected in the standard deviation. We find that LaGoNN and LaGoNN\({}_{exp}\) exhibit stronger predictive power with fewer examples than RoBERTa\({}_{full}\) despite having fewer trainable parameters. For example, on the first step of Insincere Questions under the extreme setting, LaGoNN's performance is more than 10 points higher.
\begin{table}
\begin{tabular}{l c c c c|c c c c} \multicolumn{1}{l}{**Method**} & \multicolumn{3}{c|}{**InsincereQs**} & \multicolumn{3}{c}{**AmazonCF**} \\ _Extreme_ & \(1^{st}\) & \(5^{th}\) & \(10^{th}\) & Average & \(1^{st}\) & \(5^{th}\) & \(10^{th}\) & Average \\ \hline RoBERTa\({}_{full}\) & \(19.9_{8.4}\) & \(30.9_{7.9}\) & \(42.0_{7.4}\) & \(33.5_{6.7}\) & \(21.8_{6.6}\) & \(63.9_{10.2}\) & \(72.3_{3.0}\) & \(59.6_{16.8}\) \\ SetFit & \(24.1_{6.3}\) & \(29.2_{6.7}\) & \(36.7_{7.3}\) & \(31.7_{3.4}\) & \(22.3_{8.8}\) & \(64.2_{3.3}\) & \(68.6_{4.6}\) & \(56.8_{14.9}\) \\ LaGoNN\({}_{exp}\) & \(\mathbf{30.7_{8.9}}\) & \(37.6_{6.1}\) & \(39.0_{6.1}\) & \(36.1_{2.3}\) & \(\mathbf{26.1_{17.5}}\) & \(\mathbf{68.4_{4.4}}\) & \(\mathbf{74.9_{2.9}}\) & \(\mathbf{63.2}_{16.7}\) \\ \hline RoBERTa\({}_{freeze}\) & \(19.9_{8.4}\) & \(34.1_{5.4}\) & \(37.9_{5.9}\) & \(32.5_{5.5}\) & \(21.8_{6.6}\) & \(41.0_{12.7}\) & \(51.3_{10.7}\) & \(40.6_{8.9}\) \\ \(k\)NN & \(6.8_{0.42}\) & \(15.9_{3.4}\) & \(16.9_{4.3}\) & \(14.4_{3.0}\) & \(10.3_{0.2}\) & \(15.3_{4.2}\) & \(18.4_{3.7}\) & \(15.6_{2.4}\) \\ Log Reg & \(24.1_{6.3}\) & \(31.7_{4.9}\) & \(36.1_{5.4}\) & \(31.8_{3.6}\) & \(22.3_{8.8}\) & \(32.4_{11.5}\) & \(42.3_{8.8}\) & \(34.5_{9.9}\) \\ LaGoNN & \(\mathbf{30.7_{8.9}}\) & \(39.3_{4.9}\) & \(41.2_{4.7}\) & \(38.4_{3.0}\) & \(\mathbf{26.1_{17.5}}\) & \(31.1_{19.4}\) & \(33.0_{19.1}\) & \(30.9_{2.3}\) \\ \hline Probe & \(24.3_{8.4}\) & \(39.8_{5.6}\) & \(44.8_{4.2}\) & \(38.3_{6.2}\) & \(24.2_{9.0}\) & \(46.3_{4.4}\) & \(54.6_{2.0}\) & \(45.1_{10.3}\) \\ LaGoNN\({}_{cheap}\) & \(23.6_{7.8}\) & \(\mathbf{40.7_{5.9}}\) & \(\mathbf{45.3_{4.4}}\) & \(\mathbf{38.6_{6.6}}\) & \(20.1_{6.9}\) & \(38.3_{4.9}\) & \(47.8_{3.4}\) & \(38.2_{9.5}\) \\ \hline _Balanced_ & & & & & & & & \\ RoBERTa\({}_{full}\) & \(47.1_{4.2}\) & \(52.1_{3.6}\) & \(55.7_{2.6}\) & \(52.5_{2.9}\) & \(73.6_{2.1}\) & \(78.6_{3.9}\) & \(\mathbf{82.4_{1.1}}\) & \(78.9_{2.2}\) \\ SetFit & \(43.5_{4.2}\) & \(47.1_{4.6}\) & \(48.5_{3.9}\) & \(48.0_{1.7}\) & \(73.8_{4.4}\) & \(69.8_{4.0}\) & \(64.1_{4.6}\) & \(69.6_{3.6}\) \\ LaGoNN\({}_{exp}\) & \(42.8_{5.3}\) & \(47.6_{2.9}\) & \(47.0_{1.7}\) & \(46.2_{2.0}\) & \(\mathbf{76.0_{3.0}}\) & \(73.4_{2.6}\) & \(72.3_{2.9}\) & \(72.5_{3.4}\) \\ \hline RoBERTa\({}_{freeze}\) & \(47.1_{4.2}\) & \(52.1_{0.4}\) & \(53.3_{1.7}\) & \(51.5_{2.1}\) & \(73.6_{2.1}\) & \(76.8_{1.6}\) & \(77.9_{1.0}\) & \(76.5_{1.3}\) \\ \(k\)NN & \(22.3_{2.3}\) & \(30.2_{2.3}\) & \(30.9_{1.8}\) & \(29.5_{2.5}\) & \(41.7_{3.4}\) & \(57.9_{3.3}\) & \(58.3_{3.3}\) & \(56.8_{5.1}\) \\ Log Reg & \(43.5_{4.2}\) & \(53.8_{2.2}\) & \(55.5_{1.6}\) & \(52.8_{3.5}\) & \(73.8_{4.4}\) & \(79.2_{1.9}\) & \(80.1_{1.0}\) & \(78.6_{1.8}\) \\ LaGoNN & \(42.8_{5.3}\) & \(54.1_{2.9}\) & \(56.3_{1.3}\) & \(53.4_{3.7}\) & \(\mathbf{76.0_{3.0}}\) & \(\mathbf{80.1_{2.0}}\) & \(81.4_{1.1}\) & \(\mathbf{79.8_{1.4}}\) \\ \hline Probe & \(47.5_{1.6}\) & \(52.4_{1.7}\) & \(55.3_{1.1}\) & \(52.2_{2.5}\) & \(52.4_{3.4}\) & \(64.7_{2.5}\) & \(67.5_{0.4}\) & \(63.4_{4.4}\) \\ LaGoNN\({}_{cheap}\) & \(\mathbf{49.3_{2.6}}\) & \(\mathbf{54.4_{1.4}}\) & \(\mathbf{57.6_{0.7}}\) & \(\mathbf{54.2_{2.7}}\) & \(48.1_{3.4}\) & \(62.0_{2.0}\) & \(65.3_{0.8}\) & \(60.5_{5.0}\) \\ \hline \end{tabular}
\end{table}
Table 4: Average performance (average precision \(\times\) 100) on Insincere Questions and Amazon Counterfactual. The first, fifth, and tenth step are followed by the average over all ten steps. The average gives insight into the overall strongest performer by aggregating all steps. We group methods with a comparable number of trainable parameters together. The extreme label distribution results are followed by balanced results. We provide additional results in Appendix A.2.
Figure 3: Average performance in the imbalanced and balanced sampling regimes relative to comparable methods. We include RoBERTa\({}_{full}\) results for reference. The metric is macro-F1 for Hate Speech Offensive, average precision elsewhere.
LaGoNN\({}_{cheap}\) outperforms all other methods on the Insincere Questions dataset for all balance regimes, despite being the third fastest (see Table 5) and having the second fewest trainable parameters. We attribute this result to the fact that this dataset is composed of questions from Quora13 and our ST backbone was pretrained on similar data. This intuition is supported by Probe, the cheapest method, which despite having the fewest trainable parameters, shows comparable performance.
Footnote 13: [https://www.quora.com/](https://www.quora.com/)
### SetFit for efficient many-shot learning
Respectively comparing SetFit to Log Reg and LaGoNN\({}_{exp}\) to LaGoNN suggests that fine-tuning the ST embedding model on moderate or balanced data hurts model performance as the number of training samples grows. We therefore hypothesize that randomly sampling a subset of training data to fine-tune the encoder, freezing, embedding the remaining data, and training the classifier will result in a stronger model.
To test our hypothesis, we add two models to our experimental setup: SetFit\({}_{lite}\) and LaGoNN\({}_{lite}\). SetFit\({}_{lite}\) and LaGoNN\({}_{lite}\) are respectively equivalent to SetFit and LaGoNN\({}_{exp}\), except after the fourth step (400 samples), we freeze the encoder and only retrain the classifier on subsequent steps, similar to Log Reg and LaGoNN.
Figure 4 shows our results with these two new models. As expected, in the cases of extreme and imbalanced distributions, LaGoNN\({}_{exp}\), SetFit, and RoBERTa\({}_{exp}\), are the strongest performers on Toxic Conversations. We note very different results for both LaGoNN\({}_{lite}\) and SetFit\({}_{lite}\) compared to LaGoNN\({}_{exp}\) and SetFit on Toxic Conversations and Amazon Counterfactual under the moderate and balanced label distributions. As their expensive counterparts start to plateau or degrade on the fourth step, the predictive power of these two new models dramatically increases, showing improved or comparable performance to RoBERTa\({}_{full}\), despite being optimized on less data; for example, LaGoNN\({}_{lite}\) reaches an average precision of approximately 55 after being optimized on only 500 examples. RoBERTa\({}_{full}\) does not exhibit similar performance until the tenth step. Finally, we point out that LaGoNN-based methods generally provide a performance boost for SetFit-based classification.
### LaGoNN's computational expense
LaGoNN is more computationally expensive than Sentence Transformer- or SetFit-based text classification. LaGoNN introduces additional inference with the encoder, NN-lookup, and string modification. As the computational complexity of transformers increases with sequence length [23], additional expense is created when LaGoNN appends textual information before in
Figure 4: Average performance for all sampling regimes on Toxic Conversations and the moderate and balanced regimes for Amazon Counterfactual and Hate Speech Offensive. More expensive models, such as LaGoNN\({}_{exp}\), SetFit, and RoBERTa\({}_{full}\) perform best when the label distribution is imbalanced. As the distribution becomes more balanced, however, inexpensive models, such as LaGoNN\({}_{lite}\) or SetFit\({}_{lite}\), show similar or improved performance. The metric is macro-F1 for Hate Speech Offensive, average precision elsewhere. We provide additional results in Appendix A.3.
ference with the ST. In Table 5, we provide a speed comparison between Probe, Log Reg, SetFit, and LaGoNN classification computed on the same hardware. On average, LaGoNN introduced 24.2 additional seconds of computation compared to its relative counterpart.
## 6 Discussion
Modern research has achieved impressive results on a variety of text classification tasks and with limited training data. SetFit is one such example and can be used practically, but based on our results, the task of text classification for content moderation presents a challenge even for state-of-the-art approaches. It is imperative that we develop reliable methods that can be feasibly and quickly applied. These methods should be as inexpensive as possible such that we can re-tune them for novel forms of hate speech, toxicity, and fake news.
Our results suggest that LaGoNN\({}_{exp}\) or SetFit, relatively expensive techniques, can detect harmful content when dealing with imbalanced label distributions, as is common with realistic datasets. This finding is intuitive from the perspective that less common instances are more difficult to learn and require more effort. The exception to this would be our examination of Insincere Questions, where LaGoNN\({}_{cheap}\) excelled. This highlights the fact that we can inexpensively extract pretrained knowledge if PLMs are chosen with care.
Standard fine-tuning with SetFit does not help performance on more balanced datasets that are not few-shot. SetFit was developed for few-shot learning, but we have observed that it should not be applied "out of the box" to balanced, non-few-shot data. This can be detrimental to performance and has a direct effect on our approach. However, we have observed that LaGoNN can stabilize SetFit's predictions and reduce its performance drop. Figures 3 and 4 show that when the label distribution is moderate or balanced (see Table 3), SetFit plateaus, yet less expensive systems, such as LaGoNN, continue to learn. We believe this is due to SetFit's fine-tuning objective, which optimizes a Sentence Transformer using cosine similarity loss to separate examples belonging to different labels in feature space by assuming independence between labels. This may be too strong an assumption as we optimize with more examples, which is counter-intuitive for data-hungry transformers. RoBERTa\({}_{full}\), optimized with cross-entropy loss, generally showed improved performance as we added training data.
When dealing with balanced data, it is sufficient to fine-tune the Sentence Transformer via SetFit with 50 to 100 examples per label, while 150 to 200 instances appear to be sufficient when the training data are moderately balanced. The encoder can then be frozen and all available data embedded to train a classifier. This improves performance and is more efficient than full-model fine-tuning. LaGoNN is directly applicable to this case, boosting the performance of SetFit\({}_{title}\) without introducing trainable parameters. In this setup, all models fine-tuned on Hate Speech Offensive exhibited similar, upward-trending learning curves, but we note the speed of LaGoNN relative to RoBERTa\({}_{full}\) or SetFit (see Figure 4 and Table 5).
## 7 Conclusion
We have proposed LaGoNN, a simple and inexpensive modification to Sentence Transformer- or SetFit-based text classification. LaGoNN does not introduce any trainable parameters or new hyperparameters, but typically improves SetFit's performance. To demonstrate the merit of LaGoNN, we examined text classification systems in the context of content moderation under four label distributions on five datasets and with growing training data. To our knowledge, this is the first work to examine SetFit in this way. When the training labels are imbalanced, expensive systems, such as LaGoNN\({}_{exp}\) are performant. However, when the distribution is balanced, standard fine-tuning with SetFit can actually hurt model performance. We have therefore proposed an alternative fine-tuning procedure to which LaGoNN can be easily utilized, resulting in a powerful, but inexpensive system capable of
\begin{table}
\begin{tabular}{c|c}
**Method** & **Time in seconds** \\ \hline Probe & 22.9 \\ LaGoNN\({}_{cheap}\) & 44.2 \\ Log Reg & 42.9 \\ LaGoNN & 63.4 \\ SetFit & 207.3 \\ LaGoNN\({}_{exp}\) & 238.0 \\ \hline RoBERTa\({}_{full}\) & 446.9 \\ \end{tabular}
\end{table}
Table 5: Speed comparison between LaGoNN and comparable methods. Time includes training each method on \(1,000\) examples and performing inference on \(51,000\) examples.
detecting harmful content.
## 8 Acknowledgments
We would like to thank Derek Hommel and Nils Reimers for sharing inspiring discussions with us. We would also like to extend our gratitude to Tom Aarsen, Max Glockner, Yongxin Huang, Timour Igamberdiev, Sukannya Purkayastha, and Kexin Wang for their invaluable feedback on an early draft of our manuscript. This work was funded by the German Federal Ministry of Education and Research and the Hessian Ministry of Science and the Arts (HMWK) within the projects "The Third Wave of Artificial Intelligence - 3AI", hessian.AI, and within their joint support of the National Research Center for Applied Cybersecurity ATHENE.
## 9 Limitations
In the current work, we have only considered text data, but social media content can of course consist of text, images, and videos. As LaGoNN depends only on an embedding model, an obvious extension to our approach would be examining the modifications we suggest, but on multimodal data. This is an interesting direction that we leave for future research. We have also considered English data, but harmful content can appear in any language. The authors demonstrated that SetFit is performant on multilingual data, the only necessary modification being the underlying pretrained ST. We therefore suspect that LaGoNN would behave similarly on non-English data, but this is not something we have tested ourselves. In order to examine our system's performance under different label-balance distributions, we restricted ourselves to binary and ternary text classification tasks, and LaGoNN therefore remains untested when there are more than three labels. We did not study our method when there are fewer than 100 examples, and investigating LaGoNN in a few-shot learning setting is fascinating topic for future study.
## 10 Ethics Statement
It is our sincere goal that our work contributes to the social good multiple ways. We first hope to have furthered research on text classification that can be feasibly applied to combat undesirable content, such as misinformation, on the Internet, which could potentially cause someone harm. To this end, we have tried to describe our approach as accurately as possible and released our code, such that our work is transparent and can be easily reproduced and expanded upon. We hope that we have also created a useful but efficient system which reduces the need to expend energy in the form expensive computation. For example, LaGoNN does not rely on billion-parameter language models that demand thousand-dollar GPUs to use. LaGoNN makes use of GPUs no more than SetFit, despite being more computationally expensive. We have additionally proposed a simple method to make SetFit, an already relatively inexpensive method, even more efficient.
| 少数の学習によるテキスト分類システムには、素晴らしい能力がありますが、プロンプティングと数十億のパラメータを持つ言語モデルに依存するため、信頼できるデプロイと利用が困難です。SetFit (Tunstall et al., 2022) は、文法モデルを対照学習のパラダイムで微調整する、最近の、実用的なアプローチです。これにより、より複雑なシステムと同等の結果を得ることが可能です。テキスト分類の低コスト化は、すべての分類タスクのドメインドリフトの問題に対処する重要な要素であり、特に有害なコンテンツの検出において重要です。ソーシャルメディアプラットフォームが苦しんでいる問題です。ここでは、SetFitに改良を加えたLike a GoodNearest Neighbor (LaGoNN) を提案します。LaGoNNは、学習可能なパラメータを追加せずに、入力テキストを近傍の隣接者からの情報で変化させ、例えば、訓練データにおけるラベルと |
2303.16698 | Probabilistic inverse optimal control for non-linear partially
observable systems disentangles perceptual uncertainty and behavioral costs | Inverse optimal control can be used to characterize behavior in sequential
decision-making tasks. Most existing work, however, is limited to fully
observable or linear systems, or requires the action signals to be known. Here,
we introduce a probabilistic approach to inverse optimal control for partially
observable stochastic non-linear systems with unobserved action signals, which
unifies previous approaches to inverse optimal control with maximum causal
entropy formulations. Using an explicit model of the noise characteristics of
the sensory and motor systems of the agent in conjunction with local
linearization techniques, we derive an approximate likelihood function for the
model parameters, which can be computed within a single forward pass. We
present quantitative evaluations on stochastic and partially observable
versions of two classic control tasks and two human behavioral tasks.
Importantly, we show that our method can disentangle perceptual factors and
behavioral costs despite the fact that epistemic and pragmatic actions are
intertwined in sequential decision-making under uncertainty, such as in active
sensing and active learning. The proposed method has broad applicability,
ranging from imitation learning to sensorimotor neuroscience. | Dominik Straub, Matthias Schultheis, Heinz Koeppl, Constantin A. Rothkopf | 2023-03-29T13:51:06 | http://arxiv.org/abs/2303.16698v2 | # Probabilistic inverse optimal control with local linearization
###### Abstract
Inverse optimal control methods can be used to characterize behavior in sequential decision-making tasks. Most existing work, however, requires the control signals to be known, or is limited to fully-observable or linear systems. This paper introduces a probabilistic approach to inverse optimal control for stochastic non-linear systems with missing control signals and partial observability that unifies existing approaches. By using an explicit model of the noise characteristics of the sensory and control systems of the agent in conjunction with local linearization techniques, we derive an approximate likelihood for the model parameters, which can be computed within a single forward pass. We evaluate our proposed method on stochastic and partially observable version of classic control tasks, a navigation task, and a manual reaching task. The proposed method has broad applicability, ranging from imitation learning to sensorimotor neuroscience.
Centre for Cognitive Science, Technical University of Darmstadt, Darmstadt, Germany
## 1 Introduction
Inverse optimal control (IOC) is the problem of inferring an agent's cost function, and possibly other properties of their internal model, from observed behavior. While IOC has been a fundamental task in artificial intelligence, optimal control, and machine learning, particularly reinforcement learning and robotics, it has widespread applicability in several scientific fields including behavioral economics, psychology, and neuroscience. For example, in cognitive science and sensorimotor neuroscience optimal control models have been able to explain key properties of behavior, such as speed-accuracy trade-offs (Harris and Wolpert, 1998) or the minimum intervention principle (Todorov and Jordan, 2002). But, while researchers usually build an optimal control model and compare its predictions to behavior, certain parameters of the agent's internal processes are typically unknown. For example, an agent might experience intrinsic costs of behavior such as effort that are different between individuals. Inferring these parameters from observed behavior can help to understand the agent's goals, internal tradeoffs, cognitive processes and predict their behavior under novel conditions. Applying IOC in these sensorimotor control domains poses several challenges that make most previous methods not viable.
First, most IOC methods assume the agent's action signals to be known. This assumption, while convenient in simulations or robotics applications, where the control signals may be easily quantified, does not hold in many other real-world applications. In transfer learning or behavioral experiments, the action signals are internal quantities of an animal or human, e.g., neural activity or muscle activations, and are therefore not straightforwardly observable. Thus, it is worthwhile to consider the scenario where a researcher has observations of the system's state only, i.e., measurements of the animal's behavior.
Second, with few exceptions (e. C.g. Chen and Ziebart, 2015; Kwon et al., 2020), IOC methods do not account for partial observability from the agent's perspective and model the variability of the agent using a maximum causal entropy formulation (MCE; Ziebart et al., 2010). However, many real world control problems involve sensory uncertainty, which makes the state of the world partially observable and therefore contributes to the agent's stochasticity. As an example, in sensorimotor neuroscience the noise or uncertainty in the sensory system can be well described quantitatively so that accurate observation models can be formulated, which are helpful to understand the variability of behavior (Wolpert and Ghahramani, 2000).
Third, many IOC methods are based on matching feature expectations of the cost function between the model and observed data (e.g. Ziebart et al., 2010), and are thus not easily adapted to infer parameters of other parts of the model. The cost function is often not the only quantity of interest in a behavioral experiment,
where researchers might be interested in also inferring the noise characteristics of the motor system or other properties of the agent's internal model (e.g. Golub et al., 2013).
Fourth, in many real-world scenarios, the problem is not well modeled with linear dynamics and Gaussian noise, which would allow applying linear quadratic Gaussian (LQG) control (Anderson and Moore, 1990). First, the dynamics of the system may be non-linear. A common example comes from robotics and motor control, where joint angles in a kinematic chain need to be controlled and the physical system's dynamics involve inertia, centripetal, and Coriolis forces, as well as friction and torque in the joints. Secondly, the stochasticity of the system may not be well captured by normal distributions. A prime example is biological sensorimotor control, where the system is not only non-linear but both the sensory and action noise distributions are additionally signal dependent, i.e. the variability of sensory and control signals scale with their respective means. While iterative method for solving the optimal control problem exist (Todorov and Li, 2005), here we consider the corresponding inverse problem.
To address these issues, we adopt a probabilistic perspective of the IOC problem. We distinguish between the control problem faced by the agent and the inference problem the researcher has to solve. From the agent's perspective, the problem consists of acting in a partially observable Markov decision process (POMDP), for which the probabilistic graphical model is shown in Figure 1, left. We consider the setting of continuous states and actions, stochastic non-linear dynamics, partial observations, and finite horizon. For this setting, there are efficient approximately optimal solutions to the estimation and control problem, for which we give an overview in Section 2. The researcher, on the other hand, is interested in inferring properties of the agent's model and cost function. The IOC problem from their perspective can also be formulated using a probabilistic graphical model (Figure 1, right), in which the state of the system is observed, while quantities internal to the agent are latent variables.
Here, we unify MCE models, which are agnostic regarding the probabilistic structure causing the observed stochasitcity of the agent's policy, with IOC methods, which involve an explicit observation model. We allow for both: We employ an explicit observation model, but also allow the agent to have additional stochasticity through a MCE policy. We provide a solution to the IOC problem in this setting by approximate filtering of the agent's state estimate via local linearization, which allows marginalizing over these latent variables and deriving an approximate likelihood function for observed trajectories given parameters (Section 3). This function can be efficiently evaluated as it consists of a single forward pass. An estimate of the optimal parameters can then be determined using a gradient-based optimizer, maximizing the approximate likelihood. We evaluate our proposed method on two classical control tasks, pendulum and cartpole, as well as a navigation task, and a manual reaching task (Section 4).
### Related work
Inferring costs or utilities from behavior has been of interest for a long time in several scientific fields, such as behavioral economics, psychology, and neuroscience (Mosteller and Nogee, 1951; Kahneman and Tversky, 1979; Kording and Wolpert, 2004). More specific to the problem formulation adopted here, estimating objective functions in the field of control was first investigated by Kalman (1964) in the context of deterministic linear systems with quadratic costs. More recent formulations were developed first for discrete state and action spaces under the term inverse reinforcement learning (IRL; Ng et al., 2000; Abbeel and Ng, 2004), including formulations allowing for stochasticty in action selection (Rothkopf and Dimitrakakis, 2011). In this line, the maximum entropy (ME; Ziebart et al., 2008) and MCE formulation (Ziebart et al., 2010) gave rise to a whole string of new methods, such as addressing the IOC problem for non-linear continuous systems via linearization (e.g. Levine and Koltun, 2012) or importance sampling (Boularias et al., 2011) for the case of deterministic dynamics and full observability.
IOC methods for stochastic systems have been developed that considered the setting of affine control dynamics (Aghasadeghi and Bretl, 2011; Li et al., 2011).
Figure 1: **Left:** Decision network from the agent’s perspective (following the notational convention used in Kochenderfer et al., 2022). At each time step \(t\), the agent receives a partial or noisy observation \(\mathbf{y}_{t}\) of the actual state \(\mathbf{x}_{t}\). The agent performs an action \(\mathbf{u}_{t}\) and incurs a cost \(c_{t}\). **Right:** Probabilistic graphical model from the researcher’s perspective, who observes a trajectory \(\mathbf{x}_{1:T}\) from an agent. Quantities that are internal to the agent, i.e., their partial observations \(\mathbf{y}_{t}\), their internal beliefs \(\mathbf{b}_{t}\) and the action signals \(\mathbf{u}_{t}\) are not directly observed.
Arbitrary non-linear stochastic dynamics in the infinite horizon setting have been approached using model-free deep MCE IRL formulations (Finn et al., 2016; Garg et al., 2021). The latter approaches, however, yield no interpretable representation as the reward function is represented as a neural network. The partially observable setting for IOC has previously been addressed in the case of deterministic dynamics for the discrete state-action space (Choi and Kim, 2011) and continuous state, discrete action space (Silva et al., 2019). Schmitt et al. (2016) addressed systems with linear dynamics and continuous controls for a linear switching observation model. Other work has considered partial observability from the researcher's perspective, e.g., through occlusions (Kitani et al., 2012; Bogert et al., 2016). There are some IOC methods which are applicable to partially observable and stochastic systems: Linear-quadratic-Gaussian systems have been regarded by Schultheis et al. (2021), while the work of Chen and Ziebart (2015) can be used to estimate cost functions that depend on the state only. Non-linear dynamics in the infinite-horizon setting have been approached by Kwon et al. (2020) by training a policy network as a function of the whole parameter space. This work, however, also assumes the action signals to be given and a stationary policy.
Applications where IOC methods have been used to estimate cost functions range from human locomotion (Mombaur et al., 2010) over spatial navigation (Rothkopf and Ballard, 2013), table tennis (Muelling et al., 2014), to attention switching (Schmitt et al., 2017), and target tracking (Straub and Rothkopf, 2022). Other work has been aimed at inferring other properties of control tasks, e.g. learning the dynamics model (Golub et al., 2013), learning rules (Ashwood et al., 2020), or discount functions (Schultheis et al., 2022). Several subfields of robotics including imitation and apprenticeship learning (Taylor and Stone, 2009) as well as transfer learning (Osa et al., 2018) have also employed IOC.
## 2 Background
Before we introduce our probabilistic approach to inverse optimal control, we give an overview of the control and filtering problems faced by the agent and algorithms that can be used to solve it. For a summary of our notation in this paper, see Appendix A.
### Partially observable Markov decision processes
We consider a special case of partially observable Markov decision processes Astrom (1965); Kaelbling et al. (1998), a discrete-time stochastic non-linear dynamical system (Figure 1, left) with states \(\mathbf{x}_{t}\in\mathbb{R}^{n}\) following the dynamics equation \(\mathbf{x}_{t+1}=f(\mathbf{x}_{t},\mathbf{u}_{t},\mathbf{v}_{t})\), where \(f:\mathbb{R}^{n}\times\mathbb{R}^{u}\times\mathbb{R}^{v}\to\mathbb{R}^{n}\) is the dynamics function, \(\mathbf{u}_{t}\in\mathbb{R}^{u}\) are the controls and \(\mathbf{v}_{t}\sim\mathcal{N}(0,I)\) is \(v\)-dimensional Gaussian noise. We assume that the agent has only partial observations \(\mathbf{y}_{t}\in\mathbb{R}^{m}\) following \(\mathbf{y}_{t}=h(\mathbf{x}_{t},\mathbf{w}_{t})\), with \(h:\mathbb{R}^{n}\times\mathbb{R}^{v}\to\mathbb{R}^{m}\) the stochastic observation function and \(\mathbf{w}_{t}\sim\mathcal{N}(0,I)\)\(w\)-dimensional Gaussian noise. While \(\mathbf{v}_{t}\) and \(\mathbf{w}_{t}\) are defined as standard normal random variables, the system can incorporate general control- and state-dependent noises through non-linear transformations within the dynamics function \(f\) and observation function \(h\). The agent's goal is to minimize the expected cost over a time horizon \(T\in\mathbb{N}\), defined by \(J=\mathbb{E}[c_{T}(\mathbf{x}_{T})+\sum_{t=1}^{T-1}c_{t}(\mathbf{x}_{t},\mathbf{u}_{t})]\), consisting of a final state cost \(c_{T}(\mathbf{x}_{T})\) and a cost at each time step \(c_{t}(\mathbf{x}_{t},\mathbf{u}_{t})\).
### Iterative linear quadratic Gaussian
The fully-observable control problem analogous to Section 2.1, where we assume that the agent acts directly on the state, can be solved approximately using the method of iterative linear quadratic Gaussian (iLQG; Todorov and Li, 2005). This method iteratively linearizes the dynamics and employs a quadratic approximation of the costs around a nominal trajectory, \(\{\bar{\mathbf{x}}_{i},\bar{\mathbf{u}}_{i}\}_{i=1,\ldots,T}\), with \(\bar{\mathbf{x}}_{i}\in\mathbb{R}^{n},\bar{\mathbf{u}}_{i}\in\mathbb{R}^{u}\), and computes the optimal linear control law, \(\mathbf{u}_{t}=\pi_{t}(\mathbf{x}_{t})=L_{t}(\mathbf{x}_{t}-\bar{\mathbf{x}}_{t})+\mathbf{m}_{t}+ \bar{\mathbf{u}}_{1:T}\) for the approximated system. The quantities \(L_{t}\) and \(\mathbf{m}_{t}\) are the control gain and offset, respectively, and determined through a backward pass for the current reference trajectory. In the following iteration, the determined optimal control law is used to generate a new reference trajectory and the process is repeated until the controller converges.
### MCE reinforcement learning
The MCE reinforcement learning is to minimize the expected cost as in Section 2.2, while maximizing the conditional entropy of the applied stochastic policy \(\Pi_{t}(\mathbf{u}_{t}\mid\mathbf{x}_{t})\), i.e., to minimize \(\mathbb{E}[J(\mathbf{x}_{1:T},\mathbf{u}_{1:T})-\sum_{t=1}^{T-1}H(\Pi_{t}(\mathbf{u}_{t} \mid\mathbf{x}_{t}))]\). This formulation has been used to formulate reinforcement learning as a probabilistic inference problem (Kappen et al., 2012; Toussaint, 2009; Levine, 2018) and for inverse reinforcement learning (IRL) to model the stochasticity of the agent (e.g.,
Ziebart et al., 2008, 2010). The objective of IRL is formulated as maximizing the likelihood of given states and actions \(\{\mathbf{x}_{t},\mathbf{u}_{t}\}_{t=1,\ldots,N}\), induced by the maximum entropy policy \(\Pi_{t}(\mathbf{u}_{t}\,|\,\mathbf{x}_{t})\).
It can be shown that the resulting optimal policy is given by the distribution \(\Pi_{t}(\mathbf{u}_{t}\,|\,\mathbf{x}_{t})=\exp(Q_{t}(\mathbf{x}_{t},\mathbf{u}_{t})-V_{t}(\bm {x}_{t}))\), where \(Q_{t}\) is the soft Q-function at time \(t\), given by \(Q_{t}(\mathbf{x}_{t},\mathbf{u}_{t})=-c_{t}(\mathbf{x}_{t},\mathbf{u}_{t})-\mathbb{E}[V_{t+1}( \mathbf{x}_{t+1})]\) and \(V_{t}\) the normalization, i.e., \(V_{t}(\mathbf{x}_{t})=\log\int_{\mathbf{u}_{t}}\exp(Q_{t}(\mathbf{x}_{t},\mathbf{u}_{t}))\, \mathrm{d}\mathbf{u}_{t}\)(Gleave and Toyer, 2022). For general dynamics and reward functions, it is hard to compute the soft Q-function exactly. Approximate solutions have been derived using linearization (Levine and Koltun, 2012) or importance sampling (Boularias et al., 2011). For the case of linear dynamics and quadratic reward, the optimal policy is given by a Gaussian distribution \(\Pi_{t}(\mathbf{u}_{t}\,|\,\mathbf{x}_{t})=\mathcal{N}(\mathbf{u}_{t};L_{t}\mathbf{x}_{t},-L _{t})\), where \(L_{t}\) is the controller gain of the LQG controller (Levine and Koltun, 2013). This formulation can be extended to non-linear systems by using the control law in conjuction with the iLQG method (Section 2.2).
### Extended Kalman filter
Given the system defined in Section 2.1, the optimal filtering problem is to compute a belief distribution of the current state given past observations, i.e., \(p(\mathbf{x}_{t}\,|\,\mathbf{y}_{1:t-1})\). For linear-Gaussian systems, the solution is given in closed form and known as the Kalman filter (Kalman, 1960). In case of non-linear systems as in Section 2.1, a Gaussian approximation to the optimal belief can be computed using the extended Kalman filter via \(\mathbf{b}_{t+1}=f(\mathbf{b}_{t},\mathbf{u}_{t},0)+K_{t}(\mathbf{y}_{t}-h(\mathbf{b} _{t},0))\), where \(\mathbf{b}_{t}\in\mathbb{R}^{n}\) denotes the mean of the Gaussian belief \(p(\mathbf{x}_{t}\,|\mathbf{y}_{1},\ldots,\mathbf{y}_{t-1})\). The matrix \(K_{t}\) denotes the Kalman gain for time \(t\) and is computed by applying the Kalman filter to the system locally-linearized around the nominal trajectory obtained by the approximate optimal control law of iLQG (Section 2.2).
## 3 Probabilistic IOC
We consider an agent acting in a partially observable Markov decision process as introduced in Section 2.1. We assume that the agent acts at time \(t\) based on their belief \(\mathbf{b}_{t}\) about the state of the system \(\mathbf{x}_{t}\), which evolves according to \(\mathbf{b}_{t+1}=\beta_{t}(\mathbf{b}_{t},\mathbf{u}_{t},\mathbf{y}_{t})\). While the belief of the agent is defined commonly as a distribution over the true state, here we model \(\mathbf{b}_{t}\) as a finite-dimensional summary statistics of the distribution, i.e., \(\mathbf{b}_{t}\in\mathbb{R}^{b}\). The function \(\beta_{t}:\mathbb{R}^{b}\times\mathbb{R}^{u}\times\mathbb{R}^{m}\to\mathbb{R}^ {b}\) is called belief dynamics. We further assume that the agent follows a time-dependent policy \(\pi_{t}:\mathbb{R}^{b}\times\mathbb{R}^{j}\to\mathbb{R}^{u}\), i.e., \(\mathbf{u}_{t}=\pi_{t}(\mathbf{b}_{t},\mathbf{\xi}_{t})\), which can be stochastic with \(\mathbf{\xi}_{t}\sim\mathcal{N}(0,I)\). Note that both the belief dynamics and the policy can be time-dependent.
In the inverse optimal control problem, the goal is to estimate parameters \(\mathbf{\theta}\in\mathbb{R}^{p}\) of the agent's optimal control problem given the model and trajectory data. These parameters can include properties of the agent's cost function, the sensory and control systems of the agent, or the system's dynamics. We follow a probabilistic approach to inverse optimal control, i.e., we consider the likelihood function
\[p(\mathbf{x}_{1:T}\,|\,\mathbf{\theta})=p(\mathbf{x}_{0}\,|\,\mathbf{\theta})\prod_{t=0}^{T-1}p (\mathbf{x}_{t+1}\,|\,\mathbf{x}_{1:t},\mathbf{\theta}), \tag{1}\]
describing the probability of the observed trajectory data \(\mathbf{x}_{1:T}:=\{\mathbf{x}_{t}\}_{t=1,\ldots,T}\) given the parameters. For a set of trajectories we assume them to be independent given the parameters so that the likelihood factorizes into single trajectory likelihoods of the form in Equation (1). In this equation, generally, each state \(\mathbf{x}_{t+1}\) depends on all previous states \(\mathbf{x}_{1},\ldots,\mathbf{x}_{t}\), because the agent's internal noisy observations and control signals are not accessible to the researcher (Figure 1, right). Therefore, the Markov property does not hold from the researcher's perspective, rendering computation of the likelihood function intractable. To deal with this problem, we employ two key insights: First, the joint dynamical system of the states and the agent's belief is Markovian (Van Den Berg et al., 2011). Second, by keeping track of the distribution over the agent's belief, i.e., by performing belief tracking (Schultheis et al., 2021), we can iteratively compute the individual factors of the likelihood function in Equation (1).
We first introduce a general formulation of the IOC likelihood involving marginalization over the agent's internal beliefs in Section 3.1. Then, we show how to make the computations tractable by local linearization in Section 3.2. In Section 3.3, we provide details for suitable linearization points, which enables us to evaluate the approximate likelihood within a single forward pass.
### Likelihood formulation
We start by defining a joint dynamical system of states and beliefs (Van Den Berg et al., 2011) in which each depends only on the state and belief at the previous time step and the noises. For that, we insert the policy into the dynamics and the policy and observation function into the belief dynamics, yielding the equation
\[\begin{bmatrix}\mathbf{x}_{t+1}\\ \mathbf{b}_{t+1}\end{bmatrix} =\begin{bmatrix}f(\mathbf{x}_{t},\pi_{t}(\mathbf{b}_{t},\mathbf{\xi}_{t}),\mathbf{v}_{t})\\ \beta_{t}(\mathbf{b}_{t},\pi_{t}(\mathbf{b}_{t},\mathbf{\xi}_{t}),h(\mathbf{x}_{t},\mathbf{ w}_{t}))\end{bmatrix} \tag{2}\] \[=:g(\mathbf{x}_{t},\mathbf{b}_{t},\mathbf{v}_{t},\mathbf{w}_{t},\mathbf{\xi}_{t}). \tag{3}\]
For given values of \(\mathbf{x}_{t}\) and \(\mathbf{b}_{t}\), this equation defines the distribution \(p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\mid\mathbf{x}_{t},\mathbf{b}_{t})\), as \(\mathbf{v}_{t},\mathbf{w}_{t},\mathbf{\xi}_{t}\) are independent of \(\mathbf{x}_{t+1}\) and \(\mathbf{b}_{t+1}\). In Section 3.2 we will introduce an approximation via linearization, which leads to a closed-form expression for \(p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\mid\mathbf{x}_{t},\mathbf{b}_{t})\).
One can use this Markovian joint dynamical system to compute the likelihood factors for each time step (Schultheis et al., 2021). To this end, we first rewrite the individual likelihood terms \(p(\mathbf{x}_{t+1}\!\mid\!\mathbf{x}_{1:t})\) of Equation (1) by marginalizing over the agent's belief at each time step, i.e.,
\[p(\mathbf{x}_{t+1}\!\mid\!\mathbf{x}_{1:t})=\int p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\! \mid\!\mathbf{x}_{1:t})\,\mathrm{d}\mathbf{b}_{t+1}. \tag{4}\]
As the belief is an internal quantity of the agent and thus not observable to the researcher, we keep track of its distribution, \(p(\mathbf{b}_{t}\!\mid\!\mathbf{x}_{1:t})\). For this, we rewrite
\[p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{1:t})=\int p(\mathbf{ x}_{t+1},\mathbf{b}_{t+1},\mathbf{b}_{t}\mid\!\mathbf{x}_{1:t})\,\mathrm{d} \mathbf{b}_{t}\\ =\int p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{t},\mathbf{b} _{t})\,p(\mathbf{b}_{t}\!\mid\!\mathbf{x}_{1:t})\,\mathrm{d}\mathbf{b}_{t}, \tag{5}\]
where we have exploited the fact that the joint dynamical system of states and beliefs is Markovian. The distribution \(p(\mathbf{b}_{t}\!\mid\!\mathbf{x}_{1:t})\) acts as a summary of the past states and can be computed by conditioning on the current state, i.e.,
\[p(\mathbf{b}_{t}\!\mid\!\mathbf{x}_{1:t})=\frac{p(\mathbf{x}_{t},\mathbf{b}_{t}\!\mid \!\mathbf{x}_{1:t-1})}{p(\mathbf{x}_{t}\!\mid\!\mathbf{x}_{1:t-1})}. \tag{6}\]
After determining \(p(\mathbf{b}_{t}\!\mid\!\mathbf{x}_{1:t})\), we can propagate it through the joint dynamical system to arrive at the distribution \(p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{1:t})\). To obtain the belief distribution of the following time step, \(p(\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{1:t+1})\), we condition on the observed state \(\mathbf{x}_{t+1}\). To obtain the likelihood contribution, on the other hand, we marginalize out the \(\mathbf{b}_{t+1}\).
To summarize, starting with an initialization \(p(\mathbf{b}_{0})\), we can compute the individual terms \(p(\mathbf{x}_{t+1}\!\mid\!\mathbf{x}_{1:t})\) of the likelihood by executing Algorithm 1.
```
0: Approximate likelihood of parameters \(p(\mathbf{x}_{1:T}\!\mid\!\mathbf{\theta})\)
0: Parameters \(\mathbf{\theta}\), Data \(\mathbf{x}_{1:T}\), Model \(f,h\)
1: Determine the policy \(\pi\) using iLQG
2: Determine the belief dynamics \(\beta\) using the EKF
3:for\(t\) in \(\{1,\dots,T-1\}\)do
4: Compute \(p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{1:t})\) using Equation (5)
5: Update \(p(\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{1:t+1})\) using Equation (6)
6: Obtain \(p(\mathbf{x}_{t+1}\!\mid\!\mathbf{x}_{1:t})\) using Equation (4)
7:endfor
```
**Algorithm 1** Approximate likelihood computation
### Tractable likelihood via linearization
While the marginalization and propagating operations listed in the previous section can be done in closed form for linear-Gaussian systems, this is no longer feasible for non-linear systems. Therefore, we follow the approach of local linearization used in iLQG (Section 2.2) and the extended Kalman filter (Section 2.4). For the belief statistics, we consider the mean of the agent's belief, i.e., \(\mathbf{b}_{t}=\mathbb{E}[\mathbf{x}_{t}\!\mid\!\mathbf{y}_{1},\dots,\mathbf{y}_{t-1}]\) and initialize the distribution for the first time step as a Gaussian, \(p(\mathbf{b}_{1})=\mathcal{N}(\mu_{1}^{(b)},\Sigma_{1}^{(b)})\). We then approximate the distribution \(p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{t},\mathbf{b}_{t})\) as a Gaussian by applying a first-order Taylor expansion of \(g\).
In order to obtain a closed-form expression for \(g\), which we can linearize, we model the agent's belief dynamics using the extended Kalman filter (Section 2.4) and its policy using iLQG (Section 2.2), as in the partially observable version of iLQG (Li and Todorov, 2007). This choice leads to an affine control law and belief dynamics given \(\mathbf{b}_{t}\), making linearization of \(p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{t},\mathbf{b}_{t})\) straight-forward. To allow for additional stochasticity in the agent's policy, we follow the common formulation of maximum causal entropy (MCE) reinforcement learning (Section 2.3). For the linearized dynamics, the MCE policy is - as for the fully-observable case (Section 2.3) - given by a Gaussian distribution, so that \(\pi_{t}(\mathbf{b}_{t},\mathbf{\xi}_{t})=L_{t}(\mathbf{b}_{t}-\bar{\mathbf{x}}_{1:T})+ \mathbf{m}_{t}+\tilde{\mathbf{u}}_{1:T}-\tilde{L}_{t}\mathbf{\xi}_{t}\), with \(\tilde{L}_{t}\) the Cholesky decomposition of \(L_{t}\), and can be marginalized out in closed form.
The approximations we have introduced allow us to solve the integral in Equation (5) in closed form by applying standard equations for linear transformations of Gaussians, resulting in
\[p(\mathbf{x}_{t+1},\mathbf{b}_{t+1}\!\mid\!\mathbf{x}_{1:t})\approx\mathcal{N}\left(\mu_ {t},\Sigma_{t}\right) \tag{7}\]
with
\[\mu_{t} =g(\mathbf{x}_{t},\mu_{t}^{(b)},0,0,0),\] \[\Sigma_{t} =J_{\mathbf{b}}\Sigma_{t}^{(b)}J_{\mathbf{b}}^{T}+J_{\mathbf{v}}J_{\bm {v}}^{T}+J_{\mathbf{w}}J_{\mathbf{w}}^{T}+J_{\mathbf{\xi}}J_{\mathbf{\xi}}^{T},\]
where \(J_{\bullet}\) denotes the Jacobian of \(g\) w.r.t. \(\bullet\), evaluated at \((\mathbf{x}_{t},\mu_{t}^{(b)},0,0,0)\). Under this Gaussian approximation, both remaining operations of Algorithm 1 can also be performed in closed form. A more detailed derivation and representation of these formulas can be found in Appendix B. If the agent has full observations of the system's state, the inverse optimal control problem is simplified significantly. The derivations for this special case are shown in Appendix C. Details about the implementation are provided in Appendix D.
### Data-based linearization
The described approach to evaluate the likelihood requires solving the optimal filtering and control problem for a given set of parameters. When iteratively maximizing the likelihood, we would have to solve both problems in every iteration, making the approach computationally expensive. We can make the method more efficient by using the insight that in the IOC problem, we are given a trajectory \(\mathbf{x}_{1:T}\). Instead of starting with a randomly initialized nominal trajectory and iterating between computation of the locally optimal control law and linearizing again, we can simply linearize the dynamics once around the given trajectory and keep this linearization fixed. We then need to perform only one backward-pass to compute an approximately optimal control law given the current parameters, and a forward pass to compute an approximately optimal filter. This, in particular, allows efficient computation of the gradient of the likelihood function. As we assume the actions to be unobservable, but they are needed for the linearization, we compute estimates of the actions by minimizing the squared difference of the noiseless state estimates and the actual states. Note that these estimated actions are only used for the linearization, but are not used as observed actions in the IOC likelihood itself (see Appendix E).
## 4 Experiments
We evaluated our proposed method on simulated data of two classic control tasks (Pendulum and CartPole) and two behavioral human tasks (reaching and navigation). To evaluate the accuracy of the parameter estimates obtained by our method and to compare it against a baseline, we computed absolute relative errors per parameter \(|(\theta-\hat{\theta})/\theta|\). This makes averages across parameters on different scales more interpretable compared to other metrics like root mean squared errors. For each task, we simulated 100 sets of parameters from a uniform distribution in logarithmic space. For each set of parameters, we simulated 50 trajectories. We then maximized the log likelihood using gradient-based optimization with automatic differentiation (L-BFGS algorithm; Zhu et al., 1997). See Appendix G for a summary of the hyperparameters of our experiments.
All tasks we consider have four free parameters: cost of actions \(c_{a}\), cost of velocity at the final time step \(c_{v}\), motor noise \(\sigma_{m}\), and observation noise \(\sigma_{o}\). In the fully observable case, we leave out the observation noise parameter and only infer the three remaining parameters. For concrete definitions of the parameters in each specific tasks, see Appendix F.
Figure 2: **IOC likelihood for the non-linear reaching task.****(a)** Simulated reaching trajectories for eight targets. Increasing the cost of actions and the motor noise has an effect on the trajectories, since perfectly reaching the target becomes less important to the agent and variability increases. **(b)** IOC log likelihood for two of the model parameters, action costs \(c_{a}\) and motor noise \(\sigma_{m}\). The likelihood has its maximum (pink cross) close to the ground truth parameter values (black dot). **(c)** Simulated trajectories using the MLEs from (b). The simulations are visually indistinguishable from the ground truth data.
### Baseline method
For a comparison to previously proposed methods, we applied a baseline method based on the maximum causal entropy (MCE) approach (Ziebart et al., 2010), as these formulations have been successfully used for IOC in non-linear stochastic systems. As to the best of our knowledge, no past method can be directly applied in the setting we consider (non-linear and stochastic dynamics, unknown controls, partial observability, finite horizon), we choose a straight-forward implementation of the MCE formulation for this case. The tasks we consider can be well solved by linearizing the dynamics locally, so an accurate approximation of the optimal MCE controller is given by the optimal MCE controller for the linearized dynamics (see Section 2.3). An estimate of the parameters is then obtained by maximizing the likelihood of the approximate maximum entropy policy for the given set of states and controls. To apply this baseline to the setting where control signals are missing, we use the estimates of the controls as we determine in our proposed method for the data-based linearization (Section 3.3). As past IOC methods do not have an explicit model of partial observability, except for a few exceptions which are limited to specific tasks, we follow the usual formulation of the policy acting directly on the states. To show that this approach constitutes a suitable baseline, in Appendix H.3, we provide results for the case where the true control signals are known and there is no partial observability.
### Reaching task
We evaluate the method on a manual reaching task with a non-linear biomechanical model of a two-link arm. The agent's goal is to move its arm towards a target in the horizontal plane by controlling its two joints. For a more detailed description, see Appendix F.2. Note that the cost function is non-linear in states because the positions are a non-linear function of the joint angles that comprise the state of the system. We use a fully observable version of the task (Todorov and Li, 2005) and a version in which the agent receives noisy observations (Li and Todorov, 2007). This model has been applied to reaching movements in the sensorimotor neuroscience literature (e.g., Nagengast et al., 2009; Knill et al., 2011). Figure 1(a) shows simulations from the model using iLQG with two different parameter settings. We evaluated the likelihood function for a grid of two of the model parameters (Figure 1(b)) to illustrate that it has well-defined maxima close to the true parameter values. In this example, simulated data using the maximum likelihood estimates look indistinguishable from the ground truth data (Figure 1(c)).
In Figure 3 we present maximum likelihood parameter estimates and true values for repeated runs with different random parameter settings. One can observe that the parameter estimates of our method closely align with the true parameter values, showing that our method can successfully recover the parameters from data. The baseline method, in contrast, shows considerably worse performance, in particular for estimating noises. Estimates for the fully observable case are provided in Appendix H.2. To quantify the accuracy of the maximum likelihood estimates, we computed the absolute relative errors. The results are shown separately for the fully observable and partially observable cases in Figure 4. The median absolute relative errors of our method were 0.11, while they were 0.93 for the baseline. The influence of missing control signals and of the lacking explicit observation model in the baseline can be observed by comparing the results to the fully-observable case and the case of given control signals in Appendix H.2 and Appendix H.3.
### Navigation task
In the navigation task, we consider an agent navigating to a target under non-linear dynamics while receiving noisy observations from a non-linear observation model. To reach the target, the agent can control the angular velocity of their heading direction and the acceleration with which they move forward. The agent observes noisy versions of the distance to the target and the target's bearing angle. We provide more details about the experiment in Appendix F.3.
Maximum likelihood parameter estimates for the navigation task are shown for the partially observable case in Figure S4 and for the fully observable case in Figure S8. As for the reaching task, our method provides parameter estimates close to the true ones, while the estimates of the baseline deviate for a large number of trials. Median absolute relative errors of our method were 0.31, while they were 1.99 for the baseline (Figure 4).
### Classic control tasks
Lastly, we evaluate our method on two classic control tasks (Pendulum and Cart Pole) based on the implementations in the gym library (Brockman et al., 2016). Because these tasks are neither stochastic nor partially observable in their standard formulations, we introduce noise on the dynamics and turn them into partially-observed problems by defining a stochastic observation function (see Appendix F.1). In Appendix H we show the parameter estimates for the Pendulum (Figure S2) and for the Cart Pole (Figure S3) for the partially ob
servable case, while Figure S6 and Figure S7 show the fully observable case, respectively. One can observe that the results match the ones of the reaching and navigation task, showing that our method provides accurate estimates of the parameters. Median absolute relative errors of our method were 0.12 and 0.41, while for the baseline they were 2.21 and 3.82 (Figure 4).
## 5 Conclusion
In this paper, we introduced a new method for inverse optimal control for systems with stochastic dynamics, partial observability, and missing control signals. We followed a probabilistic formulation of the problem, where the goal is formulated as maximizing the likelihood of the observed states given the parameters. As the exact evaluation of the likelihood for a general non-linear model is intractable, we developed an efficient approximation of the likelihood by linearizing the system locally around the given trajectories, as in popular approaches such as the extended Kalman filter or iLQG. By maintaining a Gaussian distribution that tracks the agent's state estimate, the proposed method is able to evaluate an approximate likelihood in closed form within a single forward pass.
Besides offering an efficient way to evaluate the likelihood, our proposed formulation is able to incorporate multiple sources of the stochasticity of the agent through an explicit model of the partial observability and by modelling control via a maximum causal entropy (MCE) policy. Our method thereby reconciles the theory of past MCE IOC algorithms (e.g., Ziebart et al., 2010) and approaches where the agent's stochasticity stems from an explicit stochastic observation model (Schultheis et al., 2021).
We have applied our method to two stochastic variations of classical control tasks, the pendulum and cart pole, and to two human behavioral tasks, a reaching and navigation tasks. In the comparison to a MCE baseline, for which missing control signals need to be estimated, we have found our method to achieve lower estimation errors across all evaluated tasks. Further, it successfully inferred noise parameters of the system, which was not possible with the baseline.
The limitations of our method are mainly due to the linearization of the dynamical system and the Gaussian approximations involved in the belief tracking formulation of the likelihood function. In more complex scenarios with belief distributions that are not well approximated by a Gaussian, e.g., multimodal beliefs, the method is likely to produce inaccurate results. This problem could be addressed by replacing the closed-form Gaussian approximation of the belief by particle-based methods (Doucet et al., 2001). Further, we focused on tasks which could be solved well by applying controllers based on linearization and Gaussian approximation (iLQG and EKF), motivated by their popularity in applications in cognitive science and neuroscience. High-dimensional problems that cannot be solved forward using iLQG, in contrast, are probably not directly solvable using our proposed method. While, in principle, our method is also applicable using other forward control methods that compute differentiable policies, it is unclear whether linearizing these policies leads to accurate approximate likelihoods and parameter estimates.
A further limitation of our method is that it requires parametric models of the dynamics and noise structure.
Figure 3: **Maximum likelihood estimates for reaching task** True parameter values plotted against the maximum likelihood parameter estimates for the partially observable reaching task. Top row: our method, bottom row: MCE baseline. The columns contain the four different model parameters (action cost \(c_{a}\), velocity cost \(c_{v}\), motor noise \(\sigma_{m}\), observation noise \(\sigma_{o}\)).
While single missing parameters can be determined using our method, in the case of completely unknown dynamics a model-free approach to IOC would be more suitable.
Lastly, while we have shown that inference of few parameters is feasible, the results probably do not scale to a large number of parameters. One reason for this is that optimization in a high-dimensional non-linear space becomes difficult, and one can potentially get stuck in local minima. This problem could be relieved by using more advanced optimization methods. A further, more fundamental, concern with a large number of parameters is that parameters are likely to become not unambiguously identifiable and there is no unique solution. However, in many scientific fields, knowledge about the structure and parametric models describing the agent's uncertainty and internal model are available or measurable, allowing our method to be used successfully. Moreover, our probabilistic approach with a closed-form likelihood opens up the possibility of using Bayesian methods to investigate the identifiability of model parameters (Acerbi et al., 2014).
Our proposed method provides a tool for researchers interested in modeling sequential behavior, e.g., in sensorimotor domains, allowing to infer an agent's subjective costs and internal uncertainties. This will enable answering novel scientific questions about how these quantities are affected by different experimental conditions, deviate from intended task goals and provided task instructions, or how they vary between individuals. This is particularly relevant to a computational understanding of naturalistic behavior (Krakauer et al., 2017; Cisek and Pastor-Bernier, 2014; Miller et al., 2022), for which subjective utilities are mostly unknown.
## Acknowledgements
The authors gratefully acknowledge the computing time provided to them on the high-performance computer Lichtenberg at the NHR Centers NHR4CES at TU Darmstadt, and financial support by the project "Whitebox" funded by the Priority Program LOEWE of the Hessian Ministry of Higher Education, Science, Research and Art.
| inverse最適制御は、決断の意思決定における行動を特徴付けるのに用いることができます。しかし、現状の多くの研究は、完全に可視化可能であるか、線形システムに限定されており、または、行動信号が知られている必要があります。ここでは、可視性が低い確率的逆最適制御アプローチを、行動信号が不明な部分的観察可能な確率的非線形システムに導入します。これにより、逆最適制御の既存の研究を最大因果 entropi の要約と組み合わせます。センサと運動システムのノイズの特徴を明確に表現したモデルと、位置線形化技術を組み合わせることで、モデルパラメータに関する近似的尤度関数を導出します。モデルパラメータを計算する際に、単一の方向伝播で計算可能です。本方法を、確率的および可視性が低いバージョンで、2つの古典的な制御タスクと2つの人間の行動タスクに |
2305.12770 | FGAM:Fast Adversarial Malware Generation Method Based on Gradient Sign | Malware detection models based on deep learning have been widely used, but
recent research shows that deep learning models are vulnerable to adversarial
attacks. Adversarial attacks are to deceive the deep learning model by
generating adversarial samples. When adversarial attacks are performed on the
malware detection model, the attacker will generate adversarial malware with
the same malicious functions as the malware, and make the detection model
classify it as benign software. Studying adversarial malware generation can
help model designers improve the robustness of malware detection models. At
present, in the work on adversarial malware generation for byte-to-image
malware detection models, there are mainly problems such as large amount of
injection perturbation and low generation efficiency. Therefore, this paper
proposes FGAM (Fast Generate Adversarial Malware), a method for fast generating
adversarial malware, which iterates perturbed bytes according to the gradient
sign to enhance adversarial capability of the perturbed bytes until the
adversarial malware is successfully generated. It is experimentally verified
that the success rate of the adversarial malware deception model generated by
FGAM is increased by about 84\% compared with existing methods. | Kun Li, Fan Zhang, Wei Guo | 2023-05-22T06:58:34 | http://arxiv.org/abs/2305.12770v1 | # FGAM:Fast Adversarial Malware Generation Method Based on Gradient Sign
###### Abstract
Malware detection models based on deep learning have been widely used, but recent research shows that deep learning models are vulnerable to adversarial attacks. Adversarial attacks are to deceive the deep learning model by generating adversarial samples. When adversarial attacks are performed on the malware detection model, the attacker will generate adversarial malware with the same malicious functions as the malware, and make the detection model classify it as benign software. Studying adversarial malware generation can help model designers improve the robustness of malware detection models. At present, in the work on adversarial malware generation for byte-to-image malware detection models, there are mainly problems such as large amount of injection perturbation and low generation efficiency. Therefore, this paper proposes FGAM (Fast Generate Adversarial Malware), a method for fast generating adversarial malware, which iterates perturbed bytes according to the gradient sign to enhance adversarial capability of the perturbed bytes until the adversarial malware is successfully generated. It is experimentally verified that the success rate of the adversarial malware deception model generated by FGAM is increased by about 84% compared with existing methods.
Adversarial malware, Adversarial examples, Adversarial attacks, Deep learning, Malware detection
## I Introduction
Malware is the primary method of network attacks, which can infect users' computers without the user's permission, posing considerable threats to users' information and property security [1]. With the rapid development of Internet technology, the cost of spreading malware is further reduced. In recent years, the amount of malware has proliferated. To deal with the rapidly growing malware, researchers began to use deep learning techniques to detect malware [2][3][4]. Traditional malware detection methods [5] detect malware by manually setting filter rules. There are problems such as detection rules being too complex, detection rule setting requires a lot of expert knowledge, and detection rules need to be updated in real time. Different from it, the detection method based on deep learning automatically obtains malware features through data learning. Research shows that this detection method can have a high accuracy rate of malware detection, and does not rely on the quality of manual rules.
However, deep learning is vulnerable to adversarial examples due to the uneven distribution of training data and insufficient robustness of model design [6]. Adversarial attack was first proposed by Szegedy et al. [6], and it is defined as fooling the neural network model by generating pair samples. In the work of Szegedy et al. [6], perturbations are added to the pixels of the original image to generate adversarial images (adversarial examples). In addition, the perturbation size is limited to ensure that the original image is consistent with the human visual observations of the adversarial image. When the adversarial image (adversarial example) and the original image are input to the neural network, the neural network outputs different results. Adversarial attacks in malware are different from images. Malware has semantics, so perturbations must be limited to ensure that the generated adversarial malware has the same functionality as the original malware. In addition, adversarial malware should be able to fool the detection model, i.e., adversarial malware is classified as benign software by the detection model.
The malware detection model based on byte-to-image was first proposed by Cui et al. [7], and it was further improved by Tekerek et al. [8], and now it can have a high detection accuracy rate. The method first converts malware bytes into images, which are then classified using convolutional neural networks. Note that the malware detection model based on byte-to-image is also vulnerable to adversarial malware. However, there are some limitations to the adversarial malware generative approach to this model due to the discrete properties and semantics of PE(Portable Executable) files. Such as the generated adversarial malware losing their original functions [9], the generation efficiency of the adversarial malware being low [10][11], and the injection of the generated adversarial malware being too much perturbation [12]. Therefore, in this paper, FGAM (Fast Generate Adversarial Malware) is proposed, a method for rapidly generating adversarial malware based on gradient signs, which injects perturbation through functionality-preserving manipulations to ensure that the adversarial malware has the same function as the original malware. In addition, the method iterates byte perturbation to enhance perturbation adversarial capability. The enhanced adversarial capability of byte perturbation can reduce the amount of perturbation and improve the success rate of generating adversarial malware. The main contributions of this paper are as follows:
1. We use a reverse gradient sign to enhance its adversarial capabilities with iteration byte perturbation, and use function-preserving manipulations to inject perturbations to generate adversarial malware. In addition, we use the least square method to detect whether there is an oscillation in the malicious fraction drop rate of the model output, and shorten the
process of generating adversarial malware by ending the oscillation process early.
2. We calculate the entropy distribution of the original sample and adversarial malware and find that the perturbation injection ratio is limited, and the perturbation injection ratio and the hiding ability of adversarial malware are mutually restrictive.
3. We conduct the transfer experiment of adversarial malware, and analyze the transfer attack capability of adversarial malware under different perturbation injection ratios, perturbation injection locations, and perturbation adversarial strengths.
In this paper, we focus on the malware detection model based on byte-to-image and propose a method to generate adversarial malware quickly. In Section 2, we introduce the research background of adversarial malware and related work. In Section 3, the adversarial malware generation method proposed in this paper is presented. In Section 4, the effectiveness of the method in this paper is verified through experiments. Finally, the work of this paper is concluded, and future research directions are proposed in Section 5.
## II Background and Related Work
### _Background_
#### Ii-A1 PE format
The structure of PE files in the Windows system is shown in Figure11, which is mainly composed of DOS header, PE header, section table and section. (1) DOS header: This contains the mark MZ of the PE file and the offset of the PE header. In addition, there is the DOS stub part, which is the data required for the PE file to be loaded in the DOS environment. (2) PE header: It contains the mark character "PE" of the PE header, and its corresponding hexadecimal is 50, 45. Also contains file headers, optional headers, etc. (3) Section table: section list, including the size, location, attributes and other information of the section (4) Section: byte block, the main content of the PE file, which stores the text, data, image and other information of the PE file.
Footnote 1: [https://docs.microsoft.com/en-us/windows/win32/debug/pe-format](https://docs.microsoft.com/en-us/windows/win32/debug/pe-format)
The structure of the PE file has redundancy and alignment features. The redundancy feature is to ensure the compatibility of the PE file with the old system, and the alignment feature is to provide the high efficiency of the software when running. Due to the structural characteristics of PE files, this paper can inject content into PE files through functionality-preserving manipulations [13], and maintain the executable and functional properties of PE files. In addition, many deep learning-based malware detection methods utilize the PE file structure, which can easily extract static feature data such as API calls and import function tables from PE files.
#### Ii-A2 Deep learning-based malware detection
Deep learning technology has been widely used in the field of malware detection. According to the type of selected features, the current main detection methods can be divided into dynamic feature-based and static feature-based, as shown in Figure2.
Dynamic features are features obtained by using dedicated tools when PE files are run in an isolated environment such as a sandbox. There are mainly API sequences, API directed cycle graphs, and software behavior topology graphs. Among them, the detection method based on API sequence [14] is to process the API calls of the software runtime into time series, and use the RNN network to analyze the context relationship of the sequence to detect malware. API-based directed cycle graph detection [4] processes API calls into graph data and uses graph neural networks for classification. Topological graph detection based on software behavior [3] is to use the behaviors of file reading, memory reading, and file writing during software runtime as graph nodes, and build a topology graph according to the program running process, and then use graph neural network to classify.
Static features are features directly obtained from PE files, mainly including byte-to-image, structure entropy flow graph, bytecode, etc. Based on the byte-to-image detection method [7][8], the binary bytes of PE files are converted into images, and then classified by convolutional neural networks. The flow graph detection method based on structure entropy [15] calculates the entropy value of each part according to the PE file structure, and uses wavelet transform to process the entropy value sequence into a manifold graph, and then uses the convolutional neural network to classify. Bytecode-based detection method (MalConv) [2] firstly performs word embedding on binary bytes of PE files, and then uses the convolutional neural network for classification. Compared with the detection method based on byte-to-image, this detection method both uses the binary bytes of the PE file as the original data, but there are two differences. First, MalConv performs word embedding on binary bytes, rather than converting them to images, making it necessary to process sequences of file size lengths [16]. Specifically, when the input PE file is 2 MB, this detection method needs to process a sequence of about 2,000,000 steps. Therefore, this detection method consumes more resources than the detection method based on byte-to-image. In addition, the convolutional neural network layer used by MalConv is shallow, and the input volume of MalConv is too large. In order to reduce resource occupation, a deeper network layer cannot be used.
In practical applications, to avoid malware evading the detection model, it is necessary to combine a variety of static feature detection with dynamic feature detection. However, dynamic feature detection has problems such as long detection time, malware deliberately evading sandboxes, and high
Fig. 1: The Windows PE file format
resource consumption. Therefore, when faced with a lot of malware in the external environment, static detection is the first and most important line of defense.
### _Related Work_
For the adversarial malware attack of the malware byte-to-image detection model, Liu et al. [9] first proposed ATMPA, and directly used the FGSM(Fast Gradient Sign Method) [17], DeepFool [18], and C&W [19] methods to generate adversarial malware. It has been verified that the generated adversarial malware can obtain a very high rate of model misclassification, but the malware has a semantic feature, which directly adds perturbations to bytes of the malware, resulting in the generated adversarial malware losing its malicious function and executability. Therefore, Khormali et al. [12] propose an improvement to generate non-executable adversarial malware using ATMPA, take the non-executable adversarial malware as the perturbation, and append the perturbation to the end of the malware to generate the final adversarial malware, making the final adversarial malware adversarial, executable, and malicious. However, when this method generates adversarial malware, it needs to add a perturbation of the size of the malware, which will cause the generated adversarial malware to lose its concealment ability.
Benkraouda et al. [11] injected perturbation by inserting empty assembly instructions (such as NOP instructions) and combined the C&W method to generate adversarial malware. The adversarial malware generated by this method has the slightest difference from the malware, but there are problems such as excessive perturbation injection restrictions and the high cost of generating adversarial malware.
Xiao Mao et al. [10] proposed a legal injection method (Bytecode Attack Remained Availability and Functionality, BARAF) that preserves the executable and functionality of the adversarial malware, adds specific bytes (such as 0x00, 0xFF) to the malware to destroy the texture feature of the sampled gray image, and finally successfully generates the adversarial malware. This method can generate executable and malicious adversarial malware with less perturbation. However, this study did not pay attention to the structure, gradient, output and other information of the attack target model, resulting in a low success rate of the generated adversarial malware to deceive the target model.
## III Method
This section first introduces the rule settings for adversarial malware adversarial attacks, then describes the adversarial malware generation method FGAM proposed in this paper, and finally introduces the functionality-preserving manipulations, which ensures that adversarial malware is malicious and executable.
### _Malware Adversarial Attacks_
#### Iii-A1 Attack settings
First, the ultimate goal of the attacker in this paper is to generate adversarial malware so that it can evade the malware detection model, i.e. the adversarial malware is classified as benign by the detection model. Second, the attacker must ensure that the adversarial malware has executable, malicious, and concealment capabilities. Finally, in this paper, the ability of the attacker is limited as follows. The attacker can only input the malware detection model, and obtain the gradient information and output results of the detection model. It is forbidden for attackers to pollute the data set of the model, obtain the data set of the model, change the network structure of the model, and inject network backdoors.
#### Iii-A2 Attack target
The target of the adversarial attack in this paper is the malware detection model based on byte-to-image, and its complete detection process is shown in Figure3. First, the Windows PE file is converted into a grayscale image through the Binary2img algorithm. Due to the different sizes of the PE files, the size of the converted grayscale image is also different. In order to facilitate the training of the neural network, the image needs to be adjusted to a uniform size. The algorithm used for Resize is the bilinear interpolation method. The adjusted image is input into the neural network, and the classification result of the detection model for the PE file can be obtained.
The specific content of the Binary2img algorithm is shown in Algorithm 1. When inputting, in order to preserve the characteristics of the PE file to the greatest extent, the width
Fig. 2: The main method of malware detection based on deep learning
of the grayscale image needs to be set according to the size of the PE file. The corresponding relationship is shown in TableI. This correspondence comes from the Malimg[21] dataset, but the original correspondence does not consider PE file over 1 MB. In this paper, the correspondence is extended to 15 MB according to the requirements. After inputting the binary file, first get its file size, and obtain the corresponding grayscale image width and height (line1, line2, line3) according to the file size. We convert the 8-bit unsigned binary number in the binary file to a decimal number between 0 and 255, and get the data list of the binary file, and get the length of the list (line4, line5). We calculate the size difference between the list and the image array, and perform zero-padding operations (line6, line7, line8, line9). Resize the list to an m*n image array, the target grayscale image (line11).
### _Fgam_
This section first describes the problem of generating adversarial examples by mathematical methods as a minimization problem, and gives the objective function of the minimization problem, and then introduces the solution method of this objective function and the functionality-preserving manipulations used in FGAM.
#### Iii-B1 Problem function
Let the malware detection model be \(f\), the output label of the model be \(y\), the input is \(x\), and the output of the model \(f(x)\) (the malicious score) range from [0, 1]. In this paper, the classification threshold of malware is set to 0.5, and the relationship between the model input and output is shown in Equation 1. According to the definition of adversarial samples, let the adversarial malware be \(x^{\prime}\) and the perturbation is \(s\), then inject the perturbation into the original sample \(x\) to get the adversarial malware \(x^{\prime}\) (Equation 2). If the generated adversarial malware can deceive the detection model, the requirements of Equation 3 need to be satisfied. Therefore, the problem of generative adversarial examples in this paper can be described as fixing the magnitude of the injected perturbation and updating the perturbation through gradient signs to minimize the maliciousness score of adversarial malware (Equation 4). The perturbation injection amount is jointly determined by the file size and the perturbation injection ratio (Equation 5). The perturbation injection amount of different files are different, so the injection ratio is used to represent the perturbation injection amount.
\[y\left\{\begin{array}{l}\text{benign, }f(x)<0.5\\ \text{malware, }f(x)\geq 0.5\end{array}\right. \tag{1}\]
\[x^{\prime}=H(s+x) \tag{2}\]
\[f(x^{\prime})=f(H(s+x))<0.5 \tag{3}\]
\[\text{minimize }f(x^{\prime})=f(H(x+s))\text{ }update\text{ }s\text{ }with\text{ }gradient \tag{4}\]
\[s_{size}=rate*x_{size} \tag{5}\]
Fig. 3: The process of malware detection method based on byte-to-image
\begin{table}
\begin{tabular}{l l l l} \hline \hline File size & Width & File size & Width \\ \hline
0-10 KB & 32 & 100-200 KB & 384 \\
10-30 KB & 64 & 200-500 KB & 512 \\
30-60 KB & 128 & 500-1024 KB & 768 \\
60-100 KB & 256 & 1-15 MB & 1024 \\ \hline \hline \end{tabular}
\end{table} TABLE I: Misclassification rate and cost time
#### Iv-A2 Fgam
The process of FGAM generating adversarial malware is shown in Figure4. First, inject perturbation into the malware to generate adversarial malware, input it into the detection model, and generate a visual image of the adversarial malware according to the output results and gradient information. Convert the image into adversarial malware. The adversarial malware is not executable at this time, so it is necessary to separate the perturbation from it, and inject the separated perturbation into the malware again to generate executable adversarial malware. The above process is repeated, and if the generated executable adversarial malware can fool the detection model. That is, the output result of the model is benign software, the process will be stopped.
The specific algorithm of FGAM is shown in Algorithm 2. First, the injected perturbation amount is determined according to the original malware size and injection rate, and the injected perturbation is initialized to random values (line1-line3). The perturbation is injected into the malware to generate adversarial malware, and the adversarial malware is adjusted into a visualized image. The iterative descending speed and model output initial value is given, so that it can satisfy the iteration start conditions (line5-line8). The iterative loop begins, first to judge whether the adversarial malware successfully deceives the detection model, and the classification threshold of the detection model in this paper is detailed in Section 3.2.1. If the deception is successful, stop the iteration and return the adversarial malware and the number of iterations (line9-ine10). If it fails, the adversarial visualization images are generated repeatedly and iteratively using the fast gradient sign method to ensure that it can fool the detection model (line11-line15). Converting adversarial visualization images into adversarial malware, although adversarial malware can deceive the detection model, but lose its executable and malicious. Therefore, it is necessary to separate adversarial perturbations from them and inject the separated perturbations into the original malware to generate executable and malicious adversarial malware. But its adversarial weakens, and it needs to be tested to whether it can fool the detection model. (line17-line18). At the end of each iteration, the malicious score of the adversarial malware (executable) is recorded, and the malicious score curve is fitted by the least square method, and the slope of the curve is used to draw the score decline curve, and the slope of the score decline curve is used as the decline speed of the malicious score. (line19). The decline speed of malicious scores can be used to judge whether the iteration process is in oscillation. Falling into oscillation does not mean that effective adversarial malware cannot be generated by continuous iteration, but the purpose of the FGAM proposed in this paper is to generate adversarial malware quickly, so it is necessary to reduce the number of iterations used and avoid the generation process falling into oscillation.
```
0:\(x\), the input binary malware sample; \(y\),the targets associated with x; \(\theta\), the parameters of the model; \(\varepsilon\), the perturbation step; \(rate\),the proportion of injected perturbation; \(model\),the malware detection model; \(T\), the maximum number of iterations;
0:\(x^{\prime}\),the adversarial malware; \(t\), the actual number of iterations;
1:\(size\gets getsize()\)
2:while\(i<size*rate\)do
3:\(s_{i}\gets random(0,255),i\gets i+1\)
4:endwhile
5:\(speed\gets 1,model_{out}\gets 1,t\gets 0\)
6:while\(t<T\)and\(speed>0.001\)and\(model_{out}>0.5\)do
7:\(x^{\prime}\gets inject\;s\;to\;x\)
8:\(x_{img}\gets Rsize(Binary2img(x^{\prime}))\)
9:\(model_{out}\gets model(x_{img}),t\gets t+1\)
10:if\(model_{out}>0.5\)then
11:\(model_{advout}\gets 1\)
12:while\(model_{advout}>0.5\)do
13:\(x_{advimg}\gets x_{img}+\varepsilon sign(\nabla_{x_{img}}J(\theta,x_{img},y))\)
14:\(model_{advout}\gets model(x_{advimg})\)
15:\(x_{img}\gets x_{advimg}\)
16:endwhile
17:\(x^{\prime}\gets Img2binary(Resize(x_{advimg}))\)
18:\(s\gets separate\;s\;from\;x^{\prime}\)
19:\(speed\gets LeastSquares(model_{out})\)
20:endif
21:endwhile
22:return\(x^{\prime},t\)
```
**Algorithm 2** FGAM
#### Iv-A3 Functionality-preserving manipulations
Functionality-preserving manipulations are the key to ensuring that adversarial malware is executable and malicious, and its main principle is to inject perturbations into areas not used by program execution. A simplified diagram of the PE file structure is shown in Figure5. We will introduce some function-preserving manipulations based on the PE structure. According to the injection location classification, the existing functionality-preserving manipulations are as follows. (1) Full the DOS header [20]. Since the MZ flag and the PE offset part of the DOS header cannot be changed, the perturbation can be injected into other parts of the DOS header to fill the space of the DOS header. (2) Extend DOS header [20]. By rewriting PE offset, inject a segment of byte perturbation between I and II, and ensure that it is aligned with the PE file, which is equivalent to increasing the space of the DOS header. (3) Injecting section [20], by rewriting the offset address of the section, injecting a section perturbation before the first section of IV. Section perturbations can also be injected into other locations of the IV, but in order to avoid too much intrusion into the PE file, it is preferred to inject before the first section or after the last section. (4) Padding [21][22] Adding byte perturbation directly at the end of the PE file (after IV), this method is the least intrusive and the operation is the easiest.
In conclusion, functionality-preserving manipulations is about injecting perturbations into the PE file, and the injected perturbations will not be executed and will not affect the execution. However, different functionality-preserving manipulations have different injection locations and different intrusiveness. Most of the existing PE files cannot run in the DOS environment, and the content of the DOS header of the PE files is relatively fixed. When adversarial malware is generated using the full DOS header and extend DOS to inject perturbation, the DOS header part of the generated adversarial malware is quite different from that of benign software. Therefore, the
adversarial malware generated by the above two methods have poor concealment and are not used in this paper. This paper takes injecting section and Padding at the end as injection operation.
## IV Experiment
The experiment uses PyTorch as the deep learning framework, Python as the programming language, and uses the NVIDIA T4 (16G) graphics card to provide acceleration. The specific physical device is a high-performance server equipped with Centos7, the CPU is Intel(R) Xeon(R) Gold 5218, and it is equipped with 256G RAM.
### _Dataset and Models_
Since there is currently no open-source malware binary dataset, it needs to be collected through VirusShare2. In addition, there is currently no open-source malware detection model, so the malware detection model needs to be trained using the dataset in this paper, and the dataset is divided into the training set and the test set with a ratio of 7:3. The composition of this dataset is shown in TableII.
Footnote 2: [https://virusshare.com/](https://virusshare.com/)
The existing byte-to-image malware detection models have the same preprocessing process, but the difference lies in the convolutional neural network used for classification. After literature research, this paper selects the latest detection model as the attack target. Tekerek et al. [8] proposed to use DenseNet121 as the classification network. When training the network, a dynamic learning rate was used, with an initial learning rate of 0.1, and after every five iterations, the learning rate was reduced to 0.5 times the original. After 50 rounds, the best performance in the test set is selected as the final detection model of this paper. The model test results are shown in TableIII.
In addition, in order to evaluate the transferability of adversarial malware, it is necessary to train the transfer target model with the dataset in this paper. In this paper, MalConv [2] is selected as the transfer target model. The maximum input of the model is 15MB. After 50 rounds of training, the best test set is selected as the final model in this paper. The model test results are shown in TableIII.
\begin{table}
\begin{tabular}{l l l l} \hline \hline Data & Training set & Testing set & Source \\ \hline Benign & 9489 & 4068 & Windows \\ Malware & 8473 & 3632 & VirusShare \\ \hline \hline \end{tabular}
\end{table} TABLE II: Dataset Composition
Fig. 4: The adversarial malware generation process
Fig. 5: PE file structure simplified diagram
\begin{table}
\begin{tabular}{l l l} \hline \hline Model & Accuracy & Auc \\ \hline Attack target model (DenseNet121) & 94.12\% & 97.69\% \\ Transfer target model (MalConv) & 94.18\% & 98.12\% \\ \hline \hline \end{tabular}
\end{table} TABLE III: Malware detection model testing results
### _Adversarial malware attack evaluation_
In TableIV, we evaluate adversarial malware generated by existing work. We evaluate by the functionality of adversarial malware and the minimal proportion of injection perturbations. Functionality indicates whether the generated adversarial malware retains the functionality of the original malware. The minimum injection ratio represents the minimum perturbation injection required when adversarial malware is effective. Since the sizes of different files are different, the ratio of the perturbation amount to the original file is used to indicate the injection size. The adversarial malware generated by ATMPA loses functionality, and the amount of perturbation required for adversarial malware generated by COPYCAT is too large, so we use BARAF as a comparison work in this paper.
To verify the effectiveness of FGAM, the adversarial malware generated by this method are evaluated for adversarial attack capability. The experiment selects 500 malwares from the dataset as the original samples, and uses three methods such as FGAM, Random (injecting random perturbation), and BARAF [10] to generate adversarial malware. This paper evaluates the attack ability of adversarial malware by the misclassification rate (MR) of the model. The selected malware samples can be correctly classified by the detection model, so the initial misclassification rate of the model is 0. Therefore, the higher the misclassification rate of the model, the higher the success rate of adversarial attacks and the more effective the method of generating adversarial malware. The experimental results are shown in TableV, where MR(0) represents the misclassification rate of the detection model when the number of iterations is 0. MR(20) indicates the misclassification rate of the detection model when iteratively 20 times, but the Random and BARAF methods cannot iterate, so there is no experimental result.
The above experimental results show that FGAM enhances perturbation adversarial ability through iteration, and improves the success rate of generating adversarial malware, but the adversarial malware generated under different injection rates and injection methods are different. To further study this problem, we set up various control experimental groups, selected 500 malware samples, and used various functionality-preserving manipulations and injection rates to generate adversarial malware. The experimental results are shown in Figure6. The injection rates were set to 5%, 10%, 20%, and 50%, respectively. The experimental results show that at the beginning of the iterative process, the number of iterations is the decisive factor to generate effective adversarial malware, and the effect of increasing the injection rate is not significant at this time. However, as the number of iterations increases, its marginal effect diminishes, and the injection rate becomes the decisive factor. In short, the injection rate determines the upper bound on the quality of adversarial examples. When increasing the number of iterations fails to generate adversarial malware effectively, it is necessary to consider increasing the injection rate. But there are also limitations on the injection rate, which are described in Section 4.3.
### _Static information statistics_
To verify the concealment ability of adversarial examples generated by FGAM under different injection rates. The file size and entropy values of benign software, malware, adversarial malware (0.1), and adversarial malware (0.5) were counted respectively. The statistical sample size is 500. The statistical results of file size are shown in Figure7. The statistical results show that there is no significant difference in the size distribution of files with the increase of the injection rate. In conclusion, the file size will not be the limit of the injection rate.
The statistical results of software entropy are shown in Figure8. The results show that the entropy distributions of malware and benign software have overlapping areas. When the injection rate is 10%, the entropy distribution of adversarial malware shifts upward, but there is still an overlap region with the distribution of benign software. However, when the injection rate is 50%, the entropy distribution of adversarial malware shifts further up, and there is a clear demarcation from the distribution of benign software. However, when the injection ratio is 50%, the entropy distribution of adversarial malware is further shifted upward, and there is a clear distinction with the distribution of benign software.
Therefore, there is a limit to the perturbation injection rate, and injecting too much perturbation will lead to a significant increase in file entropy. When generating adversarial examples, increasing the injection rate can enhance the adversarial ability of the adversarial examples, making it easier to fool the detection model. However, at the same time, the file entropy increases significantly, and there is a clear distinction between adversarial malware and benign software in entropy, which leads to the failure of the adversarial malware deception detection model.
### _Adversarial attack transfer capability_
To evaluate the transfer ability of adversarial malware, the adversarial malware generated by FGAM are input into the transfer target model, and the misclassification rate (MR) of the transfer target model is used to evaluate the transfer ability of the adversarial malware. In this paper, MalConv is selected as the transfer attack target model. Both MalConv and the attack
\begin{table}
\begin{tabular}{l l l l} \hline \hline Method & Inject operation & MR(0) & MR(20) \\ \hline Random & Injecting section & 2.6\% & No iteration \\ & Padding & 2.8\% & No iteration \\ & Injecting section & 8\% & No iteration \\ & Padding & 2.2\% & No iteration \\ FGAM & Injecting section & 2.6\% & **87.2\%** \\ & Padding & 2.8\% & **91.6\%** \\ \hline \hline \end{tabular}
\end{table} TABLE V: The model misclassification rate
\begin{table}
\begin{tabular}{l l l} \hline \hline Method & functionality & perturbation rate \\ \hline ATMPA [9] & NO & 100\% \\ COPYCAT [12] & YES & 100\% \\ BARAF [10] & YES & 5\% \\ \hline \hline \end{tabular}
\end{table} TABLE IV: Assessment of existing methods
target model in this paper use bytes as the original features. The training details and test results of the target transfer model are shown in 5.1. First, evaluate the transferability of adversarial examples generated by existing methods. In the experiment, the functionality-preserving manipulation was selected as the inject section, and the injection rate was 10%. The experimental results are shown in TableVI. The experimental results show that the adversarial malware generated by FGAM have a strong adversarial attack ability to the malware detection model based on byte-to-image. However, its attack ability on the transfer model is similar to the adversarial malware generated by Random.
In order to explore whether the transfer ability of adversarial malware is related to different functionality-preserving manipulations, different injection rates, and different adversarial strengths. In the experiment, 500 malware were selected, and under different conditions, FGAM was used to generate adversarial malware, and the transfer ability of the adversarial malware was tested. When the inject section is used as the functionality-preserving manipulation, the experimental results are shown in TableVII. The different adversarial strengths represent different model thresholds for stopping iteration in the FGAM algorithm. The original threshold is 0.5, and the enhancement threshold is set to 0.9. The experimental results show that enhancing the malicious score and injection rate cannot improve the transfer attack ability of adversarial malware. Conversely, reducing the injection rate can improve the ability of adversarial malware transfer attacks. When padding is used as the functionality-preserving manipulation, the experimental
\begin{table}
\begin{tabular}{l c c} \hline \hline \multirow{2}{*}{Sample} & Attack target model & Transfer target model \\ & (DenseNet121) & (MalConv) \\ \hline AM(Random) & 2.6\% & 25.6\% \\ AM(BARAF) & 8\% & 29\% \\ AM(FGAM) & 90.6\% & 25.8\% \\ \hline \hline \end{tabular}
\end{table} TABLE VI: The model misclassification rate
Fig. 8: Statistics of software entropy
Fig. 6: Model misclassification rate. The left picture is the experimental result of inject section, and the right picture is the experimental result of padding
Fig. 7: Statistics of software file size
results are shown in TableVIII. The experimental results show that the transfer ability of adversarial malware is completely lost, which is consistent with the interpretability conclusion of MalConv. Demetrio et al. [23] conducted an interpretability analysis of MalConv by CAM(Class Activation Mapping) [24], and the results showed that the head position information of the PE file has a significant impact on the classification results of the MalConv model. Therefore, when padding is used as the functionality-preserving manipulation, the perturbation is fully injected into the end of the malware, and the header information of the malware does not change, so the ability of the adversarial malware to deceive the model is weakened.
The perturbation injection position of the inject section is close to the head of the PE file, so the generated adversarial malware can transfer attacks. In PE files, sections occupy most of the file content. So when the injection section position is before the first section, the injection position is close to the head of the PE file. When the injection section position is before the middle section, the injection position is the middle position of the PE file.
The experimental results of the adversarial malware transfer attack generated by the above two functionality-preserving manipulations show that when the perturbation injection position is close to the PE file header, the adversarial malware has the ability to transfer attack; When the perturbation is injected at the end of the PE file, the transfer ability of the adversarial malware is almost lost, and the experimental results are mutually confirmed with the interpretability conclusion of MalConv. The experimental results in Section 4.2 show that under the same perturbation injection rate and number of iterations, the padding operation is more efficient to generate adversarial malware. Therefore, the impact of PE file header information on the byte-to-image detection model is not as significant as MalConv.
In conclusion, both models are end-to-end malware detection models, but the decision-making basis of the two is not the same. First, the adversarial malware generated by FGAM have a deception rate of about 90% for the byte-to-image detection model, but the deception effect on the MalConv model is similar to the adversarial malware generated by Random, and there is no obvious advantage. Second, when the two models detect the same PE file, the classification decisions rely on different regions. Therefore, the two detection models can be integrated through ensemble learning, etc., to improve the accuracy and robustness of the malware end-to-end detection model.
## V Conclusion
In this paper, we propose FGAM, a method for fast generation of adversarial malware. This method uses iterative perturbation to enhance perturbation adversarial ability, and the enhancement of perturbation adversarial ability can reduce perturbation injection amount and improve the success rate of adversarial attacks. The goal of the iteration is to minimize the adversarial malware score. FGAM detects the rate of decline of malicious scores, stops meaningless iterations in advance, and limits the number of iterations to a certain range. In addition, perturbation injection through function-preserving manipulations ensures that adversarial malware is malicious and executable. In the experiment, the static information of the adversarial malware is counted, and it is proposed that the perturbation injection rate has an entropy limit, and the perturbation injection rate and the concealment ability of the adversarial malware are mutually restricted. Finally, under different functionality-preserving manipulations and different perturbation injection rates, the efficiency of the method to generate adversarial examples and the transfer attack ability of adversarial examples are analyzed.
The work in this paper mainly realizes the rapid generation of adversarial malware, but there are still some shortcomings, which will be improved in future work. First, increase the variety of functionality-preserving manipulations. The generated adversarial malware consists of malware with adversarial perturbations, where the input of the adversarial perturbation causes the detection model to produce false outputs. Therefore, increasing the diversity of functionality-preserving manipulations can make it more challenging to separate adversarial perturbation from malware, and enhance the ability of adversarial malware attacks. Second, combine the sensitivity analysis with perturbation injection. The experimental results show that different injection locations have different efficiencies in generating adversarial examples. Therefore, sensitivity analysis can be used to obtain the region in the sample that has a significant influence on the classification decision, and then inject perturbation into this region to improve the efficiency of generating adversarial malware.
| Malware検出モデルは深層学習に基づいて広く利用されてきましたが、最近の研究は、深層学習モデルが攻撃的な攻撃に脆弱であることを示しています。攻撃的な攻撃は、深層学習モデルを欺くために攻撃的サンプルを生成します。攻撃的な攻撃をマルウェア検出モデルに施すと、攻撃者はマルウェアと同じ悪意のある機能を持つ攻撃的マルウェアを生成し、検出モデルがそれを良質なソフトウェアとして分類する可能性があります。攻撃的マルウェアの生成を研究することは、マルウェア検出モデルの robustness を向上させるのに役立ちます。現状では、バイト-to-画像マルウェア検出モデルに対する攻撃的マルウェアの生成に関する研究では、大量の注入 Perturbation と生成効率の低さが課題となっています。そのため、この論文では、FGAM（Fast Generate Adversarial Malware）という、高速な攻撃的マルウェア生成方法を提案しています。FGAMは、勾 |
2301.11956 | On the Connection Between MPNN and Graph Transformer | Graph Transformer (GT) recently has emerged as a new paradigm of graph
learning algorithms, outperforming the previously popular Message Passing
Neural Network (MPNN) on multiple benchmarks. Previous work (Kim et al., 2022)
shows that with proper position embedding, GT can approximate MPNN arbitrarily
well, implying that GT is at least as powerful as MPNN. In this paper, we study
the inverse connection and show that MPNN with virtual node (VN), a commonly
used heuristic with little theoretical understanding, is powerful enough to
arbitrarily approximate the self-attention layer of GT.
In particular, we first show that if we consider one type of linear
transformer, the so-called Performer/Linear Transformer (Choromanski et al.,
2020; Katharopoulos et al., 2020), then MPNN + VN with only O(1) depth and O(1)
width can approximate a self-attention layer in Performer/Linear Transformer.
Next, via a connection between MPNN + VN and DeepSets, we prove the MPNN + VN
with O(n^d) width and O(1) depth can approximate the self-attention layer
arbitrarily well, where d is the input feature dimension. Lastly, under some
assumptions, we provide an explicit construction of MPNN + VN with O(1) width
and O(n) depth approximating the self-attention layer in GT arbitrarily well.
On the empirical side, we demonstrate that 1) MPNN + VN is a surprisingly
strong baseline, outperforming GT on the recently proposed Long Range Graph
Benchmark (LRGB) dataset, 2) our MPNN + VN improves over early implementation
on a wide range of OGB datasets and 3) MPNN + VN outperforms Linear Transformer
and MPNN on the climate modeling task. | Chen Cai, Truong Son Hy, Rose Yu, Yusu Wang | 2023-01-27T19:15:31 | http://arxiv.org/abs/2301.11956v4 | # On the Connection Between MPNN and Graph Transformer
###### Abstract
Graph Transformer (GT) recently has emerged as a new paradigm of graph learning algorithms, outperforming the previously popular Message Passing Neural Network (MPNN) on multiple benchmarks. Previous work (Kim et al., 2022) shows that with proper position embedding, GT can approximate MPNN arbitrarily well, implying that GT is at least as powerful as MPNN. In this paper, we study the inverse connection and show that MPNN with virtual node (VN), a commonly used heuristic with little theoretical understanding, is powerful enough to arbitrarily approximate the self-attention layer of GT.
In particular, we first show that if we consider one type of linear transformer, the so-called Performer/Linear Transformer (Choromanski et al., 2020; Katharopoulos et al., 2020), then MPNN + VN with only \(\mathcal{O}(1)\) depth and \(\mathcal{O}(1)\) width can approximate a self-attention layer in Performer/Linear Transformer. Next, via a connection between MPNN + VN and DeepSets, we prove the MPNN + VN with \(\mathcal{O}(n^{d})\) width and \(\mathcal{O}(1)\) depth can approximate the self-attention layer arbitrarily well, where \(d\) is the input feature dimension. Lastly, under some assumptions, we provide an explicit construction of MPNN + VN with \(\mathcal{O}(1)\) width and \(\mathcal{O}(n)\) depth approximating the self-attention layer in GT arbitrarily well. On the empirical side, we demonstrate that 1) MPNN + VN is a surprisingly strong baseline, outperforming GT on the recently proposed Long Range Graph Benchmark (LRGB) dataset, 2) our MPNN + VN improves over early implementation on a wide range of OGB datasets and 3) MPNN + VN outperforms Linear Transformer and MPNN on the climate modeling task.
Machine Learning, ICML
## 1 Introduction
MPNN (Message Passing Neural Network) (Gilmer et al., 2017) has been the leading architecture for processing graph-structured data. Recently, transformers in natural language processing (Vaswani et al., 2017; Kalyan et al., 2021) and vision (d'Ascoli et al., 2021; Han et al., 2022) have extended their success to the domain of graphs. There have been several pieces of work (Ying et al., 2021; Wu et al., 2021; Kreuzer et al., 2021; Rampasek et al., 2022; Kim et al., 2022) showing that with careful position embedding (Lim et al., 2022), graph transformers (GT) can achieve compelling empirical performances on large-scale datasets and start to challenge the dominance of MPNN.
MPNN imposes a sparsity pattern on the computation graph and therefore enjoys linear complexity. It however suffers from well-known over-smoothing (Li et al., 2018; Oono & Suzuki, 2019; Cai & Wang, 2020) and over-squashing (Alon & Yahav, 2020; Topping et al., 2021) issues, limiting its usage on long-range modeling tasks where the label of one node depends on features of nodes far away. GT relies purely on position embedding to encode the graph structure and uses vanilla transformers on top. 1 It models all pairwise interactions directly in one layer, making it computationally more expensive. Compared to MPNN, GT shows promising results on tasks where modeling long-range interaction is the key, but the quadratic complexity of self-attention in GT
Figure 1: MPNN + VN and Graph Transformers.
limits its usage to graphs of medium size. Scaling up GT to large graphs remains an active research area (Wu et al., 2022).
Theoretically, it has been shown that graph transformers can be powerful graph learners (Kim et al., 2022), i.e., graph transformers with appropriate choice of token embeddings have the capacity of approximating linear permutation equivariant basis, and therefore can approximate 2-IGN (Invariant Graph Network), a powerful architecture that is at least as expressive as MPNN (Maron et al., 2018). This raises an important question that _whether GT is strictly more powerful than MPNN_. Can we approximate GT with MPNN?
One common intuition of the advantage of GT over MPNN is its ability to model long-range interaction more effectively. However, from the MPNN side, one can resort to a simple trick to escape locality constraints for effective long-range modeling: the use of an additional _virtual node (VN)_ that connects to all input graph nodes. On a high level, MPNN + VN augments the existing graph with one virtual node, which acts like global memory for every node exchanging messages with other nodes. Empirically this simple trick has been observed to improve the MPNN and has been widely adopted (Gilmer et al., 2017; Hu et al., 2020, 2021) since the early beginning of MPNN (Gilmer et al., 2017; Battaglia et al., 2018). However, there is very little theoretical study of MPNN + VN (Hwang et al., 2022).
In this work, we study the theoretical property of MPNN + VN, and its connection to GT. We systematically study the representation power of MPNN + VN, both for certain approximate self-attention and for the full self-attention layer, and provide a depth-width trade-off, summarized in Table 1. In particular,
* With \(\mathcal{O}(1)\) depth and \(\mathcal{O}(1)\) width, MPNN + VN can approximate one self-attention layer of Performer (Choromanski et al., 2020) and Linear Transformer (Katharopoulos et al., 2020), a type of linear transformers (Tay et al., 2020).
* Via a link between MPNN + VN with DeepSets (Zhaeer et al., 2017), we prove MPNN + VN with \(\mathcal{O}(1)\) depth and \(\mathcal{O}(n^{d})\) width (\(d\) is the input feature dimension) is permutation equivariant universal, implying it can approximate self-attention layer and even full-transformers.
* Under certain assumptions on node features, we prove an explicit construction of \(\mathcal{O}(n)\) depth \(\mathcal{O}(1)\) width MPNN + VN approximating 1 self-attention layer arbitrarily well on graphs of size \(n\). Unfortunately, the assumptions on node features are rather strong, and whether we can alleviate them will be an interesting future direction to explore.
* Empirically, we show 1) that MPNN + VN works surprisingly well on the recently proposed LRGB (long-range graph benchmarks) datasets (Dwivedi et al., 2022), which arguably require long-range interaction reasoning to achieve strong performance 2) our implementation of MPNN + VN is able to further improve the early implementation of MPNN + VN on OGB datasets and 3) MPNN + VN outperforms Linear Transformer (Katharopoulos et al., 2020) and MPNN on the climate modeling task.
## 2 Related Work
**Virtual node in MPNN.** The virtual node augments the graph with an additional node to facilitate the information exchange among all pairs of nodes. It is a heuristic proposed in (Gilmer et al., 2017) and has been observed to improve the performance in different tasks (Hu et al., 2021, 2020). Surprisingly, its theoretical properties have received little study. To the best of our knowledge, only a recent paper (Hwang et al., 2022) analyzed the role of the virtual node in the link prediction setting in terms of 1) expressiveness of the learned link representation and 2) the potential impact on under-reaching and over-smoothing.
**Graph transformer.** Because of the great successes of Transformers in natural language processing (NLP) (Vaswani et al., 2017; Wolf et al., 2020) and recently in computer vision (Dosovitskiy et al., 2020; d'Ascoli et al., 2021; Liu et al., 2021), there is great interest in extending transformers for graphs. One common belief of advantage of graph transformer over MPNN is its capacity in capturing long-range interactions while alleviating over-smoothing (Li et al., 2018; Oono and Suzuki, 2019; Cai and Wang, 2020) and over-squashing in MPNN (Alon and Yahav, 2020; Topping et al., 2021).
Fully-connected Graph transformer (Dwivedi and Bresson, 2020) was introduced with eigenvectors of graph Laplacian as the node positional encoding (PE). Various follow-up works proposed different ways of PE to improve GT, ranging from an invariant aggregation of Laplacian?s eigenvectors in SAN (Kreuzer et al., 2021), pair-wise graph distances in Graphormer (Ying et al., 2021), relative PE derived from diffusion kernels in GraphiT (Mialon et al., 2021), and recently Sign and Basis Net (Lim et al., 2022) with a principled way of handling sign and basis invariance. Other lines of research in GT include combining MPNN and GT (Wu et al., 2021; Rampasek et al., 2022), encoding the substructures (Chen et al., 2022), and efficient graph transformers for large graphs (Wu et al., 2022).
## 3 Preliminaries
We denote \(\mathbf{X}\in\mathbb{R}^{n\times d}\) the concatenation of graph node features and positional encodings, where node \(i\) has feature \(\mathbf{x}_{i}\in\mathbb{R}^{d}\). When necessary, we use \(\mathbf{x}_{j}^{(l)}\) to denote the node \(j\)'s feature at depth \(l\). Let \(\mathcal{M}\) be the space of multisets of vectors in \(\mathbb{R}^{d}\). We use \(\mathcal{X}\subseteq\mathbb{R}^{n\times d}\) to denote the space of node features and the \(\mathcal{X}_{i}\) be the projection of \(\mathcal{X}\) on \(i\)-th coordinate. \(\|\cdot\|\) denotes the 2-norm. \([\mathbf{x},\mathbf{y},\mathbf{z}]\) denotes the concatenation of \(\mathbf{x},\mathbf{y},\mathbf{z}\). \([n]\) stands for the set \(\{1,2,...,n\}\).
**Definition 3.1** (attention).: We denote key and query matrix as \(\mathbf{W}_{K},\mathbf{W}_{Q}\in\mathbb{R}^{d\times d^{\prime}}\), and value matrix as \(\mathbf{W}_{V}\in\mathbb{R}^{d\times d}\)2. Attention score between two vectors \(\mathbf{u},\mathbf{v}\in\mathbb{R}^{d\times 1}\) is defined as \(\alpha(\mathbf{u},\mathbf{v})=\text{softmax}(\mathbf{u}^{T}\mathbf{W}_{Q}(\mathbf{W}_{K})^{T}\mathbf{ v})\). We denote \(\mathcal{A}\) as the space of attention \(\alpha\) for different \(\mathbf{W}_{Q},\mathbf{W}_{K},\mathbf{W}_{V}\). We also define unnormalized attention score \(\alpha^{\prime}(\cdot,\cdot)\) to be \(\alpha^{\prime}(\mathbf{u},\mathbf{v})=\mathbf{u}^{T}\mathbf{W}_{Q}(\mathbf{W}_{K})^{T}\mathbf{v}\). Self attention layer is a matrix function \(\mathbf{L}:\mathbb{R}^{n\times d}\rightarrow\mathbb{R}^{n\times d}\) of the following form: \(\mathbf{L}(\mathbf{X})=\text{softmax}(\mathbf{X}\mathbf{W}_{Q}(\mathbf{X}\mathbf{W}_{K})^{T})\mathbf{X} \mathbf{W}_{V}\).
Footnote 2: For simplicity, we assume the output dimension of self-attention is the same as the input dimension. All theoretical results can be extended to the case where the output dimension is different from \(d\).
### MPNN Layer
**Definition 3.2** (MPNN layer (Gilmer et al., 2017)).: An MPNN layer on a graph \(G\) with node features \(\mathbf{x}^{(k)}\) at \(k\)-th layer and edge features \(\mathbf{e}\) is of the following form
\[\mathbf{x}_{i}^{(k)}=\gamma^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\tau_{j\in\mathcal{N}(i )}\phi^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\mathbf{x}_{j}^{(k-1)},\mathbf{e}_{j,i}\right)\right)\]
Here \(\gamma:\mathbb{R}^{d}\times\mathbb{R}^{d^{\prime}}\rightarrow\mathbb{R}^{d}\) is update function, \(\phi:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{d_{e}}\rightarrow \mathbb{R}^{d^{\prime}}\) is message function where \(d_{e}\) is the edge feature dimension, \(\tau:\mathcal{M}\rightarrow\mathbb{R}^{d}\) is permutation invariant aggregation function and \(\mathcal{N}(i)\) is the neighbors of node \(i\) in \(G\). Update/message/aggregation functions are usually parametrized by neural networks. For graphs of different types of edges and nodes, one can further extend MPNN to the heterogeneous setting. We use \(1,...,n\) to index graph nodes and vn to denote the virtual node.
**Definition 3.3** (heterogeneous MPNN + VN layer).: The heterogeneous MPNN + VN layer operates on two types of nodes: 1) virtual node and 2) graph nodes, denoted as vn and gn, and three types of edges: 1) vn-gn edge and 2) gn-gn edges and 3) gn-vn edges. It has the following form
\[\mathbf{x}_{\text{vn}}^{(k)} =\gamma_{\text{vn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\tau_{j\in[n]} \phi_{\text{vn-gn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\mathbf{x}_{j}^{(k-1)},\mathbf{e}_{ j,i}\right)\right) \tag{1}\]
for the virtual node, and
\[\mathbf{x}_{i}^{(k)} =\gamma_{\text{gn}}^{(k)}(\mathbf{x}_{i}^{(k-1)},\tau_{j\in\mathcal{N }_{1}(i)}\phi_{\text{gn-vn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\mathbf{x}_{j}^{(k-1)}, \mathbf{e}_{j,i}\right) \tag{2}\]
for graph node. Here \(\mathcal{N}_{1}(i)\) for graph node \(i\) is the virtual node and \(\mathcal{N}_{2}(i)\) is the set of neighboring graph nodes.
Our proof of approximating self-attention layer \(\mathbf{L}\) with MPNN layers does not use the graph topology. Next, we introduce a simplified heterogeneous MPNN + VN layer, which will be used in the proof. It is easy to see that setting \(\phi_{\text{gn}}^{(k)}\) to be 0 in Definition 3.3 recovers the simplified heterogeneous MPNN + VN layer.
**Definition 3.4** (simplified heterogeneous MPNN + VN layer).: A simplified heterogeneous MPNN + VN layer is the same as a heterogeneous MPNN + VN layer in Definition 3.3 except we set \(\theta_{\text{gn-gn}}\) to be 0. I.e., we have
\[\mathbf{x}_{\text{vn}}^{(k)}=\gamma_{\text{vn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\tau_{ j\in[n]}\phi_{\text{vn-gn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\mathbf{x}_{j}^{(k-1)}, \mathbf{e}_{j,i}\right)\right)\]
for the virtual node, and
\[\mathbf{x}_{i}^{(k)}=\gamma_{\text{gn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\tau_{j\in \mathcal{N}_{1}(i)}\phi_{\text{gn-vn}}^{(k)}\left(\mathbf{x}_{i}^{(k-1)},\mathbf{x}_{j} ^{(k-1)},\mathbf{e}_{j,i}\right)\right)\]
for graph nodes.
Intuitively, adding the virtual node (VN) to MPNN makes it easy to compute certain quantities, for example, the mean of node features (which is hard for standard MPNN unless the depth is proportional to the diameter of the graph). Using VN thus makes it easy to implement for example the mean subtraction, which helps reduce over-smoothing and improves the performance of GNN. (Yang et al., 2020; Zhao & Akoglu, 2019)
\begin{table}
\begin{tabular}{l l l l l} \hline \hline & Depth & Width & Self-Attention & Note \\ \hline Theorem 4.1 & \(\mathcal{O}(1)\) & \(\mathcal{O}(1)\) & Approximate & Approximate self attention in Performer (Choromanski et al., 2020) \\ Theorem 5.5 & \(\mathcal{O}(1)\) & \(\mathcal{O}(n^{d})\) & Full & Leverage the universality of equivariant DeepSets \\ Theorem 6.3 & \(\mathcal{O}(n)\) & \(\mathcal{O}(1)\) & Full & Explicit construction, strong assumption on \(\mathcal{X}\) \\ Proposition B.10 & \(\mathcal{O}(n)\) & \(\mathcal{O}(1)\) & Full & Explicit construction, more relaxed (but still strong) assumption on \(\mathcal{X}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of approximation result of MPNN + VN on self-attention layer. \(n\) is the number of nodes and \(d\) is the feature dimension of node features. The dependency on \(d\) is hidden.
### Assumptions
We have two mild assumptions on feature space \(\mathcal{X}\subset\mathbb{R}^{n\times d}\) and the regularity of target function \(\mathbf{L}\).
**AS1.**\(\forall i\in[n],\mathbf{x}_{i}\in\mathcal{X}_{i},\|\mathbf{x}_{i}\|<C_{1}\). This implies \(\mathcal{X}\) is compact.
**AS2.**\(\|\mathbf{W}_{Q}\|<C_{2},\|\mathbf{W}_{K}\|<C_{2},\|\mathbf{W}_{V}\|<C_{2}\) for target layer \(\mathbf{L}\). Combined with ASI on \(\mathcal{X}\), this means \(\alpha^{\prime}(\mathbf{x}_{i},\mathbf{x}_{j})\) is both upper and lower bounded, which further implies \(\sum_{j}e^{\alpha^{\prime}(\mathbf{x}_{i},\mathbf{x}_{j})}\) be both upper bounded and lower bounded.
\(\mathcal{O}(1)\)-depth \(\mathcal{O}(1)\)-width MPNN + VN for unbiased approximation of attention
The standard self-attention takes \(\mathcal{O}(n^{2})\) computational time, therefore not scalable for large graphs. Reducing the computational complexity of self-attention in Transformer is active research (Tay et al., 2020). In this section, we consider self-attention in a specific type of efficient transformers, Performer (Choromanski et al., 2020) and Linear Transformer (Katharopoulos et al., 2020).
One full self-attention layer \(\mathbf{L}\) is of the following form
\[\mathbf{x}_{i}^{(l+1)}=\sum_{j=1}^{n}\frac{\kappa\left(\mathbf{W}_{Q}^{(l)}\mathbf{x}_{i} ^{(l)},\mathbf{W}_{K}^{(l)}\mathbf{x}_{j}^{(l)}\right)}{\sum_{k=1}^{n}\kappa\left(\mathbf{ W}_{Q}^{(l)}\mathbf{x}_{i}^{(l)},\mathbf{W}_{K}^{(l)}\mathbf{x}_{k}^{(l)}\right)}\cdot \left(\mathbf{W}_{V}^{(l)}\mathbf{x}_{j}^{(l)}\right) \tag{3}\]
where \(\kappa:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}\) is the softmax kernel \(\kappa(\mathbf{x},\mathbf{y}):=\exp(\mathbf{x}^{T}\mathbf{y})\). The kernel function can be approximated via \(\kappa(\mathbf{x},\mathbf{y})=\langle\Phi(\mathbf{x}),\Phi(\mathbf{y})\rangle_{\mathcal{V}} \approx\phi(\mathbf{x})^{T}\phi(\mathbf{y})\) where the first equation is by Mercer's theorem and \(\phi(\cdot):\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}\) is a low-dimensional feature map with random transformation. For Performer (Choromanski et al., 2020), the choice of \(\phi\) is taken as \(\phi(\mathbf{x})=\frac{\exp\left(\frac{-|\mathbf{x}|^{2}}{2}\right)}{\sqrt{m}}\left[ \exp\left(\mathbf{w}_{1}^{T}\mathbf{x}\right),\cdots,\exp\left(\mathbf{w}_{m}^{T}\mathbf{x} \right)\right]\) where \(\mathbf{w}_{k}\sim\mathcal{N}\left(0,I_{d}\right)\) is i.i.d sampled random variable. For Linear Transformer (Katharopoulos et al., 2020), \(\phi(\mathbf{x})=\mathrm{elu}(\mathbf{x})+1\).
By switching \(\kappa(\mathbf{x},\mathbf{y})\) to be \(\phi(\mathbf{x})^{T}\phi(\mathbf{y})\), and denote \(\mathbf{q}_{i}=\mathbf{W}_{Q}^{(l)}\mathbf{x}_{i}^{(l)},\mathbf{k}_{i}=\mathbf{W}_{K}^{(l)}\mathbf{x} _{i}^{(l)}\) and \(\mathbf{v}_{i}=\mathbf{W}_{V}^{(l)}\mathbf{x}_{i}^{(l)}\), the approximated version of Equation (3) by Performer and Linear Transformer becomes
\[\mathbf{x}_{i}^{(l+1)} =\sum_{j=1}^{n}\frac{\phi\left(\mathbf{q}_{i}\right)^{T}\phi\left(\bm {k}_{j}\right)}{\sum_{k=1}^{n}\phi\left(\mathbf{q}_{i}\right)^{T}\phi\left(\mathbf{k}_ {k}\right)}\cdot\mathbf{v}_{j} \tag{4}\] \[=\frac{\left(\phi\left(\mathbf{q}_{i}\right)^{T}\sum_{j=1}^{n}\phi \left(\mathbf{k}_{j}\right)\otimes\mathbf{v}_{j}\right)^{T}}{\phi\left(\mathbf{q}_{i} \right)^{T}\sum_{k=1}^{n}\phi\left(\mathbf{k}_{k}\right)}.\]
where we use the matrix multiplication association rule to derive the second equality.
The key advantage of Equation (4) is that \(\sum_{j=1}^{n}\phi\left(\mathbf{k}_{j}\right)\) and \(\sum_{j=1}^{n}\phi(\mathbf{k}_{j})\otimes\mathbf{v}_{j}\) can be approximated by the virtual node, and shared for all graph nodes, using only \(\mathcal{O}(1)\) layers of MPNNs. We denote the self-attention layer of this form in Equation (4) as \(\mathbf{L}_{\text{Performer}}\). Linear Transformer differs from Performer by choosing a different form of \(\phi(\mathbf{x})=\mathrm{Relu}(\mathbf{x})+1\) in its self-attention layer \(\mathbf{L}_{\text{Linear-Transformer}}\).
In particular, the VN will approximate \(\sum_{j=1}^{n}\phi\left(\mathbf{k}_{j}\right)\) and \(\sum_{j=1}^{n}\phi\left(\mathbf{k}_{j}\right)\otimes\mathbf{v}_{j}\), and represent it as its feature. Both \(\phi\left(\mathbf{k}_{j}\right)\) and \(\phi\left(\mathbf{k}_{j}\right)\otimes\mathbf{v}_{j}\) can be approximated arbitrarily well by an MLP with constant width (constant in \(n\) but can be exponential in \(d\)) and depth. Note that \(\phi(\mathbf{k}_{j})\otimes\mathbf{v}_{j}\in\mathbb{R}^{dm}\) but can be reshaped to 1 dimensional feature vector.
More specifically, the initial feature for the virtual node is \(\mathbf{1}_{(d+1)m}\), where \(d\) is the dimension of node features and \(m\) is the number of random projections \(\omega_{i}\). Message function + aggregation function for virtual node \(\tau\phi_{\text{vn-gn}}:\mathbb{R}^{(d+1)m}\times\mathcal{M}\rightarrow\mathbb{R }^{(d+1)m}\) is
\[\tau_{j\in[n]}\phi_{\text{vn-gn}}^{(k)}(\cdot,\{\mathbf{x}_{i}\}_{i})=[ \sum_{j=1}^{n}\phi\left(\mathbf{k}_{j}\right), \tag{5}\] \[\texttt{ReshapeTo1D}(\sum_{j=1}^{n}\phi\left(\mathbf{k}_{j}\right) \otimes\mathbf{v}_{j})]\]
where \(\texttt{ReshapeTo1D}(\cdot)\) flattens a 2D matrix to a 1D vector in raster order. This function can be arbitrarily approximated by MLP. Note that the virtual node's feature dimension is \((d+1)m\) (where recall \(m\) is the dimension of the feature map \(\phi\) used in the linear transformer/Performer), which is larger than the dimension of the graph node \(d\). This is consistent with the early intuition that the virtual node might be overloaded when passing information among nodes. The update function for virtual node \(\gamma_{\text{vn}}:\mathbb{R}^{(d+1)m}\times\mathbb{R}^{(d+1)m}\rightarrow\mathbb{ R}^{(d+1)m}\) is just coping the second argument, which can be exactly implemented by MLP.
VN then sends its message back to all other nodes, where each graph node \(i\) applies the update function \(\gamma_{\text{gn}}:\mathbb{R}^{(d+1)m}\times\mathbb{R}^{d}\rightarrow\mathbb{R }^{d}\) of the form
\[\gamma_{\text{gn}}(\mathbf{x}_{i},[\sum_{j=1}^{n}\phi\left(\mathbf{k}_{j} \right),\texttt{ReshapeTo1D}(\sum_{j=1}^{n}\phi\left(\mathbf{k}_{j}\right)\otimes \mathbf{v}_{j})]) \tag{6}\] \[=\frac{\left(\phi\left(\mathbf{q}_{i}\right)\sum_{j=1}^{n}\phi\left(\bm {k}_{j}\right)\otimes\mathbf{v}_{j}\right)^{T}}{\phi\left(\mathbf{q}_{i}\right)^{T} \sum_{k=1}^{n}\phi\left(\mathbf{k}_{k}\right)}\]
to update the graph node feature.
As the update function \(\gamma_{\text{gn}}\) can not be computed exactly in MLP, what is left is to show that error induced by using MLP to approximate \(\tau\phi_{\text{vn-gn}}\) and \(\gamma_{\text{gn}}\) in Equation (5) and Equation (6) can be made arbitrarily small.
**Theorem 4.1**.: _Under the ASI and AS2, MPNN + VN of \(\mathcal{O}(1)\) width and \(\mathcal{O}(1)\) depth can approximate \(\mathbf{L}_{\text{Performer}}\) and \(\mathbf{L}_{\text{Linear-Transformer}}\) arbitrarily well._
Proof.: We first prove the case of \(\mathbf{L}_{\text{Performer}}\). We can decompose our target function as the composition of \(\tau_{j\in[n]}\phi_{\text{un-gn}}^{(k)}\), \(\gamma_{\text{gn}}\) and \(\phi\). By the uniform continuity of the functions, it suffices to show that 1) we can approximate \(\phi\), 2) we can approximate operations in \(\gamma_{\text{gn}}\) and \(\tau\phi_{\text{un-gn}}\) arbitrarily well on the compact domain, and 3) the denominator \(\phi\left(\mathbf{q}_{i}\right)^{T}\sum_{k=1}^{n}\phi\left(\mathbf{k}_{k}\right)\) is uniformly lower bounded by a positive number for any node features in \(\mathcal{X}\).
For 1), each component of \(\phi\) is continuous and all inputs \(\mathbf{k}_{j},\mathbf{q}_{j}\) lie in the compact domain so \(\phi\) can be approximated arbitrarily well by MLP with \(\mathcal{O}(1)\) width and \(\mathcal{O}(1)\) depth (Cybenko, 1989).
For 2), we need to approximate the operations in \(\gamma_{\text{gn}}\) and \(\tau\phi_{\text{un-gn}}\), i.e., approximate multiplication, and vector-scalar division arbitrarily well. As all those operations are continuous, it boils down to showing that all operands lie in a compact domain. By assumption AS1 and AS2 on \(\mathbf{W}_{Q},\mathbf{W}_{K},\mathbf{W}_{V}\) and input feature \(\mathcal{X}\), we know that \(\mathbf{q}_{i},\mathbf{k}_{i},\mathbf{v}_{i}\) lies in a compact domain for all graph nodes \(i\). As \(\phi\) is continuous, this implies that \(\phi(\mathbf{q}_{i}),\sum_{j=1}^{n}\phi(\mathbf{k}_{j})\otimes\mathbf{v}_{j}\) lies in a compact domain (\(n\) is fixed), therefore the numerator lies in a compact domain. Lastly, since all operations do not involve \(n\), the depth and width are constant in \(n\).
For 3), it is easy to see that \(\phi\left(\mathbf{q}_{i}\right)^{T}\sum_{k=1}^{n}\phi\left(\mathbf{k}_{k}\right)\) is always positive. We just need to show that the denominator is bound from below by a positive constant. For Performer, \(\phi(\mathbf{x})=\frac{\exp\left(\frac{-\|\mathbf{x}\|^{2}_{2}}{2}\right)}{\sqrt{m}} \left[\exp\left(\mathbf{w}_{1}^{T}\mathbf{x}\right),\cdots,\exp\left(\mathbf{w}_{m}^{T}\bm {x}\right)\right]\) where \(\mathbf{w}_{k}\sim\mathcal{N}\left(0,I_{d}\right)\). As all norm of input \(\mathbf{x}\) to \(\phi\) is upper bounded by AS1, \(\exp(\frac{-\|\mathbf{x}\|^{2}_{2}}{2})\) is lower bounded. As \(m\) is fixed, we know that \(\|\mathbf{w}_{i}^{T}\mathbf{x}\|\leq\|\mathbf{w}_{i}\|\|\mathbf{x}\|\), which implies that \(\mathbf{w}_{i}^{T}\mathbf{x}\) is lower bounded by \(-\|\mathbf{w}_{i}\|\|\|\mathbf{x}\|\) which further implies that \(\exp(\mathbf{w}_{i}^{T}\mathbf{x})\) is lower bounded. This means that \(\phi\left(\mathbf{q}_{i}\right)^{T}\sum_{k=1}^{n}\phi\left(\mathbf{k}_{k}\right)\) is lower bounded.
For Linear Transformer, the proof is essentially the same as above. We only need to show that \(\phi(\mathbf{x})=\mathrm{elu}(\mathbf{x})+1\) is continuous and positive, which is indeed the case.
Besides Performers, there are many other different ways of obtaining linear complexity. In Appendix C.2, we discuss the limitation of MPNN + VN on approximating other types of efficient transformers such as Linformer (Wang et al., 2020) and Sparse Transformer (Child et al., 2019).
## 5 \(\mathcal{O}(1)\) depth \(\mathcal{O}(n^{d})\) width MPNN + VN
We have shown that the MPNN + VN can approximate self-attention in Performer and Linear Transformer using only \(\mathcal{O}(1)\) depth and \(\mathcal{O}(1)\) width. One may naturally wonder whether MPNN + VN can approximate the self-attention layer in the _full_ transformer. In this section, we show that MPNN + VN with \(O(1)\) depth (number of layers), but with \(\mathcal{O}(n^{d})\) width, can approximate 1 self-attention layer (and full transformer) arbitrarily well.
The main observation is that MPNN + VN is able to exactly simulate (not just approximate) equivariant DeepSets (Zaheer et al., 2017), which is proved to be universal in approximating any permutation invariant/equivariant maps (Zaheer et al., 2017; Segol and Lipman, 2019). Since the self-attention layer is permutation equivariant, this implies that MPNN + VN can approximate the self-attention layer (and full transformer) with \(\mathcal{O}(1)\) depth and \(\mathcal{O}(n^{d})\) width following a result on DeepSets from Segol and Lipman (2019).
We first introduce the permutation equivariant map, equivariant DeepSets, and permutation equivariant universality.
**Definition 5.1** (permutation equivariant map).: A map \(\mathbf{F}:\mathbb{R}^{n\times k}\rightarrow\mathbb{R}^{n\times l}\) satisfying \(\mathbf{F}(\sigma\cdot\mathbf{X})=\sigma\cdot\mathbf{F}(\mathbf{X})\) for all \(\sigma\in S_{n}\) and \(\mathbf{X}\in\mathbb{R}^{n\times d}\) is called permutation equivariant.
**Definition 5.2** (equivariant DeepSets of Zaheer et al. (2017)).: Equivariant DeepSets has the following form \(\mathbf{F}(\mathbf{X})=\mathbf{L}_{\text{in}}^{\text{ds}}\circ\nu\circ\cdots\circ\nu\circ \mathbf{L}_{1}^{\text{ds}}(\mathbf{X})\), where \(\mathbf{L}_{i}^{\text{ds}}\) is a linear permutation equivariant layer and \(\nu\) is a nonlinear layer such as ReLU. The linear permutation equivariant layer in DeepSets has the following form \(\mathbf{L}_{i}^{\text{ds}}(\mathbf{X})=\mathbf{X}\mathbf{A}+\frac{1}{n}\mathbf{1}\mathbf{1}^{T}\mathbf{X} \mathbf{B}+\mathbf{1}\mathbf{c}^{T}\), where \(\mathbf{A},\mathbf{B}\in\mathbb{R}^{d_{i}\times d_{i+1}}\), \(\mathbf{c}\in\mathbb{R}^{d_{i+1}}\) is the weights and bias in layer \(i\), and \(\nu\) is ReLU.
**Definition 5.3** (permutation equivariant universality).: Given a compact domain \(\mathcal{X}\) of \(\mathbb{R}^{n\times d_{\text{in}}}\), permutation equivariant universality of a model \(\mathbf{F}:\mathbb{R}^{n\times d_{\text{in}}}\rightarrow\mathbb{R}^{n\times d_{\text {out}}}\) means that for every permutation equivariant continuous function \(\mathbf{H}:\mathbb{R}^{n\times d_{\text{in}}}\rightarrow\mathbb{R}^{n\times d_{ \text{out}}}\) defined over \(\mathcal{X}\), and any \(\epsilon>0\), there exists a choice of \(m\) (i.e., network depth), \(d_{i}\) (i.e., network width at layer \(i\)) and the trainable parameters of \(\mathbf{F}\) so that \(\|\mathbf{H}(\mathbf{X})-\mathbf{F}(\mathbf{X})\|_{\infty}<\epsilon\) for all \(\mathbf{X}\in\mathcal{X}\).
The universality of equivariant DeepSets is stated as follows.
**Theorem 5.4** (Segol and Lipman (2019)).: _DeepSets with constant layer is universal. Using ReLU activation the width \(\omega:=\text{max}_{i}d_{i}\)\((d_{i}\) is the width for \(i\)-th layer of DeepSets) required for universal permutation equivariant network satisfies \(\omega\leq d_{\text{out}}+d_{\text{in}}+\left(\begin{array}{c}n+d_{\text{in}} \\ d_{\text{in}}\end{array}\right)=\mathcal{O}(n^{d_{\text{in}}})\)._
We are now ready to state our main theorem.
**Theorem 5.5**.: _MPNN + VN can simulate (not just approximate) equivariant DeepSets: \(\mathbb{R}^{n\times d}\rightarrow\mathbb{R}^{n\times d}\). The depth and width of MPNN + VN needed to simulate DeepSets is up to a constant factor of the depth and width of DeepSets. This implies that MPNN + VN of \(\mathcal{O}(1)\) depth and \(\mathcal{O}(n^{d})\) width is permutation equivariant universal, and can approximate self-attention layer and transformers arbitrarily well._
Proof.: Equivariant DeepSets has the following form \(\mathbf{F}(\mathbf{X})=\mathbf{L}_{\text{in}}^{\text{ds}}\circ\nu\circ\cdots\circ\nu\circ \mathbf{L}_{1}^{\text{ds}}(\mathbf{X})\), where \(\mathbf{L}_{i}^{\text{ds}}\) is the
linear permutation equivariant layer and \(\nu\) is an entrywise nonlinear activation layer. Recall that the linear equivariant layer has the form \(\mathbf{L}_{i}^{\text{ds}}(\mathbf{X})=\mathbf{X}\mathbf{A}+\frac{1}{n}\mathbf{1}\mathbf{1}^{T}\mathbf{X} \mathbf{B}+\mathbf{1}\mathbf{c}^{T}\). As one can use the same nonlinear entrywise activation layer \(\nu\) in MPNN + VN, it suffices to prove that MPNN + VN can compute linear permutation equivariant layer \(\mathbf{L}^{\text{ds}}\). Now we show that 2 layers of MPNN + VN can exactly simulate any given linear permutation equivariant layer \(\mathbf{L}^{\text{ds}}\).
Specifically, at layer 0, we initialized the node features as follows: The VN node feature is set to 0, while the node feature for the \(i\)-th graph node is set up as \(\mathbf{x}_{i}\in\mathbb{R}^{d}\).
At layer 1: VN node feature is \(\frac{1}{n}\mathbf{1}\mathbf{1}^{T}\mathbf{X}\), average of node features. The collection of features over \(n\) graph node feature is \(\mathbf{X}\mathbf{A}\). We only need to transform graph node features by a linear transformation, and set the VN feature as the average of graph node features in the last iteration. Both can be exactly implemented in Definition 3.4 of simplified heterogeneous MPNN + VN.
At layer 2: VN node feature is set to be 0, and the graph node feature is \(\mathbf{X}\mathbf{A}+\frac{1}{n}\mathbf{1}\mathbf{1}^{T}\mathbf{X}\mathbf{B}+\mathbf{1}\mathbf{c}^{T}\). Here we only need to perform the matrix multiplication of the VN feature with \(\mathbf{B}\), as well as add a bias \(\mathbf{c}\). This can be done by implementing a linear function for \(\gamma_{\text{gn}}\).
It is easy to see the width required for MPNN + VN to simulate DeepSets is constant. Thus, one can use 2 layers of MPNN + VN to compute linear permutation equivariant layer \(\mathbf{L}_{i}^{\text{ds}}\), which implies that MPNN + VN can simulate 1 layer of DeepSets exactly with constant depth and constant width (independent of \(n\)). Then by the universality of DeepSets, stated in Theorem 5.4, we conclude that MPNN + VN is also permutation equivariant universal, which implies that the constant layer of MPNN + VN with \(\mathcal{O}(n^{d})\) width is able to approximate any continuous equivariant maps. As the self-attention layer \(\mathbf{L}\) and full transformer are both continuous and equivariant, they can be approximated by MPNN + VN arbitrarily well.
Thanks to the connection between MPNN + VN with DeepSets, there is no extra assumption on \(\mathcal{X}\) except for being compact. The drawback on the other hand is that the upper bound on the computational complexity needed to approximate the self-attention with wide MPNN + VN is worse than directly computing self-attention when \(d>2\).
## 6 \(\mathcal{O}(n)\) depth \(\mathcal{O}(1)\) width MPNN + VN
The previous section shows that we can approximate a full attention layer in Transformer using MPNN with \(\mathcal{O}(1)\) depth but \(\mathcal{O}(n^{d})\) width where \(n\) is the number of nodes and \(d\) is the dimension of node features. In practice, it is not desirable to have the width depend on the graph size.
In this section, we hope to study MPNN + VNs with \(\mathcal{O}(1)\) width and their ability to approximate a self-attention layer in the Transformer. However, this appears to be much more challenging. Our result in this section only shows that for a rather restrictive family of input graphs (see Assumption 3 below), we can approximate a full self-attention layer of transformer with an MPNN + VN of \(\mathcal{O}(1)\) width and \(\mathcal{O}(n)\) depth. We leave the question of MPNN + VN's ability in approximate transformers for more general families of graphs for future investigation.
We first introduce the notion of \((\mathbf{V},\delta)\) separable node features. This is needed to ensure that VN can approximately select one node feature to process at each iteration with attention \(\alpha_{\text{vn}}\), the self-attention in the virtual node.
**Definition 6.1** (\((\mathbf{V},\delta)\) separable by \(\bar{\alpha}\)).: Given a graph \(G\) of size \(n\) and a fixed \(\mathbf{V}\in\mathbb{R}^{n\times d}=[\mathbf{v}_{1},...,\mathbf{v}_{n}]\) and \(\bar{\alpha}\in\mathcal{A}\), we say node feature \(\mathbf{X}\in\mathbb{R}^{n\times d}\) of \(G\) is \((\mathbf{V},\delta)\) separable by some \(\bar{\alpha}\) if the following holds. For any node feature \(\mathbf{x}_{i}\), there exist weights \(\mathbf{W}_{K}^{\bar{\alpha}},\mathbf{W}_{Q}^{\bar{\alpha}}\) in attention score \(\bar{\alpha}\) such that \(\bar{\alpha}(\mathbf{x}_{i},\mathbf{v}_{i})>\max_{j\neq i}\bar{\alpha}(\bar{\mathbf{x}}_{j },\mathbf{v}_{i})+\delta\). We say set \(\mathcal{X}\) is \((\mathbf{V},\delta)\) separable by \(\bar{\alpha}\) if every element \(\mathbf{X}\in\mathcal{X}\) is \((\mathbf{V},\delta)\) separable by \(\bar{\alpha}\).
The use of \((\mathbf{V},\delta)\) separability is to approximate hard selection function arbitrarily well, which is stated below and proved in Appendix B.1.
**Lemma 6.2** (approximate hard selection).: _Given \(\mathbf{X}\) is \((\mathbf{V},\delta)\) separable by \(\bar{\alpha}\) for some fixed \(\mathbf{V}\in\mathbb{R}^{n\times d}\), \(\bar{\alpha}\in\mathcal{A}\)
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**\# Params.**} & \multicolumn{2}{c}{Peptides-func} & \multicolumn{2}{c}{Peptides-struct} \\ \cline{3-6} & & **Test AP before VN** & **Test AP after VN** \(\uparrow\) & **Test MAE before VN** & **Test MAE after VN** \(\downarrow\) \\ \hline GCN & 508k & 0.5930\(\pm\)0.0023 & 0.6623\(\pm\)0.0038 & 0.3496\(\pm\)0.0013 & **0.2488\(\pm\)0.0021** \\ GINE & 476k & 0.5498\(\pm\)0.0079 & 0.6346\(\pm\)0.0071 & 0.3547\(\pm\)0.0045 & 0.2584\(\pm\)0.0011 \\ GatedGCN & 509k & 0.5864\(\pm\)0.0077 & 0.6635\(\pm\)0.0024 & 0.3420\(\pm\)0.0013 & 0.2523\(\pm\)0.0016 \\ GatedGCN+RWSE & 506k & 0.6069\(\pm\)0.0035 & **0.6685\(\pm\)0.0062** & 0.3357\(\pm\)0.0006 & 0.2529\(\pm\)0.0009 \\ \hline Transformer+LapPE & 488k & 0.6326\(\pm\)0.0126 & - & 0.2529\(\pm\)0.0016 & - \\ SAN+LapPE & 493k & 0.6384\(\pm\)0.0121 & - & 0.2683\(\pm\)0.0043 & - \\ SAN+RWSE & 500k & 0.6439\(\pm\)0.0075 & - & 0.2545\(\pm\)0.0012 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Baselines for Peptides-func (graph classification) and Peptides-struct (graph regression). The performance metric is Average Precision (AP) for classification and MAE for regression. **Bold**: Best score.
and \(\delta>0\), the following holds. For any \(\epsilon>0\) and \(i\in[n]\), there exists a set of attention weights \(\mathbf{W}_{i,Q},\mathbf{W}_{i,K}\) in \(i\)-th layer of MPNN + VN such that \(\alpha_{\text{\tiny{nn}}}(\mathbf{x}_{i},\mathbf{v}_{i})>1-\epsilon\) for any \(\mathbf{x}_{i}\in\mathcal{X}_{i}\). In other words, we can approximate a hard selection function \(f_{i}(\mathbf{x}_{1},...,\mathbf{x}_{n})=\mathbf{x}_{i}\) arbitrarily well on \(\mathcal{X}\) by setting \(\alpha_{\text{\tiny{nn}}}=\bar{\alpha}\)._
With the notation set up, We now state an extra assumption needed for deep MPNN + VN case and the main theorem.
**AS3.**\(\mathcal{X}\) is \((\mathbf{V},\delta)\) separable by \(\bar{\alpha}\) for some fixed \(\mathbf{V}\in\mathbb{R}^{n\times d}\), \(\bar{\alpha}\in\mathcal{A}\) and \(\delta>0\).
**Theorem 6.3**.: _Assume AS 1-3 hold for the compact set \(\mathcal{X}\) and \(\mathbf{L}\). Given any graph \(G\) of size \(n\) with node features \(\mathbf{X}\in\mathcal{X}\), and a self-attention layer \(\mathbf{L}\) on \(G\) (fix \(\mathbf{W}_{K},\mathbf{W}_{Q},\mathbf{W}_{V}\) in \(\alpha\)), there exists a \(\mathcal{O}(n)\) layer of heterogeneous MPNN + VN with the specific aggregate/update/message function that can approximate \(\mathbf{L}\) on \(\mathcal{X}\) arbitrarily well._
The proof is presented in the Appendix B. On the high level, we can design an MPNN + VN where the \(i\)-th layer will select \(\tilde{\mathbf{x}}_{i}\), an approximation of \(\mathbf{x}_{i}\) via attention mechanism, enabled by Lemma 6.2, and send \(\tilde{\mathbf{x}}_{i}\) to the virtual node. Virtual node will then pass the \(\tilde{\mathbf{x}}_{i}\) to all graph nodes and computes the approximation of \(e^{\alpha(\mathbf{x}_{i},\mathbf{x}_{j})},\forall j\in[n]\). Repeat such procedures \(n\) times for all graph nodes, and finally, use the last layer for attention normalization. A slight relaxation of AS3 is also provided in the appendix.
## 7 Experiments
### MPNN + VN for LRGB Datasets
We experiment with MPNN + VN for Long Range Graph Benchmark (LRGB) datasets. Original paper (Dwivedi et al., 2022) observes that GT outperforms MPNN on 4 out of 5 datasets, among which GT shows significant improvement over MPNN on Peptides-func and Peptides-struct for all MPNNs. To test the effectiveness of the virtual node, we take the original code and modify the graph topology by adding a virtual node and keeping the hyperparameters of all models unchanged.
Results are in Table 2. Interestingly, such a simple change can boost MPNN + VN by a large margin on Peptides-func and Peptides-struct. Notably, with the addition of VN, GatedGCN + RWSE (random-walk structural encoding) after augmented by VN **outperforms all transformers** on Peptides-func, and GCN outperforms transformers on Peptides-struct.
### Stronger MPNN + VN Implementation
Next, by leveraging the modularized implementation from GraphGPS (Rampasek et al., 2022), we implemented a version of MPNN + VN with/without extra positional embedding. Our goal is not to achieve SOTA but instead to push the limit of MPNN + VN and better understand the source of the performance gain for GT. In particular, we replace the GlobalAttention Module in GraphGPS with DeepSets, which is equivalent to one specific version of MPNN + VN. We tested this specific version of MPNN + VN on 4 OGB datasets, both with and without the use of positional embedding. The results are reported in Table 3. Interestingly, even without the extra position embedding, our MPNN + VN is able to further improve over the previous GCN + VN & GIN + VN implementation. The improvement on **ogbg-ppa** is particularly impressive, which is from 0.7037 to 0.8055. Furthermore, it is important to note that while MPNN + VN does not necessarily outperform GraphGPS, which is a state-of-the-art architecture using both MPNN, Position/structure encoding and Transformer, the difference is quite small - this however, is achieved by a simple MPNN + VN architecture.
We also test MPNN + VN on large-scale molecule datasets PCQMv2, which has 529,434 molecule graphs. We followed (Rampasek et al., 2022) and used the original validation set as the test set, while we left out random 150K molecules for our validation set. As we can see from Table 4, MPNN + VN + NoPE performs significantly better than the early MPNN + VN implementation: GIN + VN and GCN +
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Model** & **ogbg-molhiv** & **ogbg-molpcba** & **ogbg-ppa** & **ogbg-code2** \\ \cline{2-5} & **AUROC \(\uparrow\)** & **Avg. Precision \(\uparrow\)** & **Accuracy \(\uparrow\)** & **F1 score \(\uparrow\)** \\ \hline GCN & 0.7606 \(\pm\) 0.0097 & 0.2020 \(\pm\) 0.0024 & 0.6839 \(\pm\) 0.0084 & 0.1507 \(\pm\) 0.0018 \\ GCN+virtual node & 0.7599 \(\pm\) 0.0119 & 0.2424 \(\pm\) 0.0034 & 0.6857 \(\pm\) 0.0061 & 0.1595 \(\pm\) 0.0018 \\ GIN & 0.7558 \(\pm\) 0.0140 & 0.2266 \(\pm\) 0.0028 & 0.6892 \(\pm\) 0.0100 & 0.1495 \(\pm\) 0.0023 \\ GIN+virtual node & 0.7707 \(\pm\) 0.0149 & 0.2703 \(\pm\) 0.0023 & 0.7037 \(\pm\) 0.0107 & 0.1581 \(\pm\) 0.0026 \\ \hline SAN & 0.7785 \(\pm\) 0.2470 & 0.2765 \(\pm\) 0.0042 & – & – \\ GraphTrans (GCN-Virtual) & – & 0.2761 \(\pm\) 0.0029 & – & 0.1830 \(\pm\) 0.0024 \\ K-Subtree SAT & – & – & 0.7522 \(\pm\) 0.0056 & 0.1937 \(\pm\) 0.0028 \\ GPS & 0.7880 \(\pm\) 0.0101 & 0.2907 \(\pm\) 0.0028 & 0.8015 \(\pm\) 0.0033 & 0.1894 \(\pm\) 0.0024 \\ \hline MPNN + VN + NoPE & 0.7676 \(\pm\) 0.0172 & 0.2823 \(\pm\) 0.0026 & 0.8055 \(\pm\) 0.0038 & 0.1727 \(\pm\) 0.0017 \\ MPNN + VN + PE & 0.7687 \(\pm\) 0.0136 & 0.2848 \(\pm\) 0.0026 & 0.8027 \(\pm\) 0.0026 & 0.1719 \(\pm\) 0.0013 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Test performance in graph-level OGB benchmarks (Hu et al., 2020). Shown is the mean \(\pm\) s.d. of 10 runs.
-VN. The performance gap between GPS on the other hand is rather small: 0.0938 (GPS) vs. 0.0942 (MPNN + VN + PE) for the small model and 0.0858 (GPS) vs. 0.0867 (MPNN + VN + PE) for the medium model.
### Forecasting Sea Surface Temperature
In this experiment, we apply our MPNN + VN model to forecast sea surface temperature (SST). We are particularly interested in the empirical comparison between MPNN + VN and Linear Transformer (Katharopoulos et al., 2020) as according to Section 4, MPNN + VN theoretically can approximate Linear Transformer.
In particular, from the DOISST data proposed by (Huang et al., 2021), we construct a dataset of daily SST in the Pacific Ocean from 1982 to 2021, in the region of longitudes from \(180.125^{\circ}\)E to \(269.875^{\circ}\)E and latitudes from \(-14.875^{\circ}\)N to \(14.875^{\circ}\)N. Following the procedure from (de Berenac et al., 2018; de Berenac et al., 2019) and Wang et al. (2022), we divide the region into 11 batches of equal size with 30 longitudes and 30 latitudes at 0.5\({}^{\circ}\)-degree resolution, that can be represented as a graph of 900 nodes. The tasks are to predict the next 4 weeks, 2 weeks and 1 week of SST at each location, given 6 weeks of historical data. We train on data from years 1982-2018, validate on data from 2019 and test on data from 2020-2021. The number of training, validation, and testing examples are roughly 150K, 3K, and 7K. See details of dataset construction, model architectures, and training scheme in Appendix D.4.
We compare our model to other baselines including TF-Net (Wang et al., 2020), a SOTA method for spatiotemporal forecasting, Linear Transformer (Katharopoulos et al., 2020; Wang et al., 2020) with Laplacian positional encoding (LapPE), and Multilayer Perceptron (MLP). We use Mean Square Error (MSE) as the metric and report the errors on the test set, shown in the Table 5. We observe that the virtual node (VN) alone improves upon MPNN by \(3.8\%\), \(6.6\%\) and \(4.5\%\) in 4-, 2- and 1-week settings, respectively. Furthermore, aligned with our theory in Section 4, MPNN + VN indeed achieves comparable results with Linear Transformer and outperforms it by a margin of \(0.4\%\), \(2.8\%\) and \(4.3\%\) in 4-, 2- and 1-week settings, respectively.
## 8 Concluding Remarks
In this paper, we study the expressive power of MPNN + VN under the lens of GT. If we target the self-attention layer in Performer and Linear Transformer, one only needs \(\mathcal{O}(1)\)-depth \(\mathcal{O}(1)\) width for arbitrary approximation error. For self-attention in full transformer, we prove that heterogeneous MPNN + VN of either \(\mathcal{O}(1)\) depth \(\mathcal{O}(n^{d})\) width or \(\mathcal{O}(n)\) depth \(\mathcal{O}(1)\) width (under some assumptions) can approximate 1 self-attention layer arbitrarily well. Compared to early results (Kim et al., 2022) showing GT can approximate MPNN, our theoretical result draws the connection from the inverse direction.
On the empirical side, we demonstrate that MPNN + VN remains a surprisingly strong baseline. Despite recent efforts, we still lack good benchmark datasets where GT can outperform MPNN by a large margin. Understanding the inductive bias of MPNN and GT remains challenging. For example, can we mathematically characterize tasks that require effective long-range interaction modeling, and provide a theoretical justification for using GT over MPNN (or vice versa) for certain classes of functions on the space of graphs? We believe making processes towards answering such questions is an important future direction for the graph learning community.
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**PCQM4Mv2**} \\ \cline{2-5} & **Test-dev MAE \(\downarrow\)** & **Validation MAE \(\downarrow\)** & **Training MAE** & **\# Param.** \\ \hline GCN & 0.1398 & 0.1379 & n/a & 2.0M \\ GCN-virtual & 0.1152 & 0.1153 & n/a & 4.9M \\ GIN & 0.1218 & 0.1195 & n/a & 3.8M \\ GIN-virtual & 0.1084 & 0.1083 & n/a & 6.7M \\ \hline GRPE (Park et al., 2022) & 0.0898 & 0.0890 & n/a & 46.2M \\ EGIT (Hussain et al., 2022) & 0.0872 & 0.0869 & n/a & 89.3M \\ Graphomer (Shi et al., 2022) & n/a & 0.0864 & 0.0348 & 48.3M \\ GPS-small & n/a & 0.0938 & 0.0653 & 6.2M \\ GPS-medium & n/a & 0.0858 & 0.0726 & 19.4M \\ \hline MPNN + VN + PE (small) & n/a & 0.0942 & 0.0617 & 5.2M \\ MPNN + VN + PE (medium) & n/a & 0.0867 & 0.0703 & 16.4M \\ MPNN + VN + NoPE (small) & n/a & 0.0967 & 0.0576 & 5.2M \\ MPNN + VN + NoPE (medium) & n/a & 0.0889 & 0.0693 & 16.4M \\ \hline \hline \end{tabular}
\end{table}
Table 4: Evaluation on PCQM4Mv2 (Hu et al., 2021) dataset. For GPS evaluation, we treated the _validation_ set of the dataset as a test set, since the _test-dev_ set labels are private.
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Model** & **4 weeks** & **2 weeks** & **1 week** \\ \hline MLP & 0.3302 & 0.2710 & 0.2121 \\ TF-Net & 0.2833 & **0.2036** & **0.1462** \\ Linear Transformer + LapPE & 0.2818 & 0.2191 & 0.1610 \\ MPNN & 0.2917 & 0.2281 & 0.1613 \\ \hline MPNN + VN & **0.2806** & 0.2130 & 0.1540 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of SST prediction.
## Acknowledgement
This work was supported in part by the U.S. Department Of Energy, Office of Science, U. S. Army Research Office under Grant W911NF-20-1-0334, Google Faculty Award, Amazon Research Award, and NSF Grants #2134274, #2107256, #2134178, CCF-2217033, and CCF-2112665.
| グラフTransformer（GT）は、最近グラフ学習アルゴリズムの新たなパラダイムとしてEmerged、MPNN（Message Passing Neural Network）を多岐のベンチマークで上回っています。Kim et al. (2022)の論文は、適切な位置埋め込みを持つGTがMPNNを任意に近似できることを示し、GTはMPNNと同等の能力を持つ可能性を示唆しています。この論文では、逆接続を調査し、MPNNの仮想ノード（VN）を用いることで、GTの自注意力層を任意に近似できることを示しました。特に、私たちは、1種類のリニア変換器、いわゆるPerformer/Linear Transformer（Choromanski et al. (2020); Katharopoulos et al. (2020））を考慮した場合、MPNN + VNは、O(1)の深さおよびO(1)の幅を持つことでGTの自注意力層 |
2307.03004 | Analysis and design of model predictive control frameworks for dynamic
operation -- An overview | This article provides an overview of model predictive control (MPC)
frameworks for dynamic operation of nonlinear constrained systems. Dynamic
operation is often an integral part of the control objective, ranging from
tracking of reference signals to the general economic operation of a plant
under online changing time-varying operating conditions. We focus on the
particular challenges that arise when dealing with such more general control
goals and present methods that have emerged in the literature to address these
issues. The goal of this article is to present an overview of the
state-of-the-art techniques, providing a diverse toolkit to apply and further
develop MPC formulations that can handle the challenges intrinsic to dynamic
operation. We also critically assess the applicability of the different
research directions, discussing limitations and opportunities for further
research. | Johannes Köhler, Matthas A. Müller, Frank Allgöwer | 2023-07-06T14:07:51 | http://arxiv.org/abs/2307.03004v2 | # Analysis and design of model predictive control frameworks for dynamic operation An overview 1
###### Abstract
This article provides an overview of model predictive control (MPC) frameworks for dynamic operation of nonlinear constrained systems. Dynamic operation is often an integral part of the control objective, ranging from tracking of reference signals to the general economic operation of a plant under online changing time-varying operating conditions. We focus on the particular challenges that arise when dealing with such more general control goals and present methods that have emerged in the literature to address these issues. The goal of this article is to present an overview of the state-of-the-art techniques, providing a diverse toolkit to apply and further develop MPC formulations that can handle the challenges intrinsic to dynamic operation. We also critically assess the applicability of the different research directions, discussing limitations and opportunities for further research.
keywords: Model predictive control (MPC), Tracking MPC, Economic MPC, MPC without stabilizing terminal cost, Closed-loop stability, Dynamic system operation +
Footnote †: journal: Annual Reviews in Control
## 1 Introduction
Model predictive control (MPC), also called receding horizon control, is a modern optimization-based control method. The underlying principle is to repeatedly solving finite horizon open-loop optimal control problems online. Feedback is generated implicitly by only implementing the initial part of the optimized input trajectory and repeating the online optimization in the next time step. MPC is widely used in practice (cf. the surveys by Qin and Badgwell (2003) and Samad et al. (2020)) and actively researched in academia (Mayne, 2014). This success of MPC is primarily due to some intrinsic advantages: (i) direct consideration of state and input constraints; (ii) applicability to general nonlinear MIMO systems; (iii) optimization of general performance criteria.
There have been significant advances in academia over the last decades, resulting in a mature stability theory for MPC (Mayne et al., 2000). Much ongoing research in MPC is focused on deriving efficient implementations (Verschueren et al., 2022), accounting for model errors (Kouvaritakis and Cannon, 2016; Mayne, 2016), or learning the model online (Hewing et al., 2020). In contrast, this article focuses on the design and analysis of MPC framework that can be applied to _dynamic operation_.
### Dynamic operation
The motivation of the present article comes from many emerging control applications in which the control goal is not accurately reflected by the setpoint stabilization problem, which is classically studied in MPC. We specifically consider the challenges intrinsic in _dynamic operation_, which has received less attention in the MPC community and is rarely studied in a unified fashion. By _dynamic operation_, we primarily consider the following three challenges related to the control goal:
1. Stationary operation is not desired;
2. Desired mode of operation changes online in an unpredictable fashion;
3. Desired mode of operation cannot be directly specified in terms of a given setpoint/trajectory of the system state.
First, we illustrate the challenges (C.1)-(C.3) using applications. Consider a motion planning or trajectory optimization problem, as encountered in robotics, aerospace, or autonomous driving. (C.1): The primary goal is to track/follow some dynamic trajectory/path. (C.2): This reference is often generated online by a separate unit (e.g., using artificial intelligence and visual feedback) independent of the controller (and hence unpredictably). (C.3): The reference is primarily specified in terms of a low dimensional output of the system, which is often not physically realizable due to the dynamics or constraints on the system. As a completely different application, consider a heating, ventilation, and air conditioning (HVAC) system to regulate temperature in a building. (C.3): The control goal naturally revolves around _economic criteria_ such as the energy consumption. (C.2): The optimal mode of operation depends on external factors, such as temperature, occupancy, and price/demand, which fluctuate in an unpredictable
fashion. (C.1): The same external variables are subject to significant changes over time, e.g., due to the periodic day-night cycle, making the optimal mode of operation non-stationary. Analogous considerations apply to water distribution networks, power networks, power generation using kites and many more. Overall, in many applications the desired mode of operation is dynamic; the controller needs to change the mode of operation based on external, online changing, variables; and the optimal mode of operation is not explicitly specified in terms of a known feasible setpoint/trajectory of the system state.
### Contribution
We provide an overview of recent advances in the design and analysis of MPC formulations that can accommodate and address the challenges intrinsic to dynamic operation (C.1)-(C.3). We are specifically interested in classical system theoretic properties, such guaranteeing recursive feasibility, constraint satisfaction, and stability/performance for general nonlinear systems. We provide a set of tools and methods to design MPC schemes for such challenging control applications with guaranteed closed-loop properties. We discuss the existing work and methods in a broad context, highlighting gaps in the existing literature and discussing different strategies in a unified fashion. In general, the challenges (C.1)-(C.3) and the studied methods touch on a number of different research fields in MPC, e.g.: trajectory tracking (Faulwasser, 2012) output regulation (Kohler et al., 2022), artificial references (Limon et al., 2018), economic MPC (Faulwasser et al., 2018), and MPC without terminal constraints (Grune and Pannek, 2017). We study these formulations in a unified way, focusing on how these frameworks address the challenges outlined above.
### Outline
First, we introduce _preliminaries_ regarding the design of a stabilizing MPC scheme and explain the challenges in applying this design to dynamic operation (Sec. 2). Moreover, we present designs for the _stabilizing terminal ingredients_, primarily the local control Lyapunov function (CLF), for general tracking problems (Sec. 3). Then, we show how infeasible and online changing references can be accommodated using _artificial reference trajectories_ (Sec. 4). Furthermore, we discuss how economic performance criteria can be directly optimized using _economic_ MPC formulations (Sec. 5). Finally, we present an alternative framework, focusing on the analysis of simpler MPC schemes _without stabilizing terminal ingredients_ (Sec. 6). The paper concludes with a _discussion_ regarding the different provided tools, their benefits, limitations, and open issues in the literature (Sec. 7).
### Notation
We denote the set of integers in an interval \([a,b]\) by \(\mathbb{I}_{[a,b]}\). For \(k,N\in\mathbb{I}_{\geq 0}\), the modulo operator is denoted by \(\mathrm{mod}(k,N)\in\mathbb{I}_{[0,N-1]}\). The quadratic norm w.r.t. a positive definite matrix \(Q=Q^{\top}\) is denoted by \(\left\|x\right\|_{Q}^{2}:=x^{\top}Qx\). The identify matrix is denoted by \(I_{n}\in\mathbb{R}^{n\alpha_{n}}\). For \(x\in\mathbb{R}^{n}\), \(y\in\mathbb{R}^{n\alpha_{n}}\), we abbreviate the stacked vector as \((x,y):=[x^{\top},y^{\top}]^{\top}\in\mathbb{R}^{n_{1}+n_{2}}\). For a symmetric matrix \(A=A^{\top}\), \(A>0\) (\(A\geq 0\)) indicates that the matrix is positive definite (positive semidefinite). The interior of a set \(\mathbb{X}\subseteq\mathbb{R}^{n}\) is denoted by \(\mathrm{int}(\mathbb{X})\). By \(\mathcal{K}_{\infty}\), we denote the class of functions \(\alpha:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}\), which are continuous, strictly increasing, unbounded, and satisfy \(\alpha(0)=0\). For a continuously differentiable function \(F(x,y)\), \(F:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\rightarrow\mathbb{R}^{n}\), \(\left\|\left.\frac{\partial F}{\partial x}\right\|_{(\xi,\bar{y})}\in\mathbb{R }^{n\alpha_{1}}\right.\) denotes the Jacobian matrix of \(F\) w.r.t. \(x\) evaluated at \((\bar{x},\bar{y})\).
## 2 Preliminaries: Recursive feasibility & stability in MPC
In the following, we briefly describe the basic MPC algorithm, including its closed-loop properties, analogous to the textbooks by Rawlings et al. (2017) and Grune and Pannek (2017). We consider a nonlinear discrete-time system
\[x(t+1)=f(x(t),u(t)),\quad x(0)=x_{0}, \tag{1}\]
with the state \(x(t)\in X=\mathbb{R}^{n}\), the control input \(u(t)\in\mathbb{U}\subseteq\mathbb{R}^{m}\), and the time step \(t\in\mathbb{I}_{\geq 0}\). For a sequence \(\mathbf{u}\in\mathbb{U}^{N}\), we denote the \(k\)-th element by \(\mathbf{u}_{k}\in\mathbb{U}\) with \(k\in\mathbb{I}_{[0,N-1]}\). Given an initial state \(x\in X\) and an input sequence \(\mathbf{u}\in\mathbb{U}^{N}\), we denote the solution to (1) after \(k\in\mathbb{I}_{\geq 0}\) steps by \(x_{\mathbf{u}}(k,x)\in X\), \(k\in\mathbb{I}_{[0,N]}\) with \(x_{\mathbf{u}}(0,x)=x\). The system is subject to pointwise-in-time constraints
\[(x(t),u(t))\in\mathbb{Z}\subseteq X\times\mathbb{U},\quad\forall t\in\mathbb{ I}_{\geq 0}, \tag{2}\]
which, e.g., model actuator limitations or safety critical limits on the state. The goal is to minimize a given stage cost \(\ell:X\times\mathbb{U}\rightarrow\mathbb{R}\), resulting in the following optimal control problem
\[\mathcal{J}_{\infty}^{\star}(x) :=\lim_{N\rightarrow\infty}\inf_{\mathbf{u}\in\mathbb{U}^{N}} \sum_{k=0}^{N-1}\ell(x_{\mathbf{u}}(k,x),\mathbf{u}_{k}) \tag{3}\] \[\text{s.t.}\ (x_{\mathbf{u}}(k,x),\mathbf{u}_{k})\in\mathbb{Z}, \quad k\in\mathbb{I}_{\geq 0}.\]
### Stabilizing MPC design
Solving Problem (3) is typically computationally intractable. MPC approximately solves Problem (3) by repeatedly solving a finite-horizon open-loop optimal control problem:
\[\mathcal{J}_{N}^{\star}(x) :=\inf_{\mathbf{u}\in\mathbb{U}^{N}}\sum_{k=0}^{N-1}\ell(x_{ \mathbf{u}}(k,x),\mathbf{u}_{k})+V_{\mathrm{f}}(x_{\mathbf{u}}(N,x)) \tag{4}\] \[\text{s.t.}\ (x_{\mathbf{u}}(k,x),\mathbf{u}_{k})\in\mathbb{Z}, \ k\in\mathbb{I}_{[0,N-1]},\ x_{\mathbf{u}}(N,x)\in\mathbb{X}_{\mathrm{f}},\]
with prediction horizon \(N\in\mathbb{I}_{\geq 1}\), a terminal cost \(V_{\mathrm{f}}:X\rightarrow\mathbb{R}\), and a terminal set \(\mathbb{X}_{\mathrm{f}}\subseteq X\). Throughout this article, we assume that the constraint set \(\mathbb{Z}\) is compact, the dynamics \(f\) are continuous, and the cost functions \(\ell,V_{\mathrm{f}}\) are continuous. Hence, Problem (4) has a minimizer (Rawlings et al., 2017, Prop. 2.4), which is denoted by \(\mathbf{u}^{\star}(x)\in\mathbb{U}^{N}\).1 The closed-loop operation is defined by
Footnote 1: In case the minimizer is not unique, one minimizer can be selected. Although we assume that a minimizer is computed, most closed-loop guarantees also hold if a suboptimal feasible solution is computed (Scokaert et al., 1999; McAllister and Rawlings, 2023).
\[x(t+1)=f(x(t),\mathbf{u}_{0}^{\star}(x(t))),\quad t\in\mathbb{I}_{\geq 0}, \tag{5}\]
i.e., at each time \(t\) we apply the first part of the optimal open-loop input sequence \(\mathbf{u}^{\bullet}(x(t))\in\mathbb{U}^{N}\) computed based on the measured state \(x(t)\).
The following result recaps standard design conditions and resulting closed-loop properties of the MPC, assuming a feasible steady state \((x_{\mathrm{r}},u_{\mathrm{r}})\in\mathbb{Z}\), \(f(x_{\mathrm{r}},u_{\mathrm{r}})=x_{\mathrm{r}}\) should be stabilized.
**Assumption 1**.: _(Stabilizing stage cost) There exist functions \(\underline{\alpha}_{\mathrm{r}}\), \(\overline{\alpha}_{\mathrm{f}}\), \(\alpha_{\mathrm{f}}\in\mathcal{K}_{\infty}\) such that \(\underline{\alpha}_{\mathrm{r}}(\|x-x_{\mathrm{r}}\|)\leq\ell_{\min}(x)\leq \overline{\alpha}_{\mathrm{r}}(\|x-x_{\mathrm{r}}\|)\) for all \((x,u)\in\mathbb{Z}\), \(V_{\mathrm{f}}(x)\leq\alpha_{\mathrm{r}}(\|x-x_{\mathrm{r}}\|)\) for all \(x\in\mathbb{X}_{\mathrm{f}}\), and \(\ell(x_{\mathrm{r}},u_{\mathrm{r}})=0\), \(V_{\mathrm{f}}(x_{\mathrm{r}})=0\), with \(\ell_{\min}(x):=\min_{\mathrm{e}\in\mathbb{U}}\ell(x,u)\)._
**Assumption 2**.: _(Terminal ingredients) There exists a terminal control law \(k_{\mathrm{f}}:\mathbb{X}_{\mathrm{f}}\to\mathbb{U}\) such that for all \(x\in\mathbb{X}_{\mathrm{f}}\):_
1. _Constraint satisfaction:_ \((x,k_{\mathrm{f}}(x))\in\mathbb{Z}\)_;_
2. _Positive invariance:_ \(f(x,k_{\mathrm{f}}(x))\in\mathbb{X}_{\mathrm{f}}\)_;_
3. _Local CLF:_ \(V_{\mathrm{f}}(f(x,k_{\mathrm{f}}(x)))-V_{\mathrm{f}}(x)\leq-\ell(x,k_{ \mathrm{f}}(x))\)_._
_Furthermore, there exists a function \(\alpha_{\mathrm{V}}\in\mathcal{K}_{\infty}\) such that for any state \(x\in X\) such that Problem (4) is feasible, it holds:_
1. _Weak controllability:_ \(\mathcal{J}_{N}^{\star}(x)\leq\alpha_{\mathrm{V}}(\|x-x_{\mathrm{r}}\|)\)_._
**Theorem 1**.: _(Rawlings et al., 2017, Thm. 2.19) Suppose Problem (4) is feasible with \(x=x_{0}\). Then, Problem (4) is feasible for all \(t\in\mathbb{I}_{\geq 0}\), the constraints (2) are satisfied, \(x_{\mathrm{r}}\) is asymptotically stable, and the following performance bound holds for the resulting closed-loop system (5):_
\[\mathcal{J}_{\infty}^{\mathrm{cl}}(x_{0}):=\sum_{t=0}^{\infty}\ell(x(t),u(t)) \leq\mathcal{J}_{N}^{\star}(x_{0}). \tag{6}\]
The intuition behind the terminal set and terminal cost is to approximate the infinite-horizon tail for \(k\in\mathbb{I}_{\geq N}\)(Chen and Allgower, 1998), i.e., a feasible solution to Problem (3) is given by appending \(\mathbf{u}^{\bullet}(x)\in\mathbb{U}^{N}\) with the terminal control law \(k_{\mathrm{f}}(x)\) for \(k\in\mathbb{I}_{\geq N}\) due to (7), (8). Stability is ensured by showing that the value function \(\mathcal{J}_{N}^{\star}\) is a Lyapunov function, i.e., Condition (7) implies \(\mathcal{J}_{N}^{\star}(x(t+1))-\mathcal{J}_{N}^{\star}(x(t))\leq-\ell(x(t),u(t))\), \(t\in\mathbb{I}_{\geq 0}\). Condition (10) is a technical condition to ensure that \(\mathcal{J}_{N}^{\star}\) is a Lyapunov function, which holds trivially if \(x_{\mathrm{r}}\in\mathrm{int}(\mathbb{X}_{\mathrm{f}})\)(Rawlings et al., 2017, Sec. 2.4.2). Under additional technical conditions, the performance bound (6) also yields a suboptimality/regret bound w.r.t. the optimal performance \(\mathcal{J}_{\infty}^{\star}\)(cf., Kohler (2021, App. A) and Grune and Pannek (2017, Thm. 5.22)). Overall, Theorem 1 provides all the desired closed-loop properties and the posed conditions (Asm. 1-2) can be constructively satisfied (cf. Sec. 3).
### Challenges in dynamic operation
In the following, we explain how the challenges related to dynamic operation (7)-(8) complicate the application of this design and how the different frameworks discussed in this article approach these problems. The design of the terminal ingredients (Asm. 2) is centred around the CLF \(V_{\mathrm{f}}\) and a control law \(k_{\mathrm{f}}\) that locally stabilizes the steady state \(x_{\mathrm{r}}\). This offline design becomes challenging if the setpoint \(x_{\mathrm{r}}\) can change online (8) and non-stationary references are considered (7). Constructing designs addressing this issue are presented in Section 3.
The desired setpoint \(x_{\mathrm{r}}\) can change arbitrary online (8) and may even be physically infeasible (8). This can lead to infeasibility of Problem (4) due to the terminal set constraint \(\mathbb{X}_{\mathrm{f}}\) and hence invalidate all closed-loop guarantees. In Section 4, artificial references are included in the MPC formulation to avoid these complications.
All of these designs try to stabilize some given reference using a positive definite stage cost \(\ell\) (Asm. 1). Section 5 shows how to directly minimize an economic (indefinite) cost \(\ell\) (8).
Lastly, Section 6 explores an alternative approach: deriving system theoretic conditions and a sufficiently long prediction horizon \(N\) to ensure closed-loop properties for simpler MPC designs, which do not require \(V_{\mathrm{f}}\), \(\mathbb{X}_{\mathrm{f}}\) satisfying Assumption 2.
## 3 Terminal ingredients for nonlinear tracking MPC
In this section, we focus on constructing a terminal cost \(V_{\mathrm{f}}\) and terminal set \(\mathbb{X}_{\mathrm{f}}\) (Asm. 2) for dynamic tracking problems. We first summarize the standard linearization-based design for the regulation problem (Sec. 3.1) and discuss extensions to track online changing setpoints (Sec. 3.2). Then, we consider tracking of dynamic reference trajectories (Sec. 3.3), including the case where the full reference may change online (Sec. 3.4). Lastly, we provide some discussion (Sec. 3.5).
For the following exposition, we consider a quadratic stage cost \(\ell:=\|x-x_{\mathrm{r}}\|_{Q}^{2}+\|u-u_{\mathrm{r}}\|_{\mathrm{R}}^{2}\) with \(Q,R>0\), assume that the reference \(r=(x_{\mathrm{r}},u_{\mathrm{r}})\in\mathbb{R}^{n+m}\) lies in some constraint set \(\mathbb{Z}_{\mathrm{r}}\subseteq\mathrm{int}(\mathbb{Z})\), and suppose that the dynamics \(f\) are twice continuously differentiable.
### Regulation problem
We consider the basic stabilizing MPC formulation introduced in Section 2 and provide a constructive design to satisfy Assumptions 1-2. This method was first derived by Chen and Allgower (1998), cf. Rawlings et al. (2017, Sec. 2.5.5) for the considered discrete-time variant. We denote the Jacobians by
\[A(r):=\left[\frac{\partial f}{\partial x}\right]_{r},\quad B(r):=\left[\frac{ \partial f}{\partial u}\right]_{r} \tag{7}\]
and abbreviate the following matrix expression related to the linear quadratic regulator (LQR):
\[\mathrm{LQR}(A,B,K,P,P^{+},Q,R) \tag{8}\] \[:= (A+BK)^{\top}P^{+}(A+BK)-P+Q+K^{\top}RK.\]
The condition \(LQR\geq 0\) can enforce Condition (7) from Assumption 2 for the linearization around the setpoint \(r\in\mathbb{Z}_{\mathrm{r}}\). We account for the linearization error by imposing a stronger condition on the linearization with a tuning variable \(\epsilon>0\). In particular, we choose matrices \(P,K\) such that \(LQR(A(r),B(r),K,P,P,Q+\epsilon I_{n},R)\geq 0\). This corresponds to designing an LQR for the linearized dynamics around the setpoint \(r\in\mathbb{Z}_{\mathrm{r}}\), which is feasible if \((A(r),B(r))\) is stabilizable.
**Proposition 1**.: _There exists a sufficiently small \(\alpha>0\), such that the quadratic terminal cost \(V_{\mathrm{f}}(x)=\|x-x_{\mathrm{r}}\|_{P_{\mathrm{r}}}^{2}\), the linear terminal controller \(k_{\mathrm{f}}(x)=u_{\mathrm{r}}+K(x-x_{\mathrm{r}})\) and the terminal set \(\mathbb{X}_{\mathrm{f}}=\{x\in X|V_{\mathrm{f}}(x)\leq\alpha\}\) satisfy Assumptions 1-2._
Twice continuous differentiability of \(f\) ensures that Condition (T.3) holds for the nonlinear system for all \(\alpha\leq\alpha_{1}\), with some \(\alpha_{1}>0\). Analytic formulas for \(\alpha_{1}\) can be derived using bounds on the Hessian (Chen and Allgower, 1998; Rawlings et al., 2017) and less conservative constants can be obtained using sampling-based approaches, cf. Chen and Allgower (1998, Rk. 3.1), Kohler et al. (2020, Alg. 1), Griffith et al. (2018); Rajhans et al. (2019). Assumption 1 and Condition (T.4) are trivially satisfied with quadratic bounds. Condition (T.2) follows from Condition (T.3) with \(\ell\geq 0\) and the parametrization of the terminal set \(\mathbb{X}_{\mathrm{f}}\), i.e., sublevel sets of a Lyapunov function are positively invariant. Lastly, Condition (T.1) holds for all \(\alpha\leq\alpha_{2}\) with some \(\alpha_{2}>0\) since \((x_{\mathrm{r}},u_{\mathrm{r}})\in\mathbb{Z}_{\mathrm{r}}\subseteq\mathrm{ int}(\mathbb{Z})\). In case \(\mathbb{Z}\) is a polytope, \(\alpha_{2}\) can be exactly determined using a linear program (Conte et al., 2016, Eq. (10)) and Kohler (2021, Lemma 3.37) provide a similar procedure if \(\mathbb{Z}\) is given by Lipschitz continuous inequality constraints.
### Setpoint tracking
The provided design requires some offline computation to determine a terminal cost \(V_{\mathrm{f}}\) and a terminal set \(\mathbb{X}_{\mathrm{f}}\) to stabilize a specific setpoint \(x_{\mathrm{r}}\). This offline procedure needs to be repeated if we wish to stabilize a different setpoint \(x_{\mathrm{r}}\), which is undesirable from a practical perspective. Due to its practical relevance, many approaches have been suggested to overcome this issue. Findeisen et al. (2000) suggest a fixed matrix \(P\) for the quadratic terminal cost \(V_{\mathrm{f}}\) for different setpoints \(x_{\mathrm{r}}\) using a pseudo linearisation, however, practical application of this theory is difficult. Wan and Kothare (2003, 2004) locally describe the nonlinear system as a linear difference inclusion (LDI) and compute constant matrices \(P,K\), which are valid for a local set of steady states, i.e., enforcing \(\mathrm{LQR}(A(r),B(r),K,P,P,Q,R)\geq 0\) for all \(r\) in some region. Limon et al. (2018, App. B) partition the steady-state manifold and compute different matrices \(P,K\) for each partition using this LDI description, resulting in a piece-wise quadratic terminal cost. However, the manual partitioning can result in a cumbersome design process, especially if the system is strongly nonlinear or the steady-state manifold is high-dimensional, and the resulting discontinuity can bring additional complications.
Kohler et al. (2020) alleviate these shortcomings with a continuous parametrization: \(V_{\mathrm{f}}(x,r)=\|x-x_{\mathrm{r}}\|_{P_{\mathrm{r}}(r)}^{2}\), \(k_{\mathrm{f}}(x,r)=u_{\mathrm{r}}+K(r)(x-x_{\mathrm{r}})\), \(\mathbb{X}_{\mathrm{f}}(r)=\{x\in X|\ V_{\mathrm{f}}(x,r)\leq\alpha(r)\}\) with \(P(r),K(r),\alpha(r)\) continuous in \(r\). The local CLF condition (T.3) then holds with \(\alpha(r)\) chosen sufficiently small (cf. Sec. 4.2.5) if
\[\mathrm{LQR}(A(r),B(r),K(r),P(r),P(r),Q+\epsilon I_{n},R)\geq 0 \tag{9}\]
holds for all feasible reference setpoints \(r\). By interpreting the reference \(r\) as a parameter, this can be viewed as a special case of gain-scheduling synthesis for linear-parameter varying (LPV) systems, a classical robust control problem (Rugh and Shamma, 2000). The computation of parametrized matrices \(P,K\) satisfying Condition (9) can be reformulated as linear matrix inequalities (LMIs, Apkarian and Tuan (2000)), cf. Kohler et al. (2020) for details. As a result, suitable terminal ingredients can be obtained during runtime by simply evaluating the parametrized terminal ingredients around a new setpoint \(x_{\mathrm{r}}\).
### Trajectory tracking
In the following, we consider the more general case of tracking a time-varying reference trajectory \(r(t),t\in\mathbb{I}_{\geq 0}\). We assume that the reference is feasible, i.e., \(x_{\mathrm{r}}(t+1)=f(x_{\mathrm{r}}(t),u_{\mathrm{r}}(t))\), \((x_{\mathrm{r}}(t),u_{\mathrm{r}}(t))\in\mathbb{Z}_{\mathrm{r}},\,t\in\mathbb{ I}_{\geq 0}\). For a given state \(x\) and time \(t\), the trajectory tracking MPC is characterized by
\[\min_{\mathbf{u}\in\mathbb{U}^{N}}\sum_{k=0}^{N-1}\ell(x_{ \mathbf{u}}(k,x),\mathbf{u}_{k},t+k)+V_{\mathrm{f}}(x_{\mathbf{u}}(N,x),t+N) \tag{10}\] \[\mathrm{s.t.}\ (x_{\mathbf{u}}(k,x),\mathbf{u}_{k})\in\mathbb{Z},\,k \in\mathbb{I}_{[0,N-1]},\ x_{\mathbf{u}}(N,x)\in\mathbb{X}_{\mathrm{f}}(t+N),\]
with \(\ell(x,u,t)=\|x-x_{\mathrm{r}}(t)\|_{Q}^{2}+\|u-u_{\mathrm{r}}(t)\|_{R}^{2}\), minimizer \(\mathbf{u}^{\star}(x,t)\in\mathbb{U}^{N}\), and value function \(\mathcal{J}_{N}^{\star}(x,t)\). The closed-loop system is given by
\[x(t+1)=f(x(t),\mathbf{u}_{0}^{\star}(x(t),t)),\quad t\in\mathbb{I}_{\geq 0}. \tag{11}\]
The analysis and closed-loop properties of such a trajectory tracking MPC are analogous to Section 2.
**Assumption 3**.: _There exists a terminal control law \(k_{\mathrm{f}}:X\times\mathbb{I}_{\geq 0}\to\mathbb{U}\) such that for all \(t\in\mathbb{I}_{\geq 0}\) and all \(x\in\mathbb{X}_{\mathrm{f}}(t)\):_
1. _Constraint satisfaction:_ \((x,k_{\mathrm{f}}(x,t))\in\mathbb{Z}\)_;_
2. _Positive invariance:_ \(f(x,k_{\mathrm{f}}(x,t))\in\mathbb{X}_{\mathrm{f}}(t+1)\)_;_
3. _Local CLF:_ \[V_{\mathrm{f}}(f(x,k_{\mathrm{f}}(x,t),t+1)-V_{\mathrm{f}}(x,t)\leq-\ell(x,k_{ \mathrm{f}}(x,t),t).\]
_Furthermore, there exists a function \(\alpha_{\mathrm{V}}\in\mathcal{K}_{\infty}\), such that for any \((x,t)\) such that Problem (10) is feasible, it holds_
1. _Weak controllability:_ \(\mathcal{J}_{N}^{\star}(x,t)\leq\alpha_{\mathrm{V}}(\|x-x_{\mathrm{r}}(t)\|)\)_._
**Theorem 2**.: _Suppose Problem (10) is feasible with \((x,t)=(x_{0},0)\). Then, Problem (10) is feasible for all \(t\in\mathbb{I}_{\geq 0}\), the constraints (2) are satisfied, and \(x_{\mathrm{f}}(t)\) is uniformly asymptotically stable for the resulting closed-loop system (11)._
The proof and closed-loop properties are analogous to Theorem 1 using \(\mathcal{J}_{N}^{\star}(x,t)\) as a _uniform_ time-varying Lyapunov function (Grune and Pannek, 2017, Thm. 2.22). Next, we focus on the constructive design of terminal ingredients for such time-varying reference trajectories \(r(t)\) (Asm. 3). The following design is largely based on the work by Faulwasser and Findeisen (2011); Faulwasser (2012). Analogous to the stabilization problem, we pick the linear-quadratic _time-varying_ parametrization \(V_{\mathrm{f}}(x,t)=\|x-x_{\mathrm{r}}(t)\|_{P(t)}^{2}\), \(k_{\mathrm{f}}(x,t)=u_{\mathrm{r}}(t)+K(t)(x-x_{\mathrm{r}}(t))\)
\(\mathbb{X}_{\mathrm{f}}=\{x\in X|V_{\mathrm{f}}(x,t)\leq\alpha(t)|\}\). By linearizing Condition (T.3) and adding a slack \(\epsilon>0\), we obtain the design conditions
\[\mathrm{LQR}(A(r(t)),B(r(t)),K(t),P(t),P(t+1),Q+\epsilon I,R)\geq 0, \tag{12}\]
which need to hold for all \(t\in\mathbb{I}_{\geq 0}\). These conditions correspond to an LQR problem for a linear time-varying (LTV) system. Tractable designs can, e.g., be obtained by assuming that the reference \(r(t)\) becomes constant after some finite time. In this case, a stationary LQR is solved at the final state and then the time-varying LQR is solved backwards to obtain \(P(t),K(t)\)(Faulwasser and Findeisen, 2011). The terminal set scaling \(\alpha(t)\) can be chosen similar to the setpoint stabilization problem with the difference that \(\alpha(t+1)\) and \(\alpha(t)\) are coupled through the time-varying invariance condition (T.2), cf. Faulwasser and Findeisen (2011, Thm. 2). Aydiner et al. (2016) propose a design for periodic reference trajectories, i.e., \(r(t)=r(t+T)\), \(\forall t\in\mathbb{I}_{\geq 0}\) with some \(T\in\mathbb{I}_{\geq 1}\). By imposing the same periodicity in \(P(t),K(t)\), the design can be cast as a periodic LQR problem or a sequence of \(T\) coupled LMIs.
### Trajectory tracking with online changing trajectories
The design of a trajectory tracking MPC scheme (Sec. 3.3) requires computing the Jacobian of the nonlinear dynamics around the full reference trajectory \(r(t)\) and solving a set of LMIs or Ricatti equations to determine the LQR-based terminal ingredients. This can become computationally very expensive. As a result, if the reference trajectory changes during runtime, re-computing terminal ingredients for a new reference \(r(t)\) becomes a bottleneck in the practical application. To circumvent this issue, Kohler et al. (2020) propose a _reference generic offline computation_. The goal is to have one offline computation to obtain continuously parametrized terminal ingredients of the form \(P(r),K(r)\), such that
\[\mathrm{LQR}(A(r),B(r),K(r),P(r),P(r^{+}),Q+\epsilon I,R)\geq 0 \tag{13}\]
for all \(r,r^{+}\in\mathbb{Z}_{\tau}\) with \(x_{r}^{+}=f(x_{r},u_{t})\). This problem can be cast as a gain-scheduling problem for LPV systems by viewing the reference \(r\) as a parameter that is slowly time-varying, where a bound on the change can be deduced from the dynamics. There exists much literature on the synthesis of gain-scheduled controllers, which are used by Kohler et al. (2020, Prop. 1) to derive a finite-dimensional semidefinite program (SDP) to compute \(K(r),P(r)\) satisfying (13). By defining \(P(t)=P(r(t))\), \(K(t)=K(r(t))\), these parametrized terminal ingredients also satisfy the LQR conditions (12) for any possible reference. Hence, the trajectory tracking MPC (Thm. 2) can be implemented without a priori knowledge of the full reference and requires no repeated offline computations in case the reference changes.
### Discussion
Sections 3.1-3.4 provide offline design methods for the terminal ingredients that ensure desirable closed-loop properties for nonlinear tracking MPC schemes. The efficient update of the scaling \(\alpha>0\) characterizing the terminal set \(\mathbb{X}_{\mathrm{f}}\) and general feasibility and stability questions under online changes of the reference are revisited in Section 4. In the following, we broadly discuss the applicability of these methods and mention variations.
#### 3.5.1 Design complexity
The computation of parametrized terminal ingredients for setpoint tracking (cf. (9)) is comparatively simple, as the conditions only need to be enforced on the (typically low dimensional) steady-state manifold. On the other hand, the design conditions (13) for dynamic trajectories are enforced over the set of feasible dynamic reference trajectories \(\mathbb{Z}_{\tau}\subseteq\mathbb{R}^{n+m}\), which is a significantly higher dimensional set. Hence, direct application for large scale systems is challenging (cf. also Sec. 7.2.2).2 However, the benefits of parametrized terminal ingredients are also significantly more pronounced for dynamic reference trajectories as the design of terminal ingredients for a specific reference \(r(t)\) (Sec. 3.3) can already become cumbersome.
Footnote 2: The SDP by Köhler et al. (2020, Prop. 1) has been applied to a ball-and-plate system with \((n,m)=(8,2)\)(Köhler et al., 2020) and quadrotors \((n,m)=(8,3)\)(Wang et al., 2022), (approximately using gridding) to kinematic bicycle models of cars \((n,m)=(5,2)\)(Köhler et al., 2020) and comparable SDPs are standard in the design of contraction metrics (cf. Sec. 3.5.4).
#### 3.5.2 Terminal equality constraint
A simpler design for the terminal ingredients is given by a terminal equality constraint, historically also called zero-terminal constraint (Mayne and Michalska, 1990), with \(\mathbb{X}_{\mathrm{f}}=\{x_{\mathrm{r}}\}\), \(k_{\mathrm{f}}=u_{\mathrm{r}}\), \(V_{\mathrm{f}}=0\). This design directly satisfies Conditions (T.1)-(T.3) from Assumption 2. Condition (T.4) holds if an additional local controllability condition is satisfied and the prediction horizon \(N\) is larger than the controllability index (Köhler, 2021, Prop. 3.10). This approach is easy to apply, which makes it particularly attractive for application with online changing setpoints or dynamic trajectories (Sec. 3.2, 3.4, 4), compare Limon et al. (2018, Sec. III.A); Fagiano and Teel (2013); Müller et al. (2013); Berberich et al. (2020) and Limon et al. (2016, 2014); Pereira et al. (2017, Sec. IV.A). However, there are significant drawbacks to this design: Yu et al. (2014) show that nonlinear MPC schemes with "proper" terminal ingredients are _inherently robust_3, however, terminal equality constraints require a multi-step implementation4 to retain inherent robustness (Berberich et al., 2022, Prop. IV.1). Additional practical drawbacks include a small region of attraction and in general worse control performance, cf. the comparisons by Chen and Allgower (1998, Sec. 5), Raff et al. (2006, Sec. V), Kohler et al. (2020, Sec. 4.1), and Kohler et al. (2020, Sec. 5.2).
Footnote 3: Inherent robustness implies that recursive feasibility and some form of stability are preserved for sufficiently small model mismatch. These results also require that state constraints are relaxed using penalties.
Footnote 4: In a multi-step implementation, the first \(v\in\mathbb{I}_{>1}\) elements of the optimal input sequence \(\mathbf{u}^{\star}\) are applied to the system in open loop and the optimization problem is only solved every \(v\) steps. Berberich et al. (2022) choose \(v\) larger than the controllability index to derive robustness guarantees.
#### 3.5.3 More general stage cost
In case of a positive semidefinite stage cost \(\ell\), the stability results in Theorem 1/2 are not directly applicable since As
sumption 1 does not hold. However, in case the cost \(\ell\) satisfies a _detectability_ condition, a Lyapunov function can be constructed (Rawlings et al., 2017, Thm. 2.24), compare also Section 6.2.2. Considering the design of the terminal cost \(V_{\mathrm{f}}\) using LQR: The same procedure can be applied for non-quadratic (twice cont. differentiable) stage costs \(\ell\) by replacing \(Q+K^{\tau}RK\) by the Hessian of \(\ell\), see Amrit et al. (2011, Sec. 4.1) and Kohler et al. (2020, App. D) for details.
#### 3.5.4 LPV & incremental system properties
The design conditions (9)/(13) are comparable to analysis and synthesis conditions for LPV systems. There exists a rich history on addressing nonlinearity and non-convexity using such LPV embeddings in the context of MPC, compare the survey by Morato et al. (2020). Condition (13) ensures that any dynamically feasible reference \(r\) can be stabilized, which is also referred to as _incremental stability_ with the _contraction metric_\(P\). In fact, this design is related to the long history on the interplay between LPV systems, incremental Lyapunov functions, and contraction metrics, compare Angeli (2002); Fromion and Scorletti (2003); Wang et al. (2019); Koelewijn et al. (2019) and Kohler (2021, Appendix C). The presented quadratic terminal cost \(V_{\mathrm{f}}\) is only a _local_ CLF, however, one can also construct a "globally" valid CLF \(V_{\mathrm{f}}\) using \(P(r)\).5
Footnote 5: The Riemannian energy based on the metric \(P(r)\) provides an incremental Lyapunov function \(V_{\mathrm{f}}\)(Manchester and Slotine, 2017). Evaluating this function is computationally more expensive compared to the simple quadratic expression since it involves an additional integral. Details regarding the exact region where Condition (7.3) holds for \(\mathbb{Z}_{\tau}\neq\mathbb{R}^{n+m}\) can be found in (Safafi et al., 2022, Prop. 5).
| この記事では、非線形制約されたシステムの動的運転に関するMPCフレームワークの概観を提供しています。動的運転は、参照信号の追跡からプラントの一般的な経済的な運転まで、制御の目標としてしばしば含まれます。この特定の課題は、より一般的な制御目標に取り組む際に生じる課題です。この課題に取り組むための方法を提示し、その分野で浮上してきた方法を提示しています。この記事の目的は、最先端技術の概観を提供することで、動的運転に適応する MPC 表達法を適用し発展させるための多様なツールキットを提供することです。さらに、異なる研究方向の適用可能性について批判的に評価し、今後の研究の可能性について議論します。 |
2310.09680 | Improved Contextual Recognition In Automatic Speech Recognition Systems
By Semantic Lattice Rescoring | Automatic Speech Recognition (ASR) has witnessed a profound research
interest. Recent breakthroughs have given ASR systems different prospects such
as faithfully transcribing spoken language, which is a pivotal advancement in
building conversational agents. However, there is still an imminent challenge
of accurately discerning context-dependent words and phrases. In this work, we
propose a novel approach for enhancing contextual recognition within ASR
systems via semantic lattice processing leveraging the power of deep learning
models in accurately delivering spot-on transcriptions across a wide variety of
vocabularies and speaking styles. Our solution consists of using Hidden Markov
Models and Gaussian Mixture Models (HMM-GMM) along with Deep Neural Networks
(DNN) models integrating both language and acoustic modeling for better
accuracy. We infused our network with the use of a transformer-based model to
properly rescore the word lattice achieving remarkable capabilities with a
palpable reduction in Word Error Rate (WER). We demonstrate the effectiveness
of our proposed framework on the LibriSpeech dataset with empirical analyses. | Ankitha Sudarshan, Vinay Samuel, Parth Patwa, Ibtihel Amara, Aman Chadha | 2023-10-14T23:16:05 | http://arxiv.org/abs/2310.09680v4 | Improved contextual recognition in automatic speech recognition systems by semantic lattice rescoring
###### Abstract
Automatic Speech Recognition (ASR) has witnessed a profound research interest. Recent breakthroughs have given ASR systems different prospects such as faithfully transcribing spoken language, which is a pivotal advancement in building conversational agents. However, there is still an imminent challenge of accurately discerning context-dependent words and phrases. In this work, we propose a novel approach for enhancing contextual recognition within ASR systems via semantic lattice processing leveraging the power of deep learning models in accurately delivering spot-on transcriptions across a wide variety of vocabularies and speaking styles. Our solution consists of using Hidden Markov Models and Gaussian Mixture Models (HMM-GMM) along with Deep Neural Networks (DNN) models integrating both language and acoustic modeling for better accuracy. We infused our network with the use of a transformer-based model to properly rescore the word lattice achieving remarkable capabilities with a palpable reduction in Word Error Rate (WER). We demonstrate the effectiveness of our proposed framework on the LibriSpeech dataset with empirical analyses.
Ankitha Sudarshan\({}^{1}\), Vinay Samuel\({}^{2}\), Parth Patwa\({}^{3}\), Ibtihel Amara\({}^{4}\), Aman Chadha\({}^{5,6}\)+\({}^{1}\)Purdue University \({}^{2}\)Carnegie Mellon University \({}^{3}\)University of California Los Angeles
\({}^{4}\)McGill University \({}^{5}\)Stanford University \({}^{6}\)Amazon AI
\({}^{1}\)[email protected] \({}^{5,6}\)[email protected] speech recognition, lattice re-scoring, contextual speech recognition, word lattices
Footnote †: Work does not relate to position at Amazon.
## 1 Introduction
Recognizing spoken language accurately and efficiently is a complex task due to variability in the source of speech such as pronunciation, dialects, vocabulary, accents, articulation, etc.
Semantic interpretation of a speech is crucial in ASR systems. Let us take the following example. _"I am going to a bank to deposit a check"_. Without context, the word bank could refer to either a financial institution or either the edge of a river.
To bridge this contextual gap in ASR systems, semantic lattice processing is a key component in contributing to better recognition of situational context conditions. This technique utilizes a lattice structure to represent the relationships between words and phrases in a sentence. It is created by analyzing the audio input and identifying possible word and phrase combinations and their associated probabilities. This information is then used to create a graph-like structure, where each node represents a word or phrase, and the edges represent the relationships between them [1].
In this study, our primary emphasis centers on lattice rescoring, a technique designed to efficiently re-evaluate the likelihood of potential speech hypotheses. While numerous lattice re-scoring techniques have been documented in the literature [2, 3], our research introduces a novel approach tailored to bolster contextual information within ASR systems.
Our key contributions are as follows:
1) Lattice re-scoring, which refines recognition results through the integration of language model probabilities from different language models, enhancing transcription accuracy and overall system performance.
2) Employing a Transformer architecture for our neural language model, enhancing lattice re-scoring with state-of-the-art contextual modeling.
3) Achieving a 1.36% reduction in word error rate when compared to state-of-the-art models sharing a similar architectural framework.
## 2 Related Work
Voice assistants use a variety of end-to-end (E2E) ASR techniques, including attention-based encoder-decoder (AED) [4, 5], recurrent neural network transducer (RNN-T) [6], and connectionist temporal classification (CTC) [6]. During training, the E2E model simultaneously optimizes the whole recognition pipeline and produces the word sequence output directly. One problem with this method, though, is that it has trouble identifying terms like songs or human names that don't often appear in the training set.
ASR systems' contextual recognition is crucial, especially for voice assistants, which must identify names of contacts, musicians in a user's music collection, and other entities. Shallow fusion [7, 8], attention-based deep context [9, 10, 1], and trie-based deep biasing [11, 12] are the current contextual
biasing techniques used for various E2E ASR models.
According to [1, 10], phoneme information may be injected or training with challenging negative examples can help to decrease misunderstanding between similar words. On-the-fly re-scoring is the most popular method for introducing contextual bias into ASR. In [13], this technique was first used with hybrid ASR models. In order to allow the weights on the bias terms to be changed "on the fly" at the time of inference, it entails composing a weighted finite state transducer (WFST) that represents the ASR model with a novel WFST representation of the bias terms. The bias terms are still assembled into a WFST representation for E2E models, but the ASR model composes that WFST at each decoding time-step, determining the likelihood of the current hypothesis based on the ASR model is combined with a score from the bias WFST.
A separate re-scoring module can not adequately handle the error introduced by the upstream acoustic network. Some architectures, such as[9, 14] can integrate contextual information into acoustic model to improve the output posterior distribution corresponding to the related contextual words and fit the subsequent external LM well. As the size of the possible contextual word list grows in real applications, the accuracy and latency of the system descend rapidly due to dispersed attention score and heavy attention computation. Moreover, in practice, it is hard to obtain compact and accurate contextual information in advance. Consider music search - we may face a large contextual word list (size in thousands) containing popular songs. In this case, E2E context bias cannot work well due to the large size and low quality of the contextual word list, leading to performance degradation [15].
## 3 Methodology
### Background
For ASR systems, the initial decoding process generates a lattice--a directed acyclic graph (DAG) representing a spectrum of potential word hypotheses and their associated scores. The fundamental purpose of lattice re-scoring is to elevate the accuracy of these ASR hypotheses by post-processing and re-ranking them within the lattice. We explore the mathematical and algorithmic aspects of lattice re-scoring, showcasing its role in enhancing ASR accuracy. We use a custom Transformer model with positional encoding, multiple Transformer encoder layers (controlled by \(n\) layers), an input embedding layer, and an output linear layer for sequence prediction. Let \(A(a)\) represent the acoustic score for arc \(a\) in the lattice. This score is typically based on the acoustic model and represents how well the audio aligns with the word associated with the arc. Let \(P(w1,w2,...,wN)\) denote the language model probability for the word sequence \((w1,w2,...,wN)\) based on the Transformer model. This probability reflects the likelihood of the entire word sequence occurring in the context of the language. Given a path \(P\) through the lattice, the word sequence probability \(P(P)\) can be computed as the product of acoustic scores and language model probabilities for the words along the path: \(P(P)=\prod_{a}(A(a)\cdot P(w))\), for all arcs \(a\) and words \(w\) in the path. The path \(P*\) through the lattice that maximizes the joint probability of the word sequence: \(P*=argmax_{P}P(P)\)
### Lattice Re-scoring
We provide in Figure 1 and Figure 2, respectively, our overall framework and the proposed lattice re-scoring strategy. We use the DNN-refined predictions for creating word lattices as an intermediate representation of the ASR output. This decoding process generates lattice-like outputs that contain phone-level alignments and associated scores and eventual word alignments.
Each path in the lattice has a score based on the ASR system's confidence. However, we can improve the transcription by reevaluating these paths using a neural LM, which captures the likelihood of different word sequences based on linguistic context. We use our custom-trained transformer to perform
Figure 1: **Global overview of our framework.** Our framework includes audio input, DNN acoustic model, lattice creation, language model integration, alignment, transformer re-scoring, and transcript generation.
Figure 2: **Lattice re-scoring strategy.** Alignment is involved, n-gram language model integration, and Transformer-based re-scoring to enhance contextual features in the final transcript.
re-scoring. A Transformer-based re-scoring approach, as opposed to traditional n-gram methods, introduces novelty by leveraging advanced neural network architectures that are designed to handle sequences more effectively and capture complex language patterns.
This transformation is done by computing the conditional probability of the word sequence in each path given the language model and combining it with the original acoustic likelihood score. The result is a new lattice where paths have modified scores that reflect both acoustic and language model information, enhancing transcription accuracy.
The scores from the neural LM are
converted to log-likelihoods to be combined with the original lattice scores. Once we have the lattice paths re-scored using the neural LM and have converted the scores to log-likelihoods, we combine these scores with the original lattice scores. This step helps integrate the language model probabilities into the lattice.
By combining these scores, the lattice paths that were previously assigned lower scores by the ASR system but have higher probabilities according to the LM are promoted, resulting in a better transcription of the input audio than the original input.
A prominent mathematical formula in the lattice creation process in ASR is related to the computation of the overall likelihood or score of a path through the lattice. This score is typically calculated as the sum of individual acoustic and language model scores along the path.
\[\text{Path Score}=\sum_{i=1}^{N} \Big{(}\log\bigl{(}P(\text{word}_{i}|\text{word}_{i-1})\bigr{)} \tag{1}\] \[+\log\bigl{(}P(\text{Acoustic features}_{i}|\text{word}_{i}) \bigr{)}\Big{)}\]
where, \(N\) represents the number of words in the path, \(\text{word}_{i}\) is the \(i^{th}\) word in the path, \(P(\text{word}_{i}|\text{word}_{i-1})\) is the conditional probability of transitioning from \(\text{word}_{i-1}\) to \(\text{word}_{i}\) using the language model, and \(P(\text{acoustic features}_{i}|\text{word}_{i})\) is the probability of observing acoustic features at position \(i\) given \(\text{word}_{i}\) using the acoustic model.
## 4 Experimental Details
### Data Corpus and Preprocessing
We use the LibriSpeech dataset [16], which consists of approximately 1000 hours of read English speech with a sampling rate of 16 kHz. This dataset provides a substantial and high-quality source of audio data, ensuring the robustness and generalizability of our proposed method, For data preprocessing, we utilize the Kaldi toolkit. We prepared the data in the Kaldi format organizing it into training, validation, and test sets. The format consists of two main components: the archive file (.ark) and the corresponding index file (.scp).file contains binary data and is typically organized as a sequence of key-value pairs, where the key is a unique identifier (usually a string) associated with the data, and the value is the binary data itself.
Example: <ark>
<key1> <binarytoken1> <data1>
<key2> <binarytoken2> <data2> </ark>
This involved creating text transcriptions and corresponding acoustic features for each segment of the audio data. The.scp file provides a mapping between the keys in the archive file and their corresponding file positions. Each line contains a key followed by the offset (in bytes) of the corresponding entry in the.ark file.
Example: <key1> <offset1>
### Acoustic Model
Post preprocessing, we implemented the GMM-HMM acoustic framework. The GMM-HMM model was trained using the extracted acoustic features. The model provided posterior probabilities over subword units (e.g., phonemes or context-dependent acoustic states) for each frame of audio. These probabilities were represented as a likelihood matrix.
### Language Model: Deep Neural Network (DNN)
Following the GMM-HMM stage, we incorporate a DNN to refine and improve the predictions made by the GMM-HMM model.
The DNN model generated enhances posterior probabilities over subword units. This DNN-refined output provides more accurate representations of the spoken audio, building upon the GMM-HMM predictions.
We use a neural LM - a custom transformer trained on the same Librispeech dataset for the rescoring task. We then compose Lattice FSTs with Language Model FST. The FST created using Kaldi's tools represents the language model, lexicon, and any other components of the speech recognition system. We then compile the HCLG FST (Hidden Markov Model, Context, Lexicon, Grammar) using the trained acoustic and language models, lexicon, and other necessary components.
These word-level alignments are then converted into Finite State Transducers (FSTs). These FSTs provide a structured representation that allow for efficient manipulation of word-level alignments. To this end, we generate four types of lattices:
Type 1: DNN based lattice (with phone alignments followed by word alignments).
Type 2: GMM based lattice (with phone alignments followed by word alignments).
Type 3: DNN-based lattice (with direct word alignments).
Type 4: GMM-based lattice (with direct word alignments).
### Transformer Model For Lattice Re-scoring
We trained a custom six-layer transformer model with 512 hidden embedding on an NVIDIA h100 GPU using the LibriSpeech[16] dataset. We trained the model over 4 epochs with a learning rate of 0.1 and a dropout rate of 0.1.
## 5 Results and Discussion
We conducted experiments on the four types of lattices mentioned in Section 4.3 and observed that their performance in the pipeline was identical. Hence we decided to make a comprehensive analysis on Lattice Type 1 which is representative of the other types. The below tables represent results for Lattice Type 1 on the test sets 'test-clean' and 'test-other'. | 自動音声認識 (ASR) に関しては、深い研究興味が高まっている。最近の進歩により、ASR システムは、音声の正確な録音、つまり対話型アシスタントの構築における重要な進歩をもたらすなど、新たな可能性を示唆している。しかしながら、文脈依存する単語やフレーズを正確に識別するという課題は依然として存在する。本研究では、Deep Learningモデルの力を活用して、ASR システム内の文脈認識を強化するための革新的なアプローチを提案している。これは、セマンティックラティス処理によって実現される。このアプローチにより、様々な言語と話し方スタイルに対応し、正確なテキストを生成することができる。私たちのソリューションは、Hidden Markov Model と Gaussian Mixture Model (HMM-GMM) を使用し、Deep Neural Network (DNN) モデルも統合することで、言語と音響モデルの両方を学習させることで、より正確なテキスト生成を実現している。さらに、Transformer モデルを |
2310.18955 | Optimal Algorithms for Online Convex Optimization with Adversarial
Constraints | A well-studied generalization of the standard online convex optimization
(OCO) is constrained online convex optimization (COCO). In COCO, on every
round, a convex cost function and a convex constraint function are revealed to
the learner after the action for that round is chosen. The objective is to
design an online policy that simultaneously achieves a small regret while
ensuring a small cumulative constraint violation (CCV) against an adaptive
adversary interacting over a horizon of length $T$. A long-standing open
question in COCO is whether an online policy can simultaneously achieve
$O(\sqrt{T})$ regret and $O(\sqrt{T})$ CCV without any restrictive assumptions.
For the first time, we answer this in the affirmative and show that an online
policy can simultaneously achieve $O(\sqrt{T})$ regret and
$\tilde{O}(\sqrt{T})$ CCV. Furthermore, in the case of strongly convex cost and
convex constraint functions, the regret guarantee can be improved to $O(\log
T)$ while keeping the CCV bound the same as above. We establish these results
by effectively combining the adaptive regret bound of the AdaGrad algorithm
with Lyapunov optimization - a classic tool from control theory. Surprisingly,
the analysis is short and elegant. | Abhishek Sinha, Rahul Vaze | 2023-10-29T09:55:41 | http://arxiv.org/abs/2310.18955v2 | # Playing in the Dark: No-regret Learning with Adversarial Constraints
###### Abstract
We study a generalization of the classic Online Convex Optimization (OCO) framework by considering additional long-term adversarial constraints. Specifically, after an online policy decides its action on a round, in addition to a convex cost function, the adversary also reveals a set of \(k\) convex constraints. The cost and the constraint functions could change arbitrarily with time, and no information about the future functions is assumed to be available. In this paper, we propose a meta-policy that simultaneously achieves a sublinear cumulative constraint violation and a sublinear regret. This is achieved via a black box reduction of the constrained problem to the standard OCO problem for a recursively constructed sequence of surrogate cost functions. We show that optimal performance bounds can be achieved by solving the surrogate problem using _any_ adaptive OCO policy enjoying a standard data-dependent regret bound. A new Lyapunov-based proof technique is presented that reveals a connection between regret and certain sequential inequalities through a novel decomposition result. We conclude the paper by highlighting applications to online multi-task learning and network control problems.
## 2 Introduction
Online Convex Optimization (OCO) is a standard framework for modelling and analyzing a broad class of online decision problems under uncertainty. In this problem, an online policy selects a sequence of actions from a convex feasible set over multiple rounds. After the policy selects an action on a round, the adversary reveals a convex cost function. The goal of a no-regret policy is to choose an action sequence so that its cumulative cost is not much larger than that of any fixed feasible action chosen in hindsight. In this paper, we consider a generalization of the above standard OCO framework where, in addition to a cost function, the adversary also reveals a collection of constraints of the form \(g_{t,i}(x)\leq 0,i\in[k]\) on each round \(t\). The goal is to design an online policy to control both the regret and the cumulative violation penalty optimally. This begets a natural question - Is it possible to efficiently reduce the constrained problem to an unconstrained OCO problem while obtaining the optimal regret and cumulative violation bounds? In this paper, we answer this question affirmatively in a constructive fashion.
Variants of the above problem arise in many applications, including online portfolio optimization with risk constraints, online resource allocation (_e.g.,_ CPU, memory, bandwidth) in cloud computing with time-varying demands, pay-per-click online ad markets with budget constraints (Liakopoulos et al., 2019), online recommendation systems, dynamic pricing, revenue management, robotics and path planning problems, and multi-armed bandits with fairness constraints (Sinha, 2023). Constrained optimization problems with a huge set of constraints can also be conveniently formulated in this framework. The necessity for revealing the constraints sequentially may arise, _e.g.,_ in communication-constrained settings, where it might be infeasible to reveal all constraints which define the feasible
set (_e.g.,_ combinatorial auctions). To further motivate the problem, we now introduce a convex optimization over a hidden constraint set, which we call the Hidden Set problem.
The Hidden Set Problem:Let \(\Omega\) be an _admissible_ set of actions which is known to the policy. Let \(\Omega^{*}\), called the _feasible_ set, be a closed and convex subset of \(\Omega\). Due to a large number of defining constraints, the feasible set \(\Omega^{*}\) is too complex to communicate to the policy _a priori_. However, an efficient separation oracle for \(\Omega^{*}\) is assumed to be available. On the \(t^{\text{th}}\) round, the policy first selects an admissible action \(x_{t}\in\Omega\) and then, the adversary reveals a convex cost function \(f_{t}\) and _some_ convex constraint of the form \(g_{t}(x)\leq 0\), which contains the unknown feasible set \(\Omega^{*}\). As an example, the constraints could come from the separation oracle that, if \(x_{t}\) is infeasible, outputs a hyperplane separating the current action \(x_{t}\in\Omega\) and the hidden feasible set \(\Omega^{*}\). The objective of the policy is to perform as well as any action from the hidden feasible set \(\Omega^{*}\) in terms of the regret and the cumulative constraint violation metrics.
### Related Work
In a seminal paper, Zinkevich (2003) showed that the ubiquitous online gradient descent (OGD) policy achieves an \(O(\sqrt{T})\) regret for convex cost functions with uniformly bounded sub-gradients. A number of follow-up papers proposed adaptive and parameter-free versions of OGD that do not need to know any non-causal information (Hazan et al., 2007; Orabona and Pal, 2018; Orabona, 2019). However, these lines of work do not consider additional constraints - a problem which has been systematically explored only recently (see Table 2 for a brief summary). The constraint functions could either be known _a priori_ or revealed sequentially along with the cost functions.
Known and fixed constraints:A number of papers considered the OCO problem with time-invariant constraints assumed to be known to the policy a priori (Yuan and Lamperski, 2018; Jenatton et al., 2016; Yu et al., 2017; Yu and Neely, 2016; Mahdavi et al., 2012). These papers relax the requirement that the online policy must satisfy the constraints exactly on each step. Instead, their main objective is to design an _efficient_ policy which avoids the complicated projection step of the vanilla OGD policy while requiring that the cumulative constraint violations over the horizon grow sub-linearly. In this connection, Jenatton et al. (2016) and Yuan and Lamperski (2018) established a regret bound of \(O(T^{\max(\beta,1-\beta)})\) and a cumulative constraint violation bound of \(O(T^{1-\beta})\) for augmented Lagrangian-based primal-dual algorithms, where \(0<\beta<1\) is a free parameter. For the same regret bound, Yi et al. (2021) improved the cumulative constraint violation bound to \(O(T^{(1-\beta)/2})\), which was further improved to \(O(1)\) by Guo et al. (2022) for the case \(\beta=\nicefrac{{1}}{{2}}\). In the case of fixed constraints and strongly convex losses, Guo et al. (2022) gave an algorithm which achieves \(O(\log T)\) regret and \(O(1)\) violation bounds while assuming access to the entire constraint functions and solving a convex problem on each round.
Adversarial and dynamic constraints:Coming to the more challenging problem of time-varying constraint functions which we study in this paper, with the notable exceptions of Neely and Yu (2017)
Figure 1: Illustrating the Hidden Set problem. In this figure, the sphere \(\Omega^{*}\) is the hidden set. On every round \(t\), the adversary reveals a hyperplane supporting \(\Omega^{*}\).
and Liakopoulos et al. (2019), most of the previous papers assume that the policy has access to some non-causal information, _e.g.,_ a uniform upper bound to the norm of the future gradients (Jenatton et al., 2016; Yu et al., 2017; Mahdavi et al., 2012; Sun et al., 2017; Yi et al., 2023). This information is used by their proposed policies to construct an appropriate Lagrangian function or carefully select the step size schedule. Liu et al. (2022) consider the problem of minimizing the dynamic regret by extending the virtual queue-based policy of (Neely and Yu, 2017). However, due to the dynamic benchmark, their regret bound explicitly depends on the temporal variation of the constraint functions. Recent work by Yi et al. (2023) considers the case of convex constraint functions and convex or strongly convex cost functions in a distributed setting. In the case of convex cost functions, they establish an \(O(T^{\max(\beta,1-\beta)})\) regret and an \(O(T^{1-\beta/2})\) constraint violation bound for a primal-dual mirror descent policy without assuming Slater's condition. However, their algorithm assumes a uniform upper bound to the norm of the gradients is known.
Drift-plus-penalty-based policies:Closer to our work, Neely and Yu (2017) give a drift-plus-penalty-based policy that achieves an \(O(\sqrt{T})\) regret and \(O(\sqrt{T})\) cumulative violation penalty upon assuming Slater's condition. In brief, Slater's condition ensures the existence of a fixed feasible action \(x^{*}\) such that all constraints hold strictly with a positive margin \(\eta>0\), _i.e.,_\(g_{t,i}(x^{*})<-\eta,\forall i,t\). Clearly, this condition precludes the important case of non-negative constraint functions (_e.g.,_ constraint functions of the form \(\max(0,g_{t}(x))\)). Furthermore, a sublinear violation bound obtained upon assuming Slater's condition is loose by a quantity that increases _linearly_ with the horizon-length \(T\) compared to a sublinear violation bound obtained without this assumption. Worse, the regret bound presented in Neely and Yu (2017) diverges to infinity as \(\eta\succ 0\). In a recent paper, Yi et al. (2023) consider the same problem in a distributed setup and derive tighter bounds upon assuming Slater's condition. Liakopoulos et al. (2019) extend Neely and Yu (2017)'s result by considering a stronger comparator called the \(S\)-benchmark. The fixed action \(x^{*}\) in an \(S\)-benchmark enjoys the property that the sum of the constraint functions evaluated at \(x^{*}\) over any consecutive sequence of \(S\geq 1\) rounds is non-positive. Without assuming Slater's condition, they show that the drift-plus-penalty-based policy achieves a regret bound of \(O(ST/V+\sqrt{T})\) and a violation penalty of \(O(\sqrt{VT})\). Here, \(V\) is an adjustable parameter that can take any value in \([S,T)\). Hence, _a priori_, their algorithm needs to know the value of the parameter \(S\), which, unfortunately, depends on the online constraints. Although these results are interesting, since the previous drift-plus-penalty-based algorithms are all based on linear approximations, none can be extended to prove improved bounds for the strongly convex case. In recent work, Guo et al. (2022) consider the same problem for non-negative convex and strongly convex functions with convex constraints and obtain near-optimal performance bounds. However, their algorithm is inefficient as, instead of a single gradient-descent step per round (as in our and most of the previous algorithms), their algorithm solves a general convex optimization problem at every round. Moreover, their algorithm needs access to the full description of the constraint function \(g_{t}(\cdot)\) for the optimization step, whereas ours and most of the previous algorithms need to know only the gradient and the value of the constraint function at the current action \(x_{t}\). Lastly, their analysis is tailored for the hard constraints and, hence, their algorithm can not be extended to the more general \(S\)-benchmark, where it is necessary to compensate for constraint violations at some rounds with strictly satisfying constraints on some other rounds. Please refer to Table 2 for a brief summary of the results.
The Online Constraint Satisfaction (Ocs) Problem:We also study a special case of the above problem, called Online Constraint Satisfaction (OCS), that does not involve any cost function. In this problem, on every round \(t\), \(k\) constraints of the form \(g_{t,i}(x)\leq 0\) are revealed to the policy in an online fashion, where the function \(g_{t,i}\) comes from the \(i^{\text{th}}\) stream. The constraint functions could be unrestricted in sign - they may potentially take both positive and negative values within their admissible domain. The objective is to control the cumulative violation of each separate stream by choosing a common action sequence \(\{x_{t}\}_{t\geq 1}\). To the best of our knowledge, the OCS problem has not been previously considered in the literature. Although the OCS problem can be reduced to the previous problem with dynamic constraints upon setting the cost function \(f_{t}\equiv 0,\forall t\), this reduction turns out to be provably sub-optimal. In particular, without any assumption, the best-known bound for the cumulative violation for a single convex constraint is known to be \(O(T^{3/4})\) and no violation penalty bound is known for strongly convex constraints (see Table 1). For the first time, we show that it is possible to achieve a cumulative violation bound of \(O(\sqrt{ST})\) for convex constraints with the general \(S\)-benchmark and \(O(\log T)\) for strongly convex constraints with the usual \(1\)-benchmark.
### Why are the above problems non-trivial?
Let us consider the \(\mathsf{OCS}\) problem. A first attempt to solve the \(\mathsf{OCS}\) problem could be to scalarize it by taking a _fixed_ linear combination (_e.g.,_ the sum) of the constraint functions and then running a standard OCO policy on the scalarized cost functions. The above strategy immediately yields a sublinear regret guarantee on the same linear combination (_i.e.,_ the sum) of the constraint functions. However, since the constraint functions could take both positive and negative values, the constraint violation component of some streams could still be arbitrarily large even when the overall sum is small. Hence, this strategy does not yield individual cumulative violation bounds, where we need to control the more stringent \(\ell_{\infty}\)-norm of the cumulative violation vector. Hence, to meet the objective with this scalarization strategy, the "correct" linear combination of the constraints must be learned adaptively in an online fashion. This is exactly what our online meta-policy, which we propose in Section 4.3, does.
To get around the above issue, one may alternatively attempt to scalarize the \(\mathsf{OCS}\) problem by considering a non-negative _surrogate_ cost function, _e.g.,_ the hinge loss function, defined as \(\hat{g}_{t,i}(x)=\max(0,g_{t,i}(x))\), for each constraint \(i\in[k]\). However, it can be easily seen that this transformation does not preserve the strong convexity as the function \(\hat{g}_{t,i}\) is _not_ strongly convex even when the original constraint function \(g_{t,i}\) is strongly convex. Furthermore, the above strategy does not work even for convex functions for \(S\)-feasible benchmarks with \(S\geq 2\). This is because, due to the impossibility of cancellation of positive violations by strictly feasible constraints on different rounds, an \(S\)-feasible benchmark for the original constraints does not remain feasible for the transformed non-negative surrogate constraints (see Section 6). Finally, the above transformation fails in the case of stochastic constraints where the constraint is satisfied only in expectation, i.e., \(\mathbb{E}g_{t}(x)\lesssim 0\), \(\forall t\geq 1\)(Yu et al., 2017). The above discussion shows why designing an efficient and universal policy for the \(\mathsf{OCS}\) problem, and consequently, for the generalized OCO problem - which generalizes \(\mathsf{OCS}\), is highly non-trivial.
### Our contribution
In this paper, we consider two closely related online learning problems, namely (1) Online Constraint Satisfaction (\(\mathsf{OCS}\)) and (2) Generalized OCO. In particular, we claim the following contributions.
1. We propose a general framework for designing efficient online learning policies for the Online Constraint Satisfaction (\(\mathsf{OCS}\)) and the Generalized OCO problems with optimal regret and cumulative violation bounds. In contrast with the prior works, that establish performance
\begin{table}
\begin{tabular}{l l c c c l} \hline \hline Reference & Constraints type & Benchmark & Violation & Algorithm & Assumption \\ \hline Jenatton et al. (2016) & Fixed, convex & \(1\) & \(O(T^{3/4})\) & Primal-Dual GD & Known \(G\) \\ Yuan and Lamerski (2018) & Fixed, convex & \(1\) & \(O(1)\) & Primal-Dual MD & Fixed constraints \\ Yu et al. (2017) & Stochastic, convex & \(T\) & \(O(\sqrt{T})\) & OGD+drift+penalty & Slater \\ Yi et al. (2023) & Adversarial, convex & \(1\) & \(O(\sqrt{T})\) & Primal-Dual MD & Known \(G\) \\ Sun et al. (2017) & Adversarial, convex & \(1\) & \(O(T^{3/4})\) & OMD & Known \(G\) \\ Neely and Yu (2017) & Adversarial, convex & \(1\) & \(O(\sqrt{T})\) & OGD & Slater \\ Liakopoulos et al. (2019) & Adversarial, convex, one constraint & \(S\) & \(O(\sqrt{ST})\) & OGD & Known \(S\) \\ Guo et al. (2022) & Adversarial, convex & \(1\) & \(O(T^{3/4})\) & Convex opt. each round & Full access to \(\{g_{t}\}_{t\geq 1}\) \\
**This paper** & Adversarial, convex & \(S\) & \(O(\sqrt{ST})\) & Any adaptive & - \\
**This paper** & Adversarial, strongly-convex & \(1\) & \(O(\log T)\) & Any adaptive & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the results for the \(\mathsf{OCS}\) problem. Results from other papers, which consider only the generalized OCO problem, have been appropriately adapted for the \(\mathsf{OCS}\) problem by taking the cost function to be identically equal to zero and quoting the best violation bound. Excepting our paper, all other papers referenced above bound the cumulative violation over the entire horizon of length \(T\), which is weaker than (4) for constraints assuming both positive and negative signs. \(G\) is an upper bound to the norm of the gradients, \(\alpha\) is the strong-convexity parameter; Abbreviations: MD= Mirror Descent, OMD = Online Mirror Descent, OGD = Online Gradient Descent.
bounds only for specific learning policies, we give a _policy-agnostic_ black box reduction of the constrained problem to the standard OCO using _any_ adaptive learning policy with a standard data-dependent regret bound. Consequently, efficient policies can be obtained by using known OCO sub-routines that exploit the special structure of the admissible action set (_e.g.,_ FTPL for combinatorial problems).
2. Our meta-policy for the OCS problem is _parameter-free_ and does not need any non-causal information including, uniform upper bounds to the gradients or the strong-convexity parameters of the future constraint functions. Yet, the proposed policy achieves the optimal violation bounds without making any extraneous assumptions, such as Slater's condition (see Table 1).
3. Tighter cumulative violation bounds for the Generalized OCO problem are obtained by using a new criterion - the non-negativity of the worst-case regret on a round (see Table 2). This replaces the stronger and often infeasible assumption on Slater's condition in the literature. Surprisingly, it leads to an \(O(\log T)\) violation bound for convex constraints and strongly convex cost functions.
4. As a by-product of our algorithm for the OCS problem, we obtain a new class of stabilizing control policies for the classic input-queued switching problem with adversarial arrival and service processes (see Section 7).
5. These results are established by introducing a new potential-based technique for simultaneously bounding the regret and constraint violation penalty. In brief, all performance guarantees in this paper are obtained by solving a recursive inequality which arises from plugging in off-the-shelf regret bounds in a new regret decomposition result. To the best of our knowledge, this is the first such separability result for this problem and might be of independent interest.
6. Thanks to the regret decomposition result, our proof arguments are compact requiring only a few lines of elementary algebra. This should be contrasted with long and intricate arguments in many of the papers cited above.
On the methodological contribution:Lyapunov or the potential function method has been extensively used in the literature for designing and analyzing control policies for linear and non-linear systems. The stochastic variant of the Lyapunov method, and especially the Foster-Lyapuov theorem, has played a pivotal role in designing stabilizing control policies for stochastic queueing networks (Meyn, 2008; Neely, 2010). However, to the best of our knowledge, the application of these analytical
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline Reference & Constraints type & Regret & Violation & Algorithm & Assumption \\ \hline Jenatton et al. (2016) & Fixed & \(O\big{(}T^{\max(\beta,1-\beta)}\big{)}\) & \(O(T^{1-\beta/2})\) & Primal-Dual GD & Fixed constraints \\ Yuan and Lamperski (2018) & Fixed & \(O\big{(}T^{\max(\beta,1-\beta)}\big{)}\) & \(O(T^{1-\beta/2})\) & Primal-Dual MD & Fixed constraints \\ Yi et al. (2021) & Fixed, convex & \(O\big{(}T^{\max(\beta,1-\beta)}\big{)}\) & \(O(T^{(1-\beta/2)})\) & OGD+drift+penalty & Fixed constraints \\ Yi et al. (2023) & Adversarial & \(O\big{(}T^{\max(\beta,1-\beta)}\big{)}\) & \(O(T^{1-\beta/2})\) & Primal-Dual MD & Known \(G\) \\ Yi et al. (2023) & Adversarial, str-convex cost & \(O(\log T)\) & \(O(\sqrt{T\log T})\) & Primal-Dual MD & Known \(G,\alpha\) \\ Sun et al. (2017) & Adversarial & \(O(\sqrt{T})\) & \(O(T^{3/4})\) & OMD & Known \(G\) \\ Neely and Yu (2017) & Adversarial & \(O(\sqrt{T})\) & \(O(\sqrt{T})\) & OGD+drift+penalty & Slater condition \\ Guo et al. (2022) & Adversarial & \(O(\sqrt{T})\) & \(O(T^{3/4})\) & Convex opt. each round & Full access to \(\{g_{t}\}\) \\ Guo et al. (2022) & Adversarial, str-convex cost & \(O(\log T)\) & \(O(\sqrt{T\log T})\) & Convex opt. each round & Full access to \(\{g_{t}\}\) \\
**This paper** & Adversarial & \(O(\sqrt{T})\) & \(O(T^{3/4})\) & Any adaptive & - \\
**This paper** & Adversarial & \(O(\sqrt{T})\) & \(O(\sqrt{T})\) & Any adaptive & Regret\({}_{T}\geq 0\) \\
**This paper** & Adversarial, str-convex cost & \(O(\log T)\) & \(O(\sqrt{T\log T})\) & Any adaptive & - \\
**This paper** & Adversarial, str-convex cost & \(O(\log T)\) & \(O(\frac{\log T}{\alpha})\) & Any adaptive & Regret\({}_{T}\geq 0\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of the results for the Generalized OCO problem. Unless mentioned otherwise, we assume \(1\)-feasibility, general convex constraints, and convex cost functions while stating the bounds. In the above, \(0\leq\beta\leq 1\) is an adjustable parameter. The parameter \(\alpha\) denotes the strong convexity parameter of the cost functions. \(G\) denotes a uniform upper bound to the gradient of cost and constraint functions. Regret\({}_{T}\) denotes the worst-case regret against all static admissible actions.
tools to systems with adversarial dynamics has been quite limited in the literature. In this paper, we show how the Lyapunov and the drift-plus-penalty method Neely (2010) can be seamlessly combined with the OCO framework to design new algorithms with tight performance bounds. We expect that the analytical methods developed in this paper can be generalized to more complex problems with adversarial inputs.
Note:If a convex function \(f\) is not differentiable at the point \(x\) then by \(\nabla f(x)\) we denote any sub-gradient of the function at \(x\). Recall that any convex function could be non-differentiable only on a set of measure at most zero (Rockafellar, 1972, Theorem 25.5). Hence, by smoothly perturbing at all points of non-differentiability by an infinitesimal amount, we can alternatively and without any loss of generality assume all cost and constraint functions to be differentiable without altering the regret/constraint violation bounds presented in this paper (Hazan et al., 2007).
## 3 Preliminaries on Online Convex Optimization (OCO)
The standard online convex optimization problem can be viewed as a repeated game between an online policy and an adversary (Hazan et al., 2016). Let \(\Omega\subseteq\mathbb{R}^{d}\) be a convex set, which we call the _admissible_ set. On every round \(t\geq 1,\) an online policy selects an action \(x_{t}\) from the admissible set \(\Omega\). Upon seeing the currently selected action \(x_{t}\), the adversary reveals a convex cost function \(f_{t}:\Omega\mapsto\mathbb{R}.\) The goal of the online policy is to choose an admissible action sequence \(\{x_{t}\}_{t\geq 1}\) so that its total cost over any horizon of length \(T\) is not much larger than the total cost incurred by any fixed admissible action \(x^{*}\in\Omega\). More precisely, the objective of the policy is to minimize the static regret \(\mathcal{R}_{T}\):
\[\mathcal{R}_{T}\equiv\sup_{x^{*}\in\Omega}\mathcal{R}_{T}(x^{*}),\text{ where }\mathcal{R}_{T}(x^{*})\equiv\sum_{t=1}^{T}f_{t}(x_{t})-\sum_{t=1}^{T}f_{t}(x^{*}). \tag{1}\]
```
0: Non-empty closed convex set \(\Omega\subseteq\mathbb{R}^{d}\), Sequence of convex cost functions \(\{f_{t}\}_{t\geq 1},\) step sizes \(\eta_{1},\eta_{2},\ldots,\eta_{T}>0.\)
1: Initialize \(x_{1}\) arbitrarily
2:for each round \(t\geq 1\)do
3: Predict \(x_{t}\), observe \(f_{t}\), incur a cost of \(f_{t}(x_{t})\).
4: Compute a (sub)-gradient \(\nabla_{t}\) of \(f_{t}\) at \(x_{t}\). Let \(G_{t}=\|\nabla_{t}\|_{2}\).
5: Update \(x_{t+1}=\Pi_{\Omega}(x_{t}-\eta_{t}\nabla_{t})\).
6:endfor
```
**Algorithm 1** Online Gradient Descent (OGD)
In a seminal paper, Zinkevich (2003) showed that the online gradient descent policy, given in Algorithm 1, with an appropriately chosen constant step size schedule, achieves a sublinear regret bound \(\mathcal{R}_{T}=O(\sqrt{T})\) for convex cost functions with bounded (sub)-gradients. In this paper, we are interested in stronger adaptive regret bounds where the bound is given in terms of the norm of the gradients and the strong-convexity parameters of the online cost functions. In Theorem 1, we quote two standard results on such data-dependent adaptive regret bounds achieved by the standard OGD policy with appropriately chosen adaptive step size schedules.
**Theorem 1**.: _Consider the generic OGD policy given in Algorithm 1._
1. _(_Orabona_,_ 2019_, Theorem 4.14)_ _Let the cost functions_ \(\{f_{t}\}_{t\geq 1}\) _be convex and the step size sequence be chosen as_ \(\eta_{t}=\frac{\sqrt{2}D}{2\sqrt{\sum_{t=1}^{t-1}G_{t}^{2}}},t\geq 1,\) _where_ \(D\) _is the diameter of the admissible set_ \(\Omega\) _and_ \(G_{t}=\nabla f_{t}(x_{t}),t\geq 1.\) _Then the OGD algorithm achieves the following regret bound:_ \[\mathcal{R}_{T}\leq\sqrt{2}D\sqrt{\sum_{t=1}^{T}G_{t}^{2}}.\] (2)
2. _(_Hazan et al._,_ 2007_, Theorem 2.11)_ _Let the cost functions_ \(\{f_{t}\}_{t\geq 1}\) _be strongly convex and let_ \(H_{t}>0\) _be the strong-convexity parameter for the cost function_ \(f_{t}\)_1_. Let the step size
sequence be chosen as \(\eta_{t}=\frac{1}{\sum_{s=1}^{t-1}H_{s}},t\geq 1\). Then the OGD algorithm achieves the following regret bound:_
\[\mathcal{R}_{T}\leq\frac{1}{2}\sum_{t=1}^{T}\frac{G_{t}^{2}}{\sum_{ s=1}^{t}H_{s}}. \tag{3}\]
Note that the above adaptive bounds are mentioned for the simplicity of the regret expressions and the corresponding policy and for no other particular reason. Similar regret bounds are known for a host of other online learning policies as well. For structured domains, one can use other algorithms such as AdaFTRL (Orabona and Pal, 2018) which gives better regret bounds for high-dimensional problems. Furthermore, for problems with combinatorial structures, adaptive oracle-efficient algorithms, _e.g.,_ Follow-the-Perturbed-Leader (FTPL)-based policies, may be used (Abernehty et al., 2014, Theorem 11) (see Section 7). Our proposed meta-policy is oblivious to the particular online learning sub-routine being used to solve the surrogate OCO problem - all that is expected from the sub-routine are adaptive regret bounds comparable to (2) and (3). This property can be exploited to immediately extend the scope of our proposed algorithm to various other non-standard settings, _e.g.,_ delayed feedback (Joulani et al., 2016).
## 4 The Online Constraint Satisfaction (\(\mathsf{OCS}\)) Problem
We begin our discourse with the Online Constraint Satisfaction (\(\mathsf{OCS}\)) problem, which does not contain any cost function. In this problem, the online learner is presented with an ordered \(k\)-tuple of convex constraints on each round. At each round, the constraints could be adversarially chosen, which are then revealed to the learner _after_ it selects its action for that round. The objective is to choose actions sequentially so that the cumulative constraint violations corresponding to each component of the \(k\)-tuple grow sub-linearly with the horizon length. Although the \(\mathsf{OCS}\) problem can be seen to be a special case of the generalized OCO problem, it is important as a standalone problem as well. In the following, we show that the online multi-task learning problem can be formulated as an instance of the \(\mathsf{OCS}\) problem.
Example: Online Multi-task Learning:Consider the problem of online multi-task learning where we train a single model to solve a number of related tasks (Ruder, 2017; Dekel et al., 2006; Murugesan et al., 2016). See Figure 2 for a simplified schematic. In this problem, the action \(x_{t}\) naturally translates to the shared weight vector that encodes the model common to all tasks. The loss function for the \(j^{\text{th}}\) task on round \(t\) is given by the function \(g_{t,j}(\cdot),j\in[k]\). A task is assumed to be satisfactorily completed (_e.g.,_ correct prediction in the case of classification problems) on any round if the corresponding loss is non-positive. The objective of the multi-task learning problem is to sequentially predict the shared weight vectors \(\{x_{t}\}_{t=1}^{T}\) so that the maximum cumulative loss of
Figure 2: A schematic for the online multi-task learning problem
each task over any sub-interval grows sub-linearly with the length of the interval. Since the weight vector is shared across the tasks, the above goal would be impossible to achieve had the tasks not been related to each other (Ruder, 2017). Hence, we make the assumption that there exists a fixed admissible action \(x^{*}\) which could perform all the tasks satisfactorily. See Section 7 for an application of the tools and techniques developed for the \(\mathsf{OCS}\) problem to a queueing problem.
### Problem formulation
We now formalize the online constraint satisfaction (\(\mathsf{OCS}\)) problem described above. Assume that on every round \(t\geq 1\), an online policy selects an action \(x_{t}\) from an admissible set \(\Omega\subseteq\mathbb{R}^{d}\). After observing the current action \(x_{t}\), an adversary outputs \(k\) ordered constraints of the form \(g_{t,i}(x)\leq 0\) where each \(g_{t,i}:\Omega\mapsto\mathbb{R}\) is a convex function. As an example, in the problem of multi-task learning, each of the \(k\) constraint sequences can be thought to belong to a specific task. The constraint functions are accessible to the policy via a first-order oracle, which returns only the values and the gradients (sub-gradients in the case of non-smooth functions) of the constraint functions at the chosen action points. The goal of the online policy is to choose an admissible action sequence \(\{x_{t}\}_{t=1}^{T}\) so that the _maximum_ cumulative constraint violations for any task over any sub-interval grow sub-linearly with the horizon length \(T\). Let \(\mathcal{I}\) be any sub-interval of the interval \([1,T]\). The maximum cumulative constraint violation for the \(i^{\text{th}}\) constraint sequence over any sub-interval is defined as:
\[\mathbb{V}_{i}(T)=\max_{\mathcal{I}\in[1,T]}\sum_{t\in\mathcal{I}}g_{t,i}(x_{ t}),\ \forall 1\leq i\leq k. \tag{4}\]
Our objective is to design an online learning policy so that \(\max_{i=1}^{k}\mathbb{V}_{i}(T)\) is as small as possible. Note that for constraint functions that can take both positive and negative values, controlling the cumulative violations over all sub-intervals is stronger than just controlling the cumulative violations over the entire horizon.
As an aside, the definition of maximum cumulative regret is similar to the definition of interval regret or adaptive regret (Jun et al., 2017; Hazan et al., 2007). However, the best-known strongly adaptive algorithms are inefficient as they need to run \(O(\log T)\) experts algorithm on every round (Jun et al., 2017). Fortunately, we will see that our proposed meta-policy is efficient as it needs to perform only one gradient step per round.
### Assumptions
In this section, we list the general assumptions which apply to both the \(\mathsf{OCS}\) problem and the Generalized OCO problem, which will be described later in Section 5. Since the \(\mathsf{OCS}\) problem does not contain any cost function, the cost functions mentioned below necessarily refer to the Generalized OCO problem only.
**Assumption 1** (Convexity).: _Each of the cost functions \(f_{t}:\Omega\mapsto\mathbb{R}\) and the constraint functions \(g_{t}:\Omega\mapsto\mathbb{R}\) are convex for all \(t\geq 1\). The admissible set \(\Omega\subseteq\mathbb{R}^{d}\) is closed and convex and has a finite diameter of \(D\)._
**Assumption 2** (Lipschitzness).: _Each of the cost and constraint functions is Lipschitz continuous with Lipschitz constant \(G/2,\) i.e., for all \(x,y\in\Omega,\) we have_
\[|f_{t}(x)-f_{t}(y)|\leq G\|x-y\|,\ |g_{t}(x)-g_{t}(y)|\leq\frac{G}{2}\|x-y\|,\ \forall t\geq 1.\]
Unless specified otherwise, we will assume that the above Lipschitzness condition holds w.r.t. the standard Euclidean norm. Hence, the Lipschitzness condition implies that the \(\ell_{2}\)-norm of the (sub)-gradients of the cost and constraint functions are uniformly upper-bounded by \(G/2\) over the entire admissible set \(\Omega\). We emphasize that the values of the parameters \(G\) and \(D\) are _not necessarily known_ to the policy. Finally, we make the following feasibility assumption on the online constraint functions.
**Assumption 3** (Feasibility).: _There exists some feasible action \(x^{*}\in\Omega\) s.t. \(g_{t,i}(x^{*})\leq 0,\forall t\in T,\forall i\in[k]\). The set of all feasible actions is denoted by \(\Omega^{*},\) which will be called the feasible set. In other words, the feasible set is defined as_
\[\Omega^{*}=\{x\in\Omega:g_{t,i}(x)\leq 0,\forall i,t\}. \tag{5}\]
_The feasibility assumption is equivalent to the condition that \(\Omega^{*}\neq\varnothing\)._
Assumptions 1 and 2 are standard in the online learning literature. The feasibility assumption (Assumption 3), which is common in the literature on the constrained OCO problem, implies that there exists a _uniformly_ admissible action \(x^{*}\) which satisfies _all_ constraints on _every_ round. In other words, all constraint functions are non-positive over a common subset (Neely and Yu, 2017; Yu and Neely, 2016; Yuan and Lamperski, 2018; Yi et al., 2023; Liakopoulos et al., 2019). This assumption of instantaneous feasibility will be relaxed in Section 6 where we only assume that there exists a fixed admissible action \(x^{*}\) that satisfies the property that the sum of the constraint functions at \(x^{*}\) over any interval of length \(S\geq 1\) is non-positive. Note that we _do not_ assume Slater's condition as it does not hold in many problems of interest (Yu and Neely, 2016). Furthermore, we do not restrict the sign of the constraint functions, which could take both positive and negative values on its domain.
Inspired by the Lyapunov method in the control theory, in the following, we propose an online meta-policy for the OCS problem and show that it yields optimal violation bounds. Our main technical contribution is that while the classic works, such as Neely (2010), use the Lyapunov theory in a stochastic setting, we adapt it to the adversarial setting by combining the Lyapunov method with the OCO framework.
### Designing an OCS Meta-policy with Lyapunov methods
Let the process \(\{Q_{i}(t)\}_{t\geq 1}\) keep track of the cumulative constraint violation corresponding to the \(i^{\text{th}}\) constraint sequence. Formally, we define:
\[Q_{i}(t)=\left(Q_{i}(t-1)+g_{t,i}(x_{t})\right)^{+},Q_{i}(0)=0, \ \forall i\in[k],t\geq 1, \tag{6}\]
where we have used the standard notation \(y^{*}\equiv\max(0,y)\). Expanding the above queueing recursion (also known as the _Lindley process_(Asmussen, 2003, pp. 92)), and using the definition of the maximum cumulative constraint violation (4), we can write
\[\mathbb{V}_{i}(T)\equiv\max_{t=1}^{T}\max(0,\max_{\tau=0}^{t-1} \sum_{l=t-\tau}^{t}g_{l,i}(x_{l}))=\max_{t=1}^{T}Q_{i}(t). \tag{7}\]
The above equation says that to control the cumulative constraint violations, it is sufficient to control the queueing processes. We now combine the classic Lyapunov method with no-regret learning policies to control the queues.
Bounding the drift of the Lyapunov function:Define the potential (Lyapunov) function \(\Phi(t)\equiv\sum_{i=1}^{k}Q_{i}^{2}(t),t\geq 1\). Observe that for any real number \(x\), we have \(((x)^{+})^{2}=xx^{+}\). Hence, from Eqn. (6), we have that for each \(1\leq i\leq k\) :
\[Q_{i}^{2}(t) = \big{(}Q_{i}(t-1)+g_{t,i}(x_{t})\big{)}Q_{i}(t) \tag{8}\] \[= Q_{i}(t-1)Q_{i}(t)+Q_{i}(t)g_{t,i}(x_{t})\] \[\overset{(a)}{\leq} \frac{1}{2}Q_{i}^{2}(t)+\frac{1}{2}Q_{i}^{2}(t-1)+Q_{i}(t)g_{t,i}(x _{t}).\]
where in (a), we have used the AM-GM inequality. Rearranging the above, the _one-step drift_ of the potential function \(\Phi(t)\) may be upper bounded as
\[\Phi(t)-\Phi(t-1)=\sum_{i=1}^{k}\big{(}Q_{i}^{2}(t)-Q_{i}^{2}(t-1 )\big{)}\leq 2\sum_{i=1}^{k}Q_{i}(t)g_{t,i}(x_{t}). \tag{9}\]
Motivated by the above inequality, we now _define_ a surrogate cost function \(\hat{f}_{t}:\mathcal{X}\mapsto\mathbb{R}\) as a linear combination of the constraint functions where the coefficients are given by the corresponding queue lengths, _i.e.,_
\[\hat{f}_{t}(x)\equiv 2\sum_{i=1}^{k}Q_{i}(t)g_{t,i}(x). \tag{10}\]
Clearly, the surrogate cost function \(\hat{f}_{t}(\cdot)\) is convex. Our proposed meta-policy uses the surrogate cost functions to decide the actions on every round.
Note that, unlike some of the previous work based on the Lyapunov drift approach (Neely and Yu, 2017; Yu and Neely, 2016), Algorithm 2 takes full advantage of the adaptive nature of the base OCO sub-routine by exploiting the fact that the adversary is allowed to choose the surrogate cost function \(\hat{f}_{t}\)_after_ seeing the current action of the policy \(x_{t}\). In our case, the surrogate cost function \(\hat{f}_{t}(\cdot)\) depends on \(x_{t}\) via the coefficient vector \(\mathbf{Q}(t)\).
### Analysis
Regret decomposition:Fix any feasible action \(x^{*}\in\Omega^{*}\), where the feasible set \(\Omega^{*}\) has been defined in Eqn. (5). Then, from Eqn. (9), for any round \(\tau\in[T]\) we have:
\[\Phi(\tau)-\Phi(\tau-1) \leq 2\sum_{i=1}^{k}Q_{i}(\tau)g_{\tau,i}(x_{\tau}) \tag{11}\] \[\overset{(a)}{\leq} 2\sum_{i=1}^{k}Q_{i}(\tau)g_{\tau,i}(x_{\tau})-2\sum_{i=1}^{k}Q_ {i}(\tau)g_{\tau,i}(x^{*})\] \[= \hat{f}_{\tau}(x_{\tau})-\hat{f}_{\tau}(x^{*}), \tag{12}\]
where in (a), we have used the assumption that the action \(x^{*}\) is feasible and hence, \(g_{\tau,i}(x^{*})\leq 0,\forall i,\tau\). Summing up the inequalities (11) above from \(\tau=t_{1}+1\) to \(\tau=t_{2}\), we obtain the following inequality, which upper bounds the change in the potential function in the interval \([t_{1},t_{2}]\) by the regret for learning the surrogate cost functions:
\[\Phi(t_{2})-\Phi(t_{1})\leq\text{Regret}^{\prime}_{t_{1}:t_{2}}(x^{*}),\ \forall x^{*}\in\Omega^{*}. \tag{13}\]
We emphasize that the regret on the RHS depends on the queue variables \(\{Q(t)\}_{t\geq 1}\), which are implicitly controlled by the online policy through the evolution (6). In particular, by setting \(t_{1}=0,t_{2}=t\) and recalling that \(\Phi(0)=\sum_{i}Q_{i}^{2}(0)=0,\) we have that
\[\sum_{i=1}^{k}Q_{i}^{2}(t)\leq\text{Regret}^{\prime}_{t}(x^{*}),\ \forall x^{*}\in\Omega^{*},t\geq 1. \tag{14}\]
The above bound is valid for any feasible action \(x^{*}\in\Omega^{*}\). Taking the supremum of the RHS over the set of all admissible actions in \(\Omega\supseteq\Omega^{*}\), we obtain
\[\sum_{i=1}^{k}Q_{i}^{2}(t)\leq\text{Regret}^{\prime}_{t}, \tag{15}\]
where \(\text{Regret}^{t}_{t}\equiv\sup_{x^{*}\in\Omega}\text{Regret}^{t}_{t}(x^{*})\) denotes the worst-case static regret for the sequence of surrogate cost functions \(\{\hat{f}_{\tau}(x)\}_{\tau=1}^{t}\). Note that the surrogate cost functions explicitly depend on the queueing processes. Hence, the regret bound in (15) depends on the online policy employed in step 5 of Algorithm 2. Inequality (15) will be our starting point for bounding the cumulative constraint violation by using known data-dependent adaptive regret bounds achieved by base OCO policies (see Theorem 1). The following section gives violation bounds achieved by the meta-algorithm 2 when the base OCO policy is taken to be the standard online gradient descent policy with adaptive step sizes. However, we emphasize that the bound (15) is general and can be used to obtain performance bound for any base OCO policy with similar regret bounds. In the following two sections, we consider the case of convex and strongly convex constraint functions, respectively.
#### 4.4.1 Convex constraint functions
From Eqn. (10), we have that
\[\nabla\hat{f}_{t}(x)=2\sum_{i=1}^{k}Q_{i}(t)\nabla g_{t,i}(x).\]
Hence, using the triangle inequality for the Euclidean norm, we have
\[\|\nabla\hat{f}_{t}(x_{t})\|_{2}\leq 2\sum_{i=1}^{k}Q_{i}(t)\|\nabla g_{t,i}(x )\|_{2}.\]
Using the Cauchy-Schwarz inequality and the gradient bounds for the constraint functions as given in Assumption (2), the squared \(\ell_{2}\)-norm of the (sub)-gradients of the surrogate cost functions (10) can be bounded as follows:
\[\|\nabla\hat{f}_{t}(x_{t})\|_{2}^{2}\leq 4(\sum_{i=1}^{k}Q_{i}^{2}(t))(\sum_{i =1}^{k}\|\nabla g_{t,i}(x_{t})\|_{2}^{2})\leq kG^{2}\sum_{i=1}^{k}Q_{i}^{2}(t). \tag{16}\]
In the OCS meta-policy given in Algorithm 2, let us now take the base OCO policy to be the Online Gradient Descent (OGD) policy with the adaptive step sizes given in part 1 of Theorem 1. Combining Eqn. (2), (15), and (16), we obtain the following sequential inequality:
\[\sum_{i=1}^{k}Q_{i}^{2}(t)\leq c\sqrt{\sum_{\tau=1}^{t}\big{(}\sum_{i=1}^{k}Q_ {i}^{2}(\tau)\big{)}},\ t\geq 1, \tag{17}\]
where \(c\equiv GD\sqrt{2k}\) is a time-invariant problem-specific parameter that depends on the bounds of the gradient norms, the number of constraints on a round, and the diameter of the admissible set. Note that Algorithm 2 is fully _parameter-free_ as it uses only available causal information on the constraint functions and does not need to know any parametric bounds (_e.g.,_\(G\)) on the future constraint functions.
Analysis:To upper bound the queue lengths, let us now define the auxiliary variables \(Q^{2}(t)\equiv\sum_{i=1}^{k}Q_{i}^{2}(t),t\geq 1\). From Eqn. (17), the variables \(\{Q(t)\}_{t\geq 1}\) satisfy the following non-linear system of inequalities that we need to solve.
\[Q^{2}(t)\leq c\sqrt{\sum_{\tau=1}^{t}Q^{2}(\tau)}\leq c\sqrt{\sum_{\tau=1}^{ T}Q^{2}(\tau)},\ 1\leq t\leq T. \tag{18}\]
Summing up the above inequality over all rounds \(1\leq t\leq T\) and simplifying, we have
\[\sqrt{\sum_{\tau=1}^{T}Q^{2}(t)}\leq cT.\]
Substituting the above bound in inequality (18), we have
\[\max_{i}\mathbb{V}_{i}(T)\stackrel{{(a)}}{{\leq}}\max_{i=1}^{k}Q _{i}(T)\leq Q(T)\leq c\sqrt{T}, \tag{19}\]
where in inequality (a), we have used (7).
**Observation 1**.: _By replacing the original constraint function \(g_{t}(x)\) with a surrogate constraint function \(\hat{g}_{t}(x),\forall t\geq 1,\) where the function \(\hat{g}_{t}(x)\) satisfies the assumptions in Section 4.2 and enjoys the property that \(\hat{g}_{t}(x)\geq g_{t}(x),\forall x\in\Omega,\) we have_
\[\sum_{t=1}^{T}g_{t}(x_{t})\leq\sum_{t=1}^{T}\hat{g}_{t}(x_{t})=O(\sqrt{T}).\]
_This observation is particularly useful when the original constraint functions are non-convex, e.g., \(0-1\) loss, which can be upper bounded with the hinge loss function. In particular, we can also define the surrogate function as \(\hat{g}_{t}(x)\equiv\psi(g_{t}(x)),\) where \(\psi:\mathbb{R}\rightarrow\mathbb{R}_{+}\) is any non-decreasing, non-negative, and convex function with \(\psi(0)=0\). From basic convex analysis, it follows that the surrogate function is convex. This observation substantially generalizes our previous bound (19) on the cumulative constraint violations. As an example, we can recover Yuan and Lamperski (2018)'s result for time-invariant constraints by defining the surrogate function to be \(\hat{g}_{t}(x)=\psi(g_{t}(x))\equiv(\max(0,g_{t}(x)))\)2._
Footnote 2: If the sequence is identically equal to zero, then there is nothing to prove. Otherwise, by skipping the initial zero terms, one can always assume that the first term of the sequence is non-zero.
#### 4.4.2 Strongly-convex constraint functions
Next, we consider the case when the sequence of constraint functions is uniformly \(\alpha\)-strongly convex. Let the base OCO policy be taken as the OGD policy with step sizes chosen in part 2 of Theorem 1. Combining (15) with (3) and using the gradient bound (16) for the surrogate functions and the \(\alpha\)-strong-convexity of the constraint functions, we immediately conclude that the queue length sequence satisfies the following recursive inequalities:
\[\sum_{i=1}^{k}Q_{i}^{2}(t)\leq c\sum_{\tau=1}^{t}\frac{\sum_{i=1}^{k}Q_{i}^{2} (\tau)}{\sum_{s=1}^{\tau}\sum_{i=1}^{k}Q_{i}(s)},\;t\geq 1, \tag{20}\]
where \(c\equiv\frac{kG^{2}}{4\alpha}\) is a problem-specific parameter.
Analysis:For the ease of analysis, we define the auxiliary variables \(Q^{2}(t)\equiv\sum_{i=1}^{k}Q_{i}^{2}(t),t\geq 1\). Since the queue variables are non-negative, we have for any \(s\in[T]\):
\[\sum_{i=1}^{k}Q_{i}(s)\geq Q(s).\]
Hence, from Eqn. (20), we have
\[Q^{2}(t)\leq c\sum_{\tau=1}^{t}\frac{Q^{2}(\tau)}{\sum_{s=1}^{\tau}Q(s)}. \tag{21}\]
The following Proposition bounds the growth of any sequence that satisfies the above system of inequalities.
**Proposition 1**.: _Let \(\{Q(t)\}_{t\geq 1}\) be any non-negative sequence with \(Q(1)>0\). Suppose that the \(t^{\text{th}}\) term of the sequence satisfies the inequality_
\[Q^{2}(t)\leq c\sum_{\tau=1}^{t}\frac{Q^{2}(\tau)}{\sum_{s=1}^{\tau}Q(s)},\; \forall t\geq 1, \tag{22}\]
_where \(c>0\) is a constant. Then \(Q(t)\leq c\ln(t)+O(\ln\ln t),\forall t\geq 1\)._
See 10.1 in the Appendix for the proof of the above result. The results in Sections 4.4.1 and 4.4.2 lead to our main result for the \(\mathtt{OCS}\) problem.
**Theorem 2** (Bounds on the cumulative violation for the \(\mathtt{OCS}\) problem).: _The \(\mathtt{OCS}\) Meta-policy, described in Algorithm 2, achieves \(O(\sqrt{T})\) and \(O(\log T)\) cumulative constraint violations for convex and strongly-convex constraint functions, respectively. These bounds are achieved by a standard online gradient descent subroutine with an appropriate adaptive step size schedule as described in Theorem 1._
The Generalized OCO Problem
In this section, we generalize the \(\mathtt{OCS}\) problem by considering the setting where the policy receives an adversarially chosen convex cost function and a constraint function at the end of each round 3. More precisely, on every round \(t,\) the online policy first chooses an admissible action \(x_{t}\in\Omega\) and then the adversary chooses a convex cost function \(f_{t}:\Omega\rightarrow\mathbb{R}\) and a constraint of the form \(g_{t}(x)\leq 0\), where \(g_{t}:\Omega\rightarrow\mathbb{R}\) is a convex function. Let \(\Omega^{*}\) be the feasible set satisfying all constraints as defined in Eqn. (5). Our objective is to design an online policy that achieves a sublinear cumulative violation and a sublinear worst-case regret over the feasible set \(\Omega^{*}\). Specifically, we require the following conditions to be satisfied
Footnote 3: For the simplicity of notations, in this section, we assume that only one constraint function is revealed on each round (_i.e.,_\(k=1\)). The general case where more than one constraint function is revealed can be handled similarly as in Section 4.
\[\mathbb{V}(T)\equiv\sum_{t=1}^{T}(g_{t}(x_{t}))^{+}=o(T),\text{ and }\sup_{x^{*}\in\Omega^{*}} \mathcal{R}_{T}(x^{*})=o(T),\ \forall T\geq 1, \tag{23}\]
where the regret \(\mathcal{R}_{T}(x^{*})\) w.r.t. the fixed action \(x^{*}\in\Omega\) has been defined earlier in Eqn. (1). The Generalized OCO problem can be motivated by the following offline convex optimization problem where the functions \(\{f_{t},g_{t}\}_{t=1}^{T}\) are known _a priori_:
\[\min\sum_{t=1}^{T}f_{t}(x),\]
subject to the constraints
\[g_{t}(x) \leq 0,\ 1\leq t\leq T\] \[x \in \Omega.\]
Consequently, Eqn. (23) uses a stronger definition of cumulative constraint violation compared to the \(\mathtt{OCS}\) problem (c.f. Eqn. (4)). An upper bound to the above violation metric \(\mathbb{V}(T)\) ensures that the constraint violation in one round can not be compensated by a strictly feasible constraint in a different round (Guo et al., 2022). For the simplicity of exposition, we assume that the horizon length \(T\) is known. However, this assumption can be relaxed by using the standard doubling trick (Hazan et al., 2016).
Pre-processing the constraint functions by clipping:In the Generalized OCO problem, we first pre-process each of the constraint functions by clipping them below zero, _i.e.,_ we redefine the \(t^{\text{th}}\) constraint function as 4
Footnote 4: Since, in this section, we work exclusively with the pre-processed constraint functions, by a slight abuse of notations, we use the same symbol \(g_{t}\) for the pre-processed functions.
\[g_{t}(x)\leftarrow(g_{t}(x))^{+},x\in\Omega,\ \ \forall t\geq 1.\]
It immediately follows that the pre-processed constraint functions are also convex and keep the feasible set \(\Omega^{*}\) unchanged. The pre-processing step allows us to bound the hard constraint violation \(\mathbb{V}(T)\) as defined in Eqn. (23). Furthermore, the pre-processing step also offers some technical advantages for establishing tighter regret and constraint violation bounds by ensuring the monotonicity of the queue lengths as defined in Eqn. (24) below.
### Designing a Meta-Policy for the Generalized OCO Problem
As in the \(\mathtt{OCS}\) problem, we keep track of the cumulative constraint violations through the same queueing process \(\{Q(t)\}_{t\geq 1}\) that evolves as follows.
\[Q(t)=\big{(}Q(t-1)+g_{t}(x_{t})\big{)}^{+},\ Q(0)=0. \tag{24}\]
Since the pre-processed constraint functions are non-negative, the \(\max(0,\cdot)\) operation in Eqn. (24) is superfluous. Hence, we have \(\mathbb{V}(t)=Q(t),\forall t.\) As before, let us define the potential function \(\Phi(t)\equiv Q^{2}(t).\) From Eqn. (8), the _one-step drift_ of the potential function \(\Phi(t)\) can be upper bounded as
\[\Phi(t)-\Phi(t-1)=Q^{2}(t)-Q^{2}(t-1)\leq 2Q(t)g_{t}(x_{t}). \tag{25}\]
Now, motivated by the drift-plus-penalty framework in the stochastic network control theory (Neely, 2010), we _define_ a surrogate cost function \(\hat{f}_{t}:\Omega\rightarrow\mathbb{R}\) as follows:
\[\hat{f}_{t}(x)\equiv Vf_{t}(x)+2Q(t)g_{t}(x),x\in\Omega,\ \forall t\geq 1, \tag{26}\]
where the first term corresponds to the cost and the second term is an upper bound to the drift of the quadratic potential function. In the above, \(V>0\) is an adjustable parameter that will be fixed later. It can be immediately verified that the surrogate function \(\hat{f}_{t}(\cdot)\) is convex. Next, we propose an online meta-policy to solve the constrained online learning problem defined above.
```
0: Sequence of cost functions \(\{f_{t}\}_{t\geq 1}\) and constraint functions \(\{g_{t}\}_{t\geq 1}\), a base OCO sub-routine \(\Pi\) with an adaptive regret bound (see Theorem 1), \(V\)
0: Sequence of admissible actions \(\{\mathbf{x}_{t}\}_{t\geq 1}\)
0:\(x_{1}=\mathbf{0},Q(0)=0,\ \forall i\in[k]\).
1:for each round \(t\geq 1\)do
2: The adversary reveals the cost function \(f_{t}\) and the constraint function \(g_{t}\).
3:[Preprocessing]:\(g_{t}\leftarrow\max(0,g_{t})\).
4:[Queue updates]\(Q(t)=(Q(t-1)+g_{t}(x_{t}))^{+}\).
5:[Surrogate cost] Construct the surrogate cost function \(\hat{f}_{t}(x)\equiv Vf_{t}(x)+2Q(t)g_{t}(x)\).
6:[OCO step] Pass the surrogate cost function \(\hat{f}_{t}\) to the base OCO sub-routine \(\Pi\), and hence, compute an admissible action \(x_{t+1}\in\Omega\).
7:endfor
```
**Algorithm 3**Generalized OCO Meta-policy
**Remarks:** 1. As in the OCS problem, we are taking full advantage of the adaptive nature of the OCO sub-routine by exploiting the fact that the adversary is allowed to choose the surrogate cost function \(\hat{f}_{t}\)_after_ seeing the current action \(x_{t}\) on any round.
2. Since \(\nabla\hat{f}_{t}(x_{t})=V\nabla f_{t}(x_{t})+2Q(t)\nabla g_{t}(x_{t})\), the meta-policy, in reality, needs to pass only the (sub-)gradients \(\nabla f_{t}(x_{t}),\nabla g_{t}(x_{t})\), and the current violation \(g_{t}(x_{t})\) to an OGD sub-routine. The full description of the functions at every point in its domain is _not required_ by the meta-policy, and hence, the above meta-policy is much more efficient compared to the algorithm given by Guo et al. (2022).
Using the triangle inequality for the Euclidean norms, the norm of the (sub)-gradients of the surrogate cost functions can be bounded as follows:
\[\|\nabla\hat{f}_{t}(x_{t})\|_{2}\leq V\|\nabla f_{t}(x_{t})\|_{2}+2Q(t)\| \nabla g_{t}(x_{t})\|_{2}\leq(V+Q(t))G. \tag{27}\]
The above bound on the norm of the gradient of the surrogate costs depends on the queue length, and hence, it depends on the past actions of the online policy.
### Analysis
Regret Decomposition:We now derive the regret decomposition inequality that will be pivotal in bounding the regret and violation. Fix any feasible action \(x^{*}\in\Omega^{*}\). Then, for any round \(\tau\in[T]\), we
have
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#### 5.2.2 Strongly Convex Cost and Convex Constraint Functions
We now consider the case when each of the cost functions \(f_{t}:\mathcal{X}\to\mathbb{R}_{+}\) is \(\alpha\)-strongly convex for some \(\alpha>0\). The constraint functions are assumed to be convex, as above (not necessarily strongly convex). Only in this problem, we assume that the values of the parameters \(\alpha\) and \(G\) are known to the policy, which will be used for setting the parameter \(V\).
In Algorithm 3, let the base OCO policy \(\Pi\) be taken to be the Online Gradient Descent Policy with step-sizes as described in part 2 of Theorem 1, with regret bound given by Eqn. (3). Since the cost functions are given to be \(\alpha\)-strongly convex, each of the surrogate costs, as defined in Eqn. (10), is \(V\alpha\)-strongly convex. Hence, plugging in the upper bound of the gradient norm of the surrogate costs from Eqn. (27) into Eqn. (3) and simplifying, we obtain the following regret bound for the surrogate cost functions:
\[\text{Regret}^{\prime}_{t}(x^{*})\leq\frac{VG^{2}}{\alpha}\ln(t)+\frac{G^{2}}{ \alpha V}\sum_{\tau=1}^{t}\frac{Q^{2}(\tau)}{\tau},\ \forall x^{*}\in\Omega^{*}.\]
Substituting the above bound into Eqn. (30), we arrive at the following system of inequalities:
\[Q^{2}(t)+V\text{Regret}_{t}(x^{*})\leq\frac{VG^{2}}{\alpha}\ln(t)+\frac{G^{2} }{\alpha V}\sum_{\tau=1}^{t}\frac{Q^{2}(\tau)}{\tau},\ \forall x^{*}\in\Omega^{*},\forall t. \tag{32}\]
The following theorem bounds the regret and constraint violation implied by the above inequality.
**Theorem 4**.: _Consider the generalized OCO problem with \(\alpha\)-strongly convex costs and convex constraint functions for some \(\alpha>0\). Then upon setting \(V=\frac{2G^{2}\ln(T)}{\alpha}\), we have_
\[\text{Regret}_{t}(x^{*})\leq\frac{G^{2}}{\alpha}\ln(t),\ \forall(t)=Q(t)=O \big{(}\sqrt{\frac{t\log T}{\alpha}}\big{)},\forall x^{*}\in\Omega^{*},\ \forall t\in[T].\]
_Furthermore, for any round \(t\geq 1\) where the worst-case regret is non-negative (i.e., \(\sup_{x^{*}\in\Omega^{*}}\text{Regret}_{t}(x^{*})\geq 0\)), the queue-length (and hence, constraint violation) bound can be improved to \(Q(t)=\mathbb{V}(t)=O\big{(}\frac{\log T}{\alpha}\big{)}\)._
See Appendix 10.3 for the proof of the above theorem. The second part of the theorem is _surprising_ because it implies that, under certain conditions on the sign of regret, one can derive a stronger _logarithmic_ cumulative constraint violation bound for constraints which need not be strongly convex. Of course, this strong bound does not hold for the OCS problem where we set \(f_{t}=0\), \(\forall t\geq 1\). Hence, the strong convexity of the cost function helps reduce the constraint violations at a faster scale. It can be verified that the regret bounds given in Theorem 3 and 4 are optimal as they match with the corresponding lower bounds for the unconstrained OCO problem (Hazan et al., 2016, Table 3.1).
## 6 Generalizing the \(\mathsf{OCS}\) problem with the \(S\)-feasibility assumption
In the previous sections, we established performance bounds for the proposed meta-policy upon assuming the existence of a fixed action \(x^{*}\in\Omega\) that satisfies each of the online constraints on _each round_. In particular, we assumed that \(\Omega^{*}\neq\emptyset\). In this section, we revisit the \(\mathsf{OCS}\) problem by relaxing this assumption and only assuming that there is a feasible action \(x^{*}\) that satisfies the aggregate of the constraints over any consecutive \(S\) rounds6, where the parameter \(S\in[T]\) need not be known to the policy _a priori_. Towards this end, we define the set of all \(S\)-feasible actions as below:
Footnote 6: This extension is meaningful only when the range of the constraint functions includes both positive and negative values. For non-negative constraints, clearly, \(\Omega_{S}=\Omega^{*}\), \(\forall S\geq 1\).
\[\Omega_{S}=\{x^{*}:\sum_{\tau\in[\mathcal{I}]}g_{\tau,i}(x^{*})\leq 0, \forall\text{sub-intervals}\ \mathcal{I}\subseteq[1,T],\ |\mathcal{I}|=S,\forall i\in[k]\}. \tag{33}\]
We now replace Assumption 3 with the following
**Assumption 4** ((\(S\)-feasibility)).: \(\Omega_{S}\neq\emptyset\).
We now only assume that \(\Omega_{S}\neq\emptyset\) for some \(S:1\leq S\leq T\). Clearly, Assumption 4 is weaker than Assumption 3 as \(\Omega^{*}\subseteq\Omega_{S}\), \(\forall S\geq 1\). This problem was considered earlier by Liakopoulos et al. (2019)
in the generalized OCO setting with a single constraint function per round. However, since the parameter \(V\) used in their algorithm is restricted as \(S\leq V\leq T\), their algorithm naturally needs to know the value of \(S\). In reality, the value of the parameter \(S\) is generally not available _a priori_ as it depends on the online constraint functions. Fortunately, our proposed meta-policy does not need to know the value of \(S\) and hence, it can automatically adapt itself to the best feasible \(S\).
Generalized regret decomposition:Fix any action \(x^{*}\in\Omega_{S}\) that satisfies the cumulative violation constraints over any interval of length \(S\). Hence, using Eqn. (9), we have
\[\Phi(\tau)-\Phi(\tau-1) \leq 2\sum_{i=1}^{k}Q_{i}(\tau)g_{\tau,i}(x_{\tau})\] \[= 2\sum_{i=1}^{k}Q_{i}(\tau)\big{(}g_{\tau,i}(x_{\tau})-g_{\tau,i} (x^{*})\big{)}+2\sum_{i=1}^{k}Q_{i}(\tau)g_{\tau,i}(x^{*})\] \[= \hat{f}_{\tau}(x_{\tau})-\hat{f}_{\tau}(x^{*})+2\sum_{i=1}^{k}Q_ {i}(\tau)g_{\tau,i}(x^{*}).\]
Summing up the above inequalities from \(\tau=1\) to \(\tau=t\), we have
\[\sum_{i=1}^{k}Q_{i}^{2}(t)=\Phi(t)\leq\text{Regret}_{t}^{\prime}(x^{*})+2\sum _{i=1}^{k}\sum_{\tau=1}^{t}Q_{i}(\tau)g_{\tau,i}(x^{*}). \tag{34}\]
We now bound the last term by making use of the \(S\)-feasibility of the action \(x^{*}\) as given by Eqn. (33). Let us now divide the entire interval \([1,t]\) into disjoint and consecutive sub-intervals \(\{\mathcal{I}_{j}\}_{j=1}^{[t/S]}\), each of length \(S\) (excepting the last interval which could be of a smaller length). Let \(Q_{i}^{*}(j)\) be the \(i^{\text{th}}\) queue length length at the beginning of the \(j^{\text{th}}\) interval. We have
\[\sum_{\tau=1}^{t}Q_{i}(\tau)g_{\tau,i}(x^{*})=\sum_{j=1}^{\lceil t/S\rceil} \sum_{\tau\in\mathcal{I}_{j}}\big{(}Q_{i}(\tau)-Q_{i}^{*}(j)\big{)}g_{\tau,i}( x^{*})+\sum_{j=1}^{\lceil t/S\rceil}Q_{i}^{*}(j)\sum_{\tau\in\mathcal{I}_{j}}g_{ \tau,i}(x^{*}). \tag{35}\]
Let \(|g_{t}(x)|\leq F,\forall x\in\Omega,t\). Due to our assumption of bounded gradients and bounded admissible sets, we have \(F<\infty\). Using the Lipschitz property of the queueing dynamics (6), we have
\[\max_{\tau\in\mathcal{I}_{j}}|Q_{i}(\tau)-Q_{i}^{*}(j)|\leq F(S-1).\]
Substituting the above bound into Eqn. (35), we obtain
\[\sum_{\tau=1}^{t}Q_{i}(\tau)g_{\tau,i}(x^{*})\leq\big{(}1+\frac{t}{S}\big{)}F ^{2}S(S-1)+F(S-1)(Q_{i}(t)+F(S-1)\big{)}, \tag{36}\]
where in the last term, we have used the feasibility of the action \(x^{*}\) in all intervals, except possibly the last interval. Clearly, when \(S=1\), the RHS of the above bound becomes zero and we recover Eqn. (15). Substituting the bound (36) into Eqn. (34), we arrive at the following extended regret decomposition inequality:
\[\sum_{i=1}^{k}Q_{i}^{2}(t) \leq \text{Regret}_{t}^{\prime}(x^{*})+2kF^{2}St+2FS\sum_{i=1}^{k}Q_{i }(t)+4F^{2}S^{2}k. \tag{37}\]
Eqn. (37) leads to the following bound on the cumulative constraint violation.
**Theorem 5**.: _For convex constraints, using the OGD policy with adaptive step-sizes given in part 1 of Theorem 1, Algorithm 2 achieves the following cumulative constraint violation bound with the \(S\)-feasibility assumption (Assumption 4):_
\[\mathbb{V}_{i}(T)=O(\max(\sqrt{ST},S)),\ \forall i\in[k].\]
See Section 10.4 in the Appendix for the proof of the result.
Remarks:Recall that our proof of the \(O(\sqrt{T})\) regret bound for the generalized OCO problem with the \(1\)-benchmark in Theorem 3 crucially uses the non-negativity of the pre-processed constraint functions. However, with \(S\)-feasible benchmarks, pre-processing by clipping the constraints does not work as then the positive violations can not be cancelled with a strictly feasible violation on a different round. We leave the problem of obtaining an optimal \(O(\sqrt{T})\) regret bound for Algorithm 3 for the Generalized OCO problem with the \(S\)-feasibility assumption as an open problem.
## 7 Application to a canonical Scheduling problem
As an interesting application of the machinery developed in this paper, we revisit the classic problem of packet scheduling in an \(N\times N\) input-queued switch in an internet router with _adversarial_ arrival and service processes. This problem has been extensively studied in the networking literature in the stochastic setting; however, to the best of our knowledge, no provable result is known for the problem in the adversarial context. In the stochastic setting with independent arrivals and constant service rates, the celebrated Max-Weight policy is known to achieve the full capacity region (McKeown et al., 1999; Tassiulas and Ephremides, 1990). However, this result immediately breaks down when the arrival and service processes are decided by an adversary. In this section, we demonstrate that the proposed \(\mathsf{OCS}\) meta-policy, described in Algorithm 2, can achieve a sublinear queue length in the adversarial setting as well.
**Problem description:** An \(N\times N\) input-queued switch has \(N\) input ports and \(N\) output ports. Each of the \(N\) input ports maintains a separate FIFO queue for each output port. The input and output ports are connected in the form of a bipartite network using a high-speed switch fabric (see Figure 3 for a simplified schematic). At any round (also called _slots_), each input port can be connected to at most one output port for transmitting the packets. The objective of a switching policy is to choose an input-output matching at each round to route the packets from the input queues to their destinations so that the input queue lengths grow sub-linearly with time (_i.e.,_ they remain _rate-stable_(Neely, 2010)). Please refer to the standard references _e.g.,_ Hajek (2015) and the original papers (Tassiulas and Ephremides, 1990; McKeown et al., 1999) for a more detailed description of the input-queued switch architectures and constraints.
Admissible actions and the queueing process:Let \(\Omega\) be the set of all \(N\times N\) doubly stochastic matrices (_a.k.a._ the Birkhoff polytope). The set \(\Omega\) is known to coincide with the convex hull of all incidence vectors corresponding to the perfect matchings of the \(N\times N\) bipartite graph (Hajek, 2015). At each round, the policy chooses a feasible action \(x(t)\equiv\big{(}x_{ij}(t),1\leq i,j\leq N\big{)}\) from the admissible set \(\Omega\). It then randomly samples a matching \(z(t)\equiv\big{(}z_{ij}(t)\in\{0,1\},1\leq i,j\leq N\big{)}\) using the Birkhoff-Von-Neumann decomposition, such that \(\mathbb{E}z(t)=x(t)\). At the same time, the adversary reveals a packet arrival vector \(\mathbf{b}(t)\) and a service rate vector \(\mathbf{s}(t)\). The arrival and service processes could be binary-valued or could assume any non-negative integers from a bounded range. As a result, the \(i^{\text{th}}\) input queue length corresponding to the \(j^{\text{th}}\) output port evolves as follows:
\[Q_{ij}(t)=\big{(}Q_{ij}(t-1)+b_{ij}(t)-s_{ij}(t)z_{ij}(t)\big{)}^{+},Q_{ij}(0) =0. \tag{38}\]
\(S\)-Feasibility:We assume that the adversary is \(S\)-feasible, _i.e.,_\(\exists x^{*}\in\mathcal{X}\) such that
\[\sum_{t\in\mathcal{I}}(b_{ij}(t)-s_{ij}(t)x^{*}_{ij})\leq 0,\,\forall i,j, \forall\text{sub-intervals }\mathcal{I},|\mathcal{I}|=S.\]
The fixed admissible action \(x^{*}\) is unknown to the switching policy. Since we do not assume Slater's condition, the queue-length bounds derived in Neely and Yu (2017) do not apply here.
Figure 3: A \(4\times 4\) input-queued switch used in a router. Figure taken from Hajek (2015).
Reduction to the \(\mathtt{OCS}\) problem:The above scheduling problem can be straightforwardly reduced to the \(\mathtt{OCS}\) problem where we consider \(k=N^{2}\) linear constraints, where the \((i,j)^{\text{th}}\) constraint function is defined as
\[g_{t,(ij)}(x)\equiv b_{ij}(t)-s_{ij}x_{ij},\ 1\leq i,j\leq N.\]
It follows that the auxiliary queueing variables in the \(\mathtt{OCS}\) meta-policy in Algorithm 2 evolve similarly to Eqn. (38). Hence, taking expectation over the randomness of the policy and using arguments exactly similar to the proof of Theorem 5, it follows that under Algorithm 2, each of the \(N^{2}\) input queues grows sublinearly as \(\mathbb{E}Q_{ij}(t)=O(\max(\sqrt{St},S)),\) where the expectation is taken over the randomness of the policy. Hence, as long as \(S\) is a constant, the queues remain rate stable. To exploit the combinatorial structure of the problem, it is computationally advantageous to use the FTPL sub-routine (Abernethy et al., 2014; Theorem 11) as the base OCO policy in Algorithm 2, which can be implemented efficiently by a maximum-weight matching oracle and leads to the same queue length bounds.
## 8 Conclusion and open problems
In this paper, we proposed efficient algorithms for the Online Constraint Satisfaction and the Generalized OCO problems and established tight performance bounds for static regret and cumulative constraint violations. An exciting research direction would be to extend our methodologies to bound the dynamic regret. Investigating the generalized OCO problem with strongly convex constraints would be interesting. Furthermore, obtaining optimal performance bounds with the relaxed \(S\)-feasible benchmarks for the Generalized OCO problem would be nice. Finally, extending our work to the bandit feedback setting will be of interest (Sinha, 2023).
## 9 Acknowledgement
The work of AS is partially supported by a US-India NSF-DST collaborative grant coordinated by IDEAS-Technology Innovation Hub (TIH) at the Indian Statistical Institute, Kolkata. | ```japanese
標準オンライン凸最適化(OCO)の良好な研究結果に基づき、制約オンライン凸最適化(COCO)が定義されます。COCOでは、ラウンド毎に、そのラウンドの行動が選択された後、学習者に凸なコスト関数と凸な制約関数が明示されます。目的は、小規模な regret を達成しながら、T 長さのアルゴリズムを適用した時、累積的な制約違反 (CCV) を最小限に抑えるオンラインポリシーを設計することです。COCOにおいて、オンラインポリシーが $O(\sqrt{T})$ regret と $O(\sqrt{T})$ CCV を同時に達成できるかどうかという長年の課題です。この問題に、初めての回答として、オンラインポリシーが $O(\sqrt{T})$ regret と $\tilde{O}(\sqrt{T})$ CCV を同時に達成できることを示しました。さらに、強凸なコスト関数 |
2306.14651 | Nonequilibrium steady states in coupled asymmetric and symmetric
exclusion processes | We propose and study a one-dimensional (1D) model consisting of two lanes
with open boundaries. One of the lanes executes diffusive and the other lane
driven unidirectional or asymmetric exclusion dynamics, which are mutually
coupled through particle exchanges in the bulk. We elucidate the generic
nonuniform steady states in this model. We show that in a parameter regime,
where hopping along the TASEP lane, diffusion along the SEP lane and the
exchange of particles between the TASEP and SEP lanes compete, the SEP
diffusivity $D$ appears as a tuning parameter for both the SEP and TASEP
densities for a given exchange rate in the nonequilibrium steady states of this
model. Indeed, $D$ can be tuned to achieve phase coexistence in the asymmetric
exclusion dynamics together with spatially smoothly varying density in the
diffusive dynamics in the steady state. We obtain phase diagrams of the model
by using mean field theories, and corroborate and complement the results by
stochastic Monte Carlo simulations. This model reduces to an isolated open
totally asymmetric exclusion process (TASEP) and an open TASEP with bulk
particle nonconserving Langmuir kinetics (LK), respectively, in the limits of
vanishing and diverging particle diffusivity in the lane executing diffusive
dynamics. Thus this model works as an overarching general model, connecting
both pure TASEPs and TASEPs with LK in different asymptotic limits. We further
define phases in the SEP and obtain phase diagrams, and show their
correspondence with the TASEP phases. In addition to its significance as a 1D
driven, diffusive model, this model also serves as a simple reduced model for
cell biological transport by molecular motors undergoing diffusive and directed
motion inside eukaryotic cells. | Atri Goswami, Utsa Dey, Sudip Mukherjee | 2023-06-26T12:40:16 | http://arxiv.org/abs/2306.14651v2 | # Nonequilibrium steady states in coupled asymmetric and symmetric exclusion processes
###### Abstract
We propose and study a one-dimensional (1D) model consisting of two lanes with open boundaries. One of the lanes executes diffusive and the other lane driven unidirectional or asymmetric exclusion dynamics, which are mutually coupled through particle exchanges in the bulk. We elucidate the generic nonuniform steady states in this model. We show that the nonequilibrium steady states of this model can be controlled by the ratio of the diffusive and directed motion time-scales, which can be tuned to achieve phase coexistence in the asymmetric exclusion dynamics and spatially smoothly varying density in the diffusive dynamics in the steady state. We obtain phase diagrams of the model by using mean field theories, and corroborate and complement the results by stochastic Monte Carlo simulations. This model reduces to an isolated open totally asymmetric exclusion process (TASEP) and an open TASEP with bulk particle nonconserving Langmuir kinetics (LK), respectively, in the limits of vanishing and diverging particle diffusivity in the lane executing diffusive dynamics. Thus this model works as an overarching general model, connecting both pure TASEPs and TASEPs with LK in different asymptotic limits. We further define phases in the SEP and obtain phase diagrams, and show their correspondence with the TASEP phases. In addition to its significance as a 1D driven, diffusive model, this model also serves as a simple reduced model for cell biological transport by molecular motors undergoing diffusive and directed motion inside eukaryotic cells.
## I Introduction
Natural systems driven by an external field or containing collections of self-propelled particles form prominent examples of nonequilibrium systems that often evolve into stationary states carrying steady currents. The presence of steady currents distinguish these systems from their counterparts in thermal equilibrium. Understanding the general physical principles behind such nonequilibrium transport has been the subject of intense research recently. One is often particularly interested in nonequilibrium transport in the context of simple one-dimensional (1D) model systems. In order to elucidate the nature of such nonequilibrium steady states and in the absence of a general theoretical framework, it is useful to study purpose-built simple models. To this end, a variety of driven lattice gas models have been introduced and studied extensively [1].
The Totally Asymmetric Simple Exclusion Process (TASEP) with open boundaries is one of the simplest 1D nonequilibrium models, which displays boundary induced phase transitions. It was originally proposed by MacDonald _et al_[2] to model the transport of ribosomes along messenger RNA strands in cell biological context. Subsequently, it was reinvented as a paradigmatic nonequilibrium model [3]. TASEP consists of a one-dimensional (1D) lattice, along which particles can stochastically hop from left to right, at rate unity provided the target site is vacant. The interaction between the particles is through a hard core exclusion potential. The latter ensures only single occupancy per lattice site, which implies exclusion. In TASEP, particles can enter the system from the left boundary and exit through the right with certain prescribed entry (\(\alpha\)) and exit (\(\beta\)) rates. The phases of TASEP can be tuned by \(\alpha\leq 1\) and \(\beta\leq 1\), and are generically uniform in space, except for a special case when the two rates are equal and less than \(1/2\). Three distinct phases can be identified in the phase diagram of TASEP constructed in the space spanned by the control parameters \(\alpha\) and \(\beta\). These are the High density (HD) phase, Low density (LD) phase and the Maximal current (MC) phase. TASEP is one of the very few nonequilibrium models which can be exactly solved and has emerged as a simple basis to study the principles and phenomenologies of 1D transport [4; 5; 6; 7].
The Symmetric Exclusion Process (SEP) is a simple realisation of the 1D _equilibrium diffusion process_ in which particles can move, in contrast to TASEP, in either direction (left or right) symmetrically, subject to exclusion. Also unlike TASEP, the entry and exit of particles can occur at both the ends of the lattice. In the steady state, the spatial dependence of the density profile is always a straight line, either fully flat for equal biases or an inclined line in case of unequal biases [8].
More recently, TASEP has been generalised in a variety of ways, all of which reveals many new interesting macroscopic phenomena. These usually involve the presence of additional microscopic processes competing with the basic hopping process of TASEP, or presence of conservation laws. A prominent example is a model introduced in Ref. [9], which has competing 1D nonequilibrium transport (TASEP) and equilibrium on-off ex
changes with a surrounding reservoir (known as Langmuir kinetics (LK)). In LK, one studies the attachment-detachment kinetics of particles on a lattice coupled to a bulk reservoir. As a physical motivation, this provides the simplest description of binding and unbinding kinetics of enzymes to some substrate. In LK dynamics, the particles get adsorbed at an empty site or detached from an occupied one with some given rates. As in TASEP, the only interaction between the particles is the hard-core repulsion due to particle exclusion, leading to maximum occupancy one per site even in the presence of LK. The LK and TASEP are two of the simplest paradigmatic equilibrium and nonequilibrium models, which clearly contrast equilibrium and non-equilibrium dynamics, distinguishing the corresponding stationary states. For instance, LK maintains detailed balance, and evolves into an equilibrium steady state in the long time limit. In contrast, a TASEP naturally breaks the detailed balance condition due to continuous flow of particles, and the resulting stationary state is a non-equilibrium state that carries a finite current. Such non-equilibrium steady states are known to be quite sensitive to changes in the boundary conditions. In contrast, equilibrium steady states are very robust to such changes and dominated by the bulk dynamics. TASEP is a boundary condition dependent process - in the TASEP new particles can enter or leave the system only at the system boundaries, whereas in the bulk there are no sources or sinks. In contrast in LK particles can enter or leave the system at any site. As shown in Ref. [9], a combination of the two can produce nonuniform steady state densities in the TASEP when the typical _exchange rate_ of a particle moving along the TASEP lane is comparable with the entry-exit rates in the filament, which can be achieved by introducing system size-dependent LK exchange rates between the bulk and the TASEP lane. When the two rates are comparable, the resulting steady state density profiles can show coexistence phases and domain walls, in contrast to the density profiles in isolated TASEP and Langmuir kinetic processes.
Diffusive processes are ubiquitous in nature, e.g., in cell biological contexts. How diffusive and driven processes may combine to influence mass transport is a fundamentally important question in cell biology. Notable previous studies on this topic include the work on the coupled dynamics of diffusive (unbound) motors in the cell cytoplasm and motors driven along microtubules (bound motors) in tube-like cylindrical compartments (representing the cytoplasm), containing one filament along the axis (representing a microtubule) with motors being injected and extracted at the boundaries [10]. This model reveals a phase behavior similar to that of 1D TASEP. Later, an extension of the above model was studied in Ref. [11]. These are however three dimensional models, which are relatively difficult to analyse, either analytically or by computer simulations. Moreover, the competition between the time scales of diffusive and directed dynamics has also not been studied in these works. A 1D _closed_ model consisting of two equal segments with one segment executing driven dynamics and the other diffusive was studied in Ref. [12]. Interestingly, unlike an open TASEP, this model shows a single _localised domain wall_ (LDW) instead of a delocalised domain wall (DDW) found in an open TASEP. This is attributed to the overall particle number conservation in the model. Very recently, it was shown that in the limit of a large diffusive segment, an LDW in this model can get delocalised due to fluctuation effects [13].
Our motivation here is to systematically investigate the interplay between the diffusive, driven and particle-exchange time-scales in 1D, subject to exclusion. We do this by generalising 1D nonequilibrium transport by coupling TASEP with SEP via particle exchange that is reminiscent of LK. We also study effects of space-dependent exchanges on the steady states. We expect the steady states of a coupled SEP-TASEP model will be quite different from the features of the decoupled systems, i.e., of an isolated TASEP and an isolated SEP. As we shall see, for our coupled system in the steady state we find phase co-existences in TASEP and spatially non-uniform (but smooth) density profiles in SEP, depending upon relative time scales of SEP and TASEP dynamics. This is totally in contrast to the well-known spatially uniform densities in steady states of isolated TASEP and SEP. Although effects of combining driven and diffusive transport have been studied earlier [12; 14], a systematic understanding of the effects of the competition between different time scales is clearly lacking. Lastly, space dependence of the parameters which define the local dynamics are expected to play an important role in controlling the nature of the steady states of driven systems, see, e.g., Refs. [15; 16] and references therein for detailed studies on the steady states in periodic and open TASEPs with smoothly space-dependent hopping rates. Indeed, spatially smoothly varying rates of particle exchanges between the TASEP and SEP lanes naturally extend the studies reported in Refs. [15; 16].
The principal results in this work are as follows:
(i) For a finite diffusivity in the SEP channel and equal attachment-detachment rates between the TASEP and SEP channels, both the TASEP and SEP steady state density profiles acquire complex space dependence. At the location of a discontinuity in the TASEP density, the SEP density has a strong space dependence.
(ii) For a diverging SEP diffusivity, the TASEP density profiles are identical to those in the well-known model of an open TASEP with LK. In the same limit, the SEP density profiles become flat with a value of \(1/2\).
(iii) For a vanishing SEP diffusivity, the TASEP density profiles reduce to those of an open isolated TASEP, whereas the SEP density profiles in the bulk strictly follow the TASEP density profiles.
(iv) The TASEP and SEP phase diagrams are shown to have a strong correspondence with each other.
(v) As the SEP diffusivity is reduced, a domain wall in the TASEP channel gets gradually delocalised.
Apart from its significance in nonequilibrium statistical mechanics, our model in addition has a biological inspiration as well: it may be viewed as a simple model for interacting bound and unbound molecular motors inside eukaryotic cells. Molecular motors inside eukaryotic cells transport cargo and are responsible for almost all kinds of cellular motility [17]. These are inherently nonequilibrium processes, sustained by the energy released in the hydrolysis of ATP (Adenosine triphosphate), producing ADP (Adenosine diphosphate) and inorganic phosphate. The molecular motors typically hop unidirectionally along the microtubules in cells. Examples of such molecular motors are the processive motors belonging to the kinesin family. However, due to fluctuations of both thermal and non-thermal origin, in course of their unidirectional hopping motion, these molecular motors can stochastically detach off to the surrounding cytoplasm. In the cytoplasm, these molecular motors diffuse around until they again get themselves attached to a filament. The cytoplasm thus effectively acts as a reservoir for these (unbound or detached) molecular motors. On the whole, thus, the bound molecular motors hop unidirectionally along the filaments, whereas the unbound molecular motors in the cytoplasm undergo diffusive motion, and these two can stochastically exchange molecular motors between them. We here construct and study a simple one dimensional model that reproduces these collective behaviours of transport in a cell.
The rest of this article is organised as follows. In Sec. II, we introduce our model. Next, in Sec. III we set up the mean field theory for our model. Then in Sec. IV we briefly discuss the stochastic simulations performed. Next, in Sec. V we extensively present and analyse the steady state densities and the associated phase diagrams in both the TASEP and SEP channels. In this Section, we study the nonequilibrium steady states with constant attachment detachment rates, together with a rate of symmetric hopping or diffusion that remains finite relative to the attachment-detachment rates of the LK dynamics, or the unidirectional hopping rates along the TASEP lane. Results from both MFT and MCS studies are presented. In Sec. VI, we illustrate gradual delocalisation of the domain walls as the diffusivity is reduced. In Sec. VII, we summarise and discuss our results. In Appendix, we present our mean-field results in another limit of the model, _viz._, space-dependent attachment-detachment rates together with a diffusive time scale that diverges relative to the typical particle residence time due to the LK dynamics.
## II Model
In this work, we investigate the nature of the non-equilibrium steady states of a coupled two-lane model consisting of two equally sized lanes with \(L\) lattice sites each, whose dynamics is governed by a totally asymmetric exclusion and a symmetric exclusion processes, respectively; see Fig. 1. The sites in each lane are labelled by the index \(i=1,..,L\). The dynamical update rules of this model consist of the following steps.
(a) Particles on the top lane (TASEP) may enter at the left end (\(i=1\)) at rate \(q\alpha\), stochastically hop from \(i=1\) to \(L\) unidirectionally subject to exclusion at rate \(q\), and leave the lane at the right end (\(i=L\)) at rate \(q\beta\)[16]. (Note that in a conventional study on pure open TASEP, usually \(q\) is set to unity.)
(b) On the bottom lane (SEP) particles hop with equal rate \(D\) in either direction subject to exclusion, and also may leave or enter this lane at the right (\(i=L\)) and left (\(i=1\)) end at rate \(1/2\).
(c) Particles hopping on these parallel tracks may detach from one lane and attach to the other, with generally site-dependent but equal attachment and detachment rates \(\omega_{i}\).
All these hopping and exchange processes in both the lanes are allowed under the strict constraint of an _on-site exclusion principle_, which forbids double-occupancy of particles in any of the lattice sites.
Time scales:The different time scales for our model are mentioned below :
\(\bullet\)\(\tau_{\rm TASEP}=q\) : Time-scale of the directed dynamics of the particles on the TASEP lane. This sets the time-scale for our model.
\(\bullet\)\(\tau_{\rm SEP}=D\) : Time-scale of the diffusive dynamics of the particles on the SEP lane.
\(\bullet\)\(\tau_{i}^{\times}=\omega_{i}^{-1}\) : Time-scale of the lane exchange mechanism which couples the filament to the surrounding reservoir.
Symmetries:This model admits the _particle-hole symmetry_, which will prove helpful in constructing and understanding the phase diagrams for the filament and the reservoir lanes. We note that the absence of a particle at any site on the two lanes can be interpreted as the presence of a vacancy or a hole at that position. A particle hopping from the left site to the empty lattice site to its right in the bulk may be considered as a hole
Figure 1: Illustration of the two-lane model. We label the upper lane as the TASEP lane and the lower one as the SEP lane. Particles on the upper lane follow TASEP dynamics with hopping rate \(q\) in the bulk subject to exclusion and entry and exits rates \(q\alpha\) and \(q\beta\), respectively. Particles on lower lane obey SEP dynamics with hopping rate D and possess entry rates \(1/2\), and also exit rates \(1/2\), at the left and right end, respectively. The local space-dependent exchange rate between the lanes is denoted by \(\omega_{i}\) that can depend on the site \(i\).
hopping form the right to the left lattice site. Likewise, the entry of a particle from the left end of the lattice can be considered as an exit of a hole and vice-versa. Similarly for the particle exchange dynamics between the TASEP and SEP lanes, movement of a particle from (to) TASEP to (from) SEP may be viewed as a hole moving to (from) TASEP from (to) SEP. In fact, formally the model remains invariant under the transformation of all particles into holes, with a discrete transformation of sites \(i\leftrightarrow L-i\) and all pairs of the entry-exit rate, e.g. \(\alpha\leftrightarrow\beta\). These define the _particle-hole symmetry_ in this model. As a consequence of this symmetry, the phase diagram in the \(\alpha-\beta\) plane can be split into two complementary regions by the \(\alpha=\beta\) line. As a result, it suffices to understand the phase diagram for only one of the two regions. The phase behaviour of the system in the remaining region can be constructed and analysed by using the particle-hole symmetry.
## III Mean-field theory
The microscopic dynamics of TASEP is prescribed by the rate equations for every site in the SEP and TASEP lanes, as discussed in Sec. II above. These equations are _not_ closed. The MFT approximation entails neglecting the correlation effects and replacing the average of product of the densities by the product of average of densities in the steady states [18]. Although this is an uncontrolled approximation, this has been found to work with high degree of accuracy in the original TASEP problem and its many variants subsequently (see, e.g., Refs. [9; 19; 20] as representative examples); we use the MFT here as a guideline in our analysis below.
The dynamical equations of motion for \(\rho_{i}\) and \(c_{i}\), the TASEP and SEP densities at site \(i\) in the TASEP and SEP lane respectively, are
\[\partial_{t}\rho_{i} = q\rho_{i-1}(1-\rho_{i})-q\rho_{i}(1-\rho_{i+1}) \tag{1}\] \[+ \omega_{i}[c_{i}(1-\rho_{i})-\rho_{i}(1-c_{i})],\] \[\partial_{t}c_{i} = D[(c_{i-1}+c_{i+1})(1-c_{i})-c_{i}(2-c_{i-1}-c_{i+1})]\] (2) \[- \omega_{i}[c_{i}(1-\rho_{i})-\rho_{i}(1-c_{i})].\]
It is easy to see that the MFT equations (1) and (2) are invariant under the particle-hole symmetry defined above.
To proceed further, we first take the continuum approximation: we take \(L\to\infty\), which makes \(x\) a continuous variable between \(0\) and \(1\): \(x\in[0,1]\). Without any loss of generality, we assume unit geometric length for the whole lattice (both TASEP and SEP), and define a lattice constant \(\varepsilon=1/L\) that approaches zero as \(L\to\infty\). Thus, in the thermodynamic limit, \(\varepsilon\) serves as a small parameter. Further, we define \(\rho(x)=\langle\rho_{i}\rangle\) and \(c(x)=\langle c_{i}\rangle\) as the steady state densities of TASEP and SEP respectively at \(x\). In the steady state, we expand the different terms on rhs of (1) and (2) in a Taylor series in powers of \(\varepsilon\) to obtain
\[\frac{\partial\rho}{\partial t} = \omega(x)(c-\rho)+\frac{q}{L}(2\rho-1)\partial_{x}\rho+\frac{q}{2 L^{2}}\partial_{x}^{2}\rho, \tag{3}\] \[\frac{\partial c}{\partial t} = \omega(x)(\rho-c)+\frac{D}{L^{2}}\partial_{x}^{2}c. \tag{4}\]
Here, we have retained terms up to \(\mathcal{O}(\varepsilon^{2})\equiv\mathcal{O}(1/L^{2})\) in the Taylor expansions above, discarding all higher order terms. We note the different \(L\)-dependence of the terms in (3) and (4). In order to make the nonconserving Langmuir kinetics terms compete with the hopping terms in (3) and diffusion in (4), we define \(\omega\equiv\Omega/L^{2}\), where \(\Omega\sim\mathcal{O}(1)\)[9], and set \(q=1/L\). Thus with \(q=1/L\), particles enter and exist the TASEP channel at effective rates \(\alpha/L\) and \(\beta/L\), and hop along the TASEP channel at rate \(1/L\). With these parameter rescalings, we obtain the steady state in the thermodynamic dynamic limit \(L\to\infty\)
\[\Omega(x)(c-\rho)+(2\rho-1)\partial_{x}\rho=0, \tag{5}\] \[D\partial_{x}^{2}c+\Omega(x)(\rho-c)=0. \tag{6}\]
Equations (5) and (6) are the starting point of the MFT for this model, which we discuss next.
We note that the full MFT equations (5) and (6) have the following well-known limits. Indeed, there are two limits in which the MFT equations (5) and (6) reduce to two well-known models whose MFT solutions are already known. These two limits are characterised by the limiting values of \(\Omega/D\). Consider now \(\Omega(x)\) as constant, \(\Omega(x)=\Omega\) at all \(x\) together with
(i) \(D\to\infty\), i.e., \(\Omega/D\), for a given \(\Omega\sim\mathcal{O}(1)\), vanishes. In this limit from (6), assuming \(c(0)=1/2=c(1)\), \(c(x)=1/2\) everywhere. Substituting this in (5), we get
\[\frac{\Omega}{2}(1-2\rho)+(2\rho-1)\partial_{x}\rho=0. \tag{7}\]
This is identical to the MFT equation for \(\rho(x)\) in the LK-TASEP problem with an equal attachment-detachment rate of value \(\Omega/2\)[21]. Physically, as \(D\) diverges, the diffusive dynamics in the SEP lane becomes extremely fast, effectively reducing attachment events to the SEP lane insignificant relative to the in-lane diffusion, over TASEP hopping and attachment-detachment time-scales. This means the average steady state density in the SEP lane is \(1/2\), independent of the precise values of the attachment-detachment rates, as we found above. This in turn means that the attachment-detachment events at rate \(\Omega\) to/from the TASEP will take places with a background SEP density of \(1/2\), unaffected by the TASEP dynamics, as we see form (7).
(ii) \(D\to 0\), i.e., \(\Omega/D\) diverges for a given \(\Omega\sim\mathcal{O}(1)\). In this limit from (6), \(c(x)=\rho(x)\) is the solution in the bulk. Substituting this in (5), we get
\[(2\rho-1)\partial_{x}\rho=0, \tag{8}\]
which is nothing but the MFT equation for the steady state density in the standard TASEP problem, giving the
LD, HD and MC phases [18]. Actually for vanishingly small \(D\), the only dynamics in SEP are the attachment-detachment events, which have the effects of locally decreasing the differences in the densities in the TASEP and SEP lanes. Indeed, when \(D\to 0\), different sites of the SEP lane virtually decouple from each other, and only exchange particles with the corresponding site in the TASEP lane having a density \(\rho(x)\). In the steady state then \(c(x)=\rho(x)\) is achieved, no further time evolution of the SEP density.
We thus find that for a fixed \(\Omega\), a key parameter which controls the shape of the density profiles on both the TASEP and SEP lanes is the magnitude of the effective diffusion constant \(D\). If diffusion of the SEP lane is very slow, \(D\to 0\), we find from Eq. (6) that the density on that reservoir lane becomes effectively slaved to the density on the filament, \(c(x)=\rho(x)\). Hence, in this limit, the filament dynamics is independent of the reservoir and simply given by that of the TASEP. In contrast, in the opposite limit where \(D\) is large, i.e., diffusion on the reservoir lane is fast, \(c(x)\) becomes independent of \(\rho(x)\) and shows a flat profile, e.g. \(c(x)=1/2\) with \(c(x=0)=c(x=1)=1/2\), as the boundary conditions. In this case the reservoir lane simply acts as a reservoir with a constant particle density similar to the TASEP-LK model with an attachment rate \(\Omega_{A}=\Omega/2\) and a detachment rate \(\Omega_{D}=\Omega/2\), respectively [9]. Notice that independent of \(\Omega(x)\) and \(D\), \(\rho(x)=1/2=c(x)\) remain solutions of the MF equations (3) and (4).
As an aside, we also note that solving the full MFT equations (5) and (6) with space-dependent \(\Omega(x)\) and an arbitrary \(D\) is analytically challenging and also not particularly insightful. Instead, we solve (5) and (6) in the following cases: (i) \(\Omega/D\) finite but small with \(\Omega\) being independent of \(x\), (ii) \(\Omega/D\) finite but large with \(\Omega\) being independent of \(x\). We also briefly consider \(\Omega(x)\) to be space varying, but assume \(D\) diverges, \(\Omega/D\) vanishes at all \(x\). The MFT equation for \(\rho(x)\) now becomes
\[\Omega(x)(\frac{1}{2}-\rho)+(2\rho-1)\partial_{x}\rho=0. \tag{9}\]
This is the MFT equation for the LK-TASEP problem but with an equal, space varying attachment-detachment rate. This is discussed in Appendix.
## IV MCS simulations
The TASEP and SEP lanes of the model consist of \(L\) sites each, labelled by an index \(i\) with \(i\in[1,L]\). Let \(\rho_{i}(t)\), which is either 0 or 1, be the occupation at site \(i\) of the TASEP channel, and \(c_{i}(t)\), which is again either 0 or 1, be the occupation at site \(i\) of the SEP channel at time \(t\). We perform MCS studies of the model subject to the update rules (a)-(c) described above in Sec. II by using a random sequential updating scheme. The particles enter the system through the left most site (\(i=1\)) in the TASEP channel at a fixed rate \(q\alpha\), subject to exclusion, i.e., if \(\rho_{1}=0\). After hopping through the system from \(i=1\) to \(L\) at rate \(q\), subject to exclusion, the particles exit the system from \(i=L\) at a fixed rate \(q\beta\). Here, \(\alpha\) and \(\beta\) are the two simulation parameters, which are varied to produce different steady states. We have chosen \(q=1/L\) in our MCS studies. In the SEP channel, particles can enter at rate \(1/2\) if the site \(i=1\) or \(i=L\) of the SEP channel is vacant, or if it is filled, a particle either leaves the system through the left or right end respectively at rate \(1/2\), or hops to the site \(i=2\) or \(i=L-1\) at rate \(D\), if the target site is empty. In general, in the bulk of the SEP channel, a particle can hop to its left or right site with equal probability at rate \(D\), provided the target site is empty. We use \(D\leq 1\). Lastly, we allow exchange of particles between the SEP and TASEP channels at any site \(i\), subject to exclusion, at rate \(\omega\). After reaching the steady states, the density profiles are calculated and temporal averages are performed. This produces time-averaged, space-dependent density profiles, given by \(\langle\rho_{i}(t)\rangle\), and \(\langle c_{i}(t)\rangle\); here \(\langle...\rangle\) implies temporal averages over steady states. The simulations have been performed with \(L=1000\) up to \(10^{9}\) Monte-Carlo steps. Lastly, all the measurements are made in the steady states, which are reached by the system after spending certain transient times.
## V Steady state densities
In the previous Section, we have discussed that for a fixed \(\Omega\) the diffusion constant \(D\) determines the steady state density profiles in both the TASEP and SEP lanes. For intermediate values of \(D\) the density profiles on both lanes deviate from the known asymptotic results. With increasing the magnitude of \(D\) one can study the crossover from TASEP to TASEP-LK dynamics, and it will be interesting to see how the density profiles and the ensuing phase diagram change as \(D\) is varied. Before we proceed to solve the MFT equations, we note the following general result. The microscopic dynamical rules set up in Sec. II above clearly maintain overall particle conservation (i.e., in the combined TASEP and SEP) locally in the bulk of the system, since particle entry or exit events (considering the overall system) take place only at the boundaries, although individually the TASEP and SEP lanes do not conserve particles in the bulk locally due to the particle exchanges between them. This fact is clearly reflected in the MFT equations (1) and (2), or in their continuum analogues (3) and (4), which can be combined to show that the sum \(\rho(x,t)+c(x,t)\) is a conserved density. Indeed, the MFT equations (5) and (6) can be combined to produce a conservation law given by
\[(2\rho(x)-1)^{2}+4D\partial_{x}c=J=1-4j_{\rm tot} \tag{10}\]
where \(j_{\rm tot}\) is the total current through the SEP and TASEP channels combined (see also later). Equation (10) reveals the following: Since the steady state
TASEP density \(\rho(x)\) can have at most a finite discontinuity (e.g., at the location of a domain wall), so will \(\partial_{x}c\) at the same location to maintain (10). Now since \(\partial_{x}c\) can have at most a finite discontinuity, steady state SEP density \(c(x)\) must be continuous everywhere (but not necessarily spatially uniform). Nonetheless, at the location of a discontinuity in \(\rho(x)\), \(c(x)\) should have a strong space dependence, as opposed to where \(\rho(x)\) itself is continuous. We shall see below that our actual MFT solutions for \(\rho(x)\) and \(c(x)\) bear these features.
We solve the MFT equations (6) and (5) perturbatively, assuming \(\Omega/D\) to be large or small. Given the above discussions, interesting features are expected when \(\Omega/D\) is finite (can be large or small or just \(\mathcal{O}(1)\)). We then expect \(c(x)\) to be neither \(1/2\), nor equal to \(\rho(x)\) in bulk. Likewise, \(\rho(x)\) is expected to be neither one of the LK-TASEP solutions, or standard TASEP solutions in the bulk.
### MFT for large \(D\)
We first consider "large but finite \(D\)", i.e., small but non-zero \(\Omega/D\) for a given \(\Omega\sim\mathcal{O}(1)\). In this limit, we solve the MFT equations by perturbatively expanding around the limiting solutions \(\rho(x)=\rho_{\rm LK}(x)\) and \(c(x)=1/2\). For large but finite \(D\), we expect small modifications to \(\rho(x)=\rho_{\rm LK}(x)\) and \(c(x)=1/2\). We thus expect phases in the TASEP lane similar to those reported in Ref. [9] to emerge. Furthermore, the exchange of particles between the TASEP and SEP lanes should have the physical effects of _reducing_ locally the difference in the densities in the TASEP and SEP lanes. This means whenever \(\rho(x)>(<)1/2\), we expect \(c(x)>(<)1/2\). This in turn suggests that the steady state density in the SEP lane should be excess (deficit) relative to \(1/2\), the steady state density of an isolated SEP with equal entry and exit rates. This picture is consistent with the form of the MF equation (6) with \(\Omega(x)\) being assumed to be a constant. Since \(\partial_{x}^{2}c(x)\) that gives the local curvature of \(c(x)\) is less than zero for \(\rho(x)>c(x)\), expected in the HD phase, \(c(x)\) should resemble, loosely speaking, an inverted "U", whereas for \(\rho(x)<c(x)\), expected in the LD phase, \(c(x)\) should resemble, loosely speaking, an upward "U". These considerations suggest that the SEP channel can have an average density more, less or equal to \(1/2\). We call these _excess, deficit_ and _neutral_ phases of SEP. We will see below that these expectations are borne by our MFT solutions.
To proceed with our MFT solutions valid for \(\Omega/D\ll 1\), we write
\[\rho(x) =\rho_{\rm LK}(x)+\delta\rho(x), \tag{11}\] \[c(x) =\frac{1}{2}+\delta c(x). \tag{12}\]
Here, \(\rho_{\rm LK}(x)\) is the well-known solution of the LK-TASEP problem and satisfies
\[(2\rho_{\rm LK}-1)(\partial_{x}\rho_{\rm LK}-\frac{\Omega}{2})=0, \tag{13}\]
giving
\[\rho_{\rm LK}(x)=\frac{1}{2}\ \ \ {\rm or}\ \ \ \rho_{\rm LK}(x)=\frac{\Omega}{2}x+ \rho_{0}. \tag{14}\]
Here, \(\rho_{0}\) is a constant of integration, which may be evaluated by using the boundary conditions. We set
\[\rho(0) =\alpha=\rho_{\rm LK}(0), \tag{15}\] \[\rho(1) =1-\beta=\rho_{\rm LK}(1). \tag{16}\]
Furthermore, \(\delta\rho(x)\) and \(\delta c(x)\) are assumed to be "small" deviations from \(\rho_{\rm LK}(x)\) and \(c(x)=1/2\). In particular, \(\delta c(x)\) satisfies
\[\partial_{x}^{2}\delta c(x)+\frac{\Omega}{D}\left[\rho_{\rm LK}+\delta\rho- \frac{1}{2}-\delta c\right]=0. \tag{17}\]
We know that in the limit \(\Omega/D\to 0\), \(c(x)\to 1/2\) and hence \(\delta c(x)\to 0\). We can thus write
\[\delta c(x)=f(\frac{\Omega}{D}), \tag{18}\]
with \(f(0)=0\). This suggests that to the lowest order in \(\Omega/D\), \(\delta c(x)\) should scale with \(\Omega/D\). This further implies that \(\delta\rho(x)\), which vanishes as \(\delta c\to 0\), should also scale with \(\Omega/D\) to the lowest order in \(\Omega/D\). Therefore, to the lowest order in \(\Omega/D\), \(\delta c(x)\) follows
\[\partial_{x}^{2}\delta c(x)+\frac{\Omega}{D}[\rho_{\rm LK}(x)-\frac{1}{2}]=0, \tag{19}\]
where \(\rho_{\rm LK}(x)\) is given by (14). Since \(c(x)=1/2\) at \(x=0,1\), we must have \(\delta c(x)=0\) at \(x=0,1\). If we choose \(\rho_{\rm LK}(x)=1/2\), we get \(\delta c(x)=0\) trivially, giving \(c(x)=1/2\) in the bulk. This is not surprising, since \(\rho(x)=1/2=c(x)\) is a solution in the bulk. Non-trivial solution for \(\delta c(x)\) is obtained if we set \(\rho_{\rm LK}(x)=(\Omega/2)\,x+\rho_{0}\). Substituting this in (19) and integrating with respect to \(x\) twice, we obtain
\[\delta c(x)=-\frac{\Omega}{D}\!\left[\frac{\Omega x^{3}}{12}+(\rho_{0}-\frac{ 1}{2})\frac{x^{2}}{2}\right]+\overline{c}_{1}x+\overline{c}_{2}. \tag{20}\]
Constants \(\overline{c}_{1},\,\overline{c}_{2}\) are the two constants of integration, which may be evaluated by using the boundary conditions. At \(x=0\), \(c=1/2\) giving \(\delta c(0)=0\). This gives \(\overline{c}_{2}=0\). We further have at \(x=1\), \(c=1/2\), giving \(\delta c(1)=0\). From this condition we obtain
\[\overline{c}_{1}=\frac{\Omega}{D}\left[\frac{\Omega}{12}+\frac{1}{2}(\rho_{0}- 1/2)\right] \tag{21}\]
giving
\[\delta c(x)=\frac{\Omega}{D}\left[\frac{\Omega}{12}(x-x^{3})+\frac{1}{2}(\rho_ {0}-\frac{1}{2})(x-x^{2})\right]. \tag{22}\]
Notice that \(\delta c(x)\) and hence \(c(x)\) depend explicitly on the boundary conditions on \(\rho(x)\) through \(\rho_{0}\). The full solution of the steady state SEP density \(c(x)\) is given by
\[c(x)=\frac{1}{2}+\delta c(x) \tag{23}\] \[=\frac{1}{2}+\frac{\Omega}{D}\left[\frac{\Omega}{12}(x-x^{3})+ \frac{1}{2}(\rho_{0}-\frac{1}{2})(x-x^{2})\right].\]
Clearly, \(c(x)=1/2\) at \(x=0,1\). Since \(\rho_{0}\), being the boundary condition on \(\rho(x)\) either at the left or at the right end, depending on whether we are considering LD or HD phases of the TASEP, for each of \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\), the steady state density profiles in the LD and HD phases, respectively, there are distinct solutions of \(c(x)\); see below.
We now solve for \(\rho(x)\). We start from Eq. (5), which may be written as
\[(2\rho(x)-1)[\partial_{x}\rho(x)-\hat{\Omega}]=-\Omega c(x)+\frac{\Omega}{2}, \tag{24}\]
where \(\hat{\Omega}\equiv\Omega/2\). Now write \(\rho(x)=\rho_{\rm LK}(x)+\delta\rho(x)\), where \(\rho_{\rm LK}(x)\) satisfies (14). Then \(\delta\rho(x)\) satisfies the following equation.
\[(2\rho_{\rm LK}-1)\partial_{x}\delta\rho=-\Omega\frac{\Omega}{D}\left[\frac{ \Omega}{12}(x-x^{3})+\frac{1}{2}(\rho_{0}-\frac{1}{2})(x-x^{2})\right] \tag{25}\]
to the lowest order in \(\Omega/D\). Equation (25) can be solved by standard methods, which are straight forward but lengthy. We give the solution below.
\[\delta\rho(x) = -\Omega\frac{\Omega}{D}\left[\frac{k_{1}x^{3}}{3}+\frac{k_{2}x^{2 }}{2}+k_{3}x\right] \tag{26}\] \[+ k^{\prime}\frac{\Omega}{D}\ln|x+\frac{2\rho_{0}-1}{\Omega}|+k_ {0}.\]
Clearly, \(\delta\rho(x)\) depends linearly on \(\Omega/D\), and vanishes, as it should, when \(\Omega/D\) vanishes. Here, \(k_{1},k_{2},k_{3}\) are constants given by
\[k_{1}=-\frac{1}{12}, \tag{27}\] \[k_{2}=-\frac{2\rho_{0}-1}{6\Omega},\] (28) \[k_{3}=\frac{1}{\Omega}\left[\frac{\Omega}{12}+\frac{(2\rho_{0}- 1)}{4}+\frac{(2\rho_{0}-1)^{2}}{6\Omega}\right],\] (29) \[k^{\prime}=(2\rho_{0}-1)k_{3}. \tag{30}\]
Unsurprisingly, \(\delta\rho(x)\) depends on \(\rho_{0}\), which in turn is fixed by the boundary condition on \(\rho_{\rm LK}(x)\). We first focus on the LD phase. The constant of integration \(k_{0}\) can be obtained by using the boundary conditions \(\delta\rho(x)=0\) at \(x=0\). Now, using the boundary condition at \(x=0\), \(\rho(0)=\alpha=\rho_{\rm LK}(0)\) (which means \(\delta\rho(x)=0\) at \(x=0\), as we have used), we get \(\rho_{0}=\alpha\). Then using (11) we obtain
\[\rho_{\rm LD}(x)=\alpha+\frac{\Omega x}{2}-\Omega\frac{\Omega}{D}[ \frac{k_{1}x^{3}}{3}+\frac{k_{2}x^{2}}{2}+k_{3}x] \tag{31}\] \[+ \frac{\Omega k^{\prime}}{D}\ln|x+\frac{2\alpha-1}{\Omega}|-\frac {\Omega k^{\prime}}{D}\ln|\frac{2\alpha-1}{\Omega}|.\]
As discussed above, corresponding to \(\rho_{\rm LD}(x)\) as given in (31), the SEP density is given by \(c_{-}(x)\), where
\[c_{-}(x)=\frac{1}{2}+\frac{\Omega}{D}\left[\frac{\Omega}{12}(x-x^{3})+\frac{1 }{2}(\alpha-\frac{1}{2})(x-x^{2})\right]. \tag{32}\]
Likewise, we can obtain \(\rho_{\rm HD}(x)\) by using the boundary condition at \(x=1\), \(\rho(1)=1-\beta=\rho_{\rm LK}(1),\,\delta\rho(1)=0\). We get
\[\rho_{\rm HD}(x)=1-\beta+\frac{\Omega}{2}(x-1) \tag{33}\] \[- \Omega\frac{\Omega}{D}[\frac{k_{1}}{3}(x^{3}-1)+\frac{k_{2}}{2}( x^{2}-1)+k_{3}(x-1)]\] \[+ \frac{\Omega k^{\prime}}{D}\ln|x-1+\frac{1-2\beta}{\Omega}|-\frac {\Omega k^{\prime}}{D}\ln|\frac{1-2\beta}{\Omega}|.\]
Then corresponding to \(\rho_{\rm HD}(x)\) as given in (33), the SEP density is \(c_{+}(x)\) given by
\[c_{+}(x)=\frac{1}{2}+\frac{\Omega}{D}\left[\frac{\Omega}{12}(x-x^{3})+\frac{1 }{2}(1-\beta-\frac{\Omega}{2}-\frac{1}{2})(x-x^{2})\right]. \tag{34}\]
Notice that in addition to the explicitly \(x\)-dependent solutions \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\) above, the MFT equations (5) and (6) also admit spatially uniform solutions \(\rho=1/2\), \(c=1/2\); \(\rho=1/2\) obviously corresponds to the MC phase of the TASEP. With these solutions for the steady state densities \(\rho(x)\) and \(c(x)\) phase diagrams for both the TASEP and SEP lanes can be constructed in the \(\alpha-\beta\) plane. Since for large \(D\), i.e., for small \(\Omega/D\), we only expect small modifications of \(\rho(x)\) from \(\rho_{\rm LK}(x)\) and of \(c(x)\) from \(1/2\) in the bulk, we expect the TASEP phase diagram to be close to the one obtained in the LK-TASEP problem [9], albeit with an equal attachment-detachment rate \(\Omega^{\prime}\equiv\Omega/2\).
In our MCS studies with \(D=1.0\) and \(\Omega=0.3\) (\(\Omega/D=0.3<1\)), we find the so-called "pure phases" of TASEP (albeit generally space-dependent), _viz._, LD, HD and MC phases and also detect the "mixed phases", e.g., LD-MC, HD-MC, LD-MC-HD and LD-HD phases. In these mixed phases, part of \(\rho(x)\) is in one of the phases, and the remaining part is in another phase. The SEP density profiles may be characterised as follows. We define an average SEP density \(\overline{c}\) via
\[\overline{c}\equiv\int_{0}^{1}c(x)dx. \tag{35}\]
Clearly, \(\overline{c}>,<\) or \(=1/2\) would imply excess, deficit and neutral phases. Furthermore, as we have discussed above, \(c(x)\) tends to follow \(\rho(x)\) in the bulk, although partially for non-zero \(\Omega/D\). This implies, as our MCS studies on the SEP density profile reveal, \(c(x)-1/2\) can cross zero in the bulk either once, or none at all, or remain zero over a finite extent of \(x\).
We present an MFT phase diagram in Fig. 2 of the TASEP lane and an MFT phase diagram in Fig. 3 of the SEP lane for various values of \(D=1.0\) and \(\Omega=0.3\). We
discuss how to obtain the phases and the corresponding phase boundaries between the phases in the MFT. Notice that both \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\) are growing solutions of \(x\). However, unlike in Ref. [9] they do not grow linearly with \(x\); there are (small) nonlinear modifications to the linear growth profiles, which are due to the (large but) finite diffusivity in the SEP channel. Although there is no strict particle conservation in the bulk of the TASEP lane due to the attachment-detachment events, particle current is conserved _locally_, as locally the rate of particle nonconserving attachment-detachment events actually vanish in the thermodynamic limit [9]. As in the LK-TASEP model, steady state current here in the TASEP lane is space-dependent but continuous. Corresponding to the steady state densities \(\rho_{\rm LD}(x),\rho_{\rm HD}(x)\) and \(1/2\), we define currents
\[j_{\rm LD}(x) = \rho_{\rm LD}(x)(1-\rho_{\rm LD}(x)), \tag{36}\] \[j_{\rm HD}(x) = \rho_{\rm HD}(x)(1-\rho_{\rm HD}(x)),\] (37) \[j_{\rm MC}(x) = \frac{1}{4}. \tag{38}\]
Using the above expressions of the currents and their continuity across various phase boundaries [9], we determine the location of the phase boundaries in the \(\alpha-\beta\) plane. We set \(j_{\rm LD}(x_{\alpha})=1/4\), equivalently \(\rho_{\rm LD}(x_{\alpha})=1/2\), where \(x_{\alpha}\) is the coordinate separating the LD phase from the MC phase, i.e., \(\rho(x<x_{\alpha})=\rho_{\rm LD}(x<x_{\alpha})<1/2\). Similarly, we set \(j_{\rm HD}(x_{\beta})=1/4\), equivalently \(\rho_{\rm HD}(x_{\beta})=1/2\), where \(x_{\beta}\) is the coordinate separating the HD phase from the MC phase, i.e., \(\rho(x>x_{\beta})=\rho_{\rm HD}(x>x_{\beta})>1/2\). Depending upon the relative positions of \(x_{\alpha}\) and \(x_{\beta}\), various different density profiles emerge that we list briefly. (i) \(x_{\beta}>x_{\alpha}\geq 1\) means the LD phase, (ii) \(x_{\beta}>1\), \(0<x_{\alpha}<1\) means the mixed LD-MC phase, with \(\rho(x)<1/2\) for \(0\leq x<x_{\alpha}\) and \(\rho(x)=1/2\) for \(x_{\alpha}<x<1\), (iii) \(0<x_{\alpha}<x_{\beta}<1\) gives a three-phase coexistence with \(\rho(x)<1/2\) for \(0\leq x<x_{\alpha}\), \(\rho(x)=1/2\) for \(x_{\alpha}<x<x_{\beta}\) and \(\rho(x)>1/2\) for \(x_{\beta}<x<1\). Further, \(x_{\alpha}<0\), \(0<x_{\beta}<1\) gives the HD-MC phase. It is also possible to have \(x_{\alpha}>x_{\beta}\), whence one actually has the LD-HD phase with a domain wall at \(x_{w}\).The position \(x_{w}\) may be obtained from the condition \(\rho_{\rm LD}(x_{w})+\rho_{\rm HD}(x_{w})=1\). Since this condition gives a unique solution for \(x_{w}\), the domain wall in question is a _localised domain wall_ or LDW located at \(x_{w}\) with \(0<x_{w}<1\).
The various phase boundaries in the \(\alpha-\beta\) plane may be obtained in terms of the conditions on the densities as follows: setting (i) \(\rho_{\rm LD}(x_{\alpha}=1)=1/2\) gives the phase boundary between the LD and LD-MC phases, (ii) \(\rho_{\rm HD}(x_{\beta}=0)=1/2\), gives the phase boundary between the HD and HD-MC phases, (iii) \(\rho_{\rm LD}(x_{\alpha}=0)=1/2\) gives the phase boundary between the LD-MC and MC phases, (iv) \(\rho_{\rm HD}(x_{\beta}=1)=1/2\) gives the phase boundary between the HD-MC and MC phases, (v) \(\rho_{\rm LD}(x_{w}=1)+\rho_{\rm HD}(x_{w}=1)=1\) gives the boundary between the LD and LD-HD phases (since this condition means that the domain wall is just at the right boundary \(x=1\)), (vi) \(\rho_{\rm LD}(x_{w}=0)+\rho_{\rm HD}(x_{w}=0)=1\) gives the boundary between the LD-HD and HD phases (since this condition means that the domain wall is just at the left boundary \(x=0\)), (vii) \(\rho_{\rm LD}(x_{w}=x_{\alpha}=x_{\beta})+\rho_{\rm HD}(x_{w}=x_{\alpha}=x_{ \beta})=1\) together with \(\rho_{\rm LD}(x_{\alpha}=x_{\beta})=\rho_{\rm HD}(x_{\alpha}=x_{\beta})=1/2\) gives the boundary between the LD-HD and LD-HD-MC phases. These two conditions ensure that on the associated phase boundary, the size of the MC phase given by \(x_{\beta}-x_{\alpha}\) just vanishes, indicating the threshold of the three-phase coexistence. The above-listed conditions are formally equivalent to solving for \(x_{\alpha},x_{\beta},x_{w}\) and set conditions on them, as done in Ref. [9]. However, in Ref. [9] due to the linear dependence of the solutions of \(\rho(x)\) on \(x\), it was possible to explicitly solve for \(x_{\alpha},x_{\beta},x_{w}\). The far more complex \(x\)-dependence of \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\) rules out explicitly solving for \(x_{\alpha},x_{\beta},x_{w}\). Instead, we obtain the phase boundaries by means of drawing contour plots in the \(\alpha\)-\(\beta\) plane for given values of \(D\) and \(\Omega\) corresponding to the conditions on the TASEP densities listed above; see the phase diagram in Fig. 2. Notice that notwithstanding the far more complex \(x\)-dependence of \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\), the phase diagram in Fig. 2 have straight lines parallel to either the \(\alpha\)- or \(\beta\)-axis as the phase boundaries between the LD and LD-MC phases, LD-MC and MC phases, HD and HD-MC phases, and MC and HD-MC phases. This is because \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\) are independent, respectively, of \(\beta\) and \(\alpha\). This explains these phase boundaries. It is also clear that the phase diagram is invariant under the particle-hole symmetry. This may be seen by exchanging the \(\alpha\) and \(\beta\)-axes and redrawing the phase boundaries. The resulting phase diagram is identical to that in Fig. 2.
An MFT phase diagram of the SEP lane for \(\Omega=0.3,D=1\) is shown in Fig. 3. Phase space regions with excess, deficit and neutral particle numbers in SEP are shown, which are characterised by the mean SEP density \(\overline{c}\) [see Eq. (35) above]. Due to the tendency of the SEP density \(c(x)\) to follow the TASEP density \(\rho(x)\) in the steady state, the pure LD, HD and MC phases in
Figure 2: Mean-field TASEP phase diagram in the \(\alpha-\beta\) plane with \(D=1\), \(\Omega=0.3\).
the TASEP lane correspond to deficit, excess and neutral particles. Furthermore, the quantity \(c(x)-1/2\) is in the bulk in some of the phase space regions of the SEP which correspond to the pure LD and HD phases in the TASEP lane. There are however regions in the SEP phase diagram, corresponding to the LD-HD phases in the TASEP lane, where \(c(x)-1/2\) crosses zero once in the bulk. In the remaining regions of the SEP phase diagram, \(c(x)-1/2\) does not cross zero, but remains at zero in the whole or part of the bulk. These regions correspond to the pure MC phase, or mixed LD-MC or HD-MC-HD phases in the TASEP lane. The average steady state SEP density when the TASEP in its LD-MC (HD-MC) phase is less (more) than \(1/2\), implying that the SEP is in its deficit (excess) particle phase. When the TASEP is in its LD-HD phase with an LDW at \(x=x_{w}\), the quantity \(c(x)-1/2<0\) for \(0\leq x<x_{w}\) and \(c(x)-1/2>0\) for \(x_{w}<x\leq 1\). When \(x_{w}=1/2\), the LDW is located at the mid-point of the TASEP channel, when happens on the line \(\alpha=\beta\). Specifically on this line, \(c(x)-1/2<0\) for \(0\leq x<1/2\) and \(c(x)-1/2>0\) for \(1/2<x\leq 1\), giving \(\alpha=\beta\) as the phase boundary between the deficit and excess particle phases of the SEP, when the TASEP is in its LD-HD phase. When the TASEP is in its LD-MC-HD phase, \(c(x)-1/2<0\) for \(0\leq x<x_{\alpha}\), \(c(x)-1/2=0\) for \(x_{\alpha}<x<x_{\beta}\) and \(c(x)-1/2>0\) for \(x_{\beta}<x\leq 1\). Furthermore, symmetry of the model about the line \(\alpha=\beta\) (which has its origin in the particle-hole symmetry of the model; see discussions above) ensures that \(\alpha=\beta\) continues to be the boundary between the deficit and excess particle phases of TASEP. This line terminates at the multicritical point \(\alpha=1/2=\beta\), whence it meets the neutral phase. These discussions suggest that \({\cal O}_{c}\equiv\overline{c}-1/2\) may be used as an order parameter to delineate and distinguish the different phases in the SEP. Indeed, in the neutral phase \({\cal O}_{c}=0\), in the deficit phase \({\cal O}_{c}<0\) and in the excess phase \({\cal O}_{c}>0\). All the three SEP phase boundaries are second order in nature, which meet at the multicritical point (1/2,1/2).
We verify the accuracy of the above MFT by comparing with the numerical results on the steady state density profiles obtained from our extensive MCS studies with \(D=1\), \(\Omega=0.3\). See Fig. 4 for representative plots of \(\rho(x)\) as a function of \(x\) in different phases of the TASEP with \(D=1\), \(\Omega=0.3\). Analytical MFT and numerical MCS results are superposed. For MFT, we have used \(\rho(x)=\rho_{\rm LD}(x)\) [Eq. (31)] in the LD phase of the TASEP, \(\rho(x)=\rho_{\rm HD}(x)\) [Eq. (33)] in the HD phase, and \(\rho(x)=1/2\) in the MC phase. We have presented our results on the corresponding SEP density profile \(c(x)\) in Fig. 5. Both analytical MFT and numerical MCS results are shown. Reasonably good agreements are found between the MFT and MCS results. For MFT solution of the SEP density, we have used \(c(x)=c_{-}(x)\) [Eq. (32)] for \(c(x)<1/2\), corresponding to the TASEP in its LD phase, and \(c(x)=c_{+}(x)\) [Eq. (34)] for \(c(x)>1/2\), corresponding to the TASEP in its HD phase. Notice that the quantitative agreement of the MFT solutions for the SEP density \(c(x)\) with the corresponding MCS results is good when the TASEP is in its LD or HD phases, but less so when the TASEP in its LD-HD phase; see Fig. 5. We believe this is due to the fact that near the location of an LDW in TASEP, \(c(x)\) has a strong space dependence, suggesting importance of retaining the higher order corrections in the MFT solutions of \(c(x)\). Nevertheless, qualitative agreement between the MFT and MCS solutions for \(c(x)\) can be seen even in this case.
### MFT for small D
We now consider the solutions of the MFT equations (5) and (6) for small values of \(D\) with a fixed \(\Omega\), i.e., \(\Omega/D\gg 1\). As discussed above, for \(\Omega/D\to\infty\), \(\rho(x)\) reduces to \(\rho_{T}(x)\) and \(c(x)=\rho(x)\) in the bulk, where \(\rho_{T}\) is the bulk steady state density in an open isolated TASEP. This solution for \(c\) however does not match with the boundary condition except when \(\rho=1/2=\rho_{\rm MC}\). To impose the boundary conditions, for all other steady state solutions for \(\rho(x)\), there are _two_ boundary layers close to the two boundaries at \(x=0,1\), ensuring \(c(0)=1/2=c(1)\). These boundary layers are analogous to the boundary layers observed in the MCS studies on the steady state density profiles in an open TASEP. For \(\Omega/D\) large but finite, we expect small modifications to this picture.
To find the steady state densities in both the SEP and TASEP channels, we proceed as follows. We have already noted that the exchange of particles (subject to exclusion) between the TASEP and SEP channels maintains the overall particle number conservation in the combined bulk of the two lanes. This gives rise to the conservation law (10) above, which is a quadratic equation in \(\rho(x)\) in terms of \(J\) and other quantities. Now by using (10) and assuming small \(D\), we write the two solutions of \(\rho(x)\) in
terms of \(J\) and \(c(x)\) as
\[\rho_{\pm}(x) = \frac{1}{2}\left[1\pm\sqrt{J-4D\partial_{x}c(x)}\right] \tag{39}\] \[\approx \frac{1}{2}(1\pm\sqrt{J})\mp\frac{D}{\sqrt{J}}\partial_{x}c(x).\]
Clearly, \(\rho_{\pm}(x)>1/2\) and \(\rho_{-}(x)<1/2\) in the bulk of the TASEP. We now use (39) to eliminate \(\rho\) in (6) to obtain a single closed equation for \(c(x)\):
\[D\partial_{x}^{2}c-A_{\pm}\partial_{x}c-\Omega[c(x)-B_{\pm}]=0, \tag{40}\]
where using \(\rho_{\pm}(\rho_{-})\) for \(\rho\) gives (40) with
Figure 4: Steady state density \(\rho(x)\) in the TASEP lane in the LD (top), LD-HD (middle) and LD-MC (bottom) phases with \(D=1,\,\Omega=0.3\). MFT (blue line) and MCS (red points) results are shown. For MFT, we have used \(\rho(x)=\rho_{\rm LD}(x)\) [Eq. (31)] in the LD phase of the TASEP, \(\rho(x)=\rho_{\rm HD}(x)\) [Eq. (33)] in the HD phase and \(\rho(x)=1/2\) in the MC phase (see text).
Figure 5: Steady state density \(c(x)\) in the SEP lane, when the TASEP lane is in its LD (top), HD (middle) and LD-HD (bottom) phases with \(D=1,\,\Omega=0.3\). MFT (blue line) and MCS (red points) results are shown. For MFT, we have used \(c(x)=c_{-}(x)\) [Eq. (32)] for \(c(x)<1/2\), corresponding to the TASEP in its LD phase, and \(c(x)=c_{+}(x)\) [Eq. (34)] for \(c(x)>1/2\), corresponding to the TASEP in its HD phase (see text).
\(A_{+}\), \(B_{+}\left(A_{-},\,B_{-}\right)\), which depend on \(J\). Here,
\[A_{\pm}=\pm\frac{\Omega D}{\sqrt{J}_{\pm}},\,B_{\pm}=\frac{1}{2}(1\pm\sqrt{J}_{ \pm}). \tag{41}\]
What is \(J\) here? We note that in the limit of \(D\to 0\), \(\rho(x)\rightarrow\rho_{T}\), where \(\rho_{T}\) is the MFT solution of the steady state density in an open TASEP. Thus in that limit, \(J=J_{-}=(2\alpha-1)^{2}\), if \(\rho_{T}=\rho_{\rm LD}\), whereas \(J=J_{+}=(2\beta-1)^{2}\) if \(\rho_{T}=\rho_{\rm HD}=1-\beta\). These considerations give, as expected,
\[\rho_{-}(x)=\alpha=\rho_{\rm LD}, \tag{42}\] \[\rho_{+}(x)=1-\beta=\rho_{\rm HD}, \tag{43}\]
when \(D\to 0\), coinciding with the MFT solutions for an open TASEP.
Solving (40) we get two solutions for \(c(x)\):
\[c_{\pm}(x)=B_{\pm}+U_{\rm I}^{\pm}\exp(\lambda_{1}^{\pm}x)+U_{2}^{\pm}\exp( \lambda_{2}^{\pm}x), \tag{44}\]
where
\[\lambda_{1}^{+} = \frac{1}{2D}\bigg{[}A_{+}+\sqrt{A_{+}^{2}+4D\Omega}\bigg{]}, \tag{45}\] \[\lambda_{2}^{+} = \frac{1}{2D}\bigg{[}A_{+}-\sqrt{A_{+}^{2}+4D\Omega}\bigg{]} \tag{46}\]
corresponding to \(\rho=\rho_{+}\), and
\[\lambda_{1}^{-} = \frac{1}{2D}\bigg{[}A_{-}+\sqrt{A_{-}^{2}+4D\Omega}\bigg{]}, \tag{47}\] \[\lambda_{2}^{-} = \frac{1}{2D}\bigg{[}A_{-}-\sqrt{A_{-}^{2}+4D\Omega}\bigg{]} \tag{48}\]
for \(\rho=\rho_{-}\). Here, \(U_{\rm I}^{\pm},U_{2}^{\pm}\) are two sets of constants of integration, to be evaluated by using the two boundary conditions on \(c\), _viz._, \(c(0)=1/2=c(1)\). As for \(A_{\pm}\), \(B_{\pm}\), use of \(\rho_{+}(x)\left(\rho_{-}(x)\right)\) as the solution for \(\rho(x)\) correspond to the set \(U_{\rm I}^{+}\), \(U_{\rm I}^{+}\) (\(U_{\rm I}^{-}\), \(U_{\rm I}^{-}\)). We find
\[U_{1}^{\pm} = \frac{1-\exp(\lambda_{2}^{\pm})}{\exp(\lambda_{1}^{\pm})-\exp( \lambda_{2}^{\pm})}\bigg{(}\frac{1}{2}-B_{\pm}\bigg{)}, \tag{49}\] \[U_{2}^{\pm} = \frac{\exp(\lambda_{1}^{\pm})-1}{\exp(\lambda_{1}^{\pm})-\exp( \lambda_{2}^{\pm})}\bigg{(}\frac{1}{2}-B_{\pm}\bigg{)}. \tag{50}\]
Evaluation of the constants allow us to find \(c_{-}(x)\) and \(c_{+}(x)\), which in turn yield \(\rho_{\rm LD}(x)\) and \(\rho_{\rm HD}(x)\) respectively.
For finite but small \(D\), we expect weak space dependence of \(\rho_{-}(x)\) and \(\rho_{+}(x)\), i.e., weakly deviating from the constant solutions of \(\alpha\) and \(1-\beta\) respectively in the bulk. For a finite but small \(D\), identifying \(\rho_{-}(x)<1/2\) with \(\rho_{\rm LD}(x)\) and \(\rho_{+}(x)>1/2\) with \(\rho_{\rm HD}(x)\), we find
\[\rho_{\rm LD}(x) = \frac{1}{2}\bigg{[}1-\sqrt{J_{\rm LD}-4D\partial_{x}c_{-}(x)} \bigg{]}<\frac{1}{2}, \tag{51}\] \[\rho_{\rm HD}(x) = \frac{1}{2}\bigg{[}1+\sqrt{J_{\rm HD}-4D\partial_{x}c_{+}(x)} \bigg{]}>\frac{1}{2} \tag{52}\]
for small \(D\). In general, \(J_{\rm LD}\) and \(J_{\rm HD}\) should now include current contributions from the SEP channel; see Eq. (10) above. When the TASEP lane is in its LD (HD) phase, its bulk solution and the associated current is controlled by the left (right) boundary condition. We then identify
\[J_{\rm LD} = (2\alpha-1)^{2}+4D\partial_{x}c_{-}(x)|_{x=0}, \tag{53}\] \[J_{\rm HD} = (2\beta-1)^{2}+4D\partial_{x}c_{+}(x)|_{x=1}. \tag{54}\]
Here, \(c_{-}(x)\) and \(c_{+}(x)\) are the two solutions of (40), obtained by using \(\rho(x)=\rho_{-}(x)\) and \(\rho(x)=\rho_{+}(x)\) respectively.
Equations (44), (51) and (52) provide the MFT solutions for \(c(x)\) and \(\rho(x)\) valid in the limit of small \(D\). Notice that \(J_{\rm LD/HD}\) appears in \(\rho_{\rm LD/HD}(x)\). Thus knowledge of \(J_{\rm LD/HD}\) is necessary to evaluate \(\rho_{\rm LD/HD}(x)\). Now, \(J_{\rm LD/HD}\) depends upon \(c(x)\) through its spatial derivative \(\partial_{x}c(x)\) obtained at \(x=0,1\). On the other hand, enumeration of \(c(x)\) requires \(J_{\pm}\), because of the dependence of the constants \(A_{\pm},B_{\pm}\) etc on it. For simplicity while evaluating the currents, we approximate \(J_{\pm}\) by dropping the contributions from the SEP current to it, rendering it calculable from the TASEP boundary conditions only. Thus, in this approximation \(J_{-}=(2\alpha-1)^{2},\,J_{+}=(2\beta-1)^{2}\). This is a reasonable approximation, since for small \(D\), for which the current analysis holds, \(c(x)\) is largely slaved to \(\rho(x)\) in the bulk, making it rather weakly space-dependent, which in turn means the SEP current should be much smaller than the TASEP current. Lastly, notice that \(\rho(x)=1/2=c(x)\) continue to be steady state solutions for both \(c(x)\) and \(\rho(x)\) in the bulk, even when \(D\) does not vanish. Now \(\rho(x)=1/2\) implies MC phase in TASEP, which means when the TASEP is in its MC phase, the SEP and TASEP densities should overlap in the bulk for any finite \(D\).
Having found the steady states solutions of \(\rho(x)\) and \(c(x)\) for small \(D\), we now obtain the phase diagrams for both the TASEP and SEP lanes. Since for small \(D\), \(\rho(x)\) varies weakly with \(x\), we expect a TASEP phase diagram similar to that in an open TASEP, with most of the phase diagram being covered by regions with pure LD, HD or MC phases. However, due to the expected weak space dependence of \(\rho(x)\) (as opposed to constant \(\rho\) in pure open TASEPs), mixed phases other than the pure phases should also appear albeit over smaller ranges of \(\alpha,\beta\), which should go to zero as \(D\to 0\). We present an MFT phase diagram in Fig. 6 of the TASEP lane for \(D=0.01\) and \(\Omega=0.3\) (thus \(\Omega/D=30\gg 1\)). The principles to obtain the TASEP phase diagram are same as those discussed in the large \(D\) case above. We use the continuity of the currents (36)-(38) across the phase boundaries. As with \(D>>1\), the rather complex \(x\)-dependence of \(\rho(x)\) precludes explicit enumeration of \(x_{\alpha},x_{\beta},x_{w}\). Instead, we use the conditions on the densities as listed in the previous Section, and then use contour plots in the \(\alpha-\beta\) plane for fixed \(D\) and \(\Omega\) to obtain the phase boundaries.
The corresponding phase diagram of the SEP channel with \(D=0.01\) is same as that given in Fig. 5 with three
second order phase boundaries meeting at a multicritical point located at \(\alpha=1/2=\beta\). In fact, the SEP phase diagram is same for any finite value of \(D\). This may be understood as follows. From the particle-hole symmetry of the model, the LD phase regions (covering both the pure LD phase and the LD parts of the mixed phases) must have the same area as that of the HD phase regions in the \(\alpha-\beta\) plane of a TASEP phase diagram. Furthermore, these two regions are also _symmetrically_ located on the two sides of the line \(\alpha=\beta\). According to logic outlined above, these two regions correspond to the deficit and excess particle regions in a SEP phase diagram, which are also symmetrically located on the two sides of the line \(\alpha=\beta\). The remaining region in a SEP phase diagram is the neutral region. Since these arguments hold for any finite \(D\), the SEP phase diagram remains unchanged when \(D\) varies. There is however one thing that changes, _viz._, the amount of "excess" or "deficit" particles in the corresponding phase regions of TASEP. As the SEP gets increasingly slaved to the TASEP when \(D\) is progressively reduced, for the same \(\alpha,\beta\) the degree of "excess" or "deficit" rises (assuming the TASEP is not in its MC phase), reaching the maximum for \(D\to 0\). In the opposite extreme limit when \(D\to\infty\), \(c(x)\to 1/2\) for all \(\alpha,\beta\), meaning the whole SEP phase diagram now has only the neutral phase.
Plots of the steady state TASEP density \(\rho(x)\) and SEP density \(c(x)\) versus \(x\) for \(D=0.01\) and \(\Omega=0.3\) are shown respectively in Fig. 7 and Fig. 8. Both analytical MFT and numerical MCS results are shown. For MFT solutions of TASEP, we have used \(\rho(x)=\rho_{\rm LD}(x)\) [Eq. (51)] in the LD phase and \(\rho(x)=\rho_{\rm HD}(x)\) [Eq. (52)] in the HD phase of the TASEP. In addition, the solutions \(\rho(x)=1/2\) corresponds to the MC phase of the TASEP. For MFT solutions of SEP, we have used \(c(x)=c_{-}(x)<1/2\) and \(c(x)=c_{+}(x)>1/2\) as defined in Eq. (44)] above.
We again note a lower degree of quantitative agreement between the MFT and MCS solutions of \(c(x)\), when the TASEP is in its LD-HD phase, relative to when it is in its LD or HD phases, in which case the agreement is good. As for our large-\(D\) MFT solutions, we attribute this to the stronger space dependence of \(c(x)\) near the location of an LDW in the TASEP, a feature not adequately captured by our MFT for \(c(x)\). Nonetheless, the MFT and MCS solutions for \(c(x)\) agree qualitatively.
Figure 7: Steady state density \(\rho(x)\) in the TASEP lane in the LD (top), LD-HD (middle) and HD (bottom) phases with \(D=0.01\), \(\Omega=0.3\). MFT (blue line) and MCS (red points) results are shown. For MFT solutions of TASEP, we have used \(\rho(x)=\rho_{\rm LD}(x)\) [Eq. (51)] in the LD phase and \(\rho(x)=\rho_{\rm HD}(x)\) [Eq. (52)] in the HD phase of the TASEP (see text).
Figure 6: Mean-field TASEP phase diagrams with \(D=0.01\), \(\Omega=0.3\).
### Comparison of the MFTs
When \(D\) is neither too small or too large, neither of the approximations leading to the MFT solutions are expected to work well. Nonetheless, _both_ MFTs should act as guidelines to understand the MCS results. In Fig. 9, we have shown the MFT phase diagrams for \(D=0.05\), \(\Omega=0.3\), obtained in the large \(D\) approximation (solid red lines) and small \(D\) approximation (broken black lines). While the two approximations clearly differ quantitatively, the topology of the TASEP phase diagrams remains the same, independent of the approximations used to obtain the MFT. This clearly lends credence to the physical pictures that emerge from this work.
We have further presented our results on \(\rho(x)\) when the TASEP is in its LD and HD phases. Numerical results from our MCS simulations and the two MFT predictions (in the large-\(D\) and small-\(D\) approximations) are plotted together. We find that the MFT with large-\(D\) approximation underestimates \(\rho_{\rm LD}(x)\) and overestimates \(\rho_{\rm HD}(x)\) with respect to the corresponding MCS results. The trend from the MFT with small-\(D\) approximation is just the opposite. See Fig. 10 for plots of the MCS results on \(\rho(x)\) together with the corresponding MFT predictions using MFTs with both large-\(D\) and small-\(D\) approximations. The results for the SEP density profiles \(c(x)\) are shown in Fig. 11.
## VI Delocalisation of domain walls for \(D\to 0\)
We have found that in the limit of \(D\to\infty\) this model reduces to the LK-TASEP model [9], whereas in the limit \(D\to 0\) it reduces to an isolated open TASEP. A smooth crossover from LK-TASEP behaviour to an open TASEP is expected as \(D\) is reduced. This can be seen from the phase diagrams given above for various values of \(D\). As \(D\) is reduced the regions for two- and three-phase coexistence regions shrink, which are expected to vanish for \(D\to 0\), i.e., in the limit of a pure TASEP. Indeed,
Figure 8: Steady state density \(c(x)\) in the SEP lane, when the TASEP lane is in its LD (top), LD-HD (middle) and HD (bottom) phases with \(D=0.01\), \(\Omega=0.3\). MFT (blue line) and MCS (red points) results are shown. For MFT solutions of SEP, we have used \(c(x)=c_{-}(x)<1/2\) and \(c(x)=c_{+}(x)>1/2\) as defined in Eq. (44)] (see text).
Figure 9: Mean-field TASEP phase diagrams with \(D=0.05\), \(\Omega=0.3\). MFT solutions for \(\rho(x)\) with large \(D\) approximation and small \(D\) approximation are used to get the corresponding phase diagrams. Phase diagram with solid red (broken black) phase boundaries is obtained using large \(D\) (small \(D\)) approximation (see text).
the two-phase coexistence region should shrink to the limit \(\alpha=\beta\) and the three-phase coexistence to a point \(\alpha=\beta=1/2\) in the isolated, open TASEP limit with \(D\to 0\). Our MFT is consistent with these physically motivated expectations: In Fig. 12, mean-field TASEP phase diagrams for various values of \(D\) ranging from 0.1 to 0.000001 are shown. These phase diagrams are drawn with the MFT valid for small \(D\) (or, equivalently, \(\Omega/D\gg 1\)). It is evident that as \(D\) is progressively reduced, the two- and three-phase coexistence regions increasingly shrink, eventually practically vanishing for \(D=0.000001\), for which the resulting phase diagram is virtually indistinguishable from that of an isolated open TASEP.
The TASEP density profiles in the two-phase coexistence region for any \(D>0\) (including the LK-TASEP limit of \(D\to\infty\)) is a pinned or static domain wall, i.e., an LDW. This pinning of the domain walls is attributed to spatially nonuniform TASEP densities in the steady states. However, in the limit \(D\to 0\), it must be a DDW, existing on the line \(\alpha=\beta\) for \(0<\alpha=\beta<1/2\), as it is for an isolated open TASEP. While a fully delocalised domain wall is possible only for \(D\to 0\), we observe signatures of gradual delocalisation as \(D\) is reduced. To see this, we obtain the TASEP density profiles on the line \(\alpha=\beta=0.1\) with \(\Omega=0.3\) for system size \(L=1000\) and various values of \(D\). We find that as \(D\) is reduced, the long-time averaged profile of the domain wall becomes an increasingly inclined line, signifying larger fluctuations in its position. We visually measure the extent of domain wall position fluctuations or the "width" \(W\), which is roughly the projection of the inclined line of the domain wall on the \(x\)-axis (see the inset in Fig. 13), and plot them against \(D\). See Fig. 13 for plots of the domain walls with \(D=1,0.005,0.001\), and Fig 14 for a semilog plot of \(W\) versus \(D\). While our study is purely phenomenological, it does indicate increasing delocalisation as \(D\) is reduced. Note that this effect cannot be captured within MFT, as MFT by construction neglects all fluctuations. The approaches developed in Ref. [26] to study fluctuations systematically going beyond MFT may be helpful in this study.
Figure 11: Plots of SEP steady state density \(c(x)\) versus \(x\), when the TASEP is in its (top) LD phase and (bottom) HD phase with \(D=0.05\), \(\Omega=0.3\). Broken blue lines represent \(c_{\rm MFT}\) obtained in the small-\(D\) approximation, whereas the solid green lines represent \(C_{\rm MFT}\) obtained in the large-\(D\) approximation; red points represent the corresponding MCS results.
Figure 10: Plots of steady state density \(\rho(x)\) versus \(x\). (top) LD and (bottom) HD phases with \(D=0.05\), \(\Omega=0.3\). Broken blue lines represent \(\rho_{\rm MFT}\) obtained in the small-\(D\) approximation, whereas the solid green lines represent \(\rho_{\rm MFT}\) obtained in the large-\(D\) approximation; red points represent the corresponding MCS results.
## VII Summary and Outlook
In summary, we have proposed and analysed an open one-dimensional system with two lanes, one modeling a one-dimensional lattice executing TASEP dynamics and the other with diffusive or SEP kinetics, representing a reservoir, which are coupled by exchange of particles subject to exclusion. This diffusion is unbiased, that is a particle can hop to its right or left with equal probability, subject to exclusion. We show that the ratio of the effective exchange rate \(\Omega\) to the diffusion coefficient \(D\), or for a fixed \(\Omega\), \(D\) itself appears as the tuning parameter by varying which our system goes from a diffusion dominated behaviour for large \(D\) to TASEP dominated behaviour in the limit of small \(D\). We show that for a fixed non-zero \(\Omega\), with \(D\to 0\) the SEP density is slaved to the TASEP density and the resulting steady-state phase diagram of the TASEP lane is same as that of an isolated open TASEP. In the opposite extreme limit of fast diffusion, i.e., \(D\to\infty\), the density profile of the diffusive lane is spatially constant with a value \(1/2\), whereas that in the driven lane is identical to that of a TASEP with Langmuir Kinetics in the bulk. For intermediate values of \(D\), our model has nonuniform density profiles in both the TASEP and the SEP in the steady states with rather complex position-dependence. These nontrivial space dependences are entirely due to the coupling between the SEP and TASEP kinetics, for without the coupling, both SEP and TASEP generally yield flat density profiles (except for the delocalised domain wall in an open TASEP). For intermediate values of \(D\), the MFT equations cannot be solved exactly. This has led us to solve the MFT equations for small and large \(\Omega/D\) separately, and obtain two sets of solutions with one of them giving modifications of the TASEP and SEP densities for small \(\Omega/D\) and the other for large \(\Omega/D\) in perturbative approaches. We find that the MFT solutions agree reasonably with the MCS studies results for small and large \(\Omega/D\). However, unsurprisingly when \(\Omega/D\) is intermediate none of the solutions agree quantitatively with the numerical results. We have also numerically explored how a domain wall in the TASEP lane, which is obviously localised for any finite \(D\), but must be fully delocalised when \(D\to 0\), actually gradually delocalises as \(D\) is reduced. Such an effect cannot be studied within the MFT, as it neglects all fluctuations. It would be interesting to theoretically study this delocalisation by going beyond MFT descriptions and considering fluctuations. We have further discussed phase diagrams of the SEP in the plane of the control parameters of the TASEP, _viz._, \(\alpha,\beta\). We have
Figure 14: Semilog plot of \(W\) as a function of \(D\) for \(\Omega=0.3,\alpha=0.1=\beta\). Clearly, \(W\) rises as \(D\) is reduced, indicating gradual delocalisation of the domain wall (see text).
Figure 12: Mean-field TASEP phase diagrams in the \(\alpha-\beta\) plane for various values of \(D\) ranging from \(0.1\) to \(0.000001\) are shown. These phase diagrams are drawn with the MFT valid for small \(D\) (or, equivalently, \(\Omega/D\gg 1\)). Clearly for \(D=0.000001\), the phase diagram is virtually indistinguishable from its counterpart for an isolated open TASEP.
Figure 13: Gradual delocalisation of the domain wall due to increasing fluctuations as \(D\) is reduced for a fixed system size \(L=1000\) and \(\alpha=\beta=0.1\). MCS results for \(D=1.0,0.005,0.001\) are shown. (Inset) A pictorial definition of the DW width \(W\) (for \(D=0.001\)) is shown. Clearly, as \(D\) is reduced, \(W\) is increased for a fixed system size \(L\), indicating gradual delocalisation of the domain wall.
argued that the phase diagram of the SEP is identical for any finite \(D\) (including the limiting case of \(D\to 0\)): it has just three phases, _viz._ deficit, excess and neutral particle phases (measured with respect to the mean SEP density \(\overline{c}=1/2\) in an isolated SEP with unbiased entry and exit rates). We have shown that these phases have a direct correspondence with the phases of the TASEP, a principal result from this work.
The take home message from our studies here is that the mutual coupling between driven and diffusive dynamics can be tuned to make not only the TASEP lane to pick up non-trivial position dependence in its steady state densities, the diffusive lane can also maintain nonuniform steady states. This means a reservoir, which is modeled by a SEP here, can sustain spatial gradients in its densities, when it exchanges particles with a driven channel in the bulk. An interesting future study would be to impose particle number conservation at a global level, i.e., in the combined system of a TASEP and a SEP. This would be an example of a system with finite resources having an internal dynamics in the reservoir [13; 22].
We have restricted ourselves to studying a 1D model here. As we mentioned earlier, our model, in addition to its importance as a 1D nonequilibrium model with coupled driven and diffusive dynamics, also serves as a minimal model for a molecular motors - microtubules assembly inside eukaryotic cells. The SEP here models the diffusion of the molecular motors in the bulk, whereas the TASEP represents unidirectional steady motion in a force field, overdamped by viscous drag. Since we have modeled the (three-dimensional) reservoir by SEP, a 1D model, it raises an important phenomenological point: there are significant dynamical differences between the two due to single file diffusion in 1D, which gives the mean square displacement \(W\propto\sqrt{t}\) (time) [23] in unbiased diffusion. This questions applicability of our results for a realistic three-dimensional system. Nonetheless, it is known that for an infinitesimal bias \(W\propto t\) is restored [24; 25], whereas for an infinitesimal bias, our results on the steady states of the SEP should be practically same as here. This clearly allows using our 1D results to draw physical insight about the corresponding three-dimensional situations. Nonetheless, it would definitely be interesting, from both theoretical as well as phenomenological standpoints, to extend and study our model to higher dimensions. This should give a better handle to modeling intra-cellular transport more realistically. In this study, we have considered an unbiased SEP, i.e., a SEP with equal entry and exit rates. In a more general situation, the entry and exit rates could be different, which can result in a biased SEP, which has an inclined line-shaped density profile. It would be interesting to couple such a biased SEP with a TASEP via lane exchanges and investigate the resulting steady states. Our study here is restricted to equal-sized TASEP and SEP lanes. Introduction of unequal lengths is expected to give additional complex behaviour. We hope our studies here will provide impetus to address these questions theoretically in the future.
## VIII Acknowledgement
S.M. thanks SERB (DST), India for partial financial support through the CRG scheme [file: CRG/2021/001875].
## Appendix A Space-dependent exchange
We now consider the effects of side-dependent exchange rates \(\omega_{i}\). We continue to assume equal attachment and detachment rates. As before, we use scaled attachment-detachment rates defined by \(\omega_{i}=\Omega_{i}/L^{2}\) and TASEP hopping rate \(1/L\) together with \(\alpha/L\), \(\beta/L\) as the entry and exit rates in TASEP to ensure competition with the diffusion in SEP. This results into the MF equations (5) and (6). We further consider the asymptotic limit of fast diffusion given by \(D\to\infty\). In that limit, the SEP density is independent of the TASEP density with \(c(x)=1/2\) everywhere, independent of the TASEP density. In that limit, using \(c(x)=1/2\), Eq. (5) reduces to
\[(1-2\rho)\left[\frac{\partial\rho}{\partial x}-\frac{\Omega(x)}{2}\right]=0. \tag{10}\]
Equation (10) has two solutions: \(\rho(x)=1/2\), which gives the MC phase here, and \(\rho(x)=\int dx\,\Omega(x)/2+\tilde{C}\), where \(\tilde{C}\) is a constant of integration. By using the boundary conditions \(\rho(0)=\alpha\) or \(\rho(1)=1-\beta\), we can evaluate \(\tilde{C}\): Using \(\rho(0)=\alpha\)
\[\tilde{C}\equiv\tilde{C}_{\alpha}=\alpha-\left[\int dx\,\Omega(x)/2\right]_{x =0}. \tag{11}\]
Similarly, using \(\rho(1)=1-\beta\), we get
\[\tilde{C}\equiv\tilde{C}_{\beta}=1-\beta-\left[\int dx\,\Omega(x)/2\right]_{x =1}. \tag{12}\]
Thus using (11) and (12) we get two solutions
\[\rho_{\alpha}(x) = \int dx\,\Omega(x)/2+\tilde{C}_{\alpha}, \tag{13}\] \[\rho_{\beta}(x) = \int dx\,\Omega(x)/2+\tilde{C}_{\beta}. \tag{14}\]
These solutions generalise the well-known space-dependent solutions \(\rho_{\rm LK}(x)\) as mentioned earlier. Instead of linear \(x\)-dependence, we now obtain general, nonlinear \(x\)-dependent solutions. As in the original LK-TASEP model, these solutions may meet each other or with the other solution \(\rho=1/2\) in the bulk of the system giving phase coexistence of different phases in the steady states. Following the logic outlined in Ref. [9], the steady state
density profiles and ensuing phase diagram may be calculated by equating the steady state currents. It is easy to see that the resulting phase diagram generally will have the same topology as in the LK-TASEP model, although the precise locations of the phase boundaries in the \(\alpha-\beta\) plane should depend on the specific forms of \(\Omega(x)\). This reveals a degree of robustness of the phase diagrams in the LK-TASEP model, revealing universality in the topology of the phase diagrams.
| ```
1Dのモデルとして、2つのレーンを含むシンプルなモデルを提案し、その両方を開いた境界線で構成します。1つのレーンは拡散を遂げ、もう1つのレーンは unidirectionalまたは非対称排除Dynamicsを実行します。これらのレーンは、混入によって、粒子の交換を通して相互に結合しています。このモデルでは、一般的に非均一な安定状態が明らかになります。このモデルにおけるパラメータ領域では、hopping along the TASEP Lane、Diffusion along the SEP Lane、そしてTASEPとSEP Lane間の粒子交換が競合しているとき、SEP diffusivity $D$ はSEP密度とTASEP密度にTuningパラメータとして現れます。また、$D$ が非平衡安定状態におけるこのモデルの跳躍率を調整することで、非対称排除ダイナミクスの相共存と、拡散ダイナミクスの安定状態における空間 |
2302.09491 | X-Adv: Physical Adversarial Object Attacks against X-ray Prohibited Item
Detection | Adversarial attacks are valuable for evaluating the robustness of deep
learning models. Existing attacks are primarily conducted on the visible light
spectrum (e.g., pixel-wise texture perturbation). However, attacks targeting
texture-free X-ray images remain underexplored, despite the widespread
application of X-ray imaging in safety-critical scenarios such as the X-ray
detection of prohibited items. In this paper, we take the first step toward the
study of adversarial attacks targeted at X-ray prohibited item detection, and
reveal the serious threats posed by such attacks in this safety-critical
scenario. Specifically, we posit that successful physical adversarial attacks
in this scenario should be specially designed to circumvent the challenges
posed by color/texture fading and complex overlapping. To this end, we propose
X-adv to generate physically printable metals that act as an adversarial agent
capable of deceiving X-ray detectors when placed in luggage. To resolve the
issues associated with color/texture fading, we develop a differentiable
converter that facilitates the generation of 3D-printable objects with
adversarial shapes, using the gradients of a surrogate model rather than
directly generating adversarial textures. To place the printed 3D adversarial
objects in luggage with complex overlapped instances, we design a policy-based
reinforcement learning strategy to find locations eliciting strong attack
performance in worst-case scenarios whereby the prohibited items are heavily
occluded by other items. To verify the effectiveness of the proposed X-Adv, we
conduct extensive experiments in both the digital and the physical world
(employing a commercial X-ray security inspection system for the latter case).
Furthermore, we present the physical-world X-ray adversarial attack dataset
XAD. | Aishan Liu, Jun Guo, Jiakai Wang, Siyuan Liang, Renshuai Tao, Wenbo Zhou, Cong Liu, Xianglong Liu, Dacheng Tao | 2023-02-19T06:31:17 | http://arxiv.org/abs/2302.09491v1 | # _X-Adv_: Physical Adversarial Object Attacks against X-ray
###### Abstract
Adversarial attacks are valuable for evaluating the robustness of deep learning models. Existing attacks are primarily conducted on the visible light spectrum (_e.g._, pixel-wise texture perturbation). However, attacks targeting texture-free X-ray images remain underexplored, despite the widespread application of X-ray imaging in safety-critical scenarios such as the X-ray detection of prohibited items. In this paper, we take the first step toward the study of adversarial attacks targeted at X-ray prohibited item detection, and reveal the serious threats posed by such attacks in this safety-critical scenario. Specifically, we posit that successful physical adversarial attacks in this scenario should be specially designed to circumvent the challenges posed by color/texture fading and complex overlapping. To this end, we propose _X-Adv_ to generate physically printable metals that act as an adversarial agent capable of deceiving X-ray detectors when placed in luggage. To resolve the issues associated with color/texture fading, we develop a differentiable converter that facilitates the generation of 3D-printable objects with adversarial shapes, using the gradients of a surrogate model rather than directly generating adversarial textures. To place the printed 3D adversarial objects in luggage with complex overlapped instances, we design a policy-based reinforcement learning strategy to find locations eliciting strong attack performance in worst-case scenarios whereby the prohibited items are heavily occluded by other items. To verify the effectiveness of the proposed _X-Adv_, we conduct extensive experiments in both the digital and the physical world (employing a commercial X-ray security inspection system for the latter case). Furthermore, we present the physical-world X-ray adversarial attack dataset XAD. We hope this paper will draw more attention to the potential threats targeting safety-critical scenarios. Our codes and XAD dataset are available at [https://github.com/DIG-Beihang/X-adv](https://github.com/DIG-Beihang/X-adv).
## 1 Introduction
Deep neural networks (DNNs) have achieved remarkable performance across a wide area of applications [1, 19, 21]. Recently, deep learning has been introduced into safety-critical scenarios such as X-ray security inspection in public transportation hubs (_e.g._, airports). In this scenario [32, 40, 47, 42], deep-learning-based detectors are utilized to assist inspectors in identifying both the presence and location of prohibited items (_e.g._, pistols and knives) during X-ray scanning. This approach significantly reduces the amount of human labor required and helps to protect the public from severe risks.
Despite their promising performance, DNNs are vulnerable to _adversarial examples_[38]. These elaborately designed perturbations are imperceptible to human vision, but can easily mislead DNNs into making wrong predictions, thus threatening practical deep learning applications [22, 23, 24]. By contrast, adversarial examples are also beneficial for evaluating and better understanding the robustness of DNNs [51, 53, 26, 39, 50]. In the past years, extensive research has been conducted into performing adversarial attacks on natural images (visual light); however, the robustness of texture-free X-ray images
Figure 1: Illustration of physical-world adversarial attacks on X-ray security inspection. This paper proposes _X-Adv_ to generate physically realizable 3D adversarial objects. During X-ray scanning, the detector can detect prohibited items in the right image, but our adversarial objects deceive the detector into failing to detect prohibited items in the left image.
(such as in the context of X-ray prohibited item detection) remains underexplored. This sparsity of research presents a severe risk to the safety of the general public, as it increases their vulnerability to attack.
In this paper, we take the first step in physical-world adversarial attacks on the X-ray prohibited item detection scenario, _i.e._, deceive the detector to wrong predictions by strategically placing adversarial objects around the prohibited item. However, simply extending existing physical attacks that work well on natural images to the context of X-ray images is non-trivial owing to the different imaging principles, _e.g._, the wavelengths of X-rays (_0.001\(\sim\)10nm_) and visible light (_390\(\sim\)780nm_) have a huge difference. More specifically, X-ray imaging is primarily conducted by utilizing material, thickness, and attenuation coefficients, meaning that the existing physical attacks designed for a visible light context (_e.g._, interference textures [46] or patches [2]) cannot be effectively applied to X-ray imaging. Thus, X-ray attacks should be considered a new type of attack problem in visually constrained scenarios with different wavelengths. In particular, we identify two key challenges impeding successful and feasible adversarial attacks in this scenario: (1) Color/texture fading. Due to its use of special imaging principles (_i.e._, beam intensity and attenuation rule), the X-ray scanning process eliminates most of the colors/textures and projects its outputs primarily based on item materials and shapes. Thus, the commonly used perturbations utilizing color disturbances will be removed by the X-ray scanning causing them to be ineffective. (2) Complex overlap. Luggage passed through an X-ray scanner often contains a large number of objects made of different materials, and overlap between these objects can degrade the attack performance; moreover, a successful attack should not rely on the occlusion of the prohibited item. Thus, when designing an adversarial object, it is necessary to consider the worst-case scenario (complex overlapping instances within the luggage), which increases the difficulty of the task.
To address the above problems, this paper proposes an adversarial attack approach called \(\mathcal{X}\)-_Adv_ to generate physically realizable adversarial attacks for X-ray prohibited item detection (as shown in Figure 1). As for the _color/texture fading_, we generate physically realizable 3D objects with adversarial shapes, which enable our attacks to remain effective (since the shape cannot be altered after the X-ray imaging). To guide the design of the shape, we derive a differentiable converter that projects 3D objects into X-ray images so that we could update the shape of the object using the gradients of a surrogate white-box detector. As for the _complex overlap_, we aim to find the locations that achieve strong attack ability even when occluded by other objects; moreover, we ensure that the placed adversarial objects do not overlap with the prohibited item. We thus introduce a policy-based strategy to search for the location that provides optimal attacking performance in the worst-case scenario. In summary, our \(\mathcal{X}\)-_Adv_ can generate adversarial objects by jointly optimizing the shapes and locations for X-ray attacks.
Extensive experiments in both the digital and physical world using multiple benchmarks against several detectors are conducted. Specifically, we first evaluate digital-world attacks on multiple benchmarks against both one-stage and two-stage detectors. We then successfully attack a commercial X-ray security inspection system in the real world by generating adversarial metal objects using a 3D printer. Finally, we present the physical-world X-ray adversarial attack dataset XAD which contains 5,587 images (840 adversarial images). We hope this paper will draw more attention to the potential threats in safety-critical scenarios. Our **contributions** are:
* To the best of our knowledge, this paper is the first work to study the feasibility of physical-world adversarial attacks in the visually-constrained X-ray imaging scenario.
* We propose the \(\mathcal{X}\)-_Adv_ to generate physically realizable adversarial metal objects for X-ray security inspection attacks by addressing the color fading and complex occlusion challenges.
* We conduct extensive experiments on several datasets in both digital- and physical-world settings, and the results demonstrate the effectiveness of our attack.
* We present the physical-world X-ray adversarial attack dataset, XAD, consisting of 5,587 images (840 adversarial images).
## 2 Backgrounds and Related Work
**Prohibited Item Detection in X-ray Images.** X-ray imaging has been widely used due to its strong penetrative ability. In the X-ray security inspection scenario, inspectors usually adopt X-ray scanners to check passengers' luggage for the presence of prohibited items. A plethora of studies have been devoted to detecting prohibited items (_e.g._, pistols) in X-ray scanned luggage images using object detection methods [47, 42, 32, 45] to detection performance.
In addition to the X-ray image detection methods, high-quality X-ray image datasets and benchmarks are also valuable for promoting the development of the research area. Though obtaining colorful X-ray images requires high computational costs, there are still some available open-source specialized datasets for X-ray security inspection. For instance, SIXray [32] is a large-scale X-ray dataset containing millions of X-ray images collected from real-world subway stations. However, the images containing prohibited items are less than 1%, and there is no bounding box annotation provided for object detection. Some high-quality X-ray datasets for object detection have also been made available. Wei _et al._[47] first released the OPIXray dataset, which contains 8,885 artificially synthesized X-ray images of five categories of cutters. Tao _et al._[43] proposed the HiXray dataset, comprising 45,364
images containing 102,928 prohibited items. All images from the dataset are collected from X-ray scanners in airports. Recently, Tao _et al_. [41] further proposed the first few-shot object detection dataset in the X-ray security inspection scenario.
**Adversarial Attacks.** Adversarial examples are inputs with small perturbations, which are imperceptible to humans but can easily mislead DNNs into making incorrect predictions [16, 38]. Generally, we can classify them into digital and physical attacks. _Digital attacks_ usually generate adversarial perturbation at the pixel level across the whole input image. Szegedy _et al_. [38] first defined adversarial examples and proposed L-BFGS attacks. By leveraging the gradient of the target model, Goodfellow _et al_. [17] proposed FGSM to quickly generate adversarial examples. Since then, many types of adversarial attacks have been proposed, such as PGD [30], DeepFool [33], and JSMA [35]. However, due to their addition of global perturbations to the whole image, these attacks lack physical-world feasibility.
By contrast, _physical attacks_ aim to generate adversarial perturbations by perturbing the visual characteristics of real objects in the physical world. To achieve this goal, adversaries often generate adversarial perturbations in the digital world, then perform physical attacks by applying adversarial patches, painting adversarial camouflage, or directly creating adversarial objects in the real world [2, 7, 12, 46]. Brown _et al_. [2] first proposed the adversarial patch by confining the perturbations into a local patch, which could then be printed to deceive the classification models. Eykholt _et al_. [12] then modified the attacking loss function and generated strong adversarial attacks for real-world traffic sign recognition. Chen _et al_. [7] proposed Shapeshifter to attack a Faster R-CNN object detector in the physical world, specifically by attaching it to the STOP signs. In addition to the physical attacks on natural images (visible light domain), there also exist some preliminary studies on other _visually constrained scenarios_. For example, Cao _et al_. [3] investigated attacks in multi-sensor fusion scenarios, making adversarial examples invisible to both cameras and LiDAR. Recently, Zhu _et al_. [54, 55] attacked thermal infrared pedestrian detectors using small bulbs and special clothes. Mowery _et al_. [34] attacked a full-body X-ray scanner, while their proposed cyber-physical attacks did not aim at neural networks and are different from adversarial attacks.
In summary, although numerous methods of physical attacks on natural images have been proposed, relatively little is known about the physical-world X-ray security inspection attack. This paper takes the first step to study physical-world adversarial attacks for X-ray security inspection.
## 3 Threat Model
### Problem Definition
**Object detection.** An object detector \(f_{\Theta}(\mathbf{I})\rightarrow\{\mathbf{b},\mathbf{c}\}^{K}\) with parameters \(\Theta\), which takes an image \(\mathbf{I}\in[0,255]^{n}\) as input, outputs \(K\) detection boxes with location \(\mathbf{b}_{k}=[s_{k},r_{k},w_{k},h_{k}]\) and confidence \(c_{k}\). Moreover, \(f\) applies a non-maximum suppression (NMS) operation to remove redundant bounding boxes. The formulation of the training is as follows:
\[\min_{\Theta}\mathbb{E}_{(\mathbf{I},\{\mathbf{y}_{k},\mathbf{b}_{k}\}) \sim\mathbb{D}}\mathcal{L}(f_{\Theta}(\mathbf{I}),\{\mathbf{y}_{k},\mathbf{b }_{k}\}), \tag{1}\]
where \(\mathcal{L}(\cdot)\) is the loss function that measures the difference between the output of the detector \(f\) and the ground truth. \(\mathbf{y}_{k}\) denotes the true label, and \(\mathbf{b}_{k}\) denotes the true bounding box. In practice, the loss function is a weighted sum of the classification loss \(\mathcal{L}_{cls}\) and location loss \(\mathcal{L}_{loc}\):
\[\min_{\Theta}\mathbb{E}_{(\mathbf{I},\{\mathbf{y}_{k},\mathbf{b}_{k}\}) \sim\mathbb{D}}[\mathcal{L}_{cls}(f_{\Theta}^{cls}(\mathbf{I}),\mathbf{y}_{k} )+\lambda\mathcal{L}_{loc}(f_{\Theta}^{loc}(\mathbf{I}),\mathbf{b}_{k})]. \tag{2}\]
**Attacks on object detection.** Given an object detector \(f_{\Theta}\) and an input image \(\mathbf{I}\in\mathbb{I}\) with the ground truth label \(\{\mathbf{y},\mathbf{b}_{k}\}\), an adversarial example \(\mathbf{I}_{adv}\) satisfies the following:
\[f_{\Theta}(\mathbf{I}_{adv})\neq\{\mathbf{y},\mathbf{b}_{k}\}\quad s.t.\quad \|\mathbf{I}-\mathbf{I}_{adv}\|\leq\epsilon, \tag{3}\]
where \(\|\cdot\|\) is a distance metric and commonly measured via \(\ell_{p}\)-norm (\(p\in\){1,2,\(\infty\)}). Adversarial examples in visual recognition should also satisfy \(\mathbf{I}_{adv}\in[0,255]^{n}\). In this paper, we focus on deceiving the prediction class labels (_i.e_., \(\mathbf{y}\)).
**Physical attacks on X-ray prohibited item detection.** In this scenario, the items \(\mathbf{X}=\{\mathbf{x}_{1},...,\mathbf{x}_{m}\}\) in the luggage are scanned via an X-ray scanner to produce an X-ray image, where \(\mathcal{R}\) denotes the process of generating a pseudo-color image depicted in Figure 1 as \(\mathbf{I}=\mathcal{R}(\mathbf{X})\)). To perform physical attacks, we generate a 3D adversarial object \(\mathbf{x}_{adv}\) with adversarial shapes \(\mathbf{P}\) and place it at the proper location \(\mathbf{C}\) in the luggage; the luggage is then scanned by the X-ray into image \(\mathbf{I}_{adv}\), which could deceive the object detector \(f_{\Theta}(\cdot)\), _i.e_., minimizing \(\mathcal{M}\) that measures the performance of the detector:
\[\min_{\mathbf{P},\mathbf{C}}\mathcal{M}\left[f_{\Theta}(\mathcal{R}(\mathbf{ x}_{1},...,\mathbf{x}_{m},\mathbf{x}_{adv}^{\mathbf{P},\mathbf{C}}),\{\mathbf{y}_{k}, \mathbf{b}_{k}\})\right]. \tag{4}\]
### Challenges for X-ray Attacks
Existing attacks mainly aim at the visible light domain by generating adversarial textures. However, it is highly challenging to directly apply these existing attacks to the X-ray domain. Specifically, we observe two main **challenges** as follows.
**Challenge \(\mathbf{\Theta}\)**: _The significant difference between imaging principles used in the visible light and X-ray contexts (e.g., different wavelengths)._ We here first revisit the attenuation rule of X-ray photon beams. According to [31], a narrow beam of X-ray photons with energy \(E\) and initial photon intensity \(I_{0}\), on passing through an absorber of small thickness \(\Delta\mathbf{x}\), will suffer a fractional decrease of intensity \(\Delta I/I_{0}\) given by
\[\frac{\Delta I}{I_{0}}=-\mu(\rho,Z)\Delta\mathbf{x}, \tag{5}\]
where \(\mu\) is the attenuation coefficient per unit length for an item made of a material of density \(\rho\) and atomic composition \(Z\). When the same photon beam passes through a certain absorber of finite thickness \(x\), the intensity is given by
\[I=I_{0}\cdot exp(-\mu(\rho,Z)x). \tag{6}\]
This attenuated intensity then will be received by sensors in X-ray scanners, according to which we can obtain the depth profile of the X-ray images. According to Equation 5 and 6, we can conclude that an X-ray image is constructed primarily with reference to the material, the thickness of the object, and the properties of the light wave itself. Different from the perception of visible light images, X-rays tailor RGB space into a narrow color space, which means that common attacks that change pixel-wise textures will be ineffective for X-rays. To address this challenge, we need to optimize adversarial objectives to use non-color physical properties (_e.g_., shapes).
**Challenge \(\copyright\)**: _Complex overlap due to the diversity of sampling scenarios and a massive number of luggage items in the X-ray security inspection context._ Placing the adversarial object directly on top of prohibited items would appear to be a simple attack method. However, this approach is infeasible in real-world applications, since luggage may be positioned randomly during X-ray scanning, and the overlap rate between adversarial objects and prohibited items under arbitrary sampling conditions is low. Moreover, this violates the definition of adversarial examples. To guarantee a feasible attack, the attacker should consider the worst-case scenario: that is, how to achieve an effective adversarial attack without occluding prohibited items, and with the overlapping of other objects.
### Adversarial Goals
In this paper, we attempt to generate 3D adversarial objects with adversarial shapes to attack physical-world X-ray prohibited item detection models. As illustrated in Section 3.1, given an X-ray prohibited item detector \(f_{\Theta}\) that takes an X-ray scanned image **I** as input, attackers aim to deceive \(f_{\Theta}\) into making wrong predictions. This paper focuses on the more meaningful attack that deceives the detector to predict the wrong class labels rather than the wrong item locations. Specifically, we primarily study the untargeted attack, and the goal is to reduce the detection accuracy of detectors. Meanwhile, we also investigate the possibilities of the more difficult targeted attack, where we aim to force the detector predictions to the Background and make these prohibited items "invisible" (Section 5.5). For the untargeted attack, the detector predicts any other labels that are different from the ground truth should be marked as a successful attack; while for the targeted attack, the prediction must match the assigned label.
### Possible Attack Pathways
Regarding adversarial attacks, one of the most important questions that should be answered is whether they are practical. For our \(X\)-_Adv_ objects, they could be applicable to multiple X-ray image detection-related scenarios, _e.g_., security inspections in public hubs, and health examinations in hospitals. Using the \(X\)-_Adv_ approach, adversaries could perform real-world attacks simply by generating an adversarial metal object by using 3D printers, then placing the item into their luggage or bags. The proposed attacks could make detectors yield wrong class predictions with low detection accuracy. Meanwhile, it is also possible for adversaries to conceal a prohibited item and make it "invisible" to the detectors, which can be achieved by simply modifying our attacking loss.
### Adversary Constraints and Capabilities
In considering the real-world X-ray security inspection scenario, we take comprehensive conditions into account and conduct both white-box and black-box attacks. In the white-box attack setting, the adversary has full access to the target model (_e.g_., architectures, weights), and is able to generate adversarial attacks directly based on its gradients. By contrast, the black-box attack setting is more practical; here, the adversary possesses only a little knowledge about the target model. For this setting, we assume that the target model and the source model are dealing with the same task and that the adversary performs transfer-based attacks. Specifically, the adversary first generates adversarial objects based on a white-box source model from a certain dataset; the adversary then prints the adversarial objects via a 3D printer in the real world; finally, adversaries could simply place adversarial objects in the luggage and attack the deployed X-ray security inspection model. Based on this, we could guarantee that all information of target models is unavailable to the attackers in black-box settings, which helps us to implement the strictest measures for simulating the physical scenarios. Moreover, to ensure our approach is more practical, the size of our adversarial objects should be small; thus the adversarial metal generated in this paper only takes up 1.78% of the X-ray image.
## 4 \(X\)-_Adv_ Approach
Selective Search [44] proposes a heuristic strategy to discover potential objects from four perspectives: color similarity, texture similarity, shape similarity, and overlap similarity. Since it is based on the methods by which humans judge objects, this object discovery mechanism has been adopted in current deep-learning-based object detectors. Thus, to deceive the object detector, we can also optimize adversarial objectives from four perspectives: color, texture, shape, and overlap.
Due to the special nature of X-ray imaging principles and the diverse luggage sampling, physical-world adversarial at
tacks on X-ray security inspection should take the adversarial object's color/texture fading and complex overlapping problems into consideration. Therefore, this paper proposes \(\mathcal{X}\)-\(Adv\) to generate physically realizable adversarial objects for X-ray security inspection (as illustrated in Figure 2). This approach simultaneously polishes adversarial shapes and reinforces attacking locations in the worst-case scenario (no overlapping) because the color space (_e.g_., color, texture) is not available.
### Adversarial Shape Polishment
X-ray images, which are generated by X-ray security inspection machines from natural images, emphasize the shape and the material while neglecting the original color/textures of items. Thus, to successfully attack X-ray detectors, we generate objects with _adversarial shapes_ rather than _adversarial textures_ (since the adversarial colors/textures would be simply eliminated by the X-ray imaging pipeline, making such attacks ineffective). Accordingly, given a 3D object **x**, we refine its visual characteristics **P** (_i.e_., shape) into adversarial shape **P**\({}_{adv}\), such that the generated adversarial 3D object **x**\({}_{adv}\) can attack the detector \(f_{\Theta}(\mathcal{X}({\bf x}_{adv}))\) after X-ray projection.
However, the X-ray imaging pipeline is highly complex and confidential and also varies significantly across different types of X-ray machines. Meanwhile, it is rather difficult to directly perform black-box query attacks since inspectors will not allow adversaries to query the system several times. Therefore, we propose a possible attack pathway in which we derive a differentiable X-ray converter to simulate the X-ray projection pipeline from 3D objects to 2D X-ray images and then perform gradient-based transfer attacks.
However, the transformation from the depth \(d\) of a scanned object to color images \(g\) remains unknown. Based on the knowledge of X-ray machine vendors, the transformation process (\(d\to I\to g\)) can be simply represented by exponential functions, where the attenuated intensity \(I\) of X-ray beams has an exponential relationship with the object depth \(d\) (_c.f_. Eqn. 6) and the intensity \(I\) to the color \(g\) of X-ray images can be converted using a linear transformation. Therefore, we use the following exponential function to formulate the process:
\[g_{m}(d)=a\cdot exp(-b\cdot d)+q, \tag{7}\]
where \(d\) indicates object depth, \(m\) is the material, \(g_{m}(d)\) represents the pixel value of color in a certain depth and material, and \(a\), \(b\), and \(q\) are undetermined coefficients correlated to \(m\), which will be calculated from real image sampling and regression fitting. We did not use DNNs for the transformation since it is too costly or even infeasible to collect sufficient data (_i.e_., different materials with diverse thicknesses) for DNN training (_c.f_. Section 5.1). We use HSV color space rather than RGB because we found that regression in HSV space performs better in reducing the regression error (see Appendix A.2).
However, a depth image cannot represent a unique 3D object. Therefore, we use meshes as the format of our 3D adversarial object, given that meshes have been extensively used to parameterize 3D objects. Given an original mesh \({\bf x}_{ori}\), the coordinates on the XY-plane represent the shape of the image projected onto the 2D domain, while the coordinates in the \(\mathbb{Z}\)-axis denote the depth (pixel value) of the image. Thus, we can optimize the shape of the 3D object by manipulating the coordinates in the mesh, then project the 3D mesh to a 2D depth image, and finally convert it to an X-ray image (the
Figure 2: Illustration of \(\mathcal{X}\)-\(Adv\) approach. For the color/texture fading problem, we derive a differentiable converter that projects 3D objects into X-ray images; this allows us to generate 3D printable objects with adversarial shapes, which is X-ray projection invariant to X-ray imaging. We then introduce a policy-based algorithm to search for the optimal attacking location, which also shows high physical-world feasibility to address the complex occlusion problem. By jointly optimizing the combination of attack locations and shapes, our \(\mathcal{X}\)-\(Adv\) can generate physically realizable adversarial attacks for X-ray security inspection.
whole process is differentiable). The adversarial attack loss can be formalized by maximizing the classification loss \(\mathcal{L}_{cls}\) of the target model, as follows:
\[\mathcal{L}_{udv}(\mathbf{X},\mathbf{x}_{adv};f_{\Theta},\mathbb{R}_{\delta})= \arg\max_{\mathbf{P}}\mathcal{L}_{cls}(f_{\Theta}(\mathbb{R}_{\delta}(\mathbf{ X},\mathbf{x}_{adv}^{\mathbf{P}})),\{\mathbf{y}_{k},\mathbf{b}_{k}\}), \tag{8}\]
where \(\mathbb{R}_{\delta}\) denotes the differentiable converter in Eqn. 7 that can simulate the black-box X-ray scanning process \(\mathcal{R}\).
In practice, we overlap the original X-ray image \(\mathbf{I}\) and the converted output \(g(\cdot)\) by \(\mathbf{I}\odot g_{m}(d(\mathbf{x}_{adv}^{\mathbf{P}}))\), where \(d(\mathbf{x}_{adv}^{\mathbf{P}})\) refers to the 2-axis depth of adversarial mesh \(\mathbf{x}_{adv}^{\mathbf{P}}\), and \(\odot\) indicates pixel-wise multiplication.
### Attack Location Reinforcement
In the X-ray security inspection scenario, there are often a vast number of items in the luggage. The simplest attack method is to attack the target detector by forcing a high overlap of adversarial objects and prohibited items. However, because the angle at which the bag will be scanned is uncontrollable in an X-ray detection scenario, the overlapping probability of adversarial targets and prohibited items are low, while adversarial objects are often occluded by other goods. This complex occlusion problem brings challenges to adversarial attacks. Therefore, we need to study effective attack algorithms in the _worst-case_ scenario, whereby the adversarial object does not cover prohibited items and is heavily occluded by other objects. Moreover, an appropriate location would increase the effectiveness of attacks, since our _X-Adv_ can only modify shapes while DNNs are more sensitive to texture [8, 20, 37].
Thus, to perform attacks under such a constrained scenario, we make full use of the location of the adversarial object and further improve the efficiency of our attacks by searching for the optimal attack locations. Accordingly, to achieve strong attacks, it is necessary to jointly consider the combination of attacking location and shape \((\mathbf{C}_{best},\mathbf{P}_{best})\):
\[\mathcal{L}_{adv}=\arg\max_{\mathbf{C},\mathbf{P}}\mathcal{L}_{cls}(f_{\Theta }(\mathbb{R}_{\delta}(\mathbf{X},\mathbf{x}_{adv}^{\mathbf{P},\mathbf{C}})), \{\mathbf{y}_{k},\mathbf{b}_{k}\}). \tag{9}\]
Meanwhile, these two variables are mutually interactive, and the shape \(\mathbf{P}\) is often influenced by the attack locations \(\mathbf{C}\). If we determine the attack location \(\mathbf{C}_{best}\), the adversarial shape \(\mathbf{P}\) can be optimized by the gradient descent algorithm introduced in the previous subsection using the differentiable converter. However, there is no gradient information available for the attack location, which prevents us from optimizing the coordinates of adversarial objects. Also, calculating all possible conditions would result in unacceptably high computational costs. To tackle this problem, we apply a policy-based algorithm to search for the optimal attack location.
Inspired by [56], we use the REINFORCE algorithm [48] to introduce gradients between attack locations and the cost function. We define \(\mathbb{C}\) as a finite area surrounding the prohibited item, or the "available attacking area", based on the ground truth bounding box, where we can place our adversarial objects. We simulate the common suitcases scanning scenario in the fixed top-down orientation and divide \(\mathbb{C}\) into \(N\) evenly spaced grids in the 2D space. In this scenario, the searching problem is relatively simple and the trajectory (state\(\rightarrow\)input, action\(\rightarrow\)location, reward\(\rightarrow\)loss) has only one timestep, and we use \(N\) discrete actions to substitute location-choosing operations in a continuous area. We define the policy network \(\pi_{\mathbf{w}}\) with parameters \(\mathbf{w}\), which receives the original image \(\mathbf{I}\) as input and outputs the attacking location \(\mathbf{C}\). The gradient of the objective function \(J(\mathbf{w})\) with respect to \(\mathbf{w}\) is shown as:
\[\nabla_{\mathbf{w}}J(\mathbf{w})=G\cdot\nabla_{\mathbf{w}}\log\pi(\mathbf{C}| \mathbf{I};\mathbf{w}), \tag{10}\]
where \(G\) refers to the reward of the policy. To enhance the feasibility in the physical world, we expect the locations of adversarial objects to vary, which can be quantified by the standard variance \(\sigma_{\mathbf{C}}\). Therefore, \(G\) consists of two components, attack capability, and location diversity, which can be written as:
\[G=\mathcal{L}_{cls}(f_{\Theta}(\mathbb{R}_{\delta}(\mathbf{X},\mathbf{x}_{adv }^{\mathbf{P}_{ori};\mathbf{C}})),\{\mathbf{y}_{k},\mathbf{b}_{k}\})+\alpha \cdot\sigma_{\mathbf{C}}, \tag{11}\]
where \(\mathbf{P}_{ori}\) is the initial shape of the adversarial object, and \(\alpha\) balances the two terms. With our policy-based searching algorithm, we can jointly optimize the location and shape of our adversarial objects, enabling us to perform efficient and effective physical-world attacks.
### Overall Optimization
Based on the above discussions, the overall optimization function of our attacks \(\mathcal{L}\) consists of the attack loss \(\mathcal{L}_{adv}\) and perceptual loss \(\mathcal{L}_{per}\), which can be written as follows:
\[\mathcal{L}(\mathbf{X},\mathbf{x}_{adv};f_{\Theta},\mathbb{R}_{\delta})= \mathcal{L}_{adv}(\mathbf{X},\mathbf{x}_{adv};f_{\Theta},\mathbb{R}_{\delta}) +\beta\mathcal{L}_{per}(\mathbf{x}_{adv},\mathbf{x}_{ori}), \tag{12}\]
where we append the adversarial attack loss with a perceptual loss \(\mathcal{L}_{per}\) to ensure the physical feasibility of adversarial meshes, while \(\beta\) is a coefficient to balance the two loss functions. Inspired by [4, 49], we further introduce a total variation loss into our perceptual loss to restrict the shape change as
\[\mathcal{L}_{perc}(\mathbf{x}_{adv},\mathbf{x}_{ori})=\frac{1}{|\mathbf{x}|} \sum_{V\in\mathbf{x}_{adv}}\sum_{v_{i}\in V}\sum_{v_{q}\in N(v_{i})}||\Delta v _{i}-\Delta v_{q}||_{2}^{2}, \tag{13}\]
where \(V\) is the vertex set of 3D adversarial meshes, \(\Delta v_{i}\) indicates the perturbation distance of a certain vertex \(v_{i}\) between \(\mathbf{x}_{adv}\) and the original object \(\mathbf{x}_{ori}\), and \(N(v_{i})\) refers to the vertices adjacent to \(v_{i}\). The perceptual loss expects that a vertex will have similar perturbations to its neighbors, which avoids severe distortion of adversarial meshes. The pseudo-algorithm code of our \(\mathcal{X}\)-\(Adv\) can be found in Appendix A.1.
Experiments
### Experimental Setup
**Datasets.** We choose the commonly-used OPIXray [47] and HiXray [43] datasets. Specifically, the OPIXray dataset has 7,109 images in the training set and 1,776 images in the test set, with five prohibited item categories (_e.g_., Folding Knife, Straight Knife). All data are scanned from an X-ray security inspection machine to reproduce the real-world scenario in public transportation hubs. The HiXray dataset has 36,295 images in the training set and 9,069 images in the test set, with eight prohibited item categories such as lithium battery, liquid, lighter, _etc_. Images from these datasets are captured and gathered from realistic sources, which contain diverse suitcases of different sizes and shapes (_e.g_., open trays, bags, luggage), and the prohibited items are surrounded randomly by other items of different materials (_e.g_., clothes, phones, laptops). Thus, experiments on them could better verify the effectiveness of our attack.
**Target models.** To verify the effectiveness of our attacks, we train both one-stage SSD [27] and two-stage Faster R-CNN [15] to attack; we also attack the state-of-the-art and commonly used detectors in X-ray prohibited item detection scenario (DOAM [47] and LIM [43]), where we achieve similar results on clean images compared to their original papers.
**Compared baselines.** As discussed above, we are the first to study adversarial attacks for X-ray prohibited item detection, especially in the physical world. However, to better illustrate the superiority of our attacks, we transfer some adversarial attacks from prior works into the X-ray image scenario and compare our \(\mathcal{X}\)-\(\mathit{Adv}\) with them. Specifically, we use the original adversarial patch [2] (denoted as "AdvPatch") combined with our differentiable converter to generate 2D patches, which have no physical feasibility. As for 3D meshes, we apply meshAdv [49] with a certain color of the adversarial patch (denoted as "meshAdv") as a baseline. We also apply vanilla adversarial objects without shape polishment and location reinforcement (denoted as "Vanilla") to examine the capability of the attacks above. Considering the cross-task domain gap, it is reasonable to expect that these comparison methods will not perform as well as their source works on the task at hand.
**Evaluation metrics.** We select the most widely used metric, _i.e_., mAP, as the main evaluation metric. The mAP value depicts the overall performance according to precision and recall values, _i.e_., the area integral to the prediction precision \(\frac{TP}{TP+FP}\)) and the prediction recall (\(\frac{TP}{TP+FN}\)) of object detection. Note that we set the IoU value (the intersection rate of the predicted border and the real border) as 50%. In particular, the lower mAP values indicate better attack performance. For the untargeted attack, we use mAP to evaluate the attacking performance; for the targeted attack, besides mAP, we also report the False Negative (FN) values with confidence as 0.8 (the higher the better).
**Implementation details.** We define the size of the adversarial object as 20\(\times\)20 square pixel and the number of objects as 4, which takes around 2% of the whole image. _More details are shown in Appendix A.3_. All the codes are implemented with PyTorch. For all experiments, we conduct the training and testing on an NVIDIA GeForce RTX 2080Ti GPU cluster.
**X-ray converter.** We obtain the coefficient of the X-ray converter using a commercial AT6550 X-ray scanner. In practice, we have scanned 8 thicknesses (_0.2\(\sim\)8mm_) of iron objects, 22 thicknesses (_1\(\sim\)60mm_) of aluminum objects, and 6 thicknesses (_60\(\sim\)120mm_) of plastic objects using our X-ray machine. Then we sampled their color under X-ray images. We use Eqn. 7 as the convert function, the coefficients of which are acquired from regression fitting.
### Digital-world Attacks
In this part, we evaluate our \(\mathcal{X}\)-\(\mathit{Adv}\) in the digital world under both white-box and black-box settings. Specifically, for the white-box setting, we generate the adversarial object based on the model, then test its attacking ability on the same model; for the black-box setting, we first optimize the adversarial object on one model, then test its attack performance on other models via transfer-based attack. In more detail, we employ 4 models in the digital-world experiments including both one-stage and two-stage detectors, and the white-box attack results on OPIXray are shown in Table 1. _More results on HiXray and black-box attacks can be found in Appendix B._ From the results, we can **identify**:
Despite having eliminated most of the colors and textures in the X-ray images, the adversarial attacks still pose challenges in the X-ray prohibited item detection scenario. For example, on the OPIXray dataset against DOAM, the clean mAP is 74.02%, while the mAP value drops significantly to **23.05%** after being attacked by our \(\mathcal{X}\)-\(\mathit{Adv}\). It should be noted that this observation can be made for all employed models: the observed average mAP degeneration is about **50%** on OPIXray and **30%** on HiXray. Moreover, our \(\mathcal{X}\)-\(\mathit{Adv}\) outperforms other baselines by large margins.
It should be noted that \(\mathcal{X}\)-\(\mathit{Adv}\) seems to fail in some categories of HiXray, _e.g_., laptops. We hypothesize that the reason lies in the characteristic of the target object. laptops usually occupy large proportions of an image, while our patch is much smaller than these objects. Therefore, detector models can obtain much more information about objects like laptops, which supports correct classification.
Moreover, it is important to note that the vanilla patch could not successfully attack the detector, which indicates that the observed vulnerabilities of these models are not the result of poorly trained detectors.
The results for the black-box setting in Appendix B show consistent phenomena. In summary, the digital-world evaluations demonstrate that our \(\mathcal{X}\)-\(\mathit{Adv}\) could successfully attack the X-ray prohibited item detectors and outperform other
baselines by large margins.
### Physical-world Attacks
In this part, we further investigate the X-ray prohibited item detection model robustness in the physical-world setting.
We first illustrate the **attack pipeline** for our physical-world attacks (see Fig 3). In detail: we first generate adversarial objects using our _X-Adv_ based on a white-box pre-trained DOAM target model; we then transform the adversarial objects from 3D mesh format into STL format so that we can use a third-party 3D printer to print these 3D objects in metal; we then put our adversarial objects into the fabric/plastic box with other items and employ several workers to scan them into X-ray images using a commercial AT150180B X-ray scanner (which is commonly used in the train station and airport security checkpoints); finally, we test our physical-world adversarial examples (X-ray images) on black-box X-ray prohibited item detection models, specifically, DOAM, LIM, and Faster R-CNN, which are trained on the physical-world dataset proposed in Section 7. Note that, we use the commercial X-ray scanners but cannot use their detection backend because these models/strategies are business secrets, however, we adopt a similar black-box Faster R-CNN. During the experiments, we have no access to and prior knowledge of these target detectors and X-ray scanners.
Specifically, we use _X-Adv_ to generate 16 adversarial metal objects and then print them in iron using a 3D printer. We collect items (_e.g_., laptops, headphones, bags) from our staff and students under their grants. In total, we collected 80 adversarial X-ray images as the test set; some physical-world clean and adversarial X-ray images can be found in Figure 4. Note that all X-ray images are collected without personal information to avoid privacy leakage. To assess the real-world feasibility of our attacks, in addition to having our X-Adv search for the best possible attack location (denoted as "Physical best"), we also impose two attack settings to better simulate the physical-world dynamic environment of items movement in luggage: (1) slight transformations and (2) random placement. Specifically, slight transformations add shift (random 10 pixels in each direction) and rotation (-30\({}^{\circ}\)\(\sim\)+30\({}^{\circ}\)) to adversarial objects (denoted as "Physical change"), while random placement randomly places the adversarial objects in the entire suitcase (denoted as "Physical random"). For better comparison, we also provide the results of the 80 images using digital-world attacks (denoted as "Digital attack") and the physical-world results on clean examples (denoted as "Clean").
From Table 2, we can conclude that the physical attacking
\begin{table}
\end{table}
Table 1: Digital-world white-box attacks on OPIXray. “FO”, “ST”, “SC”, “UT”, and “MU” represent Folding Knife, Straight Knife, Scissor, Utility Knife, and Multi-tool Knife.
Figure 3: Illustration of the physical-world attacking pipeline. _X_-adv first generates adversarial meshes (3D objects); we then print these meshes into metal objects using 3D printers; when scanned by X-ray scanners, these metal objects will become adversarial patches in the resulting X-ray images.
ability of the proposed \(\mathcal{X}\)-\(Adv\) has a significant impact on detection accuracy,, the mAP value of DOAM on physical clean samples is 91.35%, while the mAP values on the sampled adversarial samples are 33.16% on "Physical best", 50.97% on "Physical change", and 76.17% on "Physical random", which are lower than the results on the clean counterparts. This observation also indicates that the safety problem of X-ray prohibited item detection is worth studying from a practical perspective. Moreover, we also observe that the attacking ability of "Physical best" is stronger than that of "Physical change" (lower mAP), which supports our motivation to search for the critical attack position. Furthermore, compared to the digital-world attack results, the physical-world attack results are weaker; we speculate this is because of the digital-physical domain gap [13, 46].
### Ablation Studies
In this section, we investigate the key factors that might impact the attack ability of our \(\mathcal{X}\)-\(Adv\), thereby providing comprehensive insights and promoting a deeper understanding of our strategy. In brief, we conduct thorough ablations on several factors. All the experiments conducted in this part use the DOAM target model on OPIXray and HiXray datasets.
**Attack locations.** Here, we investigate three additional location-searching strategies on the attack performance,, fixed position (denoted as "Fix"), random positions (denoted as "Random"), and greedy-search-based positions (denoted as "Greedy"). Our proposed attacking location search strategy is denoted as "Reinforce". For Fix, we place the adversarial objects on the corners of the prohibited items; for Random, we place the adversarial objects randomly around the prohibited items; for Greedy, we first greedy-search the strongest attack locations that maximize \(\mathcal{L}_{cls}\) by placing one original object at each location, then optimize the adversarial objects at the corresponding locations. The experimental results on OPIXray and HiXray can be found in Table 3, where we can observe that among all 4 attacking location searching strategies, the result under our "Reinforce" setting shows the strongest adversarial attacking performance.
Moreover, we study a more limited setting where the adversary could stick an adversarial object on the prohibited item. In particular, we add experiments on OPIXray against DOAM, where we put a 32\(\times\)32 iron rectangle or a 40\(\times\)40 adversarial object generated by \(\mathcal{X}\)-\(Adv\) on top of the target object. The attack performance (49.48 mAP and 25.28 mAP) is still worse than our original position-searching strategy (23.05 mAP). Note that directly placing the target object into an iron box or hiding it with iron plates could make it disappear, which can be easily identified in practice. Meanwhile, this would also violate the definition of adversarial attacks (cover the salient parts and change its semantics). The goal of this experiment is to illustrate the importance of location-searching.
The above studies demonstrate that avoiding overlap and occlusion can increase the attack capability.
**Number of objects.** Regarding the number of adversarial objects, we study whether the attack ability of the adversarial objects differs when this number is changed. Thus, we set the number of the adversarial objects to 1, 2, 4, and 8 respectively, while keeping the total area of each setting the same. The results can be found in Figure 5. As the results show, more objects usually result in better attack performance. However, too many adversarial objects will introduce an additional cost in terms of physical feasibility. In practice, we set the number of objects as 4.
### Discussions and Analysis
In this part, we provide more detailed discussions and analysis on the attack ability and physical feasibility of \(\mathcal{X}\)-\(Adv\). All
\begin{table}
\end{table}
Table 2: Physical-world attack experiments on different detection models. More results are shown in Appendix B.
Figure 4: Detection results of some X-ray images in our physical-world experiments (we choose images with fewer items for better visualization). Green boxes indicate correct classes and suitable locations; blue boxes represent correct classes in incorrect locations; red boxes indicate incorrect classes. We only show detection boxes with confidence >10%.
the experiments conducted in this part are using the DOAM target model based on OPIXray and HiXray datasets.
**Object materials.** Different materials are rendered on X-ray images in different colors; therefore, it is necessary to investigate their possible influence on the final attack ability of the generated adversarial objects. To this end, we select three kinds of colors (materials), namely blue, green, and orange, which respectively correspond to three materials that commonly appear in the luggage, _i.e_., iron, aluminum, and plastic. The results on OPIXray can be found in Figure 7(a). It is clear that the generated adversarial objects with blue colors (iron) show stronger attack ability, _i.e_., lower mAP values. For instance, on the OPIXray dataset, the mAP value of the iron adversarial objects is **23.05%**, while that of the green (aluminum) ones is 55.44%, and that of the orange (plastic) ones is 55.61%. We believe that this observation is reasonable since prohibited items (such as knives and guns) tend to be made of metal, thus adversarial objects made of similar materials (and rendered in similar colors) will have a higher attack ability. On the OPIXray dataset, the prohibited items are knives, meaning that blue (iron) outperforms other colors significantly. _Results on HiXray can be found in Appendix B_.
**Targeted adversarial attacks.** As shown in Eqn. 12, our \(\mathcal{X}\)-\(Adv\) maximizes the classification loss \(\mathcal{L}_{cls}\) of the detector to output wrong classes, which is the _untargeted attacks_. In addition to the untargeted attack, we here further design another reasonable attack strategy, _i.e_., perform attacks to evade the detector by confusing the predictor to classify the object of interest as Background. Thus, we apply _targeted attacks_ and set the target label of attacks to Background (one of the classes for detection). Specifically, we substitute \(\mathcal{L}_{cls}\) with a cross-entropy loss between the confidence of all predicted boxes and the background class. Since the background is the 0-th class of object detectors, performing attacks that mislead all boxes into the background class can also reduce the number of predicted boxes.
As shown in Figure 7(b), in terms of mAP, the performance of targeted attacks is weaker than that of untargeted attacks (38.82 v.s. 23.05). However, in terms of FN bounding box numbers, the performance of targeted attacks outperforms untargeted attacks largely (1632 v.s. 1274). These results demonstrate the different adversarial goals for targeted and untargeted attacks, and the targeted attack in the X-ray security inspection scenario might be more meaningful. We conjecture the main reasons for the above observations are as follows: It is more difficult for targeted attacks to reduce the mAP values in general object detection [28]. As the distribution of all bounding boxes shown in Figure 8, the untargeted attacks produce more false bounding boxes (FP) with high confidence, which could help to reduce the precision of detectors. However, targeted attacks result in fewer FP boxes, and this will help to prevent the detection of prohibited items, but the overall precision will not be too low.
**Unseen prohibited items.** Moreover, we are interested in discovering the potential of \(\mathcal{X}\)-\(Adv\) for attacks on other prohibited items with unseen materials/shapes. In other words,
\begin{table}
\end{table}
Table 3: Ablation studies on different attack locations. Our strategy achieves the best attack performance.
Figure 5: Ablations on the numbers of adversarial objects.
Figure 6: Visualization of adversarial objects with different materials/colors (_i.e_., blue, orange, and green).
Figure 7: Results using DOAM on OPIXray: (a) different materials, (b) targeted and untargeted adversarial attacks.
adversarial objects are first generated on specific types of prohibited items, and then we use them to directly attack other unseen prohibited item types without re-training. Here, we first conduct experiments in the digital world, where we train a group of adversarial objects using DOAM on HiXray (unseen prohibited items: lighter, liquids, _etc_) and then directly test them against another DOAM on OPIXray (prohibited items are knives). Overall, our attack achieves 29.40 mAP which is slightly lower than the original \(\mathcal{X}\)-_Adv_ attack (23.05 mAP) that is trained on OPIXray. We then verify this in the physical world, where we place the adversarial objects around their unseen prohibited items, and we achieve 45.52 mAP, which is also slightly lower than the original \(\mathcal{X}\)-_Adv_ attack (33.16 mAP). The above results indicate that our attack could still work for other unseen materials/shapes without re-training.
## 6 Countermeasures against \(\mathcal{X}\)-_Adv_
In this section, we propose three possible defenses and evaluate our \(\mathcal{X}\)-_Adv_ against them.
**Data Augmentation.** Data augmentation has been identified as a popular approach for improving model robustness [36]. In light of this, we introduce the data augmentation strategy as the first countermeasure to mitigate our adversarial attacks. Given the special feature space of X-ray image recognition (_i.e_., limited colors and textures), we believe that introducing additional adversarial-object-like patches might be beneficial for improving the robustness of the X-ray prohibited item detection models. Specifically, for each image, we randomly add 1-4 blue or orange patches and mix the clean examples with the additional examples during training using a ratio of 1 : 1. The results can be found in Table 3(a). Here, "V+C" denotes that the detector is trained **without** additional examples and tested **on** clean examples, "V+A" denotes that the detector is trained **without** additional examples and tested **on** adversarial examples, "D+C" denotes that the detector is trained **with** additional examples and tested **on** clean examples and "D+A" denotes that the detector is trained **with** additional examples and tested **on** adversarial examples. It can be observed that the data augmentation can to a certain extent effectively defend the proposed \(\mathcal{X}\)-adv method for X-ray prohibited item detection.
**Adversarial Detection.** Another prevailing approach to improving model robustness is adversarial detection. Rather than correctly detecting the target item under the adversarial scenario, adversarial detection aims to detect the existence of adversarial examples [5, 9, 14, 29]. Here, we build a neural classifier capable of distinguishing images containing adversarial objects from clean X-ray images. Specifically, we use a ResNet50 model as a classifier and trained on adversarial examples generated by meshAdv on different models (_i.e_., DOAM and LIM). We then test the classifier on adversarial examples generated by \(\mathcal{X}\)-_Adv_ on a different DOAM model. The training set and test set of the classifier do not overlap. The results in Table 4(b) indicate that the adversarial examples generated by different methods are quite different and the neural classifier fails to generalize.
**Adversarial Training.** We choose adversarial training (AT) as the last countermeasure for \(\mathcal{X}\)-_Adv_. Although AT for image classification has been widely studied [25, 30], only
\begin{table}
\end{table}
Table 4: Countermeasure studies. (a) “V” and “D” denote vanilla training or data augmentation; “C” and “A” refer to testing on clean or adversarial examples; (b) We first generate adversarial examples by meshAdv on DOAM/LIM and train the classifier; we then test the detection performance on \(\mathcal{X}\)-_Adv_ generated on DOAM. ACC denotes classification accuracy, and AUC is the area under the ROC curve; (c) We adversarially train a prohibited item detector using RobustDet.
Figure 8: The distribution of TP and FP bounding boxes under different targeted and untargeted adversarial attacks. “TP” represents True Positive, while “FP” denotes False Positive.
some preliminary studies have been devoted to object detection [6, 10, 52]. Here, we adopt RobustDet [10] as the AT method and use an SSD detector with a backbone of VGG-16. Specifically, we adversarially train two detectors using adversarial examples generated by (1) PGD attacks or (2) our _X-Adv_. Note that the generated adversarial objects by \(\mathcal{X}\)-_Adv_ during AT are different from those for testing. From Table 3(b), we could observe that AT trained on PGD attacks show limited defense against _X-Adv_ mainly due to the differences between perturbations and patch/object attacks; AT trained on _X-Adv_ could mitigate our _X-Adv_ attacks to a certain extent; and compared to classification, it is still comparatively difficult to adopt AT on object detection tasks.
**Summary and Discussion.** Despite facing significant challenges launched by our attack, the proposed defenses could still mitigate its negative influence to some extent. Specifically, for the strongest countermeasure (AT trained on our _X-Adv_), we could significantly improve the model robustness and achieve 53.47 mAP against _X-Adv_ attacks. Meanwhile, we could observe that the proposed countermeasures have rather high practical feasibility for defenders mainly due to the adversarial-object-like patches of data augmentation and adversarial attacks of adversarial detection can be easily generated and obtained for model training; \(\bigoplus\)_X-Adv_ adversarial objects of adversarial training can be generated either based on our open-sourced codes or the provided adversarial objects in the XAD dataset. For the physical-world implementation, defenders could simply employ a 3D printer to print out the 3D objects based on our guidelines. Moreover, defenders could combine adversarial detection with an AT model, which could further mitigate the negative impacts of _X-Adv_ attacks. _The feasibility of data augmentation and X-Adv AT are verified under the physical-world setting in Appendix B._
## 7 Physical-world X-ray Attack Dataset
A dataset is significantly beneficial for boosting research, especially for areas where professional benchmarks are lacking or the data collection is expensive. As we have observed in Section 5.3, physical-world X-ray detectors are vulnerable to our attacks. Therefore, we further present a physical-world X-ray inspection security robustness evaluation dataset to promote the design of robust X-ray prohibited item detectors.
### Construction Details
We first introduce the construction process of our Physical-world X-ray Attack Dataset (XAD), including the data collection, category selection, and quality control.
**Data collection**. We exploit one advanced X-ray security inspection machine, AT150180B, to generate the X-ray images in our dataset. We first randomly place the objects in the plastic/fabric box to mimic the similar environment in the real-world scenario; we then send these boxes through the security inspection machine, after which the machine outputs the X-ray images. To prevent privacy leakage, all images are collected legally: items are borrowed from our staff members and students, and do not contain personal information.
**Category selection**. As shown in Figure 9, we select 4 categories of prohibited items (cutters, scissors, folding knives, straight knives, and utility knives) that frequently appear in daily life. The 4 categories of cutters have different shapes and scales, which meets the category selection diversity requirements. The sufficient numbers of instances can provide a more credible evaluation for various models.
**Quality control procedure**. We followed a similar annotation quality control procedure to the famous vision dataset Pascal VOC [11]. All annotators followed the same annotation guidelines, including what to annotate, how to annotate bounding, how to treat occlusion, _etc_. Moreover, to ensure the accuracy of annotation, we divided the annotators into 3 groups. All images were randomly assigned to 2 out of the 3 groups for annotation, after which a final group was specially organized for confirmation.
### Data Properties
**Subset division.** Our XAD contains two subsets, _i.e_., a training set with clean X-ray images and a testing set with physical-world adversarial attacks. The 4,537 images in the training set simulate the real-world scenario to help models achieve satisfactory generalization performance. For the testing set, we follow [18] and generate adversarial images from 210 clean X-ray images with 4 different severity levels (0 to 4, where "0" denotes clean images). Specifically, for each item layout, we first place all the items in the box and X-ray scanned them to obtain a clean image; we then place 1\(\sim\)4
Figure 9: Illustration of the images in our XAD. (a) illustrates the prohibited item categories; (b) denotes the X-ray images of physical-world adversarial objects; (c) denotes the different severity levels in the testing set.
adversarial metals in the box respectively and scanned them to collect 4 versions of adversarial images with 4 severity levels. Thus, our testing set contains 1,050 samples which are all scanned from a real X-ray machine. See Fig 9 for samples.
**Category distribution.** Our XAD dataset contains 4,537 images and 4 categories of 4,830 instances with bounding-box annotations of prohibited cutters.
**Color Information**. The colors of objects under X-ray are determined by their chemical composition, mainly reflected in the material, which is introduced in Table 5(b).
**Instances per image.** In the training set of our XAD, each image contains at least one prohibited object. In particular, the image numbers containing 1, 2, 3, and 4 prohibited objects are 4069, 234, 23, and 1, respectively.
### Preliminary Experiments on XAD
After introducing our XAD, we further conduct experiments on XAD to demonstrate the difficulties and practicability of maintaining robust detectors. Specifically, we use the DOAM model for detection. We train the model on the training set of XAD and then evaluate it on the test set. The implementation details are the same as our main experiment. From Table 6, we can make several observations:
The detector shows weak performance on our XAD dataset with the model's performance on prohibited item recognition reducing by as much as 60% in terms of mAP.
Increasing the number of adversarial objects improves the attack and therefore increases the recognition difficulties in this scenario. We encourage researchers to design stronger training strategies or defense modules and evaluate their robustness on this benchmark.
## 8 Conclusion and Future Work
This paper takes the first step to study physical-world adversarial attacks for X-ray prohibited item detection. Specifically, we propose \(\mathcal{X}\)-_Adv_, which generates physically realizable adversarial objects to circumvent the color fading and complex occlusion problems in this scenario. Although the results presented here are promising, there are several research directions that we are interested in exploring in the future.
We hope \(\mathcal{X}\)-_Adv_ can be used as a tool to better debug and understand the nature of object detectors' robustness.
We would like to generate attacks in other soft materials which are more stealthy.
We are interested in attacks against more types of prohibited items.
Our \(\mathcal{X}\)-_Adv_ can be regarded as a general attacking framework for visually constrained scenarios. In this paper, we focus on attacking X-ray inspection scenarios; we will further extend our attacks to other complex scenarios.
## 9 Ethics Statement
As an effective way to discover safety problems, adversarial attacks will encourage researchers to pay more attention to model robustness. Based on this, this paper proposes \(\mathcal{X}\)-_Adv_ to attack X-ray prohibited item detectors. Our large-scale experiments demonstrated that existing X-ray prohibited item detection models (even commercial systems) are not infallible and can still be easily deceived. All experiments are conducted on public-available datasets, and all images for physical-world attacks are collected legally from our staff and students without personal information under their grants.
To mitigate potential real-world impacts of the attacks, this paper proposes three countermeasures and discusses their practical feasibility for defending \(\mathcal{X}\)-_Adv_ ;
presents the physical-world X-ray attack dataset XAD to promote the design and re-training of stronger detectors; and
disclose the results, countermeasures, and resources to two relevant X-ray security inspection service providers and a stakeholder user at the airport checkpoint. Based on our easy-to-use countermeasures, we help them to recognize this critical security issue and move the first step to improve the robustness of their detection backend with adversarial training. Moreover, these service providers are also suggested to utilize the white-box \(\mathcal{X}\)-_Adv_ to help reveal the vulnerabilities of their detectors and further design stronger models. Despite the threats identified in this paper, we should note that a real-world adversary would still find it difficult to pass X-ray security inspection systems carrying prohibited items without detection, as human inspectors still help with checking luggage. We thus further suggested the airport checkpoint pay attention to the employment of human inspectors, which can relieve the concerns on the potential negative abuses of \(\mathcal{X}\)-_Adv_.
\begin{table}
\end{table}
Table 6: Results on different levels of XAD.
\begin{table}
\end{table}
Table 5: Detailed data properties of our XAD dataset.
## Acknowledgement
The authors would like to thank the shepherd and all the anonymous reviewers for their tremendous efforts and insightful comments during reviewing. This work was supported by the National Key Research and Development Plan of China (2021ZD0110601), the National Natural Science Foundation of China (No. 62022009 and 62206009), the State Key Laboratory of Software Development Environment (SKLSDE-2022ZX-23), and the grants from SenseTime.
| adversarial攻撃は、深層学習モデルの robustness を評価するための価値のあるツールです。既存の攻撃は主に可視光スペクトラム（例：ピクセル単位の質感変化）に集中しています。しかし、テクスチャーのないX線の画像をターゲットにした攻撃は、安全性の高いシナリオ（例：禁止品 X線検出）の幅広い適用状況においてまだ未探索な領域です。この論文では、X線禁止品検出に特化した攻撃研究の第一歩を踏み出し、この安全性の高いシナリオにおけるそのような攻撃に与える深刻な脅威を明らかにします。具体的には、このシナリオにおける成功した物理的攻撃は、色/質感の fading と複雑な重なり合いを回避するために特殊に設計されている必要があります。このため、X-advを提案し、禁止品を装い込む荷物の中に存在する物理的に印刷可能な金属を生成します。色/ |
2301.10490 | The delayed fracture test for viscoelastic elastomers | In a recent contribution, Shrimali and Lopez-Pamies (2023) have shown that
the Griffith criticality condition that governs crack growth in viscoelastic
elastomers can be reduced -- from its ordinary form involving a historically
elusive loading-history-dependent critical tearing energy $T_c$ -- to a
fundamental form that involves exclusively the intrinsic fracture energy $G_c$
of the elastomer. The purpose of this paper is to make use of this fundamental
form to explain one of the most distinctive fracture tests for viscoelastic
elastomers, the so-called delayed fracture test. | Bhavesh Shrimali, Oscar Lopez-Pamies | 2023-01-25T09:59:00 | http://arxiv.org/abs/2301.10490v1 | # The delayed fracture test for viscoelastic elastomers
###### Abstract
In a recent contribution, Shrimali and Lopez-Pamies (2023) have shown that the Griffith criticality condition that governs crack growth in viscoelastic elastomers can be reduced -- from its ordinary form involving a historically elusive loading-history-dependent critical tearing energy \(T_{c}\) -- to a fundamental form that involves exclusively the intrinsic fracture energy \(G_{c}\) of the elastomer. The purpose of this paper is to make use of this fundamental form to explain one of the most distinctive fracture tests for viscoelastic elastomers, the so-called delayed fracture test.
Keywords:Dissipative Solids; Fracture Nucleation; Fracture Energy
## 1 Introduction
It has been long established that the Griffith criticality condition
\[-\frac{\partial\mathcal{W}}{\partial\Gamma_{0}}=T_{c} \tag{1}\]
describes the nucleation of fracture from pre-existing cracks -- as well as the propagation of fracture -- in elastomers subjected to quasi-static mechanical loads (Rivlin and Thomas, 1953; Greensmith and Thomas, 1955; Mullins, 1959; Lake and Thomas, 1967; Ahagon and Gent, 1975; Gent and Tobias, 1982; Gent, 1996; Tsunoda et al., 2000; Knauss, 2015).
The left-hand side \(-\partial\mathcal{W}/\partial\Gamma_{0}\) in expression (1) stands for the change in total (stored and dissipated) deformation energy \(\mathcal{W}\) in the elastomer with respect to an added surface area to the pre-existing crack \(\Gamma_{0}\) under conditions of fixed deformation of those parts of the boundary that are not traction-free so that no work is done by the external forces; note that the added surface area refers to the undeformed configuration.
The right-hand side \(T_{c}\) is the so-called critical tearing energy. It is a characteristic property of the elastomer. Importantly, it is _not_ a constant. Much like \(\mathcal{W}\), it is a function of the loading history. More specifically, experiments have established that \(T_{c}\) can be written in the general form
\[T_{c}=G_{c}(1+f_{c}).\]
In this expression, \(G_{c}\) denotes the intrinsic fracture energy, or critical energy release rate, associated with the creation of new surface in the given elastomer. It is a material constant, independent of time. Its value is in the relatively narrow range \(G_{c}\in[10,100]\) N/m for all common hydrocarbon elastomers (Ahagon and Gent, 1975; Gent and Tobias, 1982). On the other hand, \(f_{c}\) is a non-negative function of the loading history that scales with the viscosity of the elastomer (Mullins, 1959; Gent and Lai, 1994; Knauss, 1973; Gent, 1996). Precisely how
\(f_{c}\) -- and hence \(T_{c}\) -- depends on the loading history for arbitrary loading conditions has remained an open problem for decades rendering the Griffith criticality condition in its ordinary form (1) of limited practical utility.
In a recent contribution, Shrimali and Lopez-Pamies (2023) have uncovered a general formula for \(f_{c}\) -- and hence \(T_{c}\) -- and in so doing they have determined that (1) can in fact be reduced to a form that involves _not_ the historically elusive critical tearing energy \(T_{c}\), but only the intrinsic fracture energy \(G_{c}\) of the elastomer. The result hinges on the following two elementary observations.
1. For any viscoelastic elastomer, the total deformation energy \(\mathcal{W}\) in (1) can be partitioned into three different contributions: \[\mathcal{W}=\underbrace{\mathcal{W}^{\text{Eq}}+\mathcal{W}^{\text{NEq}}}_{ \text{stored}}+\underbrace{\mathcal{W}^{v}}_{\text{dissipated}}.\] (2) Here, \(\mathcal{W}^{v}\) represents the part of the total energy that is dissipated by the elastomer via viscous deformation. On the other hand, the combination \(\mathcal{W}^{\text{Eq}}+\mathcal{W}^{\text{NEq}}\) represents the part of the total energy that is stored by the elastomer via elastic deformation. In this combination, \(\mathcal{W}^{\text{NEq}}\) stands for the part of the stored elastic energy that will be dissipated eventually via viscous dissipation as the elastomer reaches a state of thermodynamic equilibrium. On the contrary, \(\mathcal{W}^{\text{Eq}}\) denotes the part of the stored elastic energy that the elastomer will retain at thermodynamic equilibrium.
Rheological representations of elastomers provide a helpful visualization of the partition (2). For instance, in the Zener-type rheological representation depicted in Fig. 1, \(\mathcal{W}^{\text{Eq}}\) and \(\mathcal{W}^{\text{NEq}}\) correspond to the elastic energy stored in the equilibrium and non-equilibrium springs, whereas \(\mathcal{W}^{v}\) corresponds to the viscous energy dissipated by the dashpot.
2. "Pure-shear" fracture tests of common hydrocarbon elastomers, as well as of more modern types of elastomers, consistently show -- rather remarkably -- that fracture occurs from the pre-existing crack in the specimens at a critical stretch that is independent (to within experimental error) of the stretch rate at which the test is carried out.
Precisely, by combining the above two observations, Shrimali and Lopez-Pamies (2023) have shown that the Griffith criticality condition (1) can be reduced to the fundamental form
\[-\frac{\partial\mathcal{W}^{\text{Eq}}}{\partial\Gamma_{0}}=G_{c}. \tag{3}\]
From a physical point of view, the criticality condition (3) states that whether an elastomer simply deforms or, on the other hand, creates new surface from a pre-existing crack is dictated by a competition between its stored equilibrium elastic energy and its intrinsic fracture energy, irrespective of its viscosity.
From a practical point of view, the criticality condition (3) is straightforward to employ. This is because it is based on quantities that can be measured experimentally once and for all by means of conventional tests. Indeed, on the one hand, conventional experiments suffice to characterize the viscoelasticity of the elastomer of interest from which the storage of equilibrium elastic energy can then be identified; see, e.g., Section 4 in (Shrimali and Lopez-Pamies, 2023) and also Section 4 below. On the other hand, experiments in the spirit of those carried out, e.g., by Gent and Tobias (1982) are enough to measure the intrinsic fracture energy \(G_{c}\) of the elastomer.
What is more, as already noted above, the criticality condition (3) brings resolution to the decades-old open problem of how the critical tearing energy \(T_{c}\) depends on the loading history, as it entails that
\[T_{c}=G_{c}(1+\,f_{c})=G_{c}-\frac{\partial\mathcal{W}^{\text{NEq}}}{\partial \Gamma_{0}}-\frac{\partial\mathcal{W}^{v}}{\partial\Gamma_{0}}, \tag{4}\]
where the last two terms, \(\partial\mathcal{W}^{\text{NEq}}/\partial\Gamma_{0}\) and \(\partial\mathcal{W}^{v}/\partial\Gamma_{0}\), are to be evaluated at the instance in time at which the criticality condition (3) is attained along the loading path of interest.
Remark 1: So as to provide a modicum of historical perspective, it is appropriate to make explicit mention of the various attempts at describing the
Figure 1: A rheological representation of viscoelastic elastomers.
critical tearing energy \(T_{c}\) of elastomers that have been reported in the literature prior to the discovery of the general result (4). A representative but non-exhaustive list includes the works of Knauss (1973), Schapery (1975;1984), Christensen (1979), de Gennes (1996), and Persson and Brener (2005). Invariably, all of these attempts are based on derivations that are centered around tearing (or peeling) experiments where a crack is _propagated_ at a constant velocity. Save for an exception (Schapery, 1984), all of them restrict attention to _linear viscoelasticity_. Moreover, all of them make use either of a _cohesive zone_ or of an equivalent _cutoff region_ around the crack front, a constitutive assumption that further muddies their theoretical standing. By contrast, as already recalled above, the discovery of the general formula (4) was made possible by centering its derivation around _nucleation_ of fracture in "pure-shear" fracture experiments, in particular, around the seemingly universal fact that fracture nucleation in such experiments takes places at critical stretches that are independent of the applied stretch rate. What is more, given its general status, the formula (4) applies to any _nonlinear viscoelastic_ solid and is free of the constitutive restriction of having to explicitly identify a special region (the "fracture process zone") around the crack front.
The object of this paper is to make use of the newly-minted fundamental form (3) of the Griffith criticality condition in order to explain in a detailed and quantitative manner a tell-tale fracture test for viscoelastic elastomers: the so-called delayed fracture test. In a typical delayed fracture test, a sheet of the elastomer of interest containing a pre-exiting crack is subjected to a load that is applied rapidly over a very short time interval \([0,t_{0}\ll 1]\) and then held constant. Nucleation of fracture from the pre-existing crack occurs at a critical time \(t_{c}>t_{0}\), hence the name of the test. In this work, consistent with the setup used by Knauss (1970) in his pioneering experiments, we will focus on the configuration depicted in Fig. 2, where the pre-exiting crack is located in the center of the specimen and the load is applied in a uniaxial fashion.
The organization of the paper is as follows. We begin in Section 2 by formulating the pertinent initial-boundary-value problem. In Section 3, with the objective of exposing the chief characteristics of the delayed fracture test in the most basic of settings, we present and discuss sample generic results for the canonical case of a viscoelastic elastomer with Gaussian elasticity and constant viscosity. In Section 4, we explain the experiments of Knauss (1970) on Solithane 113, a polyurethane elastomer with non-Gaussian elasticity and nonlinear viscosity. We conclude by recording a number of final comments in Section 5.
## 2 Formulation of the initial-boundary-value problem for the delayed fracture test
### Initial configuration and kinematics
Consider the rectangular specimens depicted in Fig. 2 of length \(L=101.6\) mm and height \(H=L=101.6\) mm in the \(\mathbf{e}_{3}\) and \(\mathbf{e}_{1}\) directions and constant thickness \(B=0.7938\) mm in the \(\mathbf{e}_{2}\) direction. The specimens contain a pre-existing central crack of five different lengths
\[A=2.5,5.08,10,15,20\,\mathrm{mm}\]
in the \(\mathbf{e}_{3}\) direction. As alluded to above, these specific values for \(L\), \(H\), \(B\), \(A\) are chosen because they include those utilized by Knauss (1970) in his original delayed fracture experiments. Here, \(\{\mathbf{e}_{i}\}\) stands for the laboratory frame of reference. We place its origin at the geometric center of the specimens so that, in their initial configuration at time \(t=0\), the specimens occupy the domain
\[\overline{\Omega}_{0}=\{\mathbf{X}:\mathbf{X}\in\mathcal{P}_{0}\setminus \Gamma_{0}\},\]
Figure 2: Schematic of a typical delayed fracture test for a viscoelastic elastomer. The specimen is held firmly by stiff grips. A load is applied rapidly from \(t=0\) to \(t=t_{0}\ll 1\) and then held constant. For a sufficiently large load, the nucleation of fracture from the pre-existing crack (of initial length \(A\) here) may occur at a critical time \(t_{c}>t_{0}\), hence the name of the test.
where
\[\mathcal{P}_{0}=\left\{\mathbf{X}:|X_{1}|\leq\frac{H}{2},\,|X_{2}|\leq\frac{B}{2}, \,|X_{3}|\leq\frac{L}{2}\right\}\]
and
\[\Gamma_{0}=\left\{\mathbf{X}:X_{1}=0,\,|X_{2}|\leq\frac{B}{2},\,|X_{3}|\leq \frac{A}{2}\right\}.\]
At a later time \(t\in(0,T]\), due to the applied boundary conditions described below, the position vector \(\mathbf{X}\) of a material point in the specimens will move to a new position specified by
\[\mathbf{x}=\mathbf{y}(\mathbf{X},t),\]
where \(\mathbf{y}\) is a mapping from \(\Omega_{0}\) to the current configuration \(\Omega(t)\). We consider only invertible deformations, and write the deformation gradient field at \(\mathbf{X}\) and \(t\) as
\[\mathbf{F}(\mathbf{X},t)=\nabla\mathbf{y}(\mathbf{X},t)=\frac{\partial \mathbf{y}}{\partial\mathbf{X}}(\mathbf{X},t).\]
### Constitutive behavior of the elastomer
The specimens are taken to be made of an isotropic incompressible elastomer. Making use of the two-potential formalism (Kumar and Lopez-Pamies, 2016), we describe its constitutive behavior by two thermodynamic potentials, the free energy
\[\psi(\mathbf{F},\mathbf{F}^{v})=\left\{\begin{aligned} \psi^{\text{Eq}}(I_{1})+\psi^{\text{ NEq}}(I^{e}_{1})&\text{ if }J=1\\ +\infty&\text{otherwise}\end{aligned}\right. \tag{5}\]
that describes how the elastomer stores energy through elastic deformation and the dissipation potential
\[\phi(\mathbf{F},\mathbf{F}^{v},\dot{\mathbf{F}}^{v})=\left\{\begin{aligned} &\frac{1}{2}\dot{\mathbf{F}}^{v} \mathbf{F}^{v-1}\cdot[2\,\eta(I^{e}_{1},I^{e}_{2},I^{v}_{1})\times\\ &\mathbf{\mathcal{K}}\,\dot{\mathbf{F}}^{v}\mathbf{F}^{v-1}]& \text{ if }\text{tr}(\dot{\mathbf{F}}^{v}\mathbf{F}^{v-1})=0\\ &+\infty&\text{otherwise}\end{aligned}\right. \tag{6}\]
that describes how the elastomer dissipates energy through viscous deformation. In these expressions, the second-order tensor \(\mathbf{F}^{v}\) is an internal variable of state that describes roughly the "viscous part" of the deformation gradient \(\mathbf{F}\), the "dot" notation stands for the Lagrangian time derivative (i.e., with \(\mathbf{X}\) held fixed),
\[I_{1}=\text{tr}\,\mathbf{C},\quad J=\sqrt{\det\mathbf{C}},\] \[I^{v}_{1}=\text{tr}\,\mathbf{C}^{v},\quad I^{e}_{1}=\text{tr}( \mathbf{C}\mathbf{C}^{v-1}),\] \[I^{e}_{2}=\frac{1}{2}\left[\left(\mathbf{C}\cdot\mathbf{C}^{v- 1}\right)^{2}-\mathbf{C}^{v-1}\mathbf{C}\cdot\mathbf{C}\mathbf{C}^{v-1}\right],\]
where \(\mathbf{C}=\mathbf{F}^{T}\mathbf{F}\) denotes the right Cauchy-Green deformation tensor, \(\mathbf{C}^{v}=\mathbf{F}^{v}\mathbf{T}^{v},\mathcal{K}_{ijkl}=\frac{1}{2}( \delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl})\) stands for the standard deviatoric orthogonal projection tensor, and \(\psi^{\text{Eq}}\), \(\psi^{\text{NEq}}\), \(\eta\) are any (suitably well-behaved) non-negative material functions of their arguments.
Granted the two thermodynamic potentials (5) and (6), it follows that the first Piola-Kirchhoff stress tensor \(\mathbf{S}\) at any material point \(\mathbf{X}\in\Omega_{0}\) and time \(t\in[0,T]\) is expediently given by the relation (Kumar and Lopez-Pamies, 2016)
\[\mathbf{S}(\mathbf{X},t)=\frac{\partial\psi}{\partial\mathbf{F}}(\mathbf{F}, \mathbf{F}^{v}),\]
where \(\mathbf{F}^{v}\) is implicitly defined by the evolution equation
\[\frac{\partial\psi}{\partial\mathbf{F}^{v}}(\mathbf{F},\mathbf{F}^{v})+\frac {\partial\phi}{\partial\dot{\mathbf{F}}^{v}}(\mathbf{F},\mathbf{F}^{v},\dot{ \mathbf{F}}^{v})=\mathbf{0}.\]
Making use of the specific isotropic incompressible forms (5) and (6), this relation can be rewritten more explicitly as
\[\mathbf{S}(\mathbf{X},t)=2\psi^{\text{Eq}}_{I_{1}}\mathbf{F}+2\psi^{\text{ NEq}}_{I^{e}_{1}}\mathbf{F}\mathbf{C}^{v-1}-p\mathbf{F}^{-T}, \tag{7}\]
where \(p\) stands for the arbitrary hydrostatic pressure associated with the incompressibility constraint \(J=1\) of the elastomer, \(\mathbf{C}^{v}\) is defined implicitly as the solution of the evolution equation
\[\dot{\mathbf{C}}^{v}(\mathbf{X},t)=\frac{2\psi^{\text{NEq}}_{I^{e}_{1}}}{ \eta(I^{e}_{1},I^{e}_{2},I^{v}_{1})}\left[\mathbf{C}-\frac{1}{3}\left(\mathbf{ C}\cdot\mathbf{C}^{v-1}\right)\mathbf{C}^{v}\right], \tag{8}\]
and where we have made use of the notation \(\psi^{\text{Eq}}_{I_{1}}=\text{d}\psi^{\text{Eq}}(I_{1})/\text{d}I_{1}\) and \(\psi^{\text{NEq}}_{I^{e}_{1}}=\text{d}\psi^{\text{NEq}}\)\((I^{e}_{1})/\text{d}I^{e}_{1}\). Note that the dependence on the internal variable \(\mathbf{F}^{v}\) ends up entering (7) and (8) only through the symmetric combination \(\mathbf{C}^{v}=\mathbf{F}^{v}\mathbf{T}^{v}\).
For a detailed account of the constitutive relation (7)-(8), the interested reader is referred to (Kumar and Lopez-Pamies, 2016). Here, we remark that the constitutive relation (7)-(8) corresponds to a generalization of the classical Zener or standard solid model (Zener, 1948) to the setting of
finite deformations. Accordingly, as schematically depicted by the rheological representation in Fig. 1, the function \(\psi^{\rm Eq}\) in (5) characterizes the elastic energy storage in the elastomer at states of thermodynamic equilibrium, whereas \(\psi^{\rm NeEq}\) characterizes the additional elastic energy storage at non-equilibrium states (i.e., again, the part of the energy that gets dissipated eventually). On the other hand, the function \(\eta\) in (6) characterizes the viscosity of the elastomer.
In the results that are presented in Sections 3 and 4 below, we will make use of the following specific forms for the equilibrium and non-equilibrium free-energy functions in (5) and viscosity function in (6):
\[\begin{cases}\psi^{\rm Eq}(I_{1})=\sum\limits_{r=1}^{N}\frac{3^{1-\alpha_{r} }}{2\alpha_{r}}\mu_{r}\left(I_{1}^{\alpha_{r}}-3^{\alpha_{r}}\right)\\ \psi^{\rm Neq}(I_{1}^{e})=\sum\limits_{r=1}^{N}\frac{3^{1-\beta_{r}}}{2\beta_{ r}}\nu_{r}\left(I_{1}^{e\beta_{r}}-3^{\beta_{r}}\right)\\ \eta(I_{1}^{e},I_{2}^{e},I_{1}^{e})=\eta_{\infty}+\frac{\eta_{0}-\eta_{\infty} +K_{1}\left[I_{1}^{v\gamma_{1}}-3^{\gamma_{1}}\right]}{1+\left(K_{2}\mathcal{ J}_{2}^{\rm Neq}\right)^{\gamma_{2}}}\end{cases} \tag{9}\]
with \(\mathcal{J}_{2}^{\rm Neq}=({I_{1}^{e}}^{2}/3-I_{2}^{e})(\sum_{r=1}^{N}3^{1- \beta_{r}}\nu_{r}I_{1}^{e\beta_{r}-1})^{2}\) and \(N=1,2\), which result in the constitutive relation
\[\mathbf{S}(\mathbf{X},t)= \sum\limits_{r=1}^{N}3^{1-\alpha_{r}}\mu_{r}I_{1}^{\alpha_{r}-1} \mathbf{F}+\] \[\sum\limits_{r=1}^{N}3^{1-\beta_{r}}\nu_{r}I_{1}^{e\beta_{r}-1} \mathbf{F}\mathbf{C}^{v-1}-p\mathbf{F}^{-T} \tag{10}\]
with evolution equation
\[\dot{\mathbf{C}}^{v}(\mathbf{X},t)= \frac{\sum\limits_{r=1}^{N}3^{1-\beta_{r}}\nu_{r}I_{1}^{e\beta_{ r}-1}}{\eta_{\infty}+\frac{\eta_{0}-\eta_{\infty}+K_{1}\left[I_{1}^{v_{1}}-3^{ \gamma_{1}}\right]}{1+\left(K_{2}\mathcal{J}_{2}^{\rm Neq}\right)^{\gamma_{2}} }}\left[\mathbf{C}-\frac{1}{3}\times\right.\] \[\left.\left(\mathbf{C}\cdot\mathbf{C}^{v-1}\right)\mathbf{C}^{v} \right]. \tag{11}\]
The constitutive prescription (10)-(11) includes several fundamental constitutive relations as special cases. For instance, it includes the case of a Neo-Hookean solid (\(N=1\), \(\nu_{1}=0\), \(\alpha_{1}=1\), \(\eta_{0}=\eta_{\infty}=0\), \(K_{1}=K_{2}=0\)), that of a Newtonian fluid (\(N=1\), \(\mu_{1}=0\), \(\nu_{1}=+\infty\), \(\eta_{\infty}=0\), \(K_{1}=K_{2}=0\)), as well as that of a viscoelastic elastomer with Gaussian elasticity and constant viscosity (\(N=1\), \(\alpha_{1}=\beta_{1}=1\), \(\eta_{\infty}=0\), \(K_{1}=K_{2}=0\)). What is more, the prescription (10)-(11) has been shown to be accurately descriptive and predictive of a wide range of elastomers, which typically exhibit non-Gaussian elasticity as well as nonlinear viscosity of shear-thinning type (Lopez-Pamies, 2010; Kumar and Lopez-Pamies, 2016; Ghosh and Lopez-Pamies, 2021; Chockalingam et al., 2021; Chen and Ravi-Chandar, 2022). In all, note that the constitutive prescription (10)-(11) contains \(4N+6\) material parameters. \(2N\) of them, \(\mu_{r}\) and \(\alpha_{r}\) (\(r=1,...,N\)), serve to characterize the non-Gaussian elasticity of the elastomer at states of thermodynamic equilibrium. Another \(2N\), \(\nu_{r}\) and \(\beta_{r}\) (\(r=1,...,N\)), characterize the non-Gaussian elasticity at non-equilibrium states. Finally, the last six parameters, \(\eta_{0}\), \(\eta_{\infty}\), \(K_{1}\), \(K_{2}\), \(\gamma_{1}\), \(\gamma_{2}\), serve to characterize the nonlinear shear-thinning viscosity.
### Initial and boundary conditions
In their initial configuration, we consider that the specimens are undeformed and stress-free. Therefore, we have the initial conditions
\[\begin{cases}\mathbf{y}(\mathbf{X},0)=\mathbf{X}\\ p(\mathbf{X},0)=2\psi_{I_{1}}^{\rm Eq}(3)+2\psi_{I_{1}^{e}}^{\rm Neq}(3)\, \quad\mathbf{X}\in\overline{\Omega}_{0}.\\ \mathbf{C}^{v}(\mathbf{X},0)=\mathbf{I}\end{cases} \tag{12}\]
The top
\[\partial\Omega_{0}^{\mathcal{T}}=\left\{\mathbf{X}:X_{1}=\frac{H}{2},\,|X_{2} |\leq\frac{B}{2},\,|X_{3}|\leq\frac{L}{2}\right\}\]
and the bottom boundary
\[\partial\Omega_{0}^{\mathcal{B}}=\left\{\mathbf{X}:X_{1}=-\frac{H}{2},\,|X_{2} |\leq\frac{B}{2},\,|X_{3}|\leq\frac{L}{2}\right\}\]
of the specimens are held firmly by stiff grips on which a force of magnitude
\[P(t)=\left\{\begin{aligned} &\frac{2\sigma_{0}(B\times L)t_{0}t}{t_{0}^{2}+t^{2}} &&\quad\text{if}\quad 0\leq t\leq t_{0}\\ &\sigma_{0}(B\times L)&&\quad\text{if}\quad t_{0}<t \leq T\end{aligned}\right. \tag{13}\]
is applied in the \(\pm\mathbf{e}_{1}\) directions resulting in a separation \(h(t)\) between the grips; see Fig. 2. In the results that are presented in Sections 3 and 4, consistent, once more, with the experiments of Knauss (1970), we make use of the values
\[t_{0}=0.01\,\text{s}\quad\text{and}\quad\sigma_{0}\in[0,0.3]\,\text{MPa}, \tag{14}\]
which correspond to a force \(P_{0}=\sigma_{0}(B\times L)\) that is applied rapidly over the very short time interval \([0,t_{0}]\) and then held constant. The rest of the boundary \(\partial\Omega_{0}\) of the specimens is traction free. Precisely, making use of the notation \(\mathbf{s}(\mathbf{X},t)=\mathbf{SN}\), we have that
\[\begin{cases}y_{1}(\mathbf{X},t)=\dfrac{h(t)}{2},&(\mathbf{X},t)\in\partial \Omega_{0}^{\mathcal{T}}\times[0,T]\\ y_{3}(\mathbf{X},t)=X_{3},&(\mathbf{X},t)\in\partial\Omega_{0}^{\mathcal{T}} \times[0,T]\\ \int_{\partial\Omega_{0}^{\mathcal{T}}}s_{1}(\mathbf{X},t)\mathrm{d}\mathbf{X }=P(t),&(\mathbf{X},t)\in\partial\Omega_{0}^{\mathcal{T}}\times[0,T]\\ s_{2}(\mathbf{X},t)=0,&(\mathbf{X},t)\in\partial\Omega_{0}^{\mathcal{T}} \times[0,T]\\ y_{1}(\mathbf{X},t)=-\dfrac{h(t)}{2},&(\mathbf{X},t)\in\partial\Omega_{0}^{ \mathcal{B}}\times[0,T]\\ y_{3}(\mathbf{X},t)=X_{3},&(\mathbf{X},t)\in\partial\Omega_{0}^{\mathcal{B}} \times[0,T]\\ \int_{\partial\Omega_{0}^{\mathcal{B}}}s_{1}(\mathbf{X},t)\mathrm{d}\mathbf{X }=-P(t),&(\mathbf{X},t)\in\partial\Omega_{0}^{\mathcal{B}}\times[0,T]\\ s_{2}(\mathbf{X},t)=0,&(\mathbf{X},t)\in\partial\Omega_{0}^{\mathcal{B}} \times[0,T]\\ \mathbf{s}=\mathbf{0},&(\mathbf{X},t)\in\partial\Omega_{0}\setminus\left( \partial\Omega_{0}^{\mathcal{T}}\cup\partial\Omega_{0}^{\mathcal{B}}\right) \times[0,T]\end{cases}, \tag{15}\]
where \(\mathbf{N}\) stands for the outward unit normal to the boundary \(\partial\Omega_{0}\).
Remark 2: In experiments, specimens like the ones of interest here are typically gripped in a way that complex triaxial stresses develop near the grips. Numerical experiments indicate that these localized stresses have practically no effect on the response of the specimens, thus our idealized choice of zero traction (15)\({}_{4,8}\) at the top and bottom boundaries.
Remark 3: In all the numerical solutions that are presented below, the mixed boundary conditions (15)\({}_{1,5}\) with (15)\({}_{3,7}\) are enforced by modeling explicitly the grips holding the specimens as nonlinear elastic materials with a stiffness 6 orders of magnitude larger than the elastomer being tested; see Fig. 5.
### Governing equations
Upon putting all the above ingredients together, neglecting inertia and body forces, the mechanical response of the specimens is governed by the equilibrium and incompressibility constraint equations
\[\begin{cases}\operatorname{Div}\mathbf{S}=\mathbf{0},&(\mathbf{X},t)\in \Omega_{0}\times[0,T]\\ \det\nabla\mathbf{y}=1,&(\mathbf{X},t)\in\Omega_{0}\times[0,T]\end{cases} \tag{16}\]
subject to the initial and boundary conditions (12)\({}_{1,2}\) and (15), where \(\mathbf{S}(\mathbf{X},t)=2\psi_{I_{1}}^{\mathrm{Eq}}\nabla\mathbf{y}+2\psi_{I _{1}}^{\mathrm{NEq}}\nabla\mathbf{y}\mathbf{C}^{v-1}-p\nabla\mathbf{y}^{-T}\), coupled with the evolution equation
\[\dot{\mathbf{C}}^{v}=\dfrac{2\psi_{I_{1}^{e}}^{\mathrm{NEq}}}{\eta(I_{1}^{e}, I_{2}^{e},I_{1}^{v})}\left[\nabla\mathbf{y}^{T}\nabla\mathbf{y}-\dfrac{1}{3} \left(\nabla\mathbf{y}^{T}\nabla\mathbf{y}\cdot\mathbf{C}^{v-1}\right) \mathbf{C}^{v}\right], \tag{17}\]
subject to the initial condition (12)\({}_{3}\), for the deformation field \(\mathbf{y}(\mathbf{X},t)\), the pressure field \(p(\mathbf{X},t)\), and the internal variable \(\mathbf{C}^{v}(\mathbf{X},t)\).
In the next two sections, we present numerical solutions for the initial-boundary-value problem (16)-(17) with (12)-(15) and (9) for two sets of material parameters. First, in Section 3, we generate results for the canonical case of an elastomer with Gaussian elasticity and constant viscosity. In Section 4, we generate results for the polyurethane elastomer studied by Knauss (1970). All the results that we present in the sequel are generated by a plane-stress variant of the numerical scheme introduced by Ghosh et al. (2021), which is based on a Crouzeix-Raviart finite-element discretization of space and a high-order explicit Runge-Kutta discretization of time.
The computation of the energy release rate \(-\partial\mathcal{W}^{\mathrm{Eq}}/\partial\Gamma_{0}\) under boundary conditions of
arguments in the functional description (19) of the equilibrium elastic energy, this can be accomplished as follows.
Given a specimen with initial surface area \(\Gamma_{0}\) of the pre-existing crack and given an applied force \(P(t)\), consider the addition of an increment \(\mathrm{d}\Gamma_{0}\) to \(\Gamma_{0}\), this at fixed \(P(t)\). On use of the condition \(\mathrm{d}P=0\), the associated incremental change in the equilibrium elastic energy \(\mathcal{W}^{\mathrm{Eq}}\) reads
\[\mathrm{d}\mathcal{W}^{\mathrm{Eq}}= \frac{\partial\mathcal{W}^{\mathrm{Eq}}}{\partial h}\left[\frac{ \partial h}{\partial P}\mathrm{d}P+\frac{\partial h}{\partial\Gamma_{0}} \mathrm{d}\Gamma_{0}\right]+\frac{\partial\mathcal{W}^{\mathrm{Eq}}}{\partial \Gamma_{0}}\mathrm{d}\Gamma_{0}\] \[= \frac{\partial\mathcal{W}^{\mathrm{Eq}}}{\partial h}\frac{ \partial h}{\partial\Gamma_{0}}\mathrm{d}\Gamma_{0}+\frac{\partial\mathcal{W }^{\mathrm{Eq}}}{\partial\Gamma_{0}}\mathrm{d}\Gamma_{0}.\]
After a simple algebraic manipulation, it follows that
\[-\frac{\partial\mathcal{W}^{\mathrm{Eq}}}{\partial\Gamma_{0}} \mathrm{d}\Gamma_{0}= P^{\mathrm{Eq}}\mathrm{d}h-\mathrm{d}\mathcal{W}^{\mathrm{Eq}}\] \[= \mathrm{d}\mathcal{W}^{\mathrm{Eq}*}-[h-H]\mathrm{d}P^{\mathrm{Eq }}, \tag{20}\]
where we have made use of the relation \(\mathrm{d}h=(\partial h/\partial\Gamma_{0})\;\mathrm{d}\Gamma_{0}\) and, for convenience, have introduced the notation
\[P^{\mathrm{Eq}}(P(t),\Gamma_{0}):=\frac{\partial\mathcal{W}^{\mathrm{Eq}}}{ \partial h}\left(h(P(t),\Gamma_{0}),\Gamma_{0}\right) \tag{21}\]
and
\[\mathcal{W}^{\mathrm{Eq}*}(P(t),\Gamma_{0}):= P^{\mathrm{Eq}}(P(t),\Gamma_{0})\left[h(P(t),\Gamma_{0})-H\right]\] \[-\mathcal{W}^{\mathrm{Eq}}\left(h(P(t),\Gamma_{0}),\Gamma_{0} \right). \tag{22}\]
It follows immediately from (20) that
\[-\frac{\partial\mathcal{W}^{\mathrm{Eq}}}{\partial\Gamma_{0}}= \frac{\partial\mathcal{W}^{\mathrm{Eq}*}}{\partial\Gamma_{0}}(P(t), \Gamma_{0})-\] \[[h(P(t),\Gamma_{0})-H]\frac{\partial P^{\mathrm{Eq}}}{\partial \Gamma_{0}}(P(t),\Gamma_{0}), \tag{23}\]
which is precisely the result that we are after. Indeed, given the applied force (13) in the delayed fracture tests of interest here, the result (23) allows us to expediently determine the resulting energy release rate \(-\partial\mathcal{W}^{\mathrm{Eq}}/\partial\Gamma_{0}\) in the Griffith criticality condition (3) in terms of three readily computable quantities: the deformation \(h(t)\) between the grips and the derivatives with respect to \(\Gamma_{0}\) at fixed \(P(t)\) -- or, equivalently, at fixed time \(t\) -- of the equilibrium elastic force (21) and the complementary equilibrium elastic energy (22).
## 3 Results for a canonical elastomer with Gaussian elasticity and constant viscosity
In this section, we present solutions for the initial-boundary-value problem (16)-(17) with (12)-(15) and (9) for the basic case when the specimen is made of a canonical elastomer with Gaussian elasticity and constant viscosity. Specifically, we present solutions for the case when \(N=1\), \(\alpha_{1}=\beta_{1}=1\), \(\eta_{\infty}=0\), \(K_{1}=K_{2}=0\), equilibrium and non-equilibrium initial shear moduli
\[\mu_{1}=0.2\;\mathrm{MPa}\quad\mathrm{and}\quad\nu_{1}=2\;\mathrm{MPa},\]
and viscosity
\[\eta_{0}=500\;\mathrm{MPa}\,\mathrm{s}.\]
These values are chosen here because they are comparable with those that describe the elastomer analyzed in the next section; see Table 1. Note that these material parameters correspond to an elastomer with constant relaxation time \(\tau=\eta_{0}\nu_{1}^{-1}=250\;\mathrm{s}\) and constant creep time \(\tau^{*}=\eta_{0}(\mu_{1}^{-1}+\nu_{1}^{-1})=2750\;\mathrm{s}\).
### The force-deformation and deformation-time responses
Figures 3 and 4 present solutions for the deformation \(h(t)\) between the grips that results from the applied force (13)-(14)\({}_{1}\) with global stress \(\sigma_{0}=0.3\)
Figure 3: Force-deformation response of specimens with pre-existing cracks of lengths \(A=2.5\) and \(20\;\mathrm{mm}\) for the applied force (13)-(14)\({}_{1}\) with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\;\mathrm{s}\).
MPa and total time of applied loading \(T=20000\) s in specimens with pre-existing cracks of lengths \(A=2.5\) and 20 mm. Specifically, the results are shown for the applied force \(P(t)\) as a function of \(h(t)\) in Fig. 3 and for the evolution of \(h(t)\) in time \(t\) in Fig. 4. To aid in the visualization of the results, Fig. 5 also shows contour plots over the deformed configuration of the component \(F_{11}(\mathbf{X},t)\) of the local deformation gradient at the same final time \(t=T=20000\) s of the applied load for both specimens.
As expected, the specimen with the larger crack leads to a larger deformation between the grips for the same applied force. It is also interesting to note that by approximately \(t=10000\) s -- at which point \(h\approx 178\) mm in the specimen with crack length \(A=2.5\) mm and \(h\approx 183\) mm in that with \(A=20\) mm -- the creeping process has all but concluded, this for both specimens. Finally, we remark that the results for other values of global stress \(\sigma_{0}\) in the range \((14)_{2}\) are not fundamentally different from those shown in Figs. 3 through 5 for \(\sigma_{0}=0.3\) MPa.
The total deformation energy \(\mathcal{W}\) and its partition into \(\mathcal{W}^{\text{Eq}}\), \(\mathcal{W}^{\text{NEq}}\), and \(\mathcal{W}^{v}\)
The areas under the curves in the results presented in Fig. 3 correspond to the total work done by the applied loads. By the same token, they correspond to the total deformation stored and dissipated by the elastomer. We thus have
\[\mathcal{W}=\int_{H}^{h(t)}P\,\mathrm{d}h.\]
Since for this case the elastomer is a canonical elastomer with Gaussian elasticity and constant viscosity, we also have that
\[\mathcal{W}^{\text{Eq}}=\int_{\Omega_{0}}\frac{\mu_{1}}{2}\left[\mathrm{tr} \,\mathbf{C}-3\right]\,\mathrm{d}\mathbf{X}, \tag{24}\]
\[\mathcal{W}^{\text{NEq}}=\int_{\Omega_{0}}\frac{\nu_{1}}{2}\left[\mathrm{tr} (\mathbf{CC}^{v\,-1})-3\right]\,\mathrm{d}\mathbf{X}, \tag{25}\]
and
\[\mathcal{W}^{v}= \mathcal{W}-\mathcal{W}^{\text{Eq}}-\mathcal{W}^{\text{NEq}}. \tag{26}\]
Figures 6 and 7 show results for \(\mathcal{W}^{\text{Eq}}\), \(\mathcal{W}^{\text{NEq}}\), and \(\mathcal{W}^{v}\) -- as computed from expressions (24)-(26) and the pertinent numerical solutions for the deformation field \(\mathbf{y}(\mathbf{X},t)\) and the internal variable \(\mathbf{C}^{v}(\mathbf{X},t)\) -- for the same applied force (13)-(14)\({}_{1}\), with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s, considered in Figs. 3 through 5. The results are plotted as functions of the initial crack surface \(\Gamma_{0}=A\times B\) and time \(t\). While Fig. 6 shows results for the entire duration of the loading process \(t\in[0,T]\), Fig. 7 shows results that focus on the ramping of the applied force
Figure 4: Evolution in time \(t\) of the deformation \(h(t)\) between the grips in specimens with pre-existing cracks of lengths \(A=2.5\) and 20 mm subjected to the applied force (13)-(14)\({}_{1}\) with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s.
Figure 5: Contour plots over the deformed configuration of the component \(F_{11}(\mathbf{X},t)\) of the local deformation gradient in specimens with pre-existing cracks of lengths \(A=2.5\) and 20 mm subjected to the applied force (13)-(14)\({}_{1}\) with global stress \(\sigma_{0}=0.3\) MPa. Both plots are shown at the same final time \(t=T=20000\) s of the applied load.
and immediately afterwards, over the time interval \(t\in[0,0.015]\) s.
Several comments are in order. All three parts of the deformation energy appear to depend nonlinearly on both the crack surface \(\Gamma_{0}\) and time \(t\). Distinctly, with respect to \(t\), both the equilibrium energy \(\mathcal{W}^{\rm Eq}\) and the non-equilibrium energy \(\mathcal{W}^{\rm NEq}\) are seen to increase sharply, while the viscous dissipated energy \(\mathcal{W}^{v}\) remains negligibly small, over the short duration of the ramping of the applied force \(P(t)\) up to its final constant value \(P(t)=\sigma_{0}(B\times L)\).
Figure 6: Computed values from (24)-(26) of (a) the equilibrium elastic energy \(\mathcal{W}^{\rm Eq}\), (b) the non-equilibrium elastic energy \(\mathcal{W}^{\rm NEq}\), and (c) the dissipated viscous energy \(\mathcal{W}^{v}\) in specimens subjected to the applied force (13)-(14)\({}_{1}\), with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s, plotted as functions of the initial crack surface \(\Gamma_{0}=A\times B\) and time \(t\).
Figure 7: Zoom of the time interval \(t\in[0,0.015]\) s in Fig. 6, focusing on the ramping of the applied force (13)-(14)\({}_{1}\) and immediately afterwards.
Beyond the ramping process, when \(t>t_{0}=0.01\) s, the non-equilibrium energy \(\mathcal{W}^{\rm{NEq}}\) decreases monotonically in time resulting in the increase of \(\mathcal{W}^{v}\) and the further increase of \(\mathcal{W}^{\rm{Eq}}\). Consistent with Fig. 4, the values of \(\mathcal{W}^{\rm{Eq}}\), \(\mathcal{W}^{\rm{NEq}}\), and \(\mathcal{W}^{v}\) remain practically invariant after \(t=10000\) s, since the creeping process has all but concluded by then.
### The derivative \(-\partial\mathcal{W}^{\rm{Eq}}/\partial\Gamma_{0}\)
The type of results presented in Figs. 4 and 6(a) for the deformation \(h(t)\) between the grips and for the equilibrium elastic energy \(\mathcal{W}^{\rm{Eq}}\) can be directly used to work out the corresponding results for the equilibrium elastic force (21) and, in turn, those for the complementary equilibrium elastic energy (22) in order to ultimately compute the energy release rate \(-\partial\mathcal{W}^{\rm{Eq}}/\partial\Gamma_{0}\) by making use of the identity (23). The relevant computations go as follows.
As a first step, for the same applied force (13)-(14)\({}_{1}\), with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s, considered in the figures above, we replot in Fig. 8 the equilibrium elastic energy \(\mathcal{W}^{\rm{Eq}}\), this time around, in terms of the initial crack surface \(\Gamma_{0}=A\times B\) and the deformation \(h(t)\) between the grips. From this type of 3D plot, we can readily compute the derivative (21) that defines the equilibrium elastic force \(P^{\rm{Eq}}\). The results for \(P^{\rm{Eq}}\) from such a computation are presented in Fig. 9 as a function of \(\Gamma_{0}=A\times B\) and time \(t\). Having determined \(P^{\rm{Eq}}\), we can then compute the complementary equilibrium elastic energy \(\mathcal{W}^{\rm{Eq}^{*}}\) directly from its definition (22). Figure 10 plots the results also as a function of \(\Gamma_{0}=A\times B\) and time \(t\).
Next, from the type of 3D plots presented in Figs. 9 and 10, we can readily compute the derivatives \(\partial P^{\rm{Eq}}/\partial\Gamma_{0}\) and \(\partial\mathcal{W}^{\rm{Eq}^{*}}/\partial\Gamma_{0}\) at fixed \(P(t)\) -- which, again, it is equivalent to fixed time \(t\) -- and, finally, making use of the identity (23), the energy release rate \(-\partial\mathcal{W}^{\rm{Eq}}/\partial\Gamma_{0}\). Figure 11 reports such a computation of \(-\partial\mathcal{W}^{\rm{Eq}}/\partial\Gamma_{0}\) for specimens with a pre-existing crack of length \(A=20\) mm subjected to the global stresses \(\sigma_{0}=0.1,0.2,0.3\) MPa. While part (a) of the figure shows the results as functions of time for the entire duration of the loading pro
Figure 8: Equilibrium elastic energy \(\mathcal{W}^{\rm{Eq}}\) in specimens subjected to the applied force (13)-(14)\({}_{1}\), with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s, plotted as a function of the initial crack surface \(\Gamma_{0}=A\times B\) and the deformation \(h(t)\) between the grips.
Figure 10: Complementary equilibrium elastic energy \(\mathcal{W}^{\rm{Eq}^{*}}\) in specimens subjected to the applied force (13)-(14)\({}_{1}\), with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s, plotted as a function of the initial crack surface \(\Gamma_{0}=A\times B\) and time \(t\).
Figure 9: Equilibrium elastic force \(P^{\rm{Eq}}\) in specimens subjected to the applied force (13)-(14)\({}_{1}\), with global stress \(\sigma_{0}=0.3\) MPa and total time of applied loading \(T=20000\) s, plotted as a function of the initial crack surface \(\Gamma_{0}=A\times B\) and time \(t\).
cess \(t\in[0,T]\), part (b) shows results that focus on the first 1000 s.
There are two crucial observations to be made from Fig. 11 that lay bare the key features of the phenomenon of delayed fracture in elastomers. First, irrespective of the applied global stress, the energy release rate \(-\partial\mathcal{W}^{\text{Eq}}/\partial\Gamma_{0}\) is bounded from above and increases monotonically in time until reaching an asymptotic maximum. Second, specimens subjected to larger global stresses lead to larger values of \(-\partial\mathcal{W}^{\text{Eq}}/\partial\Gamma_{0}\) at the same instance in time \(t\).
According to the Griffith criticality condition (3), the first observation entails that delayed fracture will occur -- that is, the condition \(-\partial\mathcal{W}^{\text{Eq}}/\partial\Gamma_{0}\) = \(G_{c}\) will be reached at some \(t\in(t_{0},T)\) -- if the applied load is between two threshold values, say \(\sigma_{0}^{max}\) and \(\sigma_{0}^{min}\). If the applied load is above the upper threshold \(\sigma_{0}^{max}\), fracture will take place during the ramping process of the load at some \(t\in(0,t_{0}]\), without delay. If it is below the lower threshold \(\sigma_{0}^{min}\), fracture will never occur.
On the other hand, the second observation entails that, for the same size of the pre-existing crack, specimens subjected to larger global stresses will exhibit a shorter delay for fracture to take place. The next subsection details this behavior.
### The critical time \(t_{c}\) at fracture
Having generated the type of results presented in Fig. 11 for the energy release rate \(-\partial\mathcal{W}^{\text{Eq}}/\partial\Gamma_{0}\) vs. time \(t\) -- assuming that we also have knowledge of the intrinsic fracture energy \(G_{c}\) of the elastomer -- we can readily determine from the Griffith criticality condition (3) the critical time \(t_{c}\) at which fracture will nucleate from the pre-existing crack in the specimens. This amounts to identifying the intercept of the curve \(-\partial\mathcal{W}^{\text{Eq}}/\partial\Gamma_{0}\) vs. \(t\) with the line \(G_{c}\) vs. \(t\).
For specimens with a pre-existing crack of length \(A=20\) mm, Fig. 12 presents results for \(t_{c}\) as a
Figure 11: The energy release rate \(-\partial\mathcal{W}^{\text{Eq}}/\partial\Gamma_{0}\) for specimens with a pre-existing crack of length \(A=20\) mm subjected to the applied force (13)-(14)\({}_{1}\) with global stresses \(\sigma_{0}=0.1,0.2,0.3\) MPa and total time of applied loading \(T=20000\) s, plotted as functions of time \(t\). Part (a) presents results for the entire duration of the loading process \(t\in[0,T]\), while part (b) zooms in the interval \(t\in[0,1000]\) s.
Figure 12: The critical time \(t_{c}\) at which fracture nucleates in specimens with a pre-existing crack of length \(A=20\) mm subjected to the applied force (13)-(14)\({}_{1}\). The results are shown as functions of the global stress \(\sigma_{0}\) for three representative values of the intrinsic fracture energy \(G_{c}\) of the elastomer.
function of the applied global stress \(\sigma_{0}\) for three representative values of the intrinsic fracture energy, \(G_{c}=100,200,500\) N/m.
As foretold in the general conclusions established above, note that, for a given \(G_{c}\), fracture takes shorter to nucleate in specimens subjected to larger global stresses. By the same token, for a given \(\sigma_{0}\), fracture takes shorter to nucleate in specimens made of elastomers with smaller intrinsic fracture energies.
## 4 Comparisons with the experiments of Knauss (1970) on Solithane 113
We finally turn to deploying the Griffith criticality condition (3) to explain the delayed fracture experiments of Knauss (1970) on the polyurethane elastomer Solithane 113; since the elastomer was prepared from equal amounts by volume of resin and catalyst, it is also referred to as Solithane 50/50. As noted above, these appear to be the first experiments reported in the literature that showed that elastomers can exhibit delayed fracture. The focus is on the results for specimens with the same geometry considered in the two preceding sections (\(L=101.6\) mm, \(H=L=101.6\) mm, \(B=0.7938\) mm), featuring a pre-existing central crack of length \(A=5.08\) mm, subjected to the applied global stresses1\(\sigma_{0}=0.10,0.12,0.13,0.15\) MPa at a temperature of 0 \({}^{\circ}\)C; see Fig. 9 in (Knauss, 1970).
Footnote 1: There is some uncertainty about the precise values of the global stress \(\sigma_{0}\) applied in the experiments, since the data in Fig. 9 of (Knauss, 1970) is presented normalized by a factor (\(\sigma_{\rm gsc}\)) that was not spelled out fully explicitly. The values \(\sigma_{0}=0.10,0.12,0.13,0.15\) MPa that we use here are our best estimate based on the information provided.
### The viscoelastic behavior and intrinsic fracture energy of Solithane 113
As emphasized in the Introduction, the use of the Griffith criticality condition (3) requires knowledge of only two fundamental properties of the elastomer of interest: (_i_) its viscoelastic behavior, from which the storage of equilibrium elastic energy can be identified, and (_ii_) its intrinsic fracture energy. Both of these properties can be measured experimentally once and for all by means of conventional tests.
#### 4.1.1 The viscoelastic behavior
A few years before Knauss (1970) published his findings on delayed fracture, as part of his PhD thesis work, Mueller (1968) reported a range of experimental results on the mechanical behavior of the same Solithane 113 tested by Knauss (1970). Most of these restricted attention to small deformations, but Mueller (1968) did include a handful of results involving finite deformations for the viscoelastic response of Solithane 113 under uniaxial tension applied at various constant stretch rates at a temperature of \(-5\)\({}^{\circ}\)C; see Fig. 16 in (Mueller, 1968) and also Fig. 4 in (Mueller and Knauss, 1971).
Specializing the constitutive relation (10)-(11) to such loadings -- that is, to deformation gradients of the form \({\bf F}={\rm diag}(\lambda,\lambda^{-1/2},\lambda^{-1/2})\) with \(\lambda=1+\dot{\lambda}_{0}t\) and first Piola-Kirchhoff stresses of the form \({\bf S}={\rm diag}(S,0,0)\) -- and then fitting (by least squares) its material constants to the admittedly scarce experimental data of Mueller (1968) yields the values listed in Table 1. As seen from the comparisons presented in Fig. 13, the constitutive relation (10)-(11) with such material constants describes reasonably well the viscoelastic data (solid circles) reported by Mueller (1968).
Remark 4: The material constants listed in Table 1 indicate that, at \(-5\)\({}^{\circ}\)C, Solithane 113 is an elastomer with non-Gaussian elasticity and nonlinear viscosity. This falls squarely within the behavior of the vast majority of elastomers.
Remark 5: In the sequel, because of the absence of experimental data at temperatures other than \(-5\)\({}^{\circ}\)C, we make use of the constitutive relation (10)-(11) with the material constants listed in Table 1 -- which, again, strictly apply to the behavior of Solithane 113 at \(-5\)\({}^{\circ}\)C -- to describe the viscoelastic behavior of Solithane 113 in the delayed fracture experiments of Knauss (1970) at 0 \({}^{\circ}\)C. This 5 \({}^{\circ}\)C difference in temperature should not be taken
\begin{table}
\begin{tabular}{l l} \hline \(\mu_{1}=0.2099\) MPa & \(\mu_{2}=2.040\times 10^{-5}\) MPa \\ \(\alpha_{1}=1.941\) & \(\alpha_{2}=9.344\) \\ \(\nu_{1}=2.300\) MPa & \(\nu_{2}=4.147\times 10^{-2}\) MPa \\ \(\beta_{1}=0.5353\) & \(\beta_{2}=7.108\) \\ \(\eta_{0}=150\) MPa s & \(\eta_{\infty}=0\) MPa s \\ \(K_{1}=2.653\) MPa s & \(K_{2}=0\) MPa\({}^{-2}\) \\ \(\gamma_{1}=7.977\) & \(\gamma_{2}=1\) \\ \hline \end{tabular}
\end{table}
Table 1: Values of the material constants in the viscoelastic model (10)-(11) for the polyurethane elastomer Solithane 113.
as negligible, since the viscosity of elastomers can change rapidly near their glass transition temperature \(T_{g}\) and the glass transition temperature for Solithane 113 happens to be about \(-20\)\({}^{\circ}\)C (Mueller and Knauss, 1971).
#### 4.1.2 The intrinsic fracture energy
In his PhD thesis work, Mueller (1968) also carried out experiments aimed at measuring the intrinsic fracture energy \(G_{c}\) of Solithane 113. A summary of these was later reported in (Mueller and Knauss, 1971). The experiments consisted in carrying out "pure-shear" fracture tests at various constant global stretch rates in the range \([1.7\times 10^{-4},8.3\times 10^{-3}]\) s\({}^{-1}\) and various constant temperatures in the range \([0,50]\)\({}^{\circ}\)C on specimens that have been swollen with the solvent Toluene. The presence of the solvent led to the minimization of viscous dissipation. From the results of such "pure-shear" fracture tests, it was concluded that the intrinsic fracture energy of the swollen Solithane 113 was \(G_{c}^{sw}=28\pm 7\) N/m and that this value was independent of temperature. By making use of an argument similar to that put forth by Lake and Thomas (1967), that the intrinsic fracture energy is essentially a measure of the chain-bond strength only, Mueller and Knauss (1971) then estimated that the intrinsic fracture energy of Solithane 113 in its unswollen state is
\[G_{c}=41\pm 8\,\mbox{N/m},\]
this estimate also being independent of temperature. This value falls squarely within the range \(G_{c}\in[10,100]\) N/m for common hydrocarbon elastomers.
### Computation of the derivative \(-\partial{\cal W}^{\rm Eq}/\partial\Gamma_{0}\)
Having established the pertinent deformation and fracture properties of Solithane 113, we proceed by repeating the same type of full-field analysis presented in Section 3 in order to compute the derivative \(-\partial{\cal W}^{\rm Eq}/\partial\Gamma_{0}\) entering the Griffith criticality condition (3).
Before presenting and discussing the results, the following technical remarks are in order. Since the experiments of Knauss (1970) pertain to specimens with a pre-existing crack of length \(A=5.08\) mm, we perform the simulations for specimens with three crack lengths, \(A=2.5,5.08,10\) mm. This suffices to be able to take the required derivative \(-\partial{\cal W}^{\rm Eq}/\partial\Gamma_{0}\) at \(\Gamma_{0}=A\times B=5.08\times 0.7938\) mm\({}^{2}\). Much like the loads used in the experiments, we carry out simulations at four different global stresses, \(\sigma_{0}=0.10,0.12,0.13,0.15\) MPa. Accordingly, in all, we carry out \(3\times 4=12\) simulations of the delayed fracture tests. Furthermore, since the experiments indicate that fracture nucleates from the pre-existing crack at critical times \(t_{c}<20000\) s, we use \(T=20000\) s for the total time of applied loading in each of these simulations.
Figure 14: The energy release rate \(-\partial{\cal W}^{\rm Eq}/\partial\Gamma_{0}\) at \(A=5.08\) mm computed from the simulations of delayed fracture tests on Solithane 113. The results correspond to applied global stresses \(\sigma_{0}=0.10,0.12,0.13,0.15\) MPa and are plotted as functions of time \(t\).
Figure 13: Comparison between the stress-stretch response (solid line) predicted by the viscoelastic model (10)-(11), with the material constants in Table 1, and the experimental data (solids circles) reported by Mueller (1968) for Solithane 113 subjected to uniaxial tension applied at three different constant stretch rates, \(\dot{\lambda}_{0}=3\times 10^{-4},3\times 10^{-3},3\times 10^{-2}\) s\({}^{-1}\).
Analogous to Fig. 11, Fig. 14 presents results for the energy release rate \(-\partial\mathcal{W}^{\mathrm{Eq}}/\partial\Gamma_{0}\) computed from the simulations of the delayed fracture tests on Solithane 113, at the applied global stresses \(\sigma_{0}=0.10,0.12,\,0.13,\,0.15\) MPa. Much like the results in Fig. 11 for the canonical case of an elastomer with Gaussian elasticity and constant viscosity, the results in Fig. 14 show that, irrespective of the applied global stress \(\sigma_{0}\), the energy release rate \(-\partial\mathcal{W}^{\mathrm{Eq}}/\partial\Gamma_{0}\) increases monotonically in time towards an asymptotic maximum value. The results also show that specimens subjected to larger \(\sigma_{0}\) lead to larger values of \(-\partial\mathcal{W}^{\mathrm{Eq}}/\partial\Gamma_{0}\) at the same instance in time \(t\).
### The critical time \(t_{c}\) at fracture
At this stage, we are in a position to deploy the Griffith criticality condition (3) to explain the delayed fracture experiments of Knauss (1970).
Figure 15 confronts the theoretical predictions obtained from the results in Fig. 14 -- specifically, again, the intercepts of the curves \(-\partial\mathcal{W}^{\mathrm{Eq}}/\partial\Gamma_{0}\) vs. \(t\) with the line \(G_{c}\) vs. \(t\) -- with the corresponding experimental results (solid circles) for the critical time \(t_{c}\) at which fracture nucleates. The results are presented as a function of the applied global stress \(\sigma_{0}\). For the theoretical predictions, we include two results. The first one corresponds to using the average value \(G_{c}=41\) N/m estimated by Mueller and Knauss (1971) for the intrinsic fracture energy. The second corresponds to using the somewhat larger value \(G_{c}=107\) N/m. The experimental data falls within these two results.
Two comments are in order. First and foremost, taking into account the various sources of uncertainties (on the precise values of the applied global stress \(\sigma_{0}\) and on the viscoelastic response of Solithane 113 at \(0\ ^{\circ}\)C), the Griffith criticality condition (3) appears, indeed, to determine when delayed fracture occurs. The results in Fig. 15 also make it plain that having robust experimental data for the viscoelasticity and the intrinsic fracture energy of the elastomer of interest is essential to be able to predict its delayed fracture. This is because small variations in either property may result in large changes in the critical time \(t_{c}\) at fracture, especially when dealing with small forces that lead to long creeping processes.
## 5 Final comments
Adding to the validation results presented by Shrimali and Lopez-Pamies (2023), who made use of the Griffith criticality condition (3) to explain "pureshear" fracture experiments carried out over a wide range of constant stretch rates on an acrylic elastomer (VHB 4905 from the company 3M), the comparisons with the delayed fracture experiments on a polyurethane elastomer presented in the preceding section provide further direct evidence that the Griffith criticality condition (3) may indeed be the universal condition that governs crack growth in elastomers undergoing finite deformations in response to quasi-static mechanical loads.
In this context, given the recently demonstrated ability (Kumar et al. 2018a,b, 2000, 2022; Kumar and Lopez-Pamies 2020, 2021) of the phase-field theory of fracture initiated by Kumar et al. (2018a) to describe fracture nucleation and propagation in nominally elastic brittle materials at large and given the "seamless" mathematical generalization that the Griffith criticality condition (3) provides of the classical Griffith criticality for elastic brittle materials, a next sensible step would be to successively follow in the footsteps of Francfort and Marigo (1998), Bourdin et al. (2000), and Kumar et al. (2018a) in order to:
1. turn the Griffith criticality condition (3) into a complete mathematical description of fracture
Figure 15: Comparison between the critical time \(t_{c}\) at which fracture nucleates, according to the Griffith criticality condition (3), and the corresponding experimental results reported by Knauss (1970) for Solithane 113 at \(0\ ^{\circ}\)C. The results are presented as a function of the applied global stress \(\sigma_{0}\).
nucleation from pre-existing cracks and of fracture propagation in viscoelastic elastomers,
* regularize such a description into numerically tractable phase-field-type PDEs (partial differential equations), and
* generalize those PDEs to account for nucleation of fracture at large (not just from large pre-existing cracks, but also from the bulk, smooth and non-smooth boundary points, and small pre-existing cracks)
so as to formulate a complete and numerically tractable mathematical description of the nucleation and propagation of fracture in viscoelastic materials subjected to arbitrary quasi-static mechanical loads.
## Acknowledgements
Support for this work by the National Science Foundation through the Grants CMMI-1901583 and CMMI-2132528 is gratefully acknowledged.
| Shrimali and Lopez-Pamies (2023)の最近の貢献により、Griffithの臨界条件が弾塑性 elastoomers の亀裂成長を支配すると示されています。これは、従来の形式において、歴史的に難しい 積載履歴依存的な critical tearing energy $T_c$ を含むものから、基本的な形式へと変更されました。この基本的な形式を利用して、この論文では、弾塑性 elastoomers の 1 つの最も特徴的な断裂試験、いわゆる遅延断裂試験を説明します。 |
2307.14578 | GADER: GAit DEtection and Recognition in the Wild | Gait recognition holds the promise of robustly identifying subjects based on
their walking patterns instead of color information. While previous approaches
have performed well for curated indoor scenes, they have significantly impeded
applicability in unconstrained situations, e.g. outdoor, long distance scenes.
We propose an end-to-end GAit DEtection and Recognition (GADER) algorithm for
human authentication in challenging outdoor scenarios. Specifically, GADER
leverages a Double Helical Signature to detect the fragment of human movement
and incorporates a novel gait recognition method, which learns representations
by distilling from an auxiliary RGB recognition model. At inference time, GADER
only uses the silhouette modality but benefits from a more robust
representation. Extensive experiments on indoor and outdoor datasets
demonstrate that the proposed method outperforms the State-of-The-Arts for gait
recognition and verification, with a significant 20.6% improvement on
unconstrained, long distance scenes. | Yuxiang Guo, Cheng Peng, Ram Prabhakar, Chun Pong Lau, Rama Chellappa | 2023-07-27T01:53:57 | http://arxiv.org/abs/2307.14578v1 | # GADER: GAit DEtection and Recognition in the Wild
###### Abstract
Gait recognition holds the promise of robustly identifying subjects based on their walking patterns instead of color information. While previous approaches have performed well for curated indoor scenes, they have significantly impeded applicability in unconstrained situations, e.g. outdoor, long distance scenes. We propose an end-to-end GAit DEtection and Recognition (GADER) algorithm for human authentication in challenging outdoor scenarios. Specifically, GADER leverages a Double Helical Signature to detect the fragment of human movement and incorporates a novel gait recognition method, which learns representations by distilling from an auxiliary RGB recognition model. At inference time, GADER only uses the silhouette modality but benefits from a more robust representation. Extensive experiments on indoor and outdoor datasets demonstrate that the proposed method outperforms the State-of-The-Arts for gait recognition and verification, with a significant 20.6% improvement on unconstrained, long distance scenes.
## 1 Introduction
Unconstrained biometric identification, e.g., in outdoor and far-away situations, has been a longstanding challenge [51, 50, 36, 37]. RGB-based face and body recognition systems focus on learning discriminating _spatial_ features; however, effects like challenging view angles, low face resolution, changing appearances (e.g., clothes and glasses), and long distance turbulence are observed in the real world and can significantly disrupt spatial information. Consequently, the RGB-based recognition systems tend to perform inconsistently in unconstrained scenarios.
Gait analysis provides an alternative modality for human recognition by intentionally masking away the color information and focusing on learning discriminating features in the _temporal_ domain. As such, it is potentially more robust to challenging, unconstrained situations, and has been broadly applied in many applications such as surveillance [4], health [12] and crime analysis [15], etc.
The field of gait recognition has been significantly bloomed [35, 16, 10, 26, 1, 36] by traditional methods, including template matching methods [5, 28, 48] and model-based methods [3, 46, 7, 24], but limited by dependency on the scale and viewing angle and being sensitive to video quality, respectively. Deep learning (DL)-based approaches [9, 2, 27] have made significant advances compared to traditional methods. They are able to generate robust identity embeddings by directly processing the complex temporal information present in gait sequences. This enables effective recognition under sources of variability, such as changes in viewpoint and clothing, making DL-based methods widely used in these years.
While DL-based gait recognition performs well for indoor scenes, it often fails to achieve good performance in unconstrained/outdoor scenarios. In this work, we seek to apply gait recognition to unconstrained situations with minimum data curation. The recently collected BRIAR [11] dataset contains standing, structured walking, and random walking sequences, which mimic the real-world constraints of gait recognition. Existing gait recognition methods assume that the subject is always walking with periodic move
Figure 1: Compared to other gait recognition models, GADER has significant improvement, especially increases 26.38% at 500m, in outdoor recognition at various distances. Close Range(CR), Unmanned Aerial Vehicle(UAV).
ment and that there are no standing sequences. By making such assumption, these methods tend to learn suboptimal representation, e.g. achieving only 31% on close range recognition. This highlights the need for an approach to filter out non-walking sequences and make the gait recognition inputs more consistent.
Current silhouette-based methods employ the practice of _size normalization_, where input silhouettes are resized to the same resolution regardless of the subject's distance from the camera, can lead to information loss. Particularly, the sequence of the ratio between the original body chip and the normalized one can implicitly offer important viewpoint information, which is useful for generating effective embeddings [8]. Unfortunately, this cue is lost in the resizing operation. Besides, RGB modality is often overlooked in gait recognition due to privacy concerns and potential biases introduced by clothing. However, RGB images contain rich information that can be used to build robust features that cannot be captured by silhouettes alone. By solely referring to the feature space of the RGB modality in the training step, we are able to protect the privacy of individuals while still taking advantage of the discrimination provided by RGB images.
In this paper, we aim to push the frontiers of unconstrained gait analysis through an end-to-end GAit Detection and Recognition system in the wild (GADER). To address the problem of mixing the moving and standing sequences, we introduce a novel gait detection module (Figure 4) that detects the walking and non-walking parts of a sequence. Instead of using a 3D volume to capture the dynamic movement, our gait detector uses the reliable Double Helical Signature (DHS) [34, 32], a 2D gait pattern with a lightweight classification model to distinguish the walking portion of the input sequence from others. Specifically, we do not input the entire gait pattern into the model. Instead, we split the signature into multiple windows of varying lengths to get predictions, followed by Non-Maximum Suppression (NMS), to localize the movement duration. This provides a pure gait sequence for existing gait recognition models, making it suitable for real-world scenarios.
For the gait recognition module (GAR), along with the size normalized silhouettes, we embed the ratio of change in size from the original body size as an attention signal. This _ratio attention_ helps preserve the viewpoint information that is beneficial for robust identity representation. We further introduce a _cross-modality feature distillation_ step, where we guide intermediate gait features to be more expressive by making them more similar to features generated from RGB frames. This enables the gait features to maintain their robustness against appearance changes, while also benefiting from the discriminative power of RGB features. The augmented gait features can be obtained from silhouettes and do not require the presence of RGB frames during inference.
In summary, GADER introduces three contributions:
* We introduce a light-weight gait detector to automatically detect clips with human movements and avoid learning on static information.
* We propose a novel gait recognition model, which leverages the color space and size information during training; specifically, knowledge distillation on RGB features is used to enhance feature space capacity.
* We conduct a series of evaluations, i.e. rank retrieval and verification on CASIA-B Gait3D and BRIAR datasets, and show superior performance than the state-of-the-art means.
## 2 Related Work
**Traditional Gait Representations**: Traditional gait recognition methods can be classified into appearance-based and human model-based. Appearance based gait systems use input data without making any assumptions on human body model [35, 10, 48]. Bobick and Davis [5] proposed Motion Energy Image (MEI) and Motion History Image (MHI) to model the input silhouette sequence as a template. Han and Bhanu [16] introduced Gait Energy Image (GEI) template as an average of aligned and normalized silhouette frames. In [28], Liu and Zheng improved GEI and proposed Gait History Image (GHI). GHI is generated from silhouettes that combines both static and dynamic or temporal characteristics. While appearance-based methods are simple and fast, they are view and scale dependent.
Model-based methods represent whole human body using well-defined models and use them to represent gait. The methods vary by the different techniques used for modeling human body, such as hidden Markov models [22, 30], stride length and walking tempo [3], stick figures [46], multi-part [7, 24], inter body part distances [6, 41], joint angles [40], and Velocity Hough Transform [31] among many.
**Deep Learning-based Methods**: Early deep learning-based methods learn global gait representation using information like silhouettes [9], GEI [38, 43, 18], and body pose [26, 1, 2] as input to CNNs. The GaitSet [9] model applies horizontal pyramid mapping on a set of features extracted from individual silhouette frames to obtain efficient gait representation. Alternatively, local representations extracted using part-based methods have shown success in recent years [27, 49, 29]. GaitPart [13] uses focal convolution on different body parts and aggregates them using micro-motion capture module. Several methods have been proposed to combine global and local features. One such method by Lin _et al._[27] aggregates local temporal information while retaining spatial information. Subsequently, global and local features extraction followed by temporal pooling is applied to generate robust gait features. Recently, Guo _et al._[14] utilize both RGB and silhouette data to ex
tract robust gait features for challenging indoor and outdoor sequences. In [8], Chai _et al._ used Lagrange's equation to demonstrate the importance of second order motion information in gait recognition.
**Recognition in the Wild**: Developing an accurate gait representation in the wild is a long term goal and has been broadly researched over the past decade. Towards that aim, Zhu _et al._ curated a large scale gait dataset called GREW [51]. It is a natural videos dataset consisting of 128K sequences of 26K identities captured over 882 cameras. Similarly, Zheng _et al._[50] collected Gait3D dataset that contains silhouettes, 2D/3D keypoints, and 3D meshes for 3D gait recognition. The authors [50] observed that state-of-the-art gait recognition methods do not translate similar superior performance in GREW and Gait3D as they do in indoor dataset like CASIA-B. Due to privacy concern, both GREW and Gait3D datasets are silhouette based and do not contain the corresponding RGB sequences. Also, unlike the BRIAR dataset, GREW and Gait3D do not contain sequences captured at long distances (500m). Hence, they do not address challenges due to atmospheric turbulence. Another research direction includes enhancing images captured at long distance and use the restored images for recognition [45, 23]. In [23], Lau _et al._ proposed a generative model that deblurs and removes turbulence effects for face recognition at distance. Such methods have not been explored for gait recognition.
## 3 Method
In this work, we focus on silhouette-based gait recognition, which relies on the binary masks of the subjects in a video. Formally, we denote a video as a 4D tensor, i.e. \(\mathbf{V}\in\mathbb{R}^{H\times W\times T\times 3}\), where \(H,W,T\) are the height, width, and frame index. For each frame \(t\), the subject silhouette \(\mathbf{s}_{t}\in\{0,1\}^{H\times W}\) is obtained from an off-the-shelf segmentation model, e.g. [42]. Gait recognition takes \(\mathbf{S}=[\mathbf{s}_{t}]_{t=1}^{T}\) as the input, and obtains corresponding features \(\mathbf{f}=F_{\theta}(\mathbf{S})\) from a feature extractor \(F_{\theta}\). Triplet loss [17] is used to constrain the training process of \(F_{\theta}\) with respect to ground-truth labels \(y\in\{1,2,\dots,|\mathcal{Y}|\}\) for each video in the training sets, where \(|\mathcal{Y}|\) is the cardinality of the label set. After \(F_{\theta}\) is trained, gallery gait silhouettes \(\mathbf{S}^{g}\) and probe gait silhouettes \(\mathbf{S}^{p}\) are passed into \(F_{\theta}\) to obtain the gait feature \(\mathbf{f}^{g}\) and \(\mathbf{f}^{p}\). To recognize the probe identity, a similarity metric \(\mathcal{D}\), such as Euclidean distance or cosine similarity, is used to find \(\mathcal{D}(\mathbf{f}^{g},\mathbf{f}^{p})\), where the gallery subject \(g\) and the probe \(p\) will be regarded as the same person if they are the most similar in feature space.
In the following sections, we introduce GAit DEtection and Recognition in the Wild (GADER), designed for gait recognition in the wild. GADER has two parts: a gait detector model, which uses 2D gait representations to detect human movements in a video, and a gait recognition model, which incorporates resizing information and is guided by a teacher feature space from an RGB-based recognition model during training.
### Gait Detector
Previous gait recognition methods [9, 13, 27, 25, 8] directly use silhouettes \(\mathbf{S}\) as the input, under the assumption that the video sequence captures _the entire body_ and _constant movement_. While these assumptions are for curated datasets, such as CASIA-B [47], OU-MVLP [39], and GREW [51], they are often untrue for unconstrained videos and will lead to suboptimal learning if left untreated [19]. To this end, we propose a gait detector to assess if the video sequence contains gait movement with the complete body,
Figure 2: Overview of our proposed end-to-end pipeline. GADER consists of two parts: gait detection and gait recognition, i.e. GAR. The gait detector utilizes gait representation to filter out unqualified sequences. GAR leverages ratio attention and RGB feature space to extract a more robust silhouette feature for recognition.
similar to the role of face detection in face recognition.
#### 3.1.1 Gait Representation
To make the gait detector compatible with silhouette-based methods, we extract a gait representation based on normalized silhouette \(\mathbf{S}\). An ideal gait representation should be able to discriminate moving subjects from stationary ones and partial body from full body. Since the legs of the body have periodic movement and contribute significantly to gait recognition, we use the Double Helical Signature (DHS) [34]. DHS is a classic gait representation that captures the movement of the knee to describe gait movement as a function of time. As our input \(\mathbf{S}\) is normalized to the same size, we can deduce the knee height \(\mathcal{H}_{\text{knee}}\) to be approximately a quarter of the overall height. By taking a slice from the silhouette sequence, i.e., \(\mathbf{S}_{\text{knee}}(x,t)=\mathbf{S}(x,\mathcal{H}_{\text{knee}},t),\mathbf{S}_{\text{ knee}}\in\mathbb{R}^{W\times T}\), we can obtain a double helical structure that indicates human movement.
As shown in Figure 3, the DHS pattern is rather discriminating. For the standing case, the DHS pattern shows a constant straight line since there is no movement at knees. When the subject is walking, a periodic pattern is obtained, known as a type of "Frieze pattern" [32]. Given an incomplete body, DHS appears rather different as \(\mathcal{H}_{\text{knee}}\) does not correspond to the knee position anymore; consequently, DHS becomes thicker.
#### 3.1.2 Light-weighted Classification
Unconstrained human motion naturally involves a variety of movements, including turning, standing, and walking within a short period of time. This can result in DHS sequences that contain multiple walking fragments. During the training step, we randomly select the start point and duration from the whole DHS and get \(R_{\text{knee}}\). Each segment has the same height as DHS's. After that, the fragment is processed by a five-layer Convolutional Neural Network \(\mathcal{M}_{\phi}\) to get the feature. To handle the varying window lengths, we introduce a temporal pooling module in the form of a max-pooling layer to generate a window-length invariant embedding. Then, we use a four-class multi-layer perceptron (MLP) to obtain the prediction, and the cross entropy
\[H=\text{MLP}(\mathcal{P}_{Max}^{1\times 1\times t}(\mathcal{M}_{\phi}(R_{ \text{knee}}))), \tag{1}\]
where \(\mathcal{P}_{Max}^{1\times 1\times t}\) represents temporal pooling module with size (\(1\times 1\times t\)), and \(t\) is the window's width.
To identify and localize these fragments, in the inference time, we first split the entire DHS sequence into multiple windows of varying durations \(R_{\text{knee}}^{n}\), where \(n\) represents the number of windows. Each window goes through the well-trained gait classification model and gets a corresponding prediction. We only keep the complete body gait predictions and reduce the overlapping by Non-Maximum Suppression (NMS). Finally, we check the reduced windows' inner distance and concatenate them if the distance is smaller than a predefined threshold. And the ratio of the detected movement length to the entire DHS sequence serves as an indicator for determining whether the sequence should be further utilized in the gait recognition module. The process is illustrated in Figure 4. Thus our gait detector automatically identifies the corresponding movement status of each clip.
### GAit Recognition (GAR)
With the help of the gait detector, we can obtain relatively clean gait sequences for gait recognition. In the recognition stage, we propose GAR, which incorporates prior knowledge such as the RGB feature space and the resizing information. The architecture is shown in Figure 2.
#### 3.2.1 Ratio Attention
The _resizing ratio_ is defined as the change in size from the original bounding box to the normalized silhouette shape, which is usually \(64\times 64\times T\). Intuitively, the resizing ratios are similar if two videos are recorded from the same viewpoint. As such, these ratios naturally encode viewpoint information, as shown in Figure 5. Viewpoint information has
Figure 4: Frames with gait information are not always present in a video clip. We introduce a gait detector leveraging a DHS followed by NMS and merge to locate the place where gait information is present in a sequence. The yellow bounding boxes are the detected gait clips.
Figure 3: The four cases of DHS(a-d) are shown using two variables: full/part - indicates whether the body is complete, and stand/gait - shows whether gait appears.
been shown to improve gait recognition performances [8]; however, such information is difficult to obtain in unconstrained situations, especially when the subject's movement is not monotonic. In this work, we leverage resizing ratios to represent changing views.
To effectively incorporate ratios into the gait feature extraction module, we apply it as an attention mechanism and fuse it into the network. The ratios are embedded using 1D convolution \(F_{1DConv}\), followed by a sigmoid function \(Sigmoid(*)\), i.e.:
\[r=Sigmoid(F_{1DConv}(\frac{\mathbf{S}_{raw}}{\mathbf{S}})), \tag{2}\]
where \(\mathbf{S}_{raw}\) and \(\mathbf{S}\) are the body bounding boxes in the original and standardized silhouette, and \(r\in\mathbb{R}^{1\times 1\times T}\).
#### 3.2.2 Cross Modality Distillation
Previous works [13, 9, 27, 25, 44] have shown that silhouette-based recognition is robust to appearance changes such as different clothing and low image quality. On the flip side, segmenting RGB frames into silhouettes leads to a loss in useful information, e.g. the rich texture information. While gait recognition systems do not have access to color space information at _test time_, they can be benefited by learning from the feature space learned by an RGB-based recognition system _during training_ to better separate identities. To this end, GAR introduces an auxiliary recognition branch to extract features from RGB images. We denote \(F^{f}_{3DConv},F^{s}_{3DConv}\) as the first 3D convolution layers to the RGB and silhouette feature extraction backbones. Combining with ratio attention \(r\), GAR first obtains weight features, i.e.
\[\mathbf{F}^{f}_{1}=r*F^{f}_{3DConv}(\mathbf{V}),\mathbf{F}^{s}_{1}=r*F^{s}_{3DConv}(\mathbf{S}), \tag{3}\]
After that, we apply the rest of backbones to the processed features,
\[\begin{split}\mathbf{F}^{f}_{i+1}&=F^{f}_{i}(\mathbf{F}^{ f}_{i}),\mathbf{F}^{s}_{i+1}=F^{s}_{i}(\mathbf{F}^{s}_{i}),\\ I^{f}&=F^{f}_{N}(\mathbf{F}^{f}_{N}),I^{s}=F^{s}_{N}( \mathbf{F}^{s}_{N}),\end{split} \tag{4}\]
in which \(I^{f}\) and \(I^{s}\) are identification embedding, and \(i\in\{1,2,...,N\}\) is the convolution block index. Triplet loss [17] is applied to enlarge the similarity of representations from different subjects and minimize the ones from the same identity for both modalities.
\[\begin{split}\mathcal{L}^{f}_{tri}&=[\|I^{f}-I^{ f}_{-}\|_{2}-\|I^{f}-I^{f}_{+}\|_{2}+m]_{+},\\ \mathcal{L}^{s}_{tri}&=[\|I^{s}-I^{s}_{-}\|_{2}-\|I ^{s}-I^{s}_{+}\|_{2}+m]_{+},\end{split} \tag{5}\]
where the superscript \(+\) and \(-\) in the identification embedding represent the same and different subject to \(I^{f}\) respectively. \(m\) is the margin of the triplet loss, and operation \([*]_{+}\) describes \(\max(*,0)\).
A cross-modal distillation loss is introduced within GAR, which promotes the representation power of gait features based on the learned RGB feature space. While one possible design is to directly constrain the silhouette/RGB features to be similar, we find this approach to be suboptimal. Since both modalities have their specific discriminative advantages, forcing their features to be similar leads to an averaged representation that does not benefit from the specificity from each modality. As we focus on obtaining expressive features from silhouettes, an additional convolutional layer \(C_{i}\) is introduced for the \(i\)-th intermediate silhouette feature \(\mathbf{F}^{s}_{i}\), such that the transformed features are constrained to be similar to \(\mathbf{F}^{f}_{i}\). This cross-modality distillation loss can be described as:
\[\mathcal{L}_{distill}=\sum_{i}(\mathcal{D}(\varnothing(\mathbf{F}^{f}_{i}),C_{i} (\mathbf{F}^{s}_{i}))), \tag{6}\]
where stop gradient (\(\varnothing\)) operation is used such that the RGB branch is not affected by the silhouette features in this process. Note that \(C_{i}\) is only used at training time. Overall, the training loss is
\[\mathcal{L}_{train}=\lambda_{f}\mathcal{L}^{f}_{tri}+\lambda_{s}\mathcal{L}^ {s}_{tri}+\lambda_{distill}\mathcal{L}_{distill}, \tag{7}\]
where \(\lambda_{f,s,distill},=0.425,0.425,0.15\) are the loss hyper-parameters used during training.
## 4 Experiments
### Datasets and Metric
In this work, we focus on applying gait recognition to unconstrained scenarios with minimum data curation. To this end, the **BRIAR**[11] dataset, as shown in Figure 6, is used extensively. BRIAR consists of 577 and 639 subjects used as training and test sets, respectively. In the test set, there are 354 distractors and 285 target subjects. For each identity, there are both indoor (_controlled_) and outdoor (_field_) sequences. The _controlled_ set is collected with 10 cameras from different viewing angles to record structure and random walking. In the _field_ set, the sequences are collected at varying distances and altitudes, namely close range (CR),
Figure 5: In each subfigure, the upper row of silhouettes are collected from 0\({}^{\circ}\)and the lower ones from 180\({}^{\circ}\). It is hard to distinguish normalized silhouettes from different angles (5a). But they are critical when the ratio is introduced (5b).
100m, 400m, 500m and unmanned aerial vehicle (UAV). There are also videos consisting of subjects standing all the time in the field set, as shown in Figure 6. For each subject, two garment settings are applied, i.e. set1 and set2. The duration of each video is approximately 90 seconds. Human silhouettes are obtained using Detectron2 [42]. Compared to other unconstrained datasets, BRIAR takes 1. outdoor atmospheric turbulence, 2. different walking status and 3. incomplete body shapes into consideration, making it very challenging.
Similar to previous methods for gait recognition, we also evaluate the proposed approach on **CASIA-B**[47], a controlled, indoor dataset with periodic motion obtained. CASIA-B consists of 124 subjects with three walking conditions which are normal walking (NM), walking with a bag (BG) and walking in a coat or jacket (CL). Each sequence consists of eleven view points ranging from \(0^{\circ}\) to \(180^{\circ}\) at an interval of \(18^{\circ}\). We divide the dataset into training and test sets according to the widely applied protocol outlined in [43]. To further demonstrate the effectiveness of the proposed approach in the unconstrained case, we also test GADER on the **Gait3D**[50] dataset. It consists of 25,309 sequences recorded by 39 cameras on 4,000 subjects inside a large supermarket. We follow the official data split, taking the 1,000 subjects as the test set in cross-domain evaluation. In the test set, each subject has one sequence as a query and the rest serve as the gallery.
**Evaluation Metric** For CASIA-B and BRIAR, _verification_ and _rank retrieval_ are used to evaluate gait recognition. For verification, the gallery sequences are paired with probe sequences. We measure the performance using Receiver-Operating Characteristics (ROC) curves that plot the True Accept Rate (TAR) as a function of the False Accept Rate (FAR). As for Gait3D, the evaluation protocol follows the open-set instance retrieval setting and calculates the average Rank-1 and 5 accuracies, mean Average Precision (mAP), and mean Inverse Negative Penalty (mINP) over all queries.
### Implementation Details
All experiments are implemented in Pytorch [33] with four NVIDIA A5000 GPUs. We follow [9] to obtain normalized silhouettes and RGB body chips at the resolution of \(64\times 64\) from the original segmentation masks and frames for CASIA-B and BRIAR. The resizing ratio is recorded during the normalization process. For CASIA-B, we use GaitGL [27] as the backbone for RGB and silhouette feature extraction. For the BRIAR dataset, we use a larger GaitGL architecture designed for OU-MVLP [39], considering the number of identities and scene complexity. In the training phase, the batch size for both datasets is (8,8), i.e. eight subjects with eight videos per subject for each batch. Thirty continuous frames are randomly sampled from a video.
Our gait detector is trained using DHS generated from the BRIAR training set in a supervised manner using annotations provided by BRIAR. To ensure robustness to varying time duration, we randomly select input lengths ranging from 30 to 100 for each training iteration. The detector is evaluated on DHS from the testing sets of three datasets. Specifically, we select window sizes of 33, 50, and 80. The gait detection performance is evaluated using average accuracy, where correctness means 50% or more of the sequence has predictions matching with the ground truth. CASIA-B and Gait3D are pure gait datasets, and we suppose all sequences in these datasets as complete body gait.
### Quantitative Evaluation
**Evaluation on CASIA-B**[47] We compare our proposed gait recognition method with current arts, including GaitPart [13], GaitGL [27], 3DLocal[21], CSTL [20], Lagrange [8] on CASIA-B. The Rank-1 accuracy is in Table 1.
Compared to other methods, our model achieves similar performance on NM and BG compared to Lagrange [8] and better performance on CL. Overall, our method outperforms Lagrange [8], CSTL [20], and 3DLocal [21] by 0.2%, 0.7% and 0.5%, respectively. We note that GAR improves overall performance from 91.8% to 92.6% compared to the original GaitGL with only an extra 1D convolution to extract ratio information in the test phase. GADER even further increases the accuracy to 92.9% with the help of the gait detector.
We also evaluate the _verification_ performance. The TAR(%) results are shown in Table 2. Our proposed method
Figure 6: Examples of subjects in different conditions from the BRIAR dataset at different distances attitudes and clothing. The columns represent controlled-set1, controlled-set2, CR-stand, CR-walk, 100m, 400m, 500m and UAV respectively. Samples are in the **supplementary material** section.
outperforms GaitGL and GaitPart by 0.72% and 3.03% overall, attaining 80.83%. And our method also achieves the best performance in NM, BG and CL with 91.48%, 83.07% and 67.94%. It is noticeable that although the recognition results in NM and BG are likely saturated, there is still room for improvement on the verification task.
**Evaluation on BRIAR**[11] To demonstrate the superior performance of the proposed methods in the wild, we further evaluate on BRIAR with GaitPart [13] and GaitGL [27]. The Rank-1 accuracy is shown in Table 3. Our gait recognition model achieves the highest overall recognition performance, reaching 50.37%, and is 20.6% and 30.82% higher than GaitGL and GaitPart. It is also higher than GAR by 17.67%, just showing that it is beneficial to apply the gait detector to eliminate sequences without human movements. GAR still leads to better performance than previous arts, indicating that the referred discriminative feature space from RGB and ratio information contribute to a better silhouette feature.
When it comes to _verification_, the TAR (%) results are shown Table 3. Compared to GaitGL, our method increases by 26.47%, achieving 61.68% when FAR=\(1e^{-2}\). From Table 3, we observe a big improvement when the gait detector is included. The recognition system's verification results increase from 35.28% to over 60% when FAR=\(1e^{-2}\). This indicates that gait recognition will perform reasonably well if only the qualified gait sequences are used as input.
**Evaluation on Gait3D**[50] To evaluate our model on a public outdoor dataset, we also did cross-domain evaluation on Gait3D. The results are in Table 4. We see that our proposed method achieves higher performance in all criteria. Especially, Rank-1 increases 3.5% to 25.02%. Since cross-domain evaluation is a challenging task, the results are lower than single-domain ones. It is worth noting that, we also adopt GaitGL [27] for the cross-domain evaluation using CASIA-B, OUMVLP, GREW and BRIAR for train and testing on Gait3D. The model trained on the BRIAR dataset exhibited superior performance compared to the others, indicating that the complexity of BRIAR dataset confers greater generalization ability to the trained model.
**Gait Detector Evaluation** We trained a gait detector using DHSs generated using BRIAR training set, and it reaches 91.9%, 86.1% and 88.5% accuracy on BRIAR, CASIA-B and Gait3D respectively. The high accuracy shows that the gait detector is robust to different domains. Visualizations are included in the **supplementary material** section. From
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{2}{|c|}{Gallery NM\#1-4} & \multicolumn{10}{c|}{\(0^{\circ}-180^{\circ}\)} \\ \hline \multicolumn{2}{|c|}{Probe} & \(0^{\circ}\) & \(18^{\circ}\) & \(36^{\circ}\) & \(54^{\circ}\) & \(72^{\circ}\) & \(90^{\circ}\) & \(108^{\circ}\) & \(126^{\circ}\) & \(144^{\circ}\) & \(162^{\circ}\) & \(180^{\circ}\) & mean \\ \hline \multirow{10}{*}{CL\#1-2} & GaitPart [13] & 70.7 & 85.5 & 86.9 & 83.3 & 77.1 & 72.5 & 76.9 & 82.2 & 83.8 & 80.2 & 66.5 & 78.7 \\ & GaitGL [27] & 76.6 & 90.0 & 90.3 & 87.1 & 84.5 & 79.0 & 84.1 & 87.0 & 87.3 & 84.4 & 69.5 & 83.6 \\ & 3DLocal [21] & 78.5 & 88.9 & 91.0 & 89.2 & 83.7 & **80.5** & 83.2 & 84.3 & 87.9 & **87.1** & 74.7 & 84.5 \\ & CSTL [20] & 78.1 & 89.4 & 91.6 & 86.6 & 82.1 & 79.9 & 81.8 & 86.3 & 88.7 & 86.6 & **75.3** & 84.2 \\ & Lagrange [8] & 77.4 & **90.6** & **93.2** & 90.2 & 84.7 & 80.3 & **85.2** & 87.7 & 89.3 & 86.6 & 71.0 & 85.1 \\ \cline{2-11} & **GAR** & 80.8 & 90.5 & 92.2 & 91.0 & 84.7 & 79.7 & 84.8 & 89.6 & 89.7 & **87.1** & 72.8 & 85.7 \\ & **GADER** & **82.5** & 90.5 & 92.0 & **91.9** & **85.2** & 79.3 & 84.5 & **89.8** & **89.8** & **87.1** & 72.3 & **85.9** \\ \hline \multirow{10}{*}{Overall} & GaitPart [13] & 84.6 & 93.0 & 94.3 & 92.3 & 86.5 & 83.2 & 87.2 & 91.3 & 93.0 & 90.6 & 80.9 & 88.8 \\ & GaitGL [27] & 88.4 & 94.9 & 95.3 & 93.5 & 91.7 & 87.9 & 91.1 & 94.1 & 94.9 & 93.3 & 85.0 & 91.8 \\ & 3DLocal [21] & 89.1 & 94.6 & 96.1 & 94.7 & 91.2 & 87.5 & 90.7 & 93.2 & 94.8 & **94.5** & 86.1 & 92.1 \\ & CSTL [20] & 89.0 & 94.9 & 95.9 & 93.3 & 89.7 & 87.8 & 90.3 & 93.6 & 95.0 & 93.6 & **87.3** & 91.9 \\ & Lagrange [8] & 89.1 & 94.9 & **96.3** & 94.7 & 91.8 & **88.3** & 91.4 & 94.5 & 95.5 & 94.1 & 85.6 & 92.4 \\ \cline{2-11} & **GAR** & 90.2 & 95.0 & 95.9 & 94.7 & **91.9** & 88.2 & **91.5** & **94.9** & 95.5 & 94.1 & 86.2 & 92.6 \\ & **GADER** & **91.9** & **95.2** & 96.0 & **95.6** & **91.9** & 88.0 & 91.4 & **94.9** & **95.7** & 94.1 & 86.8 & **92.9** \\ \hline \end{tabular}
\end{table}
Table 1: Rank-1 accuracy (%) on CASIA-B excluding identical-view case for CL#1-2 and overall. The complete table is included in the **supplementary material** section.
\begin{table}
\begin{tabular}{c|c|c|c|c|c} \hline \multicolumn{2}{|c|}{Probe} & CR & 100m & 400m & 500m & UAV & Mean \\ \hline GaitPart [13] & 22.48 & 31.81 & 14.42 & 14.18 & 14.85 & 19.55 \\ GaitGL [27] & 31.26 & 45.82 & 26.81 & 19.49 & 25.48 & 29.77 \\ \cline{2-6}
**GAR** & 31.88 & 49.06 & 27.81 & 23.17 & 31.57 & 32.70 \\ \cline{2-6}
**GADER** & **51.78** & **61.88** & **48.11** & **45.87** & **44.19** & **50.37** \\ \hline \hline \multicolumn{2}{|c|}{FAR} & \(1e^{-4}\) & \(1e^{-3}\) & \(1e^{-2}\) & \(1e^{-1}\) \\ \hline GaitPart [13] & 4.04 & 11.33 & 27.47 & 59.39 \\ GaitGL [27] & 9.15 & 18.64 & 35.21 & 62.67 \\ \cline{2-6}
**GAR** & 8.50 & 18.97 & 35.28 & 63.46 \\ \cline{2-6}
**GADER** & **14.72** & **31.06** & **61.68** & **90.47** \\ \hline \end{tabular}
\end{table}
Table 3: Rank-1 accuracy (%) and verification on BRIAR. CR stands for Close Range, i.e. below 100m; Unmanned Aerial Vehicle (UAV).
Table 1, 2, 3 and 4, the recognition performance increases with the introduction of the detector.
### Ablation Study
To show the impact of each part in our design, we conduct a series of experiments.
**RGB modality is sensitive to appearance change**. In Table 5, we evaluate the framework only using RGB modality with GaitGL as a feature extraction model, i.e. GaitGL\({}_{RGB}\). We observe that it has lower performance than silhouette-based one, i.e. GaitGL, decreasing from 29.77% to 15.69% even only testing on annotated walking sequences set, i.e. Pre-defined, due to its sensitivity to appearance change, but its feature has unique attributes that augment the silhouette feature with a 3.97% improvement.
**Ratio and RGB help better silhouette embedding**. In the gait recognition model, we evaluate the _ratio attention_ and _cross modality distillation_. From Table 6, when we apply cross modality distillation, the recognition accuracy on CL reaches 85% and its verification result increases by 0.52%. As for ratio attention, it improves verification and Rank-1 accuracy under all conditions. Compared to the baseline, our proposed GAR gains a remarkable improvement on verification and Rank-1 in CL from 80.11% and 83.3% to 80.83% and 85.7% respectively, which means the view angle cue from ratio and RGB's feature space help build a representative embedding.
**Mixing gait and non-walking will ruin the feature aggregation**. To demonstrate the necessity of gait detector, we first train with all BRIAR training set data, i.e. GaitGL w/ stand, including standing, random walking and structure walking. From Table 5. We see that when standing sequences are involved in the training process, the performance on gait drops, which means that static sequences disturb gait embedding construction. And it is also computationally consuming to apply stand sequence in gait recognition, since there is little temporal information, a 2D model is enough to use. This observation inspires us to develop a detector to filter out static and low-quality silhouettes.
**Gait detector purifies gait sequences and improves gait performance**. When the gait detector is applied, both models achieve higher recognition results since the detector keeps the silhouette sequences with complete-body and periodic moving patterns. Compared to the Pre-defined gait set, the Detected set further removes sequences with an incomplete body. From Table 1, 2, 3 and 4, we see the potential of detector. Using the same model, GAR, when we exploit a gait detector, almost all performances improve. And this improvement is achieved with little cost, a light-weighted classification network since the DHS is generated by extracting from the input of gait recognition. The well-trained model is robust to domain gaps among different datasets, so it can be directly applied.
## 5 Conclusion
In this paper, we present a novel end-to-end gait detection and recognition approach to address challenging unconstrained conditions. First, we introduce a gait detector to identify sequences that contain gait movement and with complete body. With the gait detector, we can obtain gait sequences without non-walking frames. And it is convenient to incorporate with most existing backbones, just with an extra step to generate the DHS. Secondly, the gait recognition pipeline utilizes both RGB and silhouette modality to learn robust representation. Notably, we address the silhouette size normalization problem with a simple yet effective ratio attention signal. Additionally, we enhance the silhouette modality embedding through feature distillation from the RGB modality. Such a design helps leverage the well-learned feature space of RGB modality with the robustness of silhouettes and does not require RGB input at test time. Through extensive experiments, we show that our proposed method improves the performance on Gait3D in cross-domain evaluation and achieves SoTA performance in the standard CASIA-B and the more challenging BRIAR datasets.
\begin{table}
\begin{tabular}{c c|c c c c} \hline \hline Ratio & Cross & \(R_{NM}\) & \(R_{BG}\) & \(R_{CL}\) & \(Veri\) \\ \hline ✗ & ✗ & 97.2 & 94.1 & 83.3 & 80.11 \\ ✓ & ✗ & 97.4 & 94.4 & 85.1 & 80.42 \\ ✗ & ✓ & 97.3 & **94.5** & 85.0 & 80.63 \\ ✓ & ✓ & **97.5** & **94.5** & **85.7** & **80.83** \\ \hline \hline \end{tabular}
\end{table}
Table 6: Ablation studies on ratio attention and cross modality distillation. The results are shown using recognition accuracy of three probes (\(R_{NM}\), \(R_{BG}\), \(R_{CL}\)) and average verification (\(Veri\)) on CASIA-B.
\begin{table}
\begin{tabular}{c|c|c|c|c|c|c} \hline \hline Dataset & Methods & Rank-1 & Rank-5 & Rank-10 & mAP & mINP \\ \hline \multirow{2}{*}{CASIA-B} & GaitSet [9] & 6.90 & 14.60 & & 4.46 & \\ & GaitGL [27] & 8.80 & 15.70 & 18.80 & 5.55 & 3.16 \\ \hline \multirow{2}{*}{OUMV-LP} & GaitSet [9] & 6.10 & 12.40 & & 4.42 & \\ & GaitGL [27] & 16.40 & 25.80 & 31.20 & 13.11 & 7.28 \\ \hline \multirow{2}{*}{GREW} & GaitSet [9] & 16.50 & 31.10 & & 11.71 & \\ & GaitGL [27] & 18.30 & 31.90 & 39.20 & 13.12 & 7.28 \\ \hline \multirow{3}{*}{BRIAR} & GaitGL [27] & 21.50 & 36.50 & 42.50 & 15.16 & 8.18 \\ & **GAR** & 23.49 & 37.40 & 43.40 & 15.90 & 8.21 \\ \cline{1-1} & **GADER** & **25.02** & **39.18** & **45.41** & **16.89** & **8.62** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Cross domain evaluation on Gait3D with detector.
\begin{table}
\begin{tabular}{c c|c c c c} \hline \hline Model & Pre-defined & Detected \\ \hline GaitGL\({}_{RGB}\)[27] & 15.69 & 14.93 \\ GaitGL w/ stand [27] & 33.56 & 36.46 \\ GaitGL [27] & 43.46 & 45.31 \\ GAR & **47.43** & **50.37** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison among different models tested on pre-defined gait set and detected set on BRIAR.
## 6 Acknowledgement
This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via [2022-21102100005]. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The US. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.
| 歩行認識は、歩くパターンに基づいて主体を robust に識別する可能性を秘めています。色情報を用いる代わりに。これまでのアプローチは、手作業で作成された室内シーンで良好なパフォーマンスを示してきたが、その適用範囲は、屋外や長距離のシーンでは大きく制限されている。そこで、私たちは、課題のある屋外環境での人間認証のために、端から端までGAit DEtection and Recognition (GADER)アルゴリズムを提案しました。具体的には、GADERは、双曲線状のサインを使用して人間の動きの一部を検出し、新しい歩行認識の方法を採用しています。この方法は、補助的なRGB認識モデルから学習された表現に基づいて動作します。推論時には、GADERは only silhouette モダリティを使用しますが、より堅牢な表現を享受できます。室内および屋外データセットにおける広範な実験により、提案された方法は歩行認識および検証におけるState-of-The-Artsを |
2303.00944 | Attention-based Graph Convolution Fusing Latent Structures and Multiple
Features for Graph Neural Networks | We present an attention-based spatial graph convolution (AGC) for graph
neural networks (GNNs). Existing AGCs focus on only using node-wise features
and utilizing one type of attention function when calculating attention
weights. Instead, we propose two methods to improve the representational power
of AGCs by utilizing 1) structural information in a high-dimensional space and
2) multiple attention functions when calculating their weights. The first
method computes a local structure representation of a graph in a
high-dimensional space. The second method utilizes multiple attention functions
simultaneously in one AGC. Both approaches can be combined. We also propose a
GNN for the classification of point clouds and that for the prediction of point
labels in a point cloud based on the proposed AGC. According to experiments,
the proposed GNNs perform better than existing methods. Our codes open at
https://github.com/liyang-tuat/SFAGC. | Yang Li, Yuichi Tanaka | 2023-03-02T03:40:05 | http://arxiv.org/abs/2303.00944v2 | Attention-based Graph Convolution Fusing Latent Structures and Multiple Features for Graph Neural Networks
###### Abstract
We present an attention-based spatial graph convolution (AGC) for graph neural networks (GNNs). Existing AGCs focus on only using node-wise features and utilizing one type of attention function when calculating attention weights. Instead, we propose two methods to improve the representational power of AGCs by utilizing 1) structural information in a high-dimensional space and 2) multiple attention functions when calculating their weights. The first method computes a local structure representation of a graph in a high-dimensional space. The second method utilizes multiple attention functions simultaneously in one AGC. Both approaches can be combined. We also propose a GNN for the classification of point clouds and that for the prediction of point labels in a point cloud based on the proposed AGC. According to experiments, the proposed GNNs perform better than existing methods. Our codes open at [https://github.com/liyang-tuat/SFAGC](https://github.com/liyang-tuat/SFAGC).
Attention-based graph convolution, graph neural network, 3D point cloud, deep learning.
## 1 Introduction
We often encounter irregularly structured data (signals) in the real world where they do not have a fixed spatial sampling frequency. Such data include opinions on social networks, the number of passengers on traffic networks, coordinates of 3D point clouds, and so on.
Deep neural networks have been widely used in recent years to detect, segment, and recognize regular structured data [1, 2, 3]. However, classical deep learning methods can not directly process the irregularly structured data mentioned above. They can be mathematically represented as data associated with a _graph_. An example of graph-structured data, that is, _graph signals_, is shown in Fig. 1. Deep neural networks for graph signals are called graph neural networks (GNNs): They have received a lot of attention [4, 5, 6, 7].
GNNs typically contain multiple graph convolution (GC) layers. The primary mechanism of GCs is to iteratively aggregate (i.e., filter) features from neighbors before integrating the aggregated information with that of the target node [4, 5, 8, 9]. In many existing GC methods, the node-wise features are typically utilized [10, 11, 12, 13]. Furthermore, it is observed that GCs are a special form of Laplacian smoothing [14]. This low-pass filtering effect often results in over-smoothing [4, 5, 14]. Over-smoothing means that the node-wise feature values become indistinguishable across nodes. Intuitively, representational power of GCs refers to the ability to distinguish different nodes [15]. Therefore, over-smoothing may negatively affect the performance of GNNs.
To improve the representational power of GCs, attention-based spatial graph convolutions (AGCs) such as graph attention networks (GATs) [16] have been proposed. AGCs are believed to have high representational power than the direct spatial methods because they can use features on neighboring nodes through the attention weights. However, there exist two major limitations in existing AGCs: 1) They may lose the _structural information_ of the surrounding neighboring nodes, especially in a high-dimensional space. 2) When calculating attention weights for each neighboring node, only one type of attention function is used, e.g., dot-production, subtraction, or concatenation. Different types of attention functions will
lead to different attention weights, which affect the representational power of AGCs.
In this paper, we propose a new AGC to overcome the above-mentioned limitations. First, we propose a _local structure projection aggregation_. This operation aggregates the structural information of neighboring nodes of a target node. Second, we also propose an AGC that utilizes multiple-type attention functions. We can simultaneously utilize these two methods to present the attention-based graph convolution fusing latent structures and multiple features (SFAGC). Our contributions are summarized as follows:
1. By using local structure projection aggregation, we can obtain a representation of the local structural information of the graph in high-dimensional space in an AGC. This allows the convolved nodes to contain richer information than existing methods.
2. By using multiple-type attention functions simultaneously, we can obtain better attention weights than the single attention function.
3. We construct GNNs for graph and node classifications based on the proposed AGC with 3D point clouds. We demonstrate that GNNs using our method present higher classification accuracies than existing methods through experiments on ModelNet [17] for graph classification and ShapeNet [18] for node classification.
_Notation:_ An undirected and unweighted graph is defined as \(\mathcal{G}:=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}\) is a set of nodes, and \(\mathcal{E}\) is a set of edges. The adjacency matrix of \(\mathcal{G}\) is denoted as \(A\). \(\widetilde{D}\) is the diagonal degree matrix. The graph Laplacian is defined as \(L:=\widetilde{D}-A\). \(I_{n}\) is the matrix whose elements in the diagonal are 1.
Here, \(h_{v}:=[\mathrm{h}_{v1},\ldots,\mathrm{h}_{vi},\ldots,\mathrm{h}_{vD}]^{ \mathsf{T}}\in\mathbb{R}^{D}\) represents a feature vector on the node \(v\in\mathcal{V}\), and \(D\) is the number of features in \(h_{v}\). \(co_{v}:=[\mathrm{co}_{v1},\ldots,\mathrm{co}_{vj},\ldots,\mathrm{co}_{vC}]^{ \mathsf{T}}\in\mathbb{R}^{C}\) represents a coordinate of the node \(v\in\mathcal{V}\), and \(C\) is the dimension of coordinate in \(co_{v}\).
The non-linearity function is denoted as \(\sigma(\cdot)\). The set of neighboring nodes is \(N(\cdot)\) in which its cardinality is denoted as \(|N(\cdot)|\). A multilayer perceptron (MLP) layer is represented as MLP(\(\cdot\)). A channel-wise avg-pooling is denoted as AvgPool(\(\cdot\)). A channel-wise max-pooling is denoted as MaxPool(\(\cdot\)). The vector concatenation operation is denoted as cat(\(\cdot\)). The SoftMax operation is represented as SoftMax(\(\cdot\)).
## II Preliminary
In this section, we present related work for the proposed GC.
### _Graph convolutions_
The mechanism of GCs is based on message passing. Message passing involves iteratively aggregating information from neighboring nodes and then integrating the aggregated information with that of the target node [4, 5].
Typically, a GC has two parts, i.e., an aggregator and an updater. The aggregator is to collect and aggregate the node-wise features of neighboring nodes. The updater merges the aggregated features into the target node to update the node-wise features of the target node. These two parts are illustrated in Fig. 2.
Existing GCs can be classified into spectral and spatial methods. Furthermore, spatial methods can be classified into direct and attention-based methods. We briefly introduce them in the following.
#### Ii-A1 Spectral methods
In the spectral methods, the aggregation operation is carried out in the graph Fourier domain. Eigenvectors of graph Laplacian are known as the graph Fourier bases [1, 8, 19]. In order to reduce the computational complexity for large graphs, polynomial approximations of graph filters are often utilized [10, 11, 20].
GCN [12] further reduces computational complexity through the first-order graph filters. It can be formulated as follows:
\[H_{\text{GCN}}=\sigma(\widetilde{D}^{-1/2}\widetilde{A}\widetilde{D}^{-1/2}HW), \tag{1}\]
where \(H:=\{h_{v}\}_{v\in\mathcal{V}}\) is the set of node-wise features, \(\widetilde{A}=A+I_{n}\) is the adjacency matrix with self-loops, and \(W\) is a learnable weight matrix.
#### Ii-A2 Spatial methods
Spatial methods are a counterpart of spectral methods. As we described above, spatial methods can be classified into direct and attention-based methods. Direct spatial methods directly use the node-wise features and the aggregation operation is carried out spatially. A representative method for direct spatial methods is GraphSAGE [13], which treats each neighbor equally with mean aggregation.
Later, attention-based spatial methods are proposed. Instead of treating all neighboring nodes equally, attention-based methods calculate an attention weight for each neighboring node. Then, they use the weighted sum to aggregate features of neighboring nodes. GAT [16] is a representative method for attention-based spatial methods. It is composed
Figure 1: An example of graph-structured data.
Figure 2: A GC has two parts, i.e., an aggregator and an updater. The aggregator is to collects and aggregates the node-wise features of neighboring nodes. The updater merges the aggregated features into the target node to update the node-wise features of the target node.
of three steps. In the first step, the learnable weights are multiplied by the node-wise features, i.e., \(h^{\prime}_{\nu}=\{W\cdot h_{v}\}_{v\in V}\), where \(W\) is the learnable weights. The second step computes attention weights as follows:
\[a_{uv}=\text{SoftMax}(\sigma(W_{a}\cdot(\text{cat}(h^{\prime}_{v},h^{\prime}_{u} )))),u\in N(v) \tag{2}\]
where \(W_{a}\) is learnable weights. In the third step, the node-wise features of a target node \(v\) is updated as follows:
\[h^{\prime\prime}_{v}=\sigma\left(\sum_{u\in N(v)}a_{uv}\cdot(h^{\prime}_{u}) \right). \tag{3}\]
However, as mentioned earlier, existing methods ignore the structural information of neighboring nodes in the high-dimensional space, and use only one type of attention function when calculating an attention weight for each neighboring node.
In contrast to the existing methods, we first focus on computing a representation of the structure of neighboring nodes in high-dimensional feature space and installing them into a spatial GC. Second, we use multiple types of attention functions simultaneously in an AGC. We use both of these two methods simultaneously to achieve SFAGC.
### Structural Features
In previous works [21, 22], three structural features, i.e., feature angle, feature distance, and relational embedding, are proposed to describe the structural information between the target node and neighboring nodes. Below, we briefly introduce them since they are also used in our attention-based GCs.
#### 2.2.1 Feature angle
_Feature angle_ describes the local structure of neighboring nodes. First, a set of structure vectors pointing from target node \(v\) to the neighboring nodes are calculated as \(\text{SV}_{N(v)}=\{\text{s}\text{v}_{uv}:=h_{u}-h_{v}\}_{u\in N(v)}\). Then, a base structure vector \(\text{sv}_{b}\) is learned from \(\text{SV}_{N(v)}\) as follows:
\[\text{sv}_{b}=\text{AvgPool}(\{\sigma(W_{b}\cdot\text{s}\text{v}_{uv})\}_{ \text{sv}_{uv}\in\text{SV}_{N(v)}}) \tag{4}\]
where \(W_{b}\) is learnable weights. An example of a base structure vector \(\text{sv}_{b}\) is shown in Fig. 3
Finally, the cosine of the angle between \(\text{sv}_{uv}\) and \(\text{sv}_{b}\) is calculated to obtain the feature angle \(\text{fa}_{uv}\) as follows:
\[\text{fa}_{uv}=\cos(\theta_{u})=\frac{\text{s}\text{s}\text{v}_{uv}\cdot\text{ s}\text{v}_{b}^{\text{T}}}{\|\text{s}\text{v}_{uv}\|\cdot\|\text{s}\text{v}_{b}\|}, \text{sv}_{uv}\in\text{SV}_{N(v)} \tag{5}\]
An example is shown in Fig. 4 (a).
#### 2.2.2 Feature distance
The second structural feature is _feature distance_. It is the absolute difference between the node-wise features of \(h_{u}\) and \(h_{v}\) represented as follows:
\[\text{fd}_{uv}=[|\text{h}_{u1}-\text{h}_{v1}|,...,|\text{h}_{uD}-\text{h}_{vD} |]|^{\text{T}}. \tag{6}\]
An example is shown in Fig. 4 (b).
#### 2.2.3 Relational embedding
The third structural feature is _relational embedding_. It can be learned from \(\{\text{s}\text{v}_{uv}\}\) as follows:
\[\text{re}_{uv}=\sigma(W_{\text{re}}\cdot\text{s}\text{v}_{uv}),u\in N(v). \tag{7}\]
where \(W_{\text{re}}\) is learnable weights. An example of it is shown in Fig. 4 (c).
Figure 4: Example of our structural features. (a) is our feature angle; (b) is our feature distance, \(\text{h}_{v1}\) is the element of \(h_{v}\), \(D\) is the dimensions of node-wise features; (c) is our relational embedding.
Figure 5: An example of a graph in high-dimensional space. A node consists of a coordinate and the node-wise features, i.e., \(\{v:=((c_{o},h_{v}))_{v\in V}\). For example, the coordinate of a node in a graph of a 3D color point cloud is \((x,y,z)\), and the node-wise features are the values of RGB. We use the coordinates to calculate structural features.
Figure 3: An example of a base structure vector. The number in the black circle are the node indices.
## III Sfaoc
In this section, we introduce SFAGC. As mentioned above, we have two goals: Utilizing 1) the structural information of neighboring nodes in a high-dimensional feature space during a single step GC and 2) multiple-type attention functions simultaneously when calculating the attention weights.
Fig. 5 illustrates an example of a graph with feature vectors in the spatial domain. Spatially distributed nodes often have their coordinates and associated node-wise features, i.e., \(\left\{v:=(co_{v},h_{v})\right\}_{v\in\mathcal{V}}\), where \(co_{v}\) is the coordinate of the node \(v\). For example, a 3D color point cloud equips a 3-D \((x,y,z)\) coordinate and its node-wise features as RGB values. In the previous structure-aware GCs [21, 22], the node-wise features are simply used as their coordinates. In contrast, the proposed GC, SFAGC, simultaneously considers the coordinates and node-wise features.
To achieve our goals, the SFAGC has four parts:
\begin{tabular}{l l} \(A\). & Local structure projection aggregation and fusing \\ \(B\). & Position embedding \\ \(C\). & Weighted sum aggregation and update \\ \(D\). & Coordinates processing \\ \end{tabular}
Figure 6: The details of our SFAGC.
SFAGC is illustrated in Fig. 6 and the algorithm of SFAGC is summarized in Algorithm 1. We sequentially introduce these parts.
### Local structure projection aggregation and Fusing
We propose a projection aggregation operation to obtain the structure representation in the feature space. We then fuse this with the node-wise features of the target node.
#### -A1 Local structure projection aggregation
The inputs of this step are the feature angle, feature distance and relational embedding, i.e., \(\text{fa}_{uv}\), \(\text{fd}_{uv}\) and \(\text{re}_{uv}\), introduced in Section II-B. We first compute structure vectors as follows:
\[\text{s}_{uv}=\text{cat}(\text{fd}_{uv},\text{re}_{uv},\sigma(\left.W_{\text{ se}}(\text{cat}(\text{fd}_{uv},\text{re}_{uv})))\right),u\in N(v), \tag{8}\]
where \(W_{\text{se}}\) is a learnable weight matrix.
Then, we project each \(\text{s}_{uv}\) as follows:
\[\hat{\text{se}}_{uv}=\text{fa}_{uv}\cdot\text{s}_{uv},u\in N(v). \tag{9}\]
Finally, we calculate the summation of the projected structure vectors as follows:
\[\text{af}_{v}=\sum_{u\in N(v)}\hat{\text{se}}_{uv}. \tag{10}\]
Fig. 7 illustrates an example of the local structure projection aggregation.
#### -A2 Fusing structure information with node-wise features
In this step, we fuse the \(\text{af}_{v}\) with the \(h_{v}\) as follows:
\[h_{v}^{\prime}=\sigma(W_{s}(\text{cat}(h_{v},\text{af}_{v}))),v\in\mathcal{V}. \tag{11}\]
where \(W_{s}\) is learnable weights.
### Position Embedding
Position encoding is crucial for a self-attention mechanism because it enables the aggregation operation to adapt to local data structure [23, 24]. Our method directly learns a position embedding, while existing methods use cosine function-based position encoding [24].
We embed the difference of the coordinates between the neighboring node \(u\) and target node \(v\) in the feature space. Our position embedding \(p_{u}\) for the node \(u\) is represented as follows:
\[p_{u}=W_{\text{P2}}\cdot(\sigma(W_{\text{P1}}\cdot(co_{u}-co_{v}))),u\in N(v) \tag{12}\]
where \(W_{\text{P1}}\) and \(W_{\text{P2}}\) are learnable weights.
### Weighted Sum Aggregation and Update
In this part, we update the node-wise features of the target node \(v\).
First, we introduce the calculation steps of the attention weights. Then, we present the weighted sum aggregation and node-wise features update step used in SFAGC.
#### -A1 Attention weights
As we described above, we simultaneously use multiple-type attention functions to calculate attention weights. In existing methods [23, 24, 25], the subtraction or dot-production attention function is often utilized to calculate attention weights. Instead of the single attention function, we use these attention functions simultaneously. The subtraction attention function is defined as
\[\text{a}_{1vu}:=W_{q1}\cdot h_{v}^{\prime}-W_{k1}\cdot h_{u}^{\prime},u\in N( v), \tag{13}\]
where \(W_{q1}\) and \(W_{k1}\) are learnable weights. The dot-production attention function is represented as
\[\text{a}_{2vu}:=(W_{q2}\cdot h_{v}^{\prime})\cdot(W_{k2}\cdot h_{u}^{\prime}) ^{T},u\in N(v) \tag{14}\]
where \(W_{q2}\) and \(W_{k2}\) are learnable weights.
Then, the two types of attention functions are added with the position embedding \(p_{u}\) as follows:
\[\text{qk}_{vu}=\text{a}_{1vu}+W_{c}\cdot\text{a}_{2vu}+p_{u},u\in N(v) \tag{15}\]
where \(W_{c}\) is learnable weights that also converts \(\text{a}_{2vu}\) into the same dimensions as \(\text{a}_{1vu}\).
Finally, \(\text{qk}_{vu}\) is input into a small network to calculate the attention weights between the target node \(v\) and the neighboring node \(u\) as follows:
\[\text{at}_{eu}=\text{SoftMax}\left(\frac{W_{a2}\cdot\sigma(W_{a1}\cdot\text{ qk}_{vu})}{\sqrt{d_{out}}}\right),u\in N(v), \tag{16}\]
where \(W_{a1}\in\mathbb{R}^{d_{eu}\times d_{out}}\) and \(W_{a2}\in\mathbb{R}^{d_{out}\times 1}\) are learnable weights, \(d_{in}\) and \(d_{out}\) are the dimensions of the \(W_{a1}\).
#### -A2 Weighted-sum aggregation
For \(h_{v}^{\prime}\) and \(h_{u}^{\prime},u\in N(v)\), weighted sum aggregation is calculated as follows:
\[\widetilde{h_{v}}=\sum_{u\in N(v)}\text{at}_{eu}\cdot(W_{v}\cdot h_{u}^{ \prime}+p_{u}), \tag{17}\]
where \(W_{v}\) is a learnable matrix.
Figure 7: An example of the local structure projection aggregation. The \(\text{s}_{uv}\) is the base structure vector defined in (4).
3 Node-wise features update \(h_{v}\), \(co_{v}\) and \(\widetilde{h_{v}}\) are integrated as follows:
\[h_{v}^{\prime\prime}=\sigma(W\cdot\text{cat}(h_{v},co_{v},\widetilde{h_{v}})). \tag{18}\]
where \(W\) is learnable weights.
### Coordinate update
Finally, we update the coordinate of the target node \(v\) as follows:
\[co_{v}^{\prime}=\text{MLP}(co_{v}),v\in\mathcal{V}. \tag{19}\]
Hereafter, we represent the set of these operations as \(\{h_{\mathcal{V}}^{\prime\prime},co_{\mathcal{V}}^{\prime}\}:=\text{SFAGC}(h_ {\mathcal{V}},co_{\mathcal{V}},A)\).
## IV Implementation
In this section, we construct classification and segmentation networks for 3D point clouds based on SFAGC. Their architectures are illustrated in Fig. 8. In the following, first, we introduce the components shared by the two GNNs. Then, specific implementations for each of the GNNs are introduced.
Here, we suppose that the input point cloud is given by \(\mathcal{X}=\{x_{i}\}_{i=1}^{N}\) where \(N\) is the number of points.
### Preprocessing
To alleviate effects on rotations of point clouds, we use the same transformation module as PointNet [26] for preprocessing.
### SFAGC module
The inputs to this module are a set of features obtained from the previous layer and a set of node coordinates in the feature space. Suppose that the set of features output from the previous layer is \(\mathcal{H}=\{h_{i}\}_{i=1}^{M}\) where \(M\) is the number of features of the input. For the first module, \(\mathcal{H}=CO=\mathcal{X}\) where \(CO=\{co_{j}\}_{j=1}^{M}\) is the set of node coordinates.
First, we construct \(\mathcal{G}\) with the \(k\)-NN graph from \(CO\) where the \(i\)th node \(v_{i}\) in \(\mathcal{G}\) corresponds to the \(i\)th feature \(h_{i}\) and the \(i\)th coordinate \(co_{i}\).
Recent studies [27, 28] have shown that dynamic graph convolution, i.e., allowing the graph structure to change at each layer, can perform better than that with a fixed graph structure. Therefore, we update coordinates of nodes at each SFAGC module, and we construct different graphs for different SFAGC modules.
### Design of 3d point cloud classification network
Fig. 8 (a) illustrates the architecture of 3D point cloud classification network based on SFAGC. In the following, we describe the details of the building blocks that are specifically designed for point cloud classification.
#### Iv-1 Multi-resolution point clouds
We use a layer of PointNet++ [29] to generate a low-resolution point cloud. Both global and local information of the point cloud can be obtained through the multi-resolution structure.
#### Iv-2 Score-based graph pooling
Effective graph pooling methods are a hot topic in GNNs and graph signal processing [30, 31]. Early work has been done by global pooling of all node-wise features or by using graph coarsening algorithms.
Recently, trainable graph pooling operations DiffPool [32], GraphU-net [33], and AttPool [34] have been proposed. Inspired by the ResNeXt [35], we extend the score-based graph pooling module proposed in SAMGC [22] by introducing the multi-branch architecture. The score-based graph pooling module has three branches: score-based sampling, integration, and SFAGC branches. The architecture of the score-based graph pooling is shown in Fig. 9. In the following, we introduce their details.
In the score-based sampling branch, we propose a score-based sampling to find the indices of the best \(t\) nodes according to their scores. The score associated with each node is first computed as follows:
\[score_{v}=\text{SoftMax}(W_{1}\cdot h_{v}),v\in\mathcal{V}, \tag{20}\]
where \(W_{1}\) is learnable weights. We then sort the nodes in descending order according to their scores. We find the indices of the top \(t\) nodes as follows:
\[\text{idx}_{select}=\text{rank}(\{score_{v}\}_{\mathcal{V}},t), \tag{21}\]
where \(\text{rank}(\cdot)\) is the ranking operation, and it finds the indices of the \(t\) highest scores.
In the integration branch, node-wise features are multiplied by the node scores as follows:
\[\hat{h}_{\mathcal{V}}=\{score_{v}\cdot h_{v}\}_{v\in\mathcal{V}}. \tag{22}\]
In the SFAGC branch, the input graph is processed using SFAGC as follows:
\[\{h_{\mathcal{V}}^{\prime},co_{\mathcal{V}}^{\prime}\}=\text{SFAGC}(h_{ \mathcal{V}},co_{\mathcal{V}}). \tag{23}\]
Finally, the subset of \(\hat{h}_{\mathcal{V}}\) and the subset of \(\{h_{\mathcal{V}}^{\prime},co_{\mathcal{V}}^{\prime}\}\) are found using \(\text{idx}_{select}\) as follows:
\[\hat{h}_{\mathcal{V}_{sub}} =\hat{h}_{\mathcal{V}}[\text{idx}_{select}] \tag{24}\] \[\{h_{\mathcal{V}_{sub}}^{\prime},co_{\mathcal{V}_{sub}}^{\prime}\} =\{h_{\mathcal{V}}^{\prime},co_{\mathcal{V}}^{\prime}\}[\text{idx}_{select}]\]
\(\hat{h}_{\mathcal{V}_{sub}}\) and \(h_{\mathcal{V}_{sub}}^{\prime}\) are merged using learnable weights as follows:
\[\hat{h}_{\mathcal{V}_{sub}}^{\prime}=\sigma(W_{pl}\cdot\text{cat}(\hat{h}_{ \mathcal{V}_{sub}},h_{\mathcal{V}_{sub}}^{\prime})), \tag{25}\]
where \(W_{pl}\) is learnable weights. The score-based graph pooling can be summarized as follows:
\[\{co_{\mathcal{V}_{sub}}^{\prime},\hat{h}_{\mathcal{V}_{sub}}^{\prime}\}= \text{GraphPool}_{s}(co_{\mathcal{V}},h_{\mathcal{V}}). \tag{26}\]
#### 3.3.3 Hierarchical prediction architecture
Here, we also use the intermediate supervision technique [36] and the hierarchical prediction architecture (Fig. 8 (a)), as in SAMGC [22]. The advantage of this architecture is that, by combining the outcomes of the different phases, more reliable and robust predictions can be produced [22]. The details are presented below.
We use two SFAGC modules in phase 1 to phase 5. One PointNet++ [29] layer is used in phase 6. Each phase connects with a max pooling layer and an average pooling layer. The outputs are then concatenated and input into a fully connected layer. We calculate the prediction and the classification loss for each phase. The total classification loss is obtained by adding the losses of several phases. Meanwhile, the total prediction is also increased by the predictions of several phases. The following is a representation of this processing:
\[prediction =\sum_{i=1}^{P}prediction_{i}, \tag{27}\] \[loss =\sum_{i=1}^{P}loss_{i}, \tag{28}\]
where \(prediction_{i}\) is the prediction of the \(i\)th phase, \(loss_{i}\) is the cross-entropy loss of the \(i\)th phase. \(prediction\) and \(loss\) are the total prediction and classification loss, respectively. \(P\) is the number of phases.
Figure 8: The architectures of our 3D point cloud classification network and our 3D point cloud segmentation network. (a) is the architecture of 3D point cloud classification network; (b) is the architecture of 3D point cloud segmentation network, L_p_points is the set of outputs of the \(j\)th phase. \(N_{\rm Lip}\) is the number of nodes of the L_p_points.
Figure 9: The details of our graph pooling operation. The score-based graph pooling is used in 3D point cloud classification network. The FPS-based graph pooling is used in 3D point cloud segmentation network. Here, we set \(t=3\).
### Design of 3D point cloud segmentation network
Fig. 8 (b) illustrates the architecture of the 3D point cloud segmentation network based on SFAGC. In the following, we describe the details of the building blocks that are specifically designed for point cloud segmentation.
#### 1.4.1 Farthest point sampling-based graph pooling
For point cloud segmentation, we use the graph pooling module to reduce the overall computation cost. Here, we propose the farthest point sampling-based graph pooling (FPS-based graph pooling) by modifying the score-based graph pooling. The architecture of the FPS-based graph pooling is illustrated in Fig. 9. In the following, we introduce its details.
The FPS-based graph pooling has a multi-branch architecture like the score-based graph pooling in Section 4.2.2. In contrast to the score-based method, the FPS-based graph pooling has two branches, i.e., the FPS and SFAGC branches.
In the FPS branch, we perform FPS on nodes to obtain indices of the best \(t\) nodes according to their coordinates. FPS algorithm is widely utilized in 3D point cloud processing [29, 37]. The mechanism of FPS is to iteratively select the node that is farthest from the existing sampled nodes. Therefore, the sampled nodes with the FPS-based sampling are expected to be more evenly distributed than those with the score-based sampling. This branch can be summarized as follows:
\[\text{idx}_{select}=\text{FPS}(\{co_{v}\}_{\mathcal{V}},t). \tag{29}\]
where \(\text{idx}_{select}\) is the indices of the \(t\) selected nodes, \(\text{FPS}(\cdot)\) is the farthest point sampling algorithm.
The SFAGC branch is the same as the one that is used in the score-based graph pooling represented as
\[\{h^{\prime}_{\mathcal{V}},co^{\prime}_{\mathcal{V}}\}=\text{ SFAGC}(h_{\mathcal{V}},co_{\mathcal{V}}). \tag{30}\]
Finally, the subset of \(\{h^{\prime}_{\mathcal{V}},co^{\prime}_{\mathcal{V}}\}\) are extracted using \(\text{idx}_{select}\) as follows:
\[\{co^{\prime}_{\mathcal{V}_{sub}},h^{\prime}_{\mathcal{V}_{sub}}\}=\{co_{ \mathcal{V}},h_{\mathcal{V}}\}[\text{idx}_{select}]. \tag{31}\]
The FPS-based graph pooling is represented as follows:
\[\{co^{\prime}_{\mathcal{V}_{sub}},h^{\prime}_{\mathcal{V}_{sub}}\}=\text{ GraphPool}_{f}(co_{\mathcal{V}},h_{\mathcal{V}}). \tag{32}\]
#### 1.4.2 Upsampling operation
Graph pooling can be regarded as a downsampling operation. For point cloud segmentation, the network also needs to perform upsampling in order to maitain the number of points. Therefore, the feature propagation used in PointNet++ [29] is also used in our network as the upsampling operation.
## 5 Experiments
In this section, we conduct experiments on 3D point cloud classification and segmentation to validate the proposed GC.
### 3D point cloud classification
Here, we present the 3D point cloud classification experiment using the 3D point cloud classification network introduced in Section 4.
#### 5.1.1 Dataset
The ModelNet dataset [17] is used in our point cloud classification experiment. 12,308 computer-aided design (CAD) models in 40 categories are included in ModelNet40. In addition, 9,840 CAD models are utilized for training and
\begin{table}
\begin{tabular}{c|l|l|l|l} \hline \hline \multicolumn{1}{c|}{Epoch} & \multicolumn{1}{c}{200} \\ \hline \multicolumn{1}{c|}{Batch size} & \multicolumn{1}{c}{16} \\ \hline \multicolumn{1}{c|}{Learning rate} & \multicolumn{1}{c}{0.001} \\ \hline \multicolumn{1}{c|}{Drop out} & \multicolumn{1}{c}{0.3} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \hline \multicolumn{1}{c|}{Phase} & \multicolumn{1}{c}{Graph convolution layer} & \multicolumn{1}{c}{Coordinates update} & \multicolumn{1}{c}{\(k\)} & \multicolumn{1}{c}{[\(\text{co}_{\text{fa}},\text{co}_{\text{out}},\text{f}_{\text{out}}\)]} \\ \hline \multicolumn{1}{c|}{Phase1} & \multicolumn{1}{c}{SFAGC(\(co_{v}=x_{v}\))} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 20 & [3,3,26,4] \\ \multicolumn{1}{c|}{} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=co_{v}\) & 20 & [32,64,-64] \\ \hline \multicolumn{1}{c|}{Phase2} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 20 & [32,64,64,64] \\ \multicolumn{1}{c|}{} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=co_{v}\) & 20 & [64,64,-128] \\ \hline \multicolumn{1}{c|}{Phase3} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 20 & [64,128,128,256] \\ \hline \multicolumn{1}{c|}{Phase4} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 20 & [64,128,64,128] \\ \multicolumn{1}{c|}{} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=\text{co}_{v}\) & 20 & [64,128,-128] \\ \hline \multicolumn{1}{c|}{Phase5} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 20 & [128,128,128,256] \\ \multicolumn{1}{c|}{} & \multicolumn{1}{c}{SFAGC} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 20 & [128,128,256] \\ \hline \multicolumn{1}{c|}{\begin{tabular}{c} Graph pooling layername & Graph convolution layer \\ \end{tabular} } & \multicolumn{1}{c}{Coordinates update} & \multicolumn{1}{c}{\(k\)} & \multicolumn{1}{c}{[\(\text{co}_{\text{fa}},\text{co}_{\text{out}},\text{f}_{\text{out}}\)]} \\ \hline \multicolumn{1}{c|}{\begin{tabular}{c} GraphPool\_s 1 \\ GraphPool\_2 \\ GraphPool\_3 \\ GraphPool\_3 \\ \end{tabular} } & \multicolumn{1}{c}{SFAGC(\(co_{v}=h_{v}\))} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 36 & 512 & [131,131,32,64] \\ \multicolumn{1}{c|}{\begin{tabular}{c} GraphPool\_3 \\ \end{tabular} } & \multicolumn{1}{c}{SFAGC(\(co_{v}=h_{v}\))} & \(co^{\prime}_{\mathcal{V}}=\text{MLP}(co_{v})\) & 64 & 128 & [320,320,64,128] \\ \hline \multicolumn{1}{c|}{
\begin{tabular}{c} Pointnet++ layername & \(\mathcal{S}\) \\ \end{tabular} } & \multicolumn{1}{c}{\(r\)} & \multicolumn{1}{c}{\(D\)} & \multicolumn{1}{c}{[input channels, output channels]} \\ \hline \multicolumn{1}{c|}{Pointnet++} & \multicolumn{1}{c}{512} & \multicolumn{1}{c}{0.2} & \multicolumn{1}{c}{32} & \multicolumn{1}{c}{[3,64]} \\ \hline \hline \end{tabular}
\end{table}
Table 1: The hyperparameters of the point cloud classification network. \(k\) is the value of \(k\)-NN, \(t\) is the number of selected nodes. For Pointnet++ layer, \(S\) is the number of sampled points, \(r\) is the radius of each group, \(D\) is the number of points of each group, \(\text{co}_{\text{fa}}\) is the number of channels of the input coordinates. \(\text{fa}_{\text{fa}}\) is the number of channels of the input node-wise features. \(\text{co}_{\text{fa}}\) is the number of channels of the output coordinates. \(\text{fa}_{\text{fa}}\) is the number of channels of the output node-wise features. The symbol \(\mathcal{V}^{-}\) indicates that the parameters are not available. The input 3D point cloud is \(\mathcal{X}=\{x_{v}\}_{v=1}^{N}\).
2,468 CAD models are used for testing. 4,899 CAD models are included in ModelNet10. They are divided into 3,991 for training and 908 for testing from ten categories. For each CAD models, the CAD mesh faces were evenly sampled with 1,024 points. Initially, all point clouds were normalized to be in a unit sphere.
#### 3.2.2 Settings and evaluation
The settings of hyperparameters are summarized in Table 1. We use the average accuracy of all test instances (OA) and the average accuracy of all shape classes (mAcc) to evaluate the performance of our network.
#### 3.2.3 Results and discussion
Table 2 summarizes the results for point cloud classification. The results of existing methods are taken from their original papers. In terms of both OA and mAcc, our method performs better than the others.
In the following, we focus on the comparison of our method and graph-based methods. The GCs utilized in DPAM [45] are GCN [12]. Therefore, it only uses node-wise features of one-hop neighboring nodes. RGCNN [46], 3DTI-Net [47], PointGCN [48], and LocalSpaceGCN [49] are the spectral methods. Although they can design the global spectral response, they may neglect the local spatial information. In comparison with the direct spatial methods [27], [28],
\begin{table}
\begin{tabular}{c|l|c c|c c} \hline \hline \multicolumn{1}{c|}{Type} & Method & \multicolumn{2}{c|}{ModelNet40} & \multicolumn{2}{c}{ModelNet10} \\ \cline{3-6} \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & OA & mAcc & OA & mAcc \\ \hline Pointwise MLP & PointNet [26] & 89.2\% & 86.2\% & - & - \\ Methods & PointNet++ [29] & 90.7\% & - & - & - \\ & SRN-PointNet++ [38] & 91.5\% & - & - & - \\ \hline Transformer-based & PointASNL [37] & 93.2\% & - & 95.9\% & - \\ Methods & PCT [39] & 93.2\% & - & - & - \\ & PointTransformer [25] & 93.7\% & 90.6 \% & - & - \\ \hline Convolution-based & PointConv [40] & 92.5\% & - & - & - \\ Methods & A-CNN [41] & 92.6\% & 90.3\% & 95.5\% & 95.3\% \\ & SFCNN [42] & 92.3\% & - & - & - \\ & InterpCNN [43] & 93.0\% & - & - & - \\ & ConvPoint [44] & 91.8\% & 88.5\% & - & - \\ \hline Graph-based & Spectral & DPAM [45] & 91.9\% & 89.9\% & 94.6\% & 94.3\% \\ Methods & Methods & RGCNN [46] & 90.5\% & 87.3\% & - & - \\ & & 3DTI-Net [47] & 91.7\% & - & - & - \\ & PointGCN [48] & 89.5\% & 86.1\% & 91.9\% & 91.6\% \\ & LocalSpecGCN [49] & 92.1\% & - & - & - \\ \hline Spatial & ECC [28] & 87.4\% & 83.2\% & 90.8\% & 90.0\% \\ Methods & KCNet [50] & 91.0\% & - & 94.4\% & - \\ & DGCNN [27] & 92.2\% & 90.2\% & - & - \\ & LDGCNN [51] & 92.9\% & 90.3\% & - & - \\ & Hassan et al. [52] & 89.1\% & - & - & - \\ & ClusterNet [53] & 87.1\% & - & - & - \\ & Grid-GCN [54] & 93.1\% & 91.3\% & 97.5\% & 97.4\% \\ & SAGConv [21] & 93.5\% & 91.3\% & 98.3\% & 97.7\% \\ & SAMGC [22] & 93.6\% & 91.4\% & 98.3\% & 97.7\% \\ \cline{2-6} & **SFAGC** & **94.0\%** & **91.6\%** & **98.6\%** & **97.8\%** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison results of the 3D shape classification on the ModelNet benchmark. OA indicates the average accuracy of all test instances, and mAcc indicates the average accuracy of all shape categories. The symbol ‘-’ indicates that the results are not available from the references.
\begin{table}
\begin{tabular}{c|l|l|l|l} \hline \hline \multicolumn{1}{c|}{Epoch} & \multicolumn{1}{c|}{251} \\ \hline \multicolumn{1}{c|}{Batch size} & \multicolumn{1}{c|}{16} \\ \hline \multicolumn{1}{c|}{Learning rate} & \multicolumn{1}{c|}{0.001} \\ \hline \multicolumn{1}{c|}{Drop out} & \multicolumn{1}{c|}{0.4} \\ \hline \multicolumn{1}{c|}{Phase} & Graph convolution layer & \multicolumn{1}{c|}{Coordinates update} & \(k\) & \([\text{co}_{n}\text{-}\text{f}_{n}\text{co}_{n}\text{-f}_{out}]\) \\ \hline \multicolumn{1}{c|}{Phase1} & SFAGC(\(cov_{n}=x_{v}\)) & \(cov_{n}^{\prime}=\text{MLP}(cov_{n})\) & 20 & \([3,3,32,64]\) \\ \hline \multicolumn{1}{c|}{Phase2} & SFAGC & \(cov_{n}^{\prime}=\text{MLP}(cov_{n})\) & 20 & \([3,64,32,64]\) \\ \multicolumn{1}{c|}{} & SFAGC & \(cov_{n}^{\prime}=\text{co}_{n}\) & 20 & \([32,64,-128]\) \\ \hline \multicolumn{1}{c|}{Phase3} & SFAGC & \(cov_{n}^{\prime}=\text{MLP}(cov_{n})\) & 20 & \([3,128,128,256]\) \\ \hline \multicolumn{1}{c|}{Graph pooling layername} & Graph convolution layer & Coordinates update & \(k\) & \(t\) & \([\text{co}_{n}\text{-f}_{n}\text{co}_{n}\text{-f}_{out}]\) \\ \hline \multicolumn{1}{c|}{GraphPool\_1} & SFAGC(\(cov_{n}=x_{v}\)) & \(cov_{n}^{\prime}=cov_{n}\) & 36 & 512 & \([3,131,36,4]\) \\ \multicolumn{1}{c|}{GraphPool\_2} & \multicolumn{1}{c|}{SFAGC(\(cov_{n}=x_{v}\))} & \(cov_{n}^{\prime}=cov_{n}\) & 64 & 128 & \([3,320,3,128]\) \\ \hline \multicolumn{1}{c|}{Feature propagation layer name} & \multicolumn{1}{c|}{[input channels, output channels]} \\ \hline \multicolumn{1}{c|}{Feature propagation1} & \multicolumn{1}{c|}{[256+256+128+128,256]} \\ \multicolumn{1}{c|}{Feature propagation2} & \multicolumn{1}{c|}{[256+64+64,128]} \\ \hline \hline \end{tabular}
\end{table}
Table 3: The hyperparameters of the point cloud segmentation network. \(k\) is the value of \(k\)-NN, \(t\) is the number of selected nodes. \(\text{co}_{n}\) is the number of channels of the input coordinates. \(\text{f}_{n}\) is the number of channels of the input node-wise features. \(\text{co}_{n}\) is the number of channels of the output coordinate. \(\text{f}_{out}\) is the number of channels of the output node-wise features. The symbol ‘-’ indicates that the parameters are not available. The input 3D point cloud is \(\mathcal{X}=\{\chi_{i}\}_{i=1}^{k}\).
[50, 51, 52, 53, 54], our method can obtain the local structural information of the graph in the feature space, and the information of neighboring nodes can be utilized efficiently using attention-based aggregation. In comparison with the direct spatial methods, i.e., SAGConv [21] and SAMGC [22], the proposed method can better aggregate the structural information of the neighboring nodes using the local structure projection aggregation. Furthermore, the information of neighboring nodes can be utilized efficiently using attention-based aggregation. These are possible reasons for the performance improvement of the proposed method.
### 3D Point Cloud Segmentation
In this subsection, we also perform a 3D point cloud segmentation experiment.
#### 3.c.1 Dataset
The ShapeNet dataset [18] is used in the experiment. It contains 16,846 computer-aided design (CAD) models in 40 categories and each point cloud has 2,048 points. 2,874 point clouds are used for testing, and 13,998 of them are taken for training.
#### 3.c.2 Settings and evaluation
Table 3 shows the hyperparameter settings. For this experiment, we re-perform existing methods with their corresponding codes available online. The hyperparameters (epoch, batch size, learning rate, and drop out) used in the existing methods experiments are the same with those shown in Table 3.
We evaluated the average mIoU of all test instances with other neural networks designed specifically for point cloud segmentation.
#### 3.c.3 Results and discussion
Experimental results for point cloud segmentation are summarized in Table 4. It is observed that our method has higher mIoU than the existing methods.
Here, we discuss the possible reasons of the improvement. We first focus on the comparison between our method and DGCNN [27]. DGCNN corresponds to a graph-based direct spatial method. In contrast to the direct method, our method can utilize the local structural information in the feature space, and it also collects the information of neighboring nodes through attention-based aggregation.
We also compare our method with transformer-based methods PointASNL [37] and PCT [39]. While the transformer basically has a large number of parameters, they are restricted to use only node-wise features and the single dot-production attention function to calculate attention weights. In contrast, SFAGC utilizes the local structural information in the feature space and the multiple-type attention functions.
### Ablation Study
We also perform extra 3D point cloud segmentation experiments to validate the effectiveness of the components in the SFAGC.
The hyperparameters are the same as those shown in Table 3. Here, we use some AGCs with different settings as follows:
1. **SFAGC**. This is the full SFAGC described in Section 3.
2. **SFAGC-nS**. To validate the effectiveness of the local structure projection aggregation and fusing part proposed in the Section 3.A, we discard this part from the SFAGC.
3. **SFAGC-nP**. To confirm the effectiveness of the position embedding proposed in Section 3.B, we bypass the position embedding part from the SFAGC.
4. **SFAGC-ndot and SFAGC-nsub**. To validate the effectiveness of the multi-type attention function proposed in Section 3.C, SFAGC-ndot is set as the SFAGC without the dot-production attention function, and SFAGC-nsub is set as the SFAGC without the subtraction attention function.
The architecture of the network and hyperparameters are the same as the previous experiment.
The results are summarized in Fig. 10. By comparing SFAGC with SFAGC-nS, use of the local structures in feature space increases 0.3 mIoU. In SFAGC vs. SFAGC-nP, the position-embedding phase increases 0.4 mIoU. In SFAGC vs. SFAGC-ndot, the multi-type attention function increases 0.2 mIoU. In SFAGC vs. SFAGC-nsub, the multi-type attention
\begin{table}
\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c} \hline \hline Method & mIoU & air- & bag & cap & car & chair & car- & guitar & knife & lamp & laptop & motor & mug & pistol & rocket & skate- & table \\ & & plane & & & & phone & & & top & & & & & board & \\ \hline PointNet [26] & 83.0 & 81.5 & 64.8 & 77.2 & 73.5 & 88.6 & 68.3 & 90.4 & 84.1 & 80.0 & 95.1 & 59.2 & 91.8 & 79.7 & 52.1 & 72.4 & 81.6 \\ Point- & 84.7 & 81.4 & 73.4 & 80.6 & **77.8** & 89.9 & 74.5 & 90.6 & 85.8 & 83.5 & 95.1 & **69.3** & **94.0** & 81.0 & 58.5 & **74.3** & 82.0 \\ DGCNN [27] & 85.0 & 82.6 & 79.8 & **85.3** & 76.9 & 90.4 & 77.1 & **91.0** & 86.9 & 84.0 & **95.6** & 61.5 & 93.0 & 79.9 & 58.2 & 73.7 & **83.2** \\ Point- & 84.6 & 82.4 & 80.3 & 83.2 & 76.8 & 89.9 & **80.6** & 90.8 & 86.7 & 83.2 & 95.3 & 60.1 & 93.5 & **81.6** & 59.1 & 73.7 & 82.3 \\ ASNL [37] & 84.7 & 83.6 & 67.6 & 83.6 & 75.4 & 90.1 & 74.5 & 90.8 & 85.8 & 82.1 & 95.4 & 64.0 & 92.1 & 81.1 & 56.2 & 72.5 & **83.2** \\ \hline
**SFAGC** & **85.5** & **83.7** & **80.9** & 83.5 & 77.4 & **90.5** & 76.2 & **91.0** & **88.7** & **84.2** & 95.5 & 67.4 & 93.7 & 81.1 & **59.4** & 74.1 & **83.2** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison results of the 3D point cloud segmentation on the ShapeNet. mIoU indicates the average mIoU of all test instances. The mIoU of each class is also shown. The results are obtained by experimenting with the same hyper-parameters.
function increases 0.3 mIoU. The effectiveness of the SFAGC modules was demonstrated in this study.
## VI Conclusion
In this paper, we propose a new attention-based graph convolution named SFAGC. It can better aggregate the structural information of the neighboring nodes in high-dimensional feature space by using local structure projection aggregation. It also can computes better attention weights by using multi-type attention functions simultaneously. Through experiments on point cloud classification and segmentation, our method outperforms existing methods.
| 私たちは、グラフ神経ネットワーク (GNN) に適用する注意に基づいた空間グラフ畳み込み (AGC) を提案します。既存の AGC は、ノードのみに特徴を適用し、注意重み計算時に単一の注意関数を用いていました。代わりに、 AGC の表現力の向上を図るために、構造情報を含む高次元空間において、 1) 構造情報を利用し、2) 重み計算時に複数の注意関数を利用する方法を提案します。 1 つの方法は、高次元空間においてグラフの局所構造情報を計算します。もう 1 つの方法は、複数の注意関数を同時に AGC に利用します。両方のアプローチは組み合わせて使用できます。提案された AGC を用いた点雲の分類と、点雲上の点ラベルの予測のための GNN を提案しました。実験の結果、提案された GNN は既存の方法と比べて性能が向上しています。私たちのコードは、https://github.com/liyang |
2303.11691 | A versatile classification tool for galactic activity using optical and
infrared colors | We use the Random Forest (RF) algorithm to develop a tool for automated
activity classification of galaxies into 5 different classes: Star-forming
(SF), AGN, LINER, Composite, and Passive. We train the algorithm on a
combination of mid-IR (WISE) and optical photometric data while the true labels
(activity classes) are based on emission line ratios. Our classifier is built
to be redshift-agnostic and it is applicable to objects up to z $\sim$0.1. It
reaches a completeness $>$80 % for SF and Passive galaxies, and $\sim$60 % for
AGN. Applying it to an all-sky galaxy catalog (HECATE) reveals a large
population of low-luminosity AGNs outside the AGN locus in the standard mid-IR
diagnostics. | Elias Kyritsis, Charalampos Daoutis, Andreas Zezas, Konstantinos Kouroumpatzakis | 2023-03-21T09:21:22 | http://arxiv.org/abs/2303.11691v1 | # A versatile classification tool for galactic activity using optical and infrared colors
###### Abstract
We use the Random Forest (RF) algorithm to develop a tool for automated activity classification of galaxies into 5 different classes: Star-forming (SF), AGN, LINER, Composite, and Passive. We train the algorithm on a combination of mid-IR (WISE) and optical photometric data while the true labels (activity classes) are based on emission line ratios. Our classifier is built to be redshift-agnostic and it is applicable to objects up to z \(\sim\)0.1. It reaches a completeness \(>\)80% for SF and Passive galaxies, and \(\sim\)60% for AGN. Applying it to an all-sky galaxy catalog (HECATE) reveals a large population of low-luminosity AGNs outside the AGN locus in the standard mid-IR diagnostics.
activity diagnostics - star-formation - machine Learning- AGN 75282
M12 and A13
## 1 Introduction
Activity classification of galaxies is of great importance for many fields of extragalactic Astrophysics, such as understanding galaxy evolution (Kewley et al., 2019) and/or AGN demographics. Traditionally, this is done using characteristic emission-line ratios which discriminate galaxies into different classes depending on the source of ionization (e.g. Kewley et al., 2019). However, the need for spectroscopic data hampers the applicability of these diagnostics to very large datasets since spectroscopic observations are available for a subset of the objects with photometric data. In addition, these diagnostics cannot be used on galaxies without emission lines rendering them inapplicable to passive galaxies. While alternative diagnostics based on mid-IR colors (Mateos et al., 2012; Assef et al., 2013, hereafter M12 and A13) are successfully used for identifying luminous AGNs, they are not as reliable in the local universe.
To address these limitations, we develop a new activity diagnostic by combining the RF machine learning algorithm (Louppe, 2014) with multi-wavelength photometric data.
## 2 Classification scheme and data
### Photometric data
Galaxies have different spectral shapes depending on their source of ionization. Previous works have shown that these differences are stronger in the UV, optical, and mid-IR bands. In order to maximize the available sample, we opted to use for training our algorithm mid-IR and optical photometric data from the AllWISE Source Catalog (Wright et al., 2010)
and the SDSS DR16 (Brinchmann et al. 2004), respectively. To avoid the need for redshift measurements, we use colors rather than luminosities. In order to mitigate aperture effects, we use a mid-IR hybrid-photometric scheme that combines custom-aperture photometry for nearby (extended) and fixed aperture photometry for more distant point-like sources. The optical data consist of \(g-r\) colors, based on SDSS DR16 fiber-photometry. In order to reduce the noise in our training data set we chose only galaxies with S/N \(>\) 5 (Signal-to-Noise) for the optical bands g,r and the mid-IR bands W1, W2. For the band W3 we used a S/N \(>\) 3. Our final optimal feature scheme comprises colors: W1-W2, W2-W3, and \(g-r\) and its distribution per activity class is presented in Fig. 1.
### Classification scheme
We adopt a 5-class classification scheme that discriminates galaxies into different activity classes: SF, AGN, LINER, Composite, and Passive. In order to construct the training sample, we use spectroscopic information from the SDSS-MPA-JHU (Brinchmann et al. 2004) catalog by selecting only the galaxies that show strong emission lines (S/N \(>\) 5). The emission-line galaxies are classified based on the 4-dimensional data-driven classification algorithm of Stampoulis et al. (2019). To define a sample of Passive galaxies without emission lines we selected objects with good spectra (continuum S/N\({}_{cont}\)\(>\) 3) and absent emission lines (S/N\({}_{line}\)\(<\) 3). Our final sample includes 40954 galaxies spanning a redshift range of \(z\) = 0.02 - 0.08. Table 1 shows the composition of our final sample classification scheme. For the training of the RF algorithm we considered 50% of the full set (20477/40954), and for the test the rest 50%. Given the strong imbalance between the different classes in the training sample, we used a stratified split in order to ensure that both training and the test set have the same proportions of each class.
\begin{table}
\begin{tabular}{l c c} \hline \hline Class & Number of objects & Percentage (\%) \\ \hline Star-forming & 35878 & 87.6 \\ Seyfert & 1337 & 3.3 \\ LINER & 1322 & 3.2 \\ Composite & 1673 & 4.1 \\ Passive & 744 & 1.8 \\ \hline \end{tabular}
\end{table}
Table 1: The composition of the training sample per galaxy activity class that was used in the training sample.
Figure 2: The confusion matrix of our classifier. The completeness for the SF and passive galaxies is very high ( \(>\)80%) while for the other 3 classes it is lower as expected given the strong mixing between them in the feature distribution.
Figure 1: Distribution of our training sample in the feature space. The 5 classes of our classification scheme are well separated with a higher mixing between the Composite and AGN activity classes.
## 3 Results
The confusion matrix (Fig. 2) shows that our classifier reaches maximum completeness of \(\sim\)82% for SF and Passive galaxies and \(\sim\)56% for AGN. This performance is expected if we consider the feature distribution of our training sample where the 5 classes are reasonably separated with higher mixing between the composite and AGN galaxies (Fig. 1). Furthermore, these high scores indicate the robustness and reliability of our classifier when it is applied to unseen data (i.e. test set).
We apply our new diagnostic to the HECATE nearby galaxy catalog (D\(\leq\)200 Mpc) (Kovlakas et al., 2021), and we compare our classifications with the mid-IR diagnostics from M13 and A12. Our new classifier reveals a large population of AGN outside their locus as defined in the other mid-IR diagnostics (green points below the dashed line in Fig.3). In particular, in a sample of 1227 spectroscopically classified AGN we find that our method recovers \(\sim 36\%\) of the initial sample, while the M13, and A12 methods recover \(\sim 5\%\) and \(\sim 6\%\), respectively. Thus our new diagnostic increases the completeness of AGN identified with mid-IR colors since the other methods are more sensitive to luminous AGN, omitting a significant fraction of lower luminosity AGN. The reason for the success of our method is that the inclusion of the optical color allows the classifier to identify more AGN and also cases of starbursts with extreme mid-IR colors that mimic obscured AGN galaxies (blue points at the top right of Fig. 3).
| ランダムフォレストアルゴリズムを用いて、星形成銀河、AGN、LINER、複合、そして非星形成銀河の5つのクラスに自動分類のためのツールを開発しました。これは、 mid-IR（WISE）と光学写真データの組み合わせでトレーニングされ、実際のラベル（活動クラス）は発光線比率に基づいています。この分類器は、赤方偏移に依存せず、z〜0.1の天体まで適用できます。SFと非星形成銀河には、 completeness >80% となり、AGNには、 $\sim$60% となります。HECATE全天銀河カタログに適用すると、標準的な mid-IR 診断において、AGNの低輝度クラスがAGN Locus外に大きな集団が存在します。 |
2305.15779 | Custom-Edit: Text-Guided Image Editing with Customized Diffusion Models | Text-to-image diffusion models can generate diverse, high-fidelity images
based on user-provided text prompts. Recent research has extended these models
to support text-guided image editing. While text guidance is an intuitive
editing interface for users, it often fails to ensure the precise concept
conveyed by users. To address this issue, we propose Custom-Edit, in which we
(i) customize a diffusion model with a few reference images and then (ii)
perform text-guided editing. Our key discovery is that customizing only
language-relevant parameters with augmented prompts improves reference
similarity significantly while maintaining source similarity. Moreover, we
provide our recipe for each customization and editing process. We compare
popular customization methods and validate our findings on two editing methods
using various datasets. | Jooyoung Choi, Yunjey Choi, Yunji Kim, Junho Kim, Sungroh Yoon | 2023-05-25T06:46:28 | http://arxiv.org/abs/2305.15779v1 | # Custom-Edit: Text-Guided Image Editing with Customized Diffusion Models
###### Abstract
Text-to-image diffusion models can generate diverse, high-fidelity images based on user-provided text prompts. Recent research has extended these models to support text-guided image editing. While text guidance is an intuitive editing interface for users, it often fails to ensure the precise concept conveyed by users. To address this issue, we propose Custom-Edit, in which we (i) customize a diffusion model with a few reference images and then (ii) perform text-guided editing. Our key discovery is that customizing only language-relevant parameters with augmented prompts improves reference similarity significantly while maintaining source similarity. Moreover, we provide our recipe for each customization and editing process. We compare popular customization methods and validate our findings on two editing methods using various datasets.
+
Footnote †: \(*\) Corresponding Authors
## 1 Introduction
Recent work on deep generative models has led to rapid advancements in image editing. Text-to-image models [19, 22] trained on large-scale databases [23] allow intuitive editing [7, 15] of images in various domains. Then, to what extent can these models support precise editing instructions? Can a unique concept of the user, especially one not encountered during large-scale training, be utilized for editing? Editing with a prompt acquired from a well-performing captioning model [13] fails to capture the appearance of reference, as shown in Fig. 1.
We propose _Custom-Edit_, a two-step approach that involves (i) customizing the model [6, 12, 21] using a few reference images and then (ii) utilizing effective text-guided editing methods [7, 15, 16] to edit images. While prior customization studies [6, 12, 21] deal with the random generation of images (noise\(\rightarrow\)image), our work focuses on image editing (image\(\rightarrow\)image). As demonstrated in Fig. 1, customization improves faithfulness to the reference's appearance by a large margin. This paper shows that customizing only language-relevant parameters with augmented prompts significantly enhances the quality of edited images. Moreover, we present our design choices for each customization and editing process and discuss the _source-reference trade-off_ in Custom-Edit.
## 2 Diffusion Models
Throughout the paper, we use Stable Diffusion [19], an open-source text-to-image model. The diffusion model [5, 8, 24, 26] is trained in the latent space of a VAE [11], which downsamples images for computation efficiency. The model is trained to reconstruct the clean latent representation \(x_{0}\) from a perturbed representation \(x_{t}\) given the text condition \(c\), which is embedded with the CLIP text encoder [18]. The diffusion model is trained with the following objective:
\[\sum_{t=1}^{T}\mathbb{E}_{x_{0},\epsilon}[||\epsilon-\epsilon_{\theta}(x_{t},t,c)||^{2}], \tag{1}\]
where \(\epsilon\) is an added noise, \(t\) is a time step indicating a perturbed noise level, and \(\epsilon_{\theta}\) is a diffusion model with a U-Net [20] architecture with attention blocks [27]. During training, the text embeddings are projected to the keys and
Figure 1: Our _Custom-Edit_ allows high-fidelity text-guided editing, given a few references. Edited images with BLIP2 [13] captions show the limitation of textual guidance in capturing the fine-grained appearance of the reference.
values of cross-attention layers, and the text encoder is kept frozen to preserve its _language understanding capability_. Imagen [22] and eDiffi [1] have shown that leveraging rich language understandings of large language models by freezing them is the key to boosting the performance.
## 3 Custom-Edit
Our goal is to edit images with complex visual instructions given as reference images (Fig. 1). Therefore, we propose a two-step approach that (i) customizes the model on given references (Sec. 3.1) and (ii) edits images with textual prompts (Sec. 3.2). Our method is presented in Fig. 2.
### Customization
**Trainable Parameters.** We optimize only the keys and values of cross-attention and the '[rare token]', following Custom-Diffusion [12]. As we discuss in Sec. 4, our results indicate that training these _language-relevant_ parameters is crucial for successfully transferring reference concepts to source images. Furthermore, training only these parameters requires less storage than Dreambooth [21].
**Augmented Prompts.** We fine-tune the abovementioned parameters by minimizing Eq. (1). We improve Custom-Diffusion for editing by augmenting the text input as '[rare token] [_modifier_] [class noun]' (e.g., 'V*_patterned_ teapot'). We find that '[modifier]' encourages the model to focus on learning the appearance of the reference.
**Datasets.** To keep the language understanding while fine-tuning on the reference, we additionally minimize prior preservation loss [21] over diverse images belonging to the same class as the reference. Thus, we use CLIP-retrieval [3] to retrieve 200 images and their captions from the LAION dataset [23] using the text query 'photo of a [modifier] [class noun]'.
### Text-Guided Image Editing
**Prompt-to-Prompt.** We use Prompt-to-Prompt [7] (P2P), a recently introduced editing framework that edits images by only modifying source prompts. P2P proposes attention injection to preserve the structure of a source image. For each denoising step \(t\), let us denote the attention maps of the source and edited image as \(M_{t}\) and \(M_{t}\)*, respectively. P2P then injects a new attention map \(Edit(M_{t},{M_{t}}^{*},t)\) into the model \(\epsilon_{\theta}\). \(Edit\) is an attention map editing operation, including _prompt refinement_ and _word swap_. Additionally, P2P enables local editing with an automatically computed mask. P2P computes the average of cross-attention \(\bar{M}_{t,w}\) and \(\bar{M}_{t,w}^{*}\) related to the word \(w\) and thresholds them to produce the binary mask \(B(\bar{M}_{t})\cup B(\bar{M}_{t}^{*})\). Before editing with P2P, we utilize Null-Text Inversion [16] to boost the source preservation. Refer to Sec. C for a more description.
**Operation Choice.** Due to the limited number of reference images, the customized words favor only a limited variety of structures. This inspired us to propose the following recipe. First, we use _prompt refinement_ for the Edit function. _Word swap_ fails when the customized words do not prefer the swapped attention map. Second, we use mask \(B(\bar{M}_{t})\) rather than \(B(\bar{M}_{t})\cup B(\bar{M}_{t}^{*})\), as the customized words are likely to generate incorrect masks.
**Source-Reference Trade-Off.** A key challenge in image editing is balancing the edited image's source and reference similarities. We refer to \(\tau/T\) as _strength_, where P2P injects self-attention from \(t=T\) to \(t=\tau\). In P2P, we observed that a critical factor in controlling the trade-off is the injection
Figure 2: Our Custom-Edit consists of two processes: the customization process and the editing process. **(a) Customization.** We customize a diffusion model by optimizing only language-relevant parameters (i.e., custom embedding V* and attention weights) on a given set of reference images. We also apply the prior preservation loss to alleviate the language drift. **(b) Editing.** We then transform the source image to the output using the customized word. We leverage the P2P and Null-text inversion methods [7, 16] for this process.
Figure 3: **Custom-Edit results.** Our method transfers the reference’s appearance to the source image with unprecedented fidelity. The structures of the source are well preserved. We obtain source prompts using BLIP2 [13]. Except for the pencil drawing example, we use local editing of P2P with automatically generated masks.
of self-attention rather than cross-attention. Higher strength denotes higher source similarity at the expense of reference similarity. In Sec. 4, we also show results with SDEdit [15], which diffuses the image from \(t=0\) to \(t=\tau\) and denoises it back. As opposed to P2P, higher strength in SDEdit means higher reference similarity.
## 4 Experiment
In this section, we aim to validate each process of Custom-Edit. Specifically, we assess our design choices for customization by using Textual Inversion [6] and Dream-booth [21] in the customization process. We compare their source-reference trade-off in the editing process. As well as P2P, we use SDEdit [15] for experiments.
**Baselines.** Textual Inversion learns a new text embedding V*, initialized with a class noun (e.g., 'pot'), by minimizing Eq. (1) for the input prompt 'V*'. Dreambooth fine-tunes the diffusion model while the text encoder is frozen. Eq. (1) is minimized over a few images given for input prompt '[rare token] [class noun]' (e.g., 'ktn teapot'). SDEdit is the simplest editing method that diffuse-and-denoise the image.
**Datasets.** We use eight references in our experiments, including two pets, five objects, and one artwork. For each reference, we used five source images on average.
**Metrics.** We measure the source and reference similarities with CLIP ViT-B/32 [18]. We use strengths [0.2, 0.4, 0.6, 0.8] for P2P and [0.5, 0.6, 0.7, 0.8] for SDEdit results. We generated two P2P samples with cross-attention injection strengths [0.2, 0.6], and three SDEdit samples for each strength and source image from different random seeds.
**Inference Details.** We employ a guidance scale of 7.5 and 50 inference steps. We acquire all source prompts using BLIP2 [13]. More details are available in Sec. B.
### Qualitative Results
Fig. 3 illustrates the selected results. Custom-Edit transfers the reference's detailed appearance to the source while preserving the overall structure. For example, Custom-Edit generates a horizontally elongated V* wooden pot from the wine bottle (first row). In the second row, Custom-Edit generates a V* tortoise plushy wearing a hat with the texture of its shell. The blue jay in the third row became a V* ceramic bird with perfectly preserved macarons. In the last row, the V* cat is sitting in a pose that does not exist in the reference set. We show qualitative comparisons in Sec. A.1.
### Quantitative Results
Fig. 4 shows average trade-off curves on P2P and SDEdit. Our improved Custom-Diffusion yields the best trade-off, while Textual Inversion shows similar source similarity but lower reference similarity. Dreambooth has higher source similarity but lower reference similarity, suggesting that it is ineffective in modifying images. SDEdit results also show a similar tendency, supporting our claim that customizing language-relevant parameters is effective for editing. Note that SDEdit shows lower source similarity than P2P, indicating the superiority of P2P and our operation choices in text-guided editing.
## 5 Discussion
We propose Custom-Edit, which allows fine-grained editing with textual prompts. We present our design choices for each process, which can benefit future customization and editing work. Additionally, we discuss the trade-off between source and reference in diffusion-based editing.
Although Custom-Edit shows various successful editing results, there are some failure cases, as presented in Sec. A.3. Custom-Edit sometimes edits undesired regions or fails to edit complex backgrounds. We hypothesize that this is due to the inaccurate attention maps of Stable Diffusion [7, 16] and the limited controllability of the text input. Potential solutions are to apply Custom-Edit on text-to-image models with larger text encoders [1, 22] or extended controllability [14, 28].
Figure 4: **Source-Reference Trade-Off. Custom-Diffusion shows the best trade-off, indicating the effectiveness of training only language-relevant parameters. We exhibit qualitative comparisons and samples with various strengths in Sec. A.2.**
**Acknowledgements:** This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (Ministry of Science and ICT, MSIT) (2022R1A3B1077720), Institute of Information & communications Technology Planning & Evaluation (IITP) grants funded by the Korea government (MSIT) (2021-0-01343: AI Graduate School Program, SNU), and the BK21 FOUR program of the Education and Research Program for Future ICT Pioneers, Seoul National University in 2023.
| 画像生成のためのテキスト対画像拡散モデルは、ユーザーが提示するテキストの指示に基づいて、多様な高品質の画像を生成できます。近年、これらのモデルはテキストガイドされた画像編集をサポートするために拡張されました。テキストガイドはユーザーにとって直感的で編集インターフェースですが、ユーザーが意図した概念を正確に表現する能力は限られています。この問題に対処するために、私たちはCustom-Editを提案します。このモデルでは、 (i) いくつかの参照画像を用いて拡散モデルをカスタマイズし、 (ii) テキストガイドされた編集を行うという方法を採用しています。私たちの重要な発見は、言語に関連するパラメータをカスタマイズしたAugmented Promptsを用いることで、参照画像の類似度を大幅に向上させながら、ソース画像の類似度を維持できることです。さらに、各カスタマイズと編集プロセスのためのレシピを提供しています。人気カスタマイズ手法を比較し、様々なデータセット |
2306.14137 | BotanicGarden: A High-Quality Dataset for Robot Navigation in
Unstructured Natural Environments | The rapid developments of mobile robotics and autonomous navigation over the
years are largely empowered by public datasets for testing and upgrading, such
as sensor odometry and SLAM tasks. Impressive demos and benchmark scores have
arisen, which may suggest the maturity of existing navigation techniques.
However, these results are primarily based on moderate structured scenario
testing. When transitioning to challenging unstructured environments,
especially in GNSS-denied, texture-monotonous, and dense-vegetated natural
fields, their performance can hardly sustain at a high level and requires
further validation and improvement. To bridge this gap, we build a novel robot
navigation dataset in a luxuriant botanic garden of more than 48000m2.
Comprehensive sensors are used, including Gray and RGB stereo cameras, spinning
and MEMS 3D LiDARs, and low-cost and industrial-grade IMUs, all of which are
well calibrated and hardware-synchronized. An all-terrain wheeled robot is
employed for data collection, traversing through thick woods, riversides,
narrow trails, bridges, and grasslands, which are scarce in previous resources.
This yields 33 short and long sequences, forming 17.1km trajectories in total.
Excitedly, both highly-accurate ego-motions and 3D map ground truth are
provided, along with fine-annotated vision semantics. We firmly believe that
our dataset can advance robot navigation and sensor fusion research to a higher
level. | Yuanzhi Liu, Yujia Fu, Minghui Qin, Yufeng Xu, Baoxin Xu, Fengdong Chen, Bart Goossens, Poly Z. H. Sun, Hongwei Yu, Chun Liu, Long Chen, Wei Tao, Hui Zhao | 2023-06-25T06:11:51 | http://arxiv.org/abs/2306.14137v2 | BotanicGarden: A high-quality and large-scale robot navigation dataset in challenging natural environments
###### Abstract
The rapid developments of mobile robotics and autonomous navigation over the years are largely empowered by public datasets for testing and upgrading, such as SLAM and localization tasks. Impressive demos and benchmark results have arisen, indicating the establishment of a mature technical framework. However, from the view point of real-world deployments, there are still critical defects of robustness in challenging environments, especially in large-scale, GNSS-denied, textural-monotonous, and unstructured scenarios. To meet the pressing validation demands in such scope, we build a novel challenging robot navigation dataset in a large botanic garden of more than 48000m\({}^{2}\). Comprehensive sensors are employed, including high-res/rate stereo Gray&RGB cameras, rotational and forward 3D LiDARs, and low-cost and industrial-grade IMUs, all of which are well calibrated and accurately hardware-synchronized. An all-terrain wheeled robot is configured to mount the sensor suite and provide odometry data. A total of 32 long and short sequences of 2.3 million images are collected, covering scenes of thick woods, riversides, narrow paths, bridges, and grasslands that rarely appeared in previous resources. Excitedly, both highly-accurate ego-motions and 3D map ground truth are provided, along with fine-annotated vision semantics. Our goal is to contribute a high-quality dataset to advance robot navigation and sensor fusion research to a higher level.
Data Sets for SLAM, Field Robots, Data Sets for Robotic Vision, Navigation, Unstructured Environments. Code and dataset: [https://github.com/robot-pesg/BotanicGarden](https://github.com/robot-pesg/BotanicGarden)
## I Introduction
Mobile robots and autonomous navigation are the pressing needs of today's social development, as well as the crucial link of productivity evolution. Various applications such as robotaxi [1], space rover [2], unmanned mining [3], and forest inventory come to the fore, proving that robots can liberate repetitive labors, work beyond the reach of human, displace dangerous operations, and reduce accidental fault. Impressive demos and benchmark results might indicate that the technical framework is mature enough. However, whenever we face up to real life and large-scale deployment, there are always deep concerns: Do we have full confidence in today's autonomy? Is robot navigation robust enough in challenging environments?
The answer could be pessimistic. Modern navigation techniques such as integrated Odometry and Simultaneous Localization and Mapping (SLAM) [4] are indeed highly dependent on good scene comfortableness and positioning assistances to avoid tracking failures and cumulative drifts. Within well textured and structured scenarios, both vision- and LiDAR-based
Fig. 1: **Top:** A bird’s eye view of the 3D map of _BotanicGarden_ generated by rigorous survey and mapping works; **Middle**: The robot walks through the challenging narrow path and riverside; **Bottom**: A detailed view of the 3D map in GNSS-denied thick woods.
navigation can work reliably by fusing inertial sensors and external positioning signals; also, the distinctive features can effectively enable place recognition thus trigger global localizations and loop-closure corrections [5], so comes the successful cases of unmanned warehouses and logistics [6]. However, in problematic scenarios of large-scale fields, GNSS rejections, monotonous textures, and especially in unstructured wilds, the robustness defects are still critical and in need of validation.
As is well known, due to the costly hardware and complicated experiments, robot navigation research relies heavily on publicly available datasets for testing and upgrading [7]. The most famous resources, including KITTI [8], TUM-RGBD [9], and EuRoC [10], have become the indispensable references in today's algorithm developments. Other newer datasets such as NCLT [11], Oxford RobotCar [12], Complex Urban [13], New College [14], and 4Seasons [15] also complement a wide scene and motion variety. However, such datasets are mostly with urbanized and indoor environments, which cannot faithfully expose the aforementioned problematic scenarios. This motivates us to build a novel dataset in challenging natural environments to promote robot navigation research.
In this paper, we introduce a high-quality and large-scale robot navigation dataset which is collected in a botanic garden of more than 48000m\({}^{2}\). An all-terrain robot, along with strictly integrated stereo visions, LiDARs, inertial sensors, and wheels odometry, traversing through diverse venues like dense woods, riversides, narrow paths, bridges, and grasslands, as shown in Fig. 1. Here GNSS is practically unworkable under the block of thick vegetations; Besides, the repetitive green features and weak-structured field may also shake the robustness of motion and recognition modules. The work most similar to ours could be Montmorey [16], while it focuses on LiDAR mapping in forest, lacking in sensor variety, scene scale and diversity, and authentic ground truth. Our main contributions are as follows:
* We build a novel multi-sensory dataset in a large botanic garden, with a total of 32 long&short sequences and \(\sim\)2.3 million images which contain diverse challenging natural factors that are rarely seen in previous resources.
* We employed comprehensive sensors, including high-res and high-rate stereo Gray&RGB cameras, rotational and forward-facing 3D LiDARs, and low-cost and industrial-grade IMUs, supporting a wide range of applications. By elaborate development of the integrated system, we have achieved high-precision hardware-synchronization. Both the sensors and sync-quality are at top-level of this field.
* We provide both highly-accurate 3D map and trajectories ground truth by dedicated surveying works and advanced map-based localization algorithm. We also provide dense vision semantics labeled by experienced annotators. This is the first field robot navigation dataset that provides such all-sided and high-quality reference truth.
## II Related Works
### _Odometry/SLAM-based navigation_
Traditional navigation systems are typically achieved with GNSS (Global Navigation Satellite System), and filtered with inertial data. GNSS can provide drift-free global positioning at meters level, while inertial data are in duty of attitude and can boost the frequency to more than 100Hz. However, as is well known, GNSS requires an open-sky to locate reliably, while is unworkable indoors and is out of precision in denied outdoor areas such as urban canyon, tunnels, and forests. These failure cases motivate the developments of modern Odometry/SLAM -based navigation which employ vision and LiDAR as centric sensors. Odometry is the process of tracking an agent's location-on incrementally over time. It has been widely researched and already formed mature deployments such as Visual/Visual-In-ertial Odometry (VO/VIO), which are compact and computationally lightweight. As a growth of Odometry (O), SLAM is a process of building a map of the environments while simultaneously keeping the track of the agent's locations within it. Compared with Odometry, SLAM could be more accurate and robust: by loop closure corrections, SLAM is able to keep optimizing the map and path to achieve global consistency; and it is also possible to re-localize after getting lost by searching the base-map. Famous O/SLAM frameworks include VINS series [17], ORB-SLAM [18, 19], LOAM and its extensions [20, 21], etc. As indicated by the benchmark results, in structured environments, the state-of-the-arts are relatively mature to provide trustable performance, and are able to overcome occasional difficulties. Whereas, in unstructured scenes where things are mainly natural and with monotonous textures, the robustness remains unauthentic, and in need of validations. Therefore, current trend is to strengthen the conventional methods in frail challenges and to explore new frameworks of multi-sensor fusion navigation, such as LVI-SAM [22] and R3LIVE [23].
### _Representative datasets_
Publicly available datasets have significantly promoted the research of robot navigation in the past 2 decades. The earliest open resource could be from the DARPA Urban Challenge [24] in 2007 recorded by a vehicle. It contains a Velodyne-64C and 12 Sick-2D LiDARs, along with 5 RGB cameras. The ground truth (GT) poses were accurately provided by a D-GPS&INS. Later in 2009, 2 robotic vision-2D LiDAR datasets Rawseeds [25] and New College [26] were released, focusing on campus daily scenario. Huge efforts were made in Rawseeds (outdoors) both on routes design and station setup, to ensure that D-GPS can output good GT-positions in travelling. Whereas, for New College, the GPS was inaccurate and cannot be used as GT. In 2012, the popular KITTI [8] dataset was proposed, providing various urban and highway sequences of synchronized stereo Gray&RGB vision and 64C-LiDAR data. It also contains high quality GT poses measured by a D-GPS&INS system, leading to a famous O/SLAM benchmark. Whereas, these datasets are known to be easy due to the good textures, structure, weathers, and the static scenes, which may be too ideal for validation.
To complement previous datasets with more temporal variations, NCLT [11], Oxford RobotCar [12], KAIST Day/Night [27], and 4-Seasons [15] collected amounts of long-term data in urban and campus, bring in diverse timeslots, weathers, and seasonal changes, and their GT trajectories were acquired and processed based on D-GNSS&INS results. However, these datasets are within comfortable urbanized environments, lacking dense dynamics and high-rise buildings where GNSS is unstable. To fill this gap, ComplexUrban [13] and UrbanLoco [28] respectively collected multi-sensory sequences in South Korea metropolis, and Hong Kong & San Francisco. They both cover diverse complex urban features such as canyon, bridge, tunnel, dense residential, crowded streets, etc., whereas exactly due to the messy GNSS, the GT trajectories are not fully authentic.
Many popular indoor and 6-DoF datasets also exist. TUM [9] collected many hand-held and robot carried RGB-D sequences in an office and a hall; EuRoC [10] recorded a stereo visual-inertial drone dataset in room and factory. Both these two
datasets provided accurate(\(\sim\)1mm) and time-synchronized GT trajectories, leading to 2 popular and high-quality benchmarks. While a pity is that, these 2 studies are aiming at small scenes, which could not fit larger benchmarking demands. Later TUM published another 2 vision datasets MonoVO [29] and VI [30], which traversed long-range data in- and outdoors, whereas, no complete GT was available here. To bring in more real-life challenges, OpenLORIS [31] and M2DGR (indoors) [32] proposed long-term and multi-sensory robot sequences in various environments with changing illuminations and human effects. However, their GT of long paths were restricted by the scenarios: OpenLORIS had to use LiDAR-SLAM to generate GT poses, which were with suspicious precision; M2DGR has make careful designs on collection routes and instrument settings, so that the Laser Tracker (LasTrack) can work in regular. Other datasets, such as Newer College [14], Hilti SLAM [33], and NTU VIRAL [34], have collected plenty of 6-DoF Multi-Sensory sequences, which were with high quality, while could be unsuitable for ground robotic and vehicular validations.
### _Datasets in unstructured environments_
After years of dedication by the community, datasets in s-structured scenes have already been abundant. In contrast, data with problematic unstructured environments are still in serious lacking. Existing works mainly cover scenarios of planet, river, underground, agriculture, forest, etc., as described below.
Planetary datasets are uniquely featured with severe texture losses&repetitions, and structurelessness, which may cause great difficulty for both visual and LiDAR motion estimations. Such scenarios are usually mimicked by sandy and rocky field in widely-open areas. Furgale et al. [35] created a long-range rover navigation dataset with stereo vision on Devon Island; Vayugundla et al. [36] recorded 2 sequences in a Moon analogue environment on Mount Etna with stereo, IMU, and odometry sensor; Hewitt et al. [37] collected a multi-sensory dataset in Katwijk beach to emulate Mars landing sites; and Meyer et al. [38] recorded diverse visual-inertial sequences in the Mor-occan desert. Benefiting from the open-sky, these datasets are able to provide accurate D-GNSS GT trajectories.
River scenarios mainly bring in challenges from the water and surrounded vegetations. To date, there are mainly 2 available river-based datasets: VI-Canoe [39] and USVInland [40]. The GT trajectories were generated by D-GNSS&INS, while could suffer from occasional drifts in sky-blocked areas.
Other related resources mainly include Chilean Mine [41] and SubT-Tunnel [42] for underground navigation, Sugar Bets [43] and Rosario [44] for agric-automation, and FinnForest [45], Wild-Places [46], and Montmorency [16] for forest navigation/mapping. In such datasets, the limitation mainly lies on the GT precision in GNSS-denied areas, e.g., in thick woods, Montmorency had to use SLAM for ground truth generation, which was not that accurate.
### _Discussions_
In summary, current navigation techniques still have robustness issues in GNSS-denied and unstructured scenes, especially in natural environments, such as rivers and forests, while datasets in such problematic venues are still very limited. This paper fills the gap by presenting a novel high-quality dataset in a large botantic garden. Table I summarizes a comparison table of partial aforementioned datasets and our work, showing that our scene scale, sensors availability, synchronization, and ground truth are all at top-level of this field. We thus believe that our work can provide strong values to the robotics community.
## III The Botanic Garden Dataset
### _Acquisition platform_
To cope with the complex field environments, we employ an all-terrain wheeled robot Scout V1.0 from AgileX for data collection. It is with a powerful 4-wheel-drive and differential steering mechanism, which can ensure the working robustness and obstacle crossing ability in the wilds. Each wheel contains a 1024-line encoder to provide ego-motions, and we have developed a set of corresponding programs to calculate the robot dead-reckoning odometers. To ensure a low latency communication, the robot is configured to link with the host via a high-speed CAN bus at 500kbps, which can lower the transmission time to less than 1ms. Besides, the host controller is performed by an Intel NUC11 running with a Real-Time Linux kernel1 to minimize the clock jitter and data buffer time. We have customized the NUC to support dual-Ethernet with Precision Time Protocol (PTP2, also known as IEEE1588) capability, which is able to be synchronized with other devices at sub-\(\upmu\)s accuracy.
Footnote 1: [https://wiki.linuxfoundation.org/realtime/start](https://wiki.linuxfoundation.org/realtime/start)
Footnote 2: [https://standards.ieee.org/iece/15884355/](https://standards.ieee.org/iece/15884355/)
On top of the robot chassis, we design a set of aluminum profiles to carry the batteries, computers, controllers, sensors, and the display, as illustrated in Fig. 2. The computer used for data collection is an Advantech MIC-7700 Industrial PC assembled with a PCIE expansion module. It houses an Intel Core i7-6700TE 4C8T processor running with Ubuntu 18.04.1 LTS and ROS Melodic systems. A total of 8 USB 3.0, 10 GigE, and a set of GPIO and serial ports are available. All the GigE ports supports PTP, available for precise synchronization. For high-speed data logging, 2\(\times\)16GB DDR4 memories (Dual-channel) and a 2TB Samsung 980 Pro NVME SSD (of 3-bit MLC, over 1.5GB/s sequential writes throughout the whole storages) are equipped for real time database. To ensure full communication bandwidth, both the GigE cards (for sensor streaming) and the SSD are fastened to the PCIE slots that directly linked to the CPU. Benefiting from our elaborate development, this system can record over 500MB/s stream without losing a single piece of image, which is a common issue in many other datasets.
### _Sensor Setup_
Our dataset focuses on robot navigation research based on conventional mainstream sensor modalities and their fusions. To this end, we have employed comprehensive sensors including stereo Gray&RGB cameras, rotational and forward-facing 3D LiDARs, and low-cost and industrial-grade IMUs. The specifications are as listed in Table II. All the sensors are accurately mounted on a compact self-designed aluminum carrier with precise 3D printing fittings, as shown in Fig. 2.
The stereo sensors are composed of two grayscale and two RGB cameras with a baseline of around 255mm. To facilitate research on robotic vision, we have chosen models from Teledyne DALSA with both high rate and resolution: M1930 and C1930, working at 1920\(\times\)1200 and 40fps in our configuration. The CMOS used for the cameras is the PYTHON 2000 from ONSemi with 2/3" format and 4.8\(\upmu\)m pixel size, which has a good performance under subnormal illuminations. However, this sensor in its nature has very strong infrared response, thus we have customized IR-cutoff filters of 400-650nm to exclude the side-effects on white-balance and exposure. The cameras use GPIO as external trigger, and GigE for data streaming, which also supports PTP synchronizations. The attended lens for imaging is Ricoh's CC0614A (6mm focus and F1.4 iris), which has been adjusted to 5-10m clear view to fit the scene.
To support different testing demands, 2 LiDARs are used in collection: Velodyne VLP-16 and Livox AVIA. VLP-16 is a cost-effective 16-beam LiDAR which has a 360\({}^{\circ}\)\(\times\)30\({}^{\circ}\) Field-of-View (FoV), suitable for ground robotic navigation. AVIA is a MEMS 3D LiDAR with non-repetitive 70\({}^{\circ}\)\(\times\)77\({}^{\circ}\) circular FoV, thus is more suitable for dense mapping and fusion with co-heading visions. They are both configured to scan at 10Hz, and can be synchronized via pulse per second (PPS) interface.
For inertial sensors, we provide a low-cost BMI088 IMU (200Hz) and an industrial-grade Xsens Mti-680G D-GNSS& INS (IMU@400Hz, GNSS not in use) for comparison usage. BMI088 is built-in and synchronized with AVIA LiDAR, and Xsens supports external triggerings via pulse rising edges.
### _Time Synchronization_
In a precise robot system with rich sensors and multi-hosts, time synchronization is extremely vital to eliminate perception delay and ensure navigation accuracy. Towards a high-quality dataset, we have taken very special cares on this problem. Our synchronization is based on a self-designed hardware trigger& timing board and a PTP-based network, as illustrated in Fig. 2. The trigger and timing board is implemented by a compact STM32 MCU. It is programmed to produce three channels of pulses 1Hz-40Hz-400Hz in the same phases. The 1Hz channel (pulse per second, PPS) is used for the synchronization of VLP -16 and AVIA accompanied with GPRMC signals: Every time the rising edge arrives, LiDAR immediately clears its internal sub-second counter, thus all the point clouds in the subsequent second can be timed cumulatively based on PPS arrival, which will then be appended with UTC integer time by GPRMC. The 40Hz signal is used to trigger the cameras, when a rising edge arrives, the global shutter will immediately start exposure until reaching a target gain, and the image timestamp is acquired by adding half the exposure time to the trigger stamp. The 400Hz signal is used for triggering the Xsens IMU: Xsens has its own internal clock, and when the rising edge arrives, Xsens will be triggered an external interruption thus feedbacks its exact time, then our program can bridge a transform thus stamp the neighboring sample instance. The UTC time is maintained by MCU based on its onboard oscillator. Note that, to maintain the timing smoothness, we will never interrupt the MCU clock during the collections, instead, an UTC stamp will be conferred at the begin of each course-day via NTP or GNSS timing. So far, the LiDAR-vision-IMU chain has been fully synchronized in hardware. With a sub-us level triggering consistency between the sensors (see Fig. 3), a high sync-precision should be obtained.
The PTP-based network is designed for MCU triggers capture and multi-host synchronizations, thus the wheel odometry
\begin{table}
\begin{tabular}{c c c} \hline
**Sensor/Device** & **Model** & **Specification** \\ \hline Gray Stereo & DALSA M1930 & 1920\({}^{\circ}\)1200, 23\({}^{\circ}\), 71\({}^{\circ}\)-56FoV, 40Hz \\ \hline RGB Stereo & DALSA C1930 & 1920\({}^{\circ}\)1200, 23\({}^{\circ}\), 71\({}^{\circ}\)-56FoV, 40Hz \\ \hline LiDAR & Velodyne VLP166 & 360\({}^{\circ}\)-36FoV, 3\({}^{\circ}\)-5200nm, 10Hz \\ \hline MEMS 3LADR & Livox AVIA & 70\({}^{\circ}\)-77FoV, 2\({}^{\circ}\)-2cm200m, 10Hz \\ \hline D-GNSS/INS & Xsens Mti-680G & 9-axis, 400Hz, GNSS not in use \\ Consumer IMU & BMI088 & 6-axis, 200Hz, Livox built-in \\ \hline Wheel Encoder & Scour V1.0 & 400, 3-axis, 200Hz \\ \hline GT 3D Sanner & Leica RTC360 & 130m range, lumm=10ppm accuracy \\ \hline \end{tabular}
\end{table} TABLE II: Specifications of Sensors and Devices
Fig.2: **Left**: The robot platform design and its base coordinate; **Middle**: The multi-sensory system and the corresponding coordinate (the camera below the VLP16 is only for visualization usage, thus is not annotated); **Right**: The synchronization system of the whole platform.
can be aligned to an identical timeline with other sensors. Our network frame is built based on LinuxTP3 library. We assign MIC-7700 as grand master, and DALSA cameras and NUC11 are configured as slaves. When the synchronization starts, the slaves will keep exchanging sync-packets with the master, and to ensure the smoothness of local clocks, we have not directly compensated the offsets, while instead employ a PID mechanism to adjust the time and frequency. During the data collection, once the camera is triggered, it will report its timestamp of PTP clock, and based on the MCU trigger stamp, our software will bridge a relation thus transforms the wheel odometry from PTP to MCU timeline. Here although the PTP network and the real-time kernel are used, there exists a latency from the CAN bus of around 1ms, which has been compensated in advance.
Footnote 3: [https://linuxtp.sourceforge.net/](https://linuxtp.sourceforge.net/)
### _Spatial Calibration_
Spatial calibration, both on the intrinsic and extrinsic parts, are the precondition of navigation and sensor fusion tasks. To ensure the calibration quality, we have adopted mature frameworks and human checks on the results. Note that, as the robot and sensors are already assembled according to the CADs, the calibration is conducted based on these mounting results.
1) Camera calibration: For camera intrinsics and extrinsics calibration, we choose the Matlab camera calibration toolbox4, which uses an interactive engine for inspecting the errors and filtering the qualified instances. Considering the standard lens FoV, we choose Pinhole imaging model (\(f_{x}\),\(f_{y}\), \(c_{x}\), \(c_{y}\)) and a 4th degree polynomial Radial distortion model (\(k_{1}\), \(k_{2}\), \(p_{1}\), \(p_{2}\)) for intrinsics. The calibration is conducted by manually posing a large checker board (11\(\times\)8, 60mm/square) at different distances and orientations in front of the cameras. To avoid possible motion blur, the exposure has been controlled to \(\leq\)10ms, and we finally achieve less than 0.1pixels mean reprojection error in all the 4 cameras. Furthermore, based upon these intrinsics, the extrinsics are finely calculated via joint optimizations, and we have checked the epipolar coherence for a verification.
Footnote 4: [https://www.mathworks.com/help/vision/camera-calibration.html](https://www.mathworks.com/help/vision/camera-calibration.html)
2) Camera-IMU calibration: The extrinsics between camera-as and IMUs are determined using the famous Kalibr5 toolbox. Thanks to our specially-designed detachable sensors suite, we were able to handheld it for 6-DoF movements. Before running the joint calibration, we recorded \(\sim\)120min IMU sequence to identify the IMU intrinsics (noise densities and random walks of the accelerometers and gyroscopes). During the calibration, we used a 6\(\times\)6 Aprilgrid as stationary target and properly moved the sensor suite to excite all IMU axes. To avoid excessive motion blur, we have conducted the calibration in good lights and limited the exposure to \(\leq\)10ms. Note that, this joint calibration can also output time offset, whereas, as the sensors have already been synchronized, thus to avoid the side effects, this workflow was limited to camera-IMU extrinsics only.
Footnote 5: [https://github.com/ethz-asl/kalibr](https://github.com/ethz-asl/kalibr)
3) Camera-LiDAR calibration: For the extrinsics of camera and LiDAR, we have developed a concise calibration toolbox based on 3D checker boards. We define the left RGB camera as center, then by sub-pixel extractions and extrinsics calculation, we can fully reconstruct the known-sized checkerboards to an accurate 3D model. At LiDAR side, we choose AVIA as reference because it works in non-repetitive scan mechanism which can integrate a dense point cloud in 1-2s. Then the two models are registered by point-to-plane ICP, and the camera-LiDAR extrinsics are thus solved, as illustrated in Fig. 4.
4) Other calibrations: Based on the aforementioned process, an arterial camera-LiDAR-IMU calibration chain has already been established. The other sensors can either be calculated from the CADs, or be concatenated from the calibration chain. For example, AVIA manufacturer has provided explicit coordinates relation between LiDAR and its built-in IMU; Xsens and VLP16 also have explicit coordinates provided. To refine a better extrinsic for VLP16, we have performed a scan registration with AVIA, and the related params were updated in the chain. For the robot base, we have observed enough data from both the CADs and external measurements, achieving sub-cm calibration and have integrated it in the main chain, also.
### _Data Collection_
Our datasets are collected at 5\({}^{\text{th}}\), 6\({}^{\text{th}}\), 8\({}^{\text{th}}\), and 18\({}^{\text{th}}\) of October, 2022 in a large botanic garden of our university. Various unstructured natural features are covered inside, such as thick woods, narrow paths, riversides, bridges, grasslands, as shown in Fig. 5. A total of 32 sequences are traversed, including short and long-distance trajectories, cloudy and sunny illumination, loop closures, sharp turns, and monotonous textures, ideal for navigational research. Around 8h&20km's travelling and 2.3 million images are recorded (see Table III for sample sequences statistics), leading to a very large-scale robot dataset.
### _Ground Truth Map_
Ground truth could be the most important part of a dataset. As indicated by Table I, most datasets fail to provide an authentic GT map, which is necessary for evaluating the mapping results and plays a key role in robot navigation. To ensure the global accuracy, we have not used any mobile-mapping based techniques (e.g., SLAM), instead we employ a tactical-grade stationary 3D laser scanner and conduct a qualified surveying and mapping job with professional colleagues. The scanner is the RTC360 from Leica, which can output very dense and colored point cloud with a 130m scan radius and mm-level rang
ing accuracy, as shown the specifications in Table II. For possible future benefits, we have arranged two independent jobs both in early summer and middle autumn, which takes around 20 workdays in total, and respectively with 515 and 400 individual scans (each scan will cost at least 3mins overall, Fig. 6 shows a work photo during the autumn survey). The scans are pre-registered by VI-SLAM and post-registered by Leica Cyclone Register360 software based on ICP and graph optimization (illustrated in Fig. 6). As the scans are quite dense (12mm @10m), accurate, and with huge overlaps, the registration has been conducted in high accuracy (11mm std. according to the report). From our calculation, the map coverage is more than 48000m2, which is at top-level among existing robot datasets.
Footnote 2: [https://github.com/wkentaro/labelme](https://github.com/wkentaro/labelme)
### _Ground Truth Pose_
Serving as the reference of navigation results, ground truth pose should be with good completeness, and as well as globally accurate. This is why GNSS is such widely used in GT generation, while incremental techniques such as SLAM are not authentic due to the cumulative drift. However, in sky-closed and large-scale environments, conventional means such as D-GNSS, LasTrack, and MoCap can hardly work in regular: our garden scenario exactly belongs to this case. To bridge the gap, we take advantage of the authentic GT map, and develop a map-based localization algorithm to calculate the trajectories using the on-robot VLP16 LiDAR. As the map is quite unstructured and with degenerated areas, and VLP16 is very sparse, naive registration approaches such as ICP cannot converge to correct poses on its own. This requires an accurate local tracking thread to provide a good initial value for registration. To this end, we have designed an algorithm fusing global initialization, SVIO odometry, and fine-registration modules: Firstly, the initializer will search current frame in a pre-build 2D database for possible candidates, and a seg-match process accumulates a segment of LiDAR clouds and registers to global map for final initialization; Then, the stereo-VIO local tracker will keep estimating the inter-frames motions for LiDAR pre-transforms; Finally, the fine-registration module will employ a point-to-plane ICP based on the initial pose for a final localization, as illustrated in Fig. 7. As the scene is really complex, we have slowed down the data playbacks and human monitored the visualization panel to make sure the poses have converged correctly. Note that, it is also difficult to evaluate the GT accuracy in such environment, however, as the GT-map is with \(\sim\)1cm precision and VLP16 with \(\sim\)3cm (usually results in 2 -5% drift in point-to-plane ICP-based LiDAR odometry [20], [47], while our motion is up to 15cm per frame), we may infer that the GT-poses are also with cm-level accuracy.
### _Semantic Annotation_
Semantic is the highest perception level of a robot. As a comprehensive and high-quality dataset, we attach great importance to the role that semantic information plays in navigation. Since our LiDARs are relatively sparse, we have arranged the annotation at 2D-image level, which is also possible to reproject to point clouds if needed. Our segmentation consists of 27 classes overall, mainly including different vegetations (bush, grass, tree, tree trunks, water plants), fixed facilities, drivable regions (roads, grassland), river, bridges, sky, etc., and is with dense pixel-level human annotations, as shown in Fig. 5. All data are provided in LabelMe6 format and support future reproducing. It is expected that these data can strengthen the abilities of robust motion estimation and semantic map paintings.
Fig. 5: **Top: Sample frames of typical scene features (riversides, thick woods, grasslands, bridges, narrow paths, etc.); Middle: The corresponding 3D map venues; Bottom: Dense semantic annotations of the corresponding frames.**
Fig. 7: GT-trajectory generation based on our map-localization algorithm.
## IV Example Dataset Usage
### _Vision/LiDAR/Multi-sensor-fusion Navigation_
To verify the versatility of our dataset in navigation research, we select 7 sample sequences and comprehensively test the state-of-the-arts performance of different sensor combinations, including visual, LiDAR, and multi-sensor fusion approaches, regarding the metrics of relative pose error (RPE) and absolute trajectory error (ATE) [9]: the evaluation results and trajectories visualization are shown in Table III and Fig. 8.
From the evaluation statistics we get mainly 3 conclusions: 1. Our dataset can support a wide range of navigation frameworks, including but not limited to stereo vision, visual-inertial, LiDAR-only, LiDAR-inertial, and visual-LiDAR-inertial fusions, which also demonstrate our calibration and sync quality.
2. Our dataset is a challenging test bench for ground robots. As shown by the results, the RPE errors are around 5-10 times larger than KITTI leaderboard (ORB-stereo even failed 2/7 of the tests); and it can be clearly identified that, most algorithms have met significant Z-axis error through the traverse, which should be paid more attention in future research. Besides, a noteworthy statistic is that, although designed loop closures in all 7 sequences, only 8/21 tests (visual methods) have succeeded in detection, indicating a high textural monotonicity of our data.
3. Multi-sensor fusion is an inevitable trend of future navigations on research. It can be clearly seen that, compared with vision- and LiDAR-centric methods, multi-sensor fusion frameworks have earned very obvious elevation on both accuracy and robustness performance: we thus expect that our dataset can serve as an incubator for novel sensor fusion mechanisms.
### _Other Possible Usage_
Although our dataset is originally designed for navigation research, benefiting from the all-rounded data and GT, it can also support applications such as 3D mapping, semantic segmentation, image localization, depth estimation, and so on. New chances and data will be released constantly on our website.
## V Conclusion
This paper proposed _BotanicGarden_, a novel robot navigation dataset in problematic large-scale, GNSS-denied, and unstructured natural environment. Compared with existing datasets, we have paid a lot of attention to data quality: The comprehensive sensors, precise time synchronization, rigorous data loggings, large scenes and sequences, and the all-rounded and high-quality ground truth, all of which are at top-level of this field. In the future, we will keep updating and extending this dataset. For example, to append data of new season and weather conditions, to boost the GT-poses to IMU rate by improving the map-based localization algorithm, etc. We believe our work can bridge the data gap in certain scenarios and trigger new breakthroughs in robot navigation.
## Acknowledgment
The authors would like to thank the colleagues from Tong Ji University and Sun Yat-sen University for their assistances in the rigorous survey works and post-processings, especially Xiaohang Shao, Chen Chen, and Kunhua Liu. We also thank A/Prof. Hangbin Wu for his guidance in data collection. Besides, we acknowledge Grace Xu from Livox for the support on Avia LiDAR, we acknowledge Claude Ng from Leica for the support on high-definition surveying, and we appreciate the colleagues of Appen for their professional works in visual semantic annotations. Yuanzhi Liu would like to thank Chenbo Gong for scene preparation work, and thank Jingxin Dong for her job-loggings and photographs during our data collection.
| ロボットナビゲーションの急速な発展と自律的な航法は、テストとアップグレードのための公開データセットによって大きく促進されています。これは、センサーオデメトリとSLAMタスクなどです。 印象的なデモとベンチマークスコアが生まれ、既存のナビゲーション技術の成熟を示唆しています。しかし、これらの結果は、主に中規模の構造化されたシナリオテストに基づいています。困難な非構造化環境に遷移する際には、特にGNSS欠落、テクスチャ単調、そして森林茂みのある自然環境では、彼らのパフォーマンスは高いレベルを維持することができず、さらなる検証と改善が求められています。このギャップを埋めるために、私たちは、48000m2を超える豪華な植物園での新しいロボットナビゲーションデータセットを作成しました。広範囲なセンサーを適用し、グレーとRGBステレオカメラ、回転型3DLiDAR、低コストで工業用 |
2308.04688 | Generating News-Centric Crossword Puzzles As A Constraint Satisfaction
and Optimization Problem | Crossword puzzles have traditionally served not only as entertainment but
also as an educational tool that can be used to acquire vocabulary and language
proficiency. One strategy to enhance the educational purpose is
personalization, such as including more words on a particular topic. This paper
focuses on the case of encouraging people's interest in news and proposes a
framework for automatically generating news-centric crossword puzzles. We
designed possible scenarios and built a prototype as a constraint satisfaction
and optimization problem, that is, containing as many news-derived words as
possible. Our experiments reported the generation probabilities and time
required under several conditions. The results showed that news-centric
crossword puzzles can be generated even with few news-derived words. We
summarize the current issues and future research directions through a
qualitative evaluation of the prototype. This is the first proposal that a
formulation of a constraint satisfaction and optimization problem can be
beneficial as an educational application. | Kaito Majima, Shotaro Ishihara | 2023-08-09T03:50:26 | http://arxiv.org/abs/2308.04688v1 | # Generating News-Centric Crossword Puzzles As A Constraint Satisfaction and Optimization Problem
###### Abstract.
Crossword puzzles have traditionally served not only as entertainment but also as an educational tool that can be used to acquire vocabulary and language proficiency. One strategy to enhance the educational purpose is personalization, such as including more words on a particular topic. This paper focuses on the case of encouraging people's interest in news and proposes a framework for automatically generating news-centric crossword puzzles. We designed possible scenarios and built a prototype as a constraint satisfaction and optimization problem, that is, containing as many news-derived words as possible. Our experiments reported the generation probabilities and time required under several conditions. The results showed that news-centric crossword puzzles can be generated even with few news-derived words. We summarize the current issues and future research directions through a qualitative evaluation of the prototype. This is the first proposal that a formulation of a constraint satisfaction and optimization problem can be beneficial as an educational application.
2023
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This study proposes a strategy to automatically generate personalized crossword puzzles that contain many words on particular _topics_. Focusing on news as a _topic_, we design a framework that can automatically generate crossword puzzles that contain as many news-derived words as possible (Figure 1). This procedure can be implemented using a combination of existing natural language processing and mathematical optimization methods. However, the combination of techniques is not obvious and requires sophisticated design, experimentation, and evaluation (Srivastava et al., 2016). We are the first to identify that a specific problem design in mathematical optimization, namely _a constraint satisfaction and optimization problem_(Beng et al., 2015), is beneficial for this application.
In summary, this research makes the following contributions:
1. We designed possible scenarios to increase interest in the news and prototyped the automatic generation of news-centric crossword puzzles as a constraint satisfaction and optimization problem (Section 3).
2. Our experiments showed that news-centric crossword puzzles can be generated even with a small number of news-derived words (Section 4).
3. We described our findings from a qualitative evaluation and outlined current issues and future directions (Section 5).
## 2. Related Work
This section reviews related studies from three perspectives.
### Puzzle Combination Exploration
There is a long history of research on automatic crossword puzzle generation (Krishna et al., 2017). Generating crossword puzzles is known to be an NP-hard problem (Beng et al., 2015; Chen et al., 2016; Chen et al., 2016). Mazlack et al. proposed a letter-by-letter fulfillment approach in 1976 (Mazlack et al., 2017) and Ginsberg et al. employed a word-by-word filling strategy in 1990 (Mazlack et al., 2017). Meehan et al. compared these two strategies and concluded that the word-by-word approach is more effective (Mazlack et al., 2017). Bulitko et al. grouped the research surrounding crossword puzzles into three categories (Chen et al., 2016): solving a puzzle (Beng et al., 2015), generating without a score (Beng et al., 2015; Chen et al., 2016), and generating for a higher score (Beng et al., 2015; Chen et al., 2016; Chen et al., 2016; Chen et al., 2016). In the third category, Douglas et al. used genetic algorithms to generate crossword puzzles (Beng et al., 2015), and most others adopted the mathematical optimization approach.
Our method of puzzle combination exploration follows prior work (Chen et al., 2016) in the third category and solves a constraint satisfaction and optimization problem. Specifically, the task is to generate crossword puzzles with a score containing many news-derived words and to fill in the answers word-by-word. We emphasize that this is the first to propose this problem formulation as an educational application aimed at increasing words derived from specific topics, in this case, news. Unlike some previous studies (Chen et al., 2016; Chen et al., 2016), the placement of black cells is not explored.
### Crossword Puzzles for Education
There is limited but gradually growing attention to automatic question generation in the context of education, with the goal of reducing costs and providing continuity. Esteche et al. proposed a method to extract words and their definitions from Spanish news and automatically create crosswords (Esteche et al., 2017). They also utilize external dictionaries and fill the gaps readily based on the score values attached to the words. Some studies have examined the educational benefits of crossword puzzles (Esteche et al., 2017; Wang et al., 2017; Wang et al., 2018).
Our study is similar to the work of Esteche et al. (Esteche et al., 2017) in that we combined multiple technologies to generate crossword puzzles from news articles for educational purposes. Recent automatic question generation systems increasingly use neural networks for end-to-end learning (Wang et al., 2018; Wang et al., 2018), but these systems face challenges in controlling question difficulty and considering different sources of knowledge. Wang et al. conducted a needs assessment survey of 11 faculty members at seven universities and argued that we need a combination of fine-grained techniques and careful design to implement question generation in practice (Srivastava et al., 2016).
### News for Education
News content has traditionally been associated with education and is considered effective for improving reading comprehension and interest in current affairs. Newspapers have been widely used as educational tools, for example, there is an international program called _Newspapers in Education_(Beng et al., 2015; Chen et al., 2016). As the issue of information overload grows and the problem of fake news has become more apparent, it has become increasingly important to develop the skill of information literacy among the general public.
However, interest in the news has been rapidly declining, particularly among young people1. While crossword puzzles are popular content in news media2, there is a limited research focus on their educational effectiveness (Esteche et al., 2017). This study provides a new role of crossword puzzles within the news media.
Footnote 1: [https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2022](https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2022)
Footnote 2: [https://www.nytvo.com/press/both-cooking-and-games-reach-1-million-subscriptions/](https://www.nytvo.com/press/both-cooking-and-games-reach-1-million-subscriptions/)
## 3. Proposed Framework
This section proposes a strategy for automatically generating personalized crossword puzzles that contain many words on particular topics. In particular, we describe a scenario for the news as a topic and a prototyping methodology.
### Possible Scenarios
This study assumes a scenario where crossword puzzles can motivate people to read news articles and thereby increase their interest in the news. Examples include: 1) encouraging people to keep up with current events based on information published regularly, such as news content found in morning and evening newspapers, and 2) reviewing the comprehension of current events based on news articles read by each individual. These examples can be customized. For example, there are options to focus on the genre (e.g., economics, politics), and limit the types of words (e.g., company, person).
### Prototyping
We describe a prototype for automatic crossword puzzle generation that can be applied to specified scenarios. The procedure consists of four steps, as shown in Figure 1: article collection, keyword detection, clue generation, and puzzle combination exploration.
#### 3.2.1. Article Collection
The first step is the gathering of articles according to the specified scenario and objective. For example,
to encourage people to grasp current events, it is a good idea to gather information published regularly in morning and evening newspapers. If individual browsing histories are available, it is also possible to check the level of understanding of current events by gathering news articles read by each individual. We also have to prepare external resources because the news articles alone do not provide a sufficient volume of words to generate crossword puzzles. We use dumped data from Wikipedia3.
Footnote 3: [https://meta.wikimedia.org/wiki/Data_dumps](https://meta.wikimedia.org/wiki/Data_dumps)
#### 3.2.2. Keyword Detection
The second step is the detection of keywords to be used as answers. Candidate keywords are generated by extracting nouns from the text body using Named Entity Recognition (NER) (Nagolov et al., 2015). The same process can be used to extract keywords from external resources. The prototype procedure proposed in this study is almost language-independent. However, if there are multiple character types, we should convert them into a specified type. For example, Japanese text contains several character types including Chinese characters, hiragana, katakana, and alphabetical letters.
#### 3.2.3. Clue Generation
The third step is the generation of clues corresponding to the answers. We mask the body text (fill-in-the-blank) as a simplified method. For the case of clue B in Figure 1, when _liberal_ is extracted, we can create the clue sentence as follows: "U.S. media coverage of the U.S. midterm elections was divided between [Answer] and conservative media."
#### 3.2.4. Puzzle Combination Exploration
Finally, we explore puzzle combinations from the set of answers, by following prior work (Bog
setting T to 50 and ensuring a minimum of 11 black cells is expected to result in a success probability of over 90 % and generation times typically within 10 seconds. Considering the gap between the number of keywords from news articles (2,006) and Wikipedia (449,895), it is a contribution of mathematical optimization that the success probability of crossword puzzles with T=50 was over 90 %.
## 5. Future Research Directions
We demonstrated the prototype to approximately 20 people of several demographic groups and asked for their reactions. The target group included researchers in natural language processing and crossword puzzle creators. This section lists future research directions based on the findings from this qualitative evaluation.
Improved Article CollectionOur study selected news as one of the topics, but there are other extensions. The proposed framework can generate topic-focused crossword puzzles even with a small number of topic-derived words. There could be educational applications, for example, choosing a textbook as a topic.
Improved Keyword DetectionThe quality of NER strongly propagates to crossword puzzles. It is important to determine whether a noun is a suitable word for an answer. A related research area is vocabulary acquisition support systems (Kang et al., 2016; Li et al., 2017; Li et al., 2018). Although this area has been studied specifically for second language learners, it is still relevant to our study in the sense that we encourage people to acquire new vocabulary words through crossword puzzles.
Improved Clue GenerationHigh-quality clue generation is another important point. Fill-in-the-blank questions are easy to create, but there is a challenge in controlling the level of difficulty (Li et al., 2018). For example, there is no guarantee that the answer is the only one that can be identified. We can generate more reliable clues by utilizing question-answer datasets. A rapidly evolving text generation approach, using pre-trained language models, is expected to improve clue quality (Li et al., 2018). However, we must be careful of output that is not faithful to the facts. We are exploring measures such as improving the quality of fine-tuning data, applying filters, and re-ranking output. In addition, acquiring a text style specific to crossword puzzle clues is expected to enhance the quality of the output (Li et al., 2018). We are attempting to characterize the text style of clues by referring to real-world crossword puzzles and interviewing their creators.
Improved Puzzle Combination ExplorationThe current prototype assumes the placement of black cells. By considering optimization with variable black cell placements (Kang et al., 2016; Li et al., 2018), we can generate crossword puzzles with more news-derived words.
Further User TestingAlthough we have performed a qualitative evaluation, larger-scale user testing would be desirable. Examples include providing users with automatically generated crossword puzzles and investigating their interest in the news and its impact on education.
## 6. Conclusion
This paper focused on the case of encouraging people's interest in news and proposed a framework for automatically generating news-centric crossword puzzles. One of the contributions is that the educational objective of including more news-derived words is achieved as a constraint satisfaction and optimization problem. Our experiments showed that news-centric crossword puzzles can be generated even with a small number of news-derived words. We also demonstrated the prototype to approximately 20 people and described current issues and future directions. We hope this paper accelerates research and practice in crossword puzzles for educational purposes.
Figure 4. Distribution of generation times by the number of black cells. The more black cells there are, the shorter the generation time.
Figure 3. Distribution of generation times when the target rate (T) is higher. When T is above 50, the distribution of generation times becomes unstable.
Figure 2. Success probability and generation time (in seconds) by target rate (T). The higher the rate, the lower the success probability and the longer the generation time. | Crosswordパズルは、伝統的に娯楽としてだけでなく、語彙習得や言語能力向上のための教育ツールとしても使用されてきました。教育目的を向上させる戦略の一つに、特定の話題に多くの単語を含めるなど、パーソナライズ化があります。この論文では、ニュースへの関心を高めるケースを扱っており、ニュース中心のクロスワードパズルを自動生成するためのフレームワークを提案しています。私たちは、可能なシナリオを設計し、コンストリントの満足と最適化の問題としてプロトタイプを構築しました。それは、可能な限りニュース由来の単語を含めるという制約に基づいています。私たちの実験では、生成される確率と、いくつかの条件下で必要な時間などの結果が出ました。結果は、ニュース中心のクロスワードパズルは、少ないニュース由来の単語でも生成可能であることを示しました。私たちは、プロトタイプの質的な評価を通じて、現在の課題と将来の研究の方向性をまとめました。この提案は |
2306.04324 | GCT-TTE: Graph Convolutional Transformer for Travel Time Estimation | This paper introduces a new transformer-based model for the problem of travel
time estimation. The key feature of the proposed GCT-TTE architecture is the
utilization of different data modalities capturing different properties of an
input path. Along with the extensive study regarding the model configuration,
we implemented and evaluated a sufficient number of actual baselines for
path-aware and path-blind settings. The conducted computational experiments
have confirmed the viability of our pipeline, which outperformed
state-of-the-art models on both considered datasets. Additionally, GCT-TTE was
deployed as a web service accessible for further experiments with user-defined
routes. | Vladimir Mashurov, Vaagn Chopurian, Vadim Porvatov, Arseny Ivanov, Natalia Semenova | 2023-06-07T10:44:13 | http://arxiv.org/abs/2306.04324v2 | # GCT-TTE: Graph Convolutional Transformer for Travel Time Estimation
###### Abstract
This paper introduces a new transformer-based model for the problem of travel time estimation. The key feature of the proposed GCT-TTE architecture is the utilization of different data modalities capturing different properties of an input path. Along with the extensive study regarding the model configuration, we implemented and evaluated a sufficient number of actual baselines for path-aware and path-blind settings. The conducted computational experiments have confirmed the viability of our pipeline, which outperformed state-of-the-art models on both considered datasets. Additionally, GCT-TTE was deployed as a web service accessible for further experiments with user-defined routes.
machine learning graph convolutional networks transformers geospatial data travel time estimation
## Introduction
Travel time estimation (TTE) is an actively developing branch of computational logistics that considers the prediction of potential time expenditures for specific types of trips Jenelius and Koutsopoulos (2013); Wu et al. (2020). With the recent growth of urban environment complexity, such algorithms have become highly demanded both in commercial services and general traffic management Xuegang et al. (2010).
Despite the applied significance of travel time estimation, it still remains a challenging task in the case of ground vehicles. The majority of the currently established algorithms Wang et al. (2021); Derrow-Pinion et al. (2021) tend to utilize specific data modalities in order to capture complex spatio-temporal dependencies influencing the traffic flow. With the recent success of multimodal approaches in adjacent areas of travel demand prediction Chu et al. (2020) and journey planning He et al. (2022), fusing the features from different sources is expected to be the next step towards better performance in TTE.
In this paper, we explored the predictive capabilities of TTE algorithms with different temporal encoders and propose a new transformer-based model GCT-TTE. The main contributions of this study are the following:
1. In order to perform the experiments with the image modality, we extended the graph-based datasets for Abakan and Omsk Porvatov et al. (2022) by the cartographic images in accordance with the provided trajectories.
Currently, the extended datasets are the only publicly available option for experiments with multimodal TTE algoritms.
2. In order to boost further research in the TTE area, we reimplemented and published the considered baselines in a unified format as well as corresponding weights and data preprocessing code. This contribution will enable the community to enhance evaluation quality in the future, as most of the TTE methods lack official implementations.
3. We proposed the GCT-TTE neural network for travel time estimation and extensively studied its generalization ability under various conditions. Obtained results allowed us to conclude that our pipeline achieved better performance regarding the baselines in terms of several metrics.
4. Conducted experiments explicitly indicate that performance of the transformer-based models is less prone to decrease with the scaling of a road network. This property remains crucial from an industrial perspective, as the classic recurrent models undergo considerably larger performance dropdowns.
5. For demonstration purposes, we deployed inference of the GCT-TTE model as the web application accessible for manual experiments.
The web application is available at [http://gctte.online](http://gctte.online) and the code is published in the GitHub repository of the project1.
Footnote 1: [https://github.com/Eighonet/GCT-TTE](https://github.com/Eighonet/GCT-TTE)
## Related work
Travel time estimation methods can be divided into two main types of approaches corresponding to the _path-blind_ and _path-aware estimation_, Table 1. The path-blind estimation refers to algorithms relying only on data about the start and end points of a route Wang et al. (2019). The path-aware models utilize intermediate positions of a moving object represented in the form of GPS sequences Wang et al. (2014), map patches Fu and Lee (2019), or a road subgraph Wang et al. (2021). Despite the certain computational complexity increase, such approaches provide significantly better results which justify the attention paid to them in the recent studies Zhang et al. (2018), Derrow-Pinion et al. (2021), Sun et al. (2021).
One of the earliest path-aware models was the WDR architecture Wang et al. (2018) which mostly inherited the concept of joint learning from recommender systems Cheng et al. (2016). In further studies, this approach was extended regarding the usage of different data modalities. In particular, the DeepIST Fu and Lee (2019) model utilizes rectangular fragments of a general reference map corresponding to elements of a route GPS sequence. Extracted images are fed into a convolutional neural network (CNN) that captures spatial patterns of depicted infrastructure. These feature representations are further concatenated into the matrix processed by the LSTM-based temporal layer Hochreiter et al. (1997).
In contrast with the other approaches, DeepTTE Wang et al. (2018) is designed to operate directly on GPS coordinates via geospatial convolutions paired with a recurrent neural network. The first part of this pipeline transforms raw GPS sequences into a series of feature maps capturing the local spatial correlation between consecutive coordinates. The final block learns the temporal relations of obtained feature maps and produces predictions for the entire route along with its separate segments.
The concept of modality fusing was first introduced in TTE as a part of the DeepI2T Lan et al. (2019) model. This architecture utilizes LINE Tang et al. (2015) to produce grid embeddings and 3-layer CNN with pooling for image
\begin{table}
\begin{tabular}{|c|c|c|c|} \hline \multicolumn{4}{|c|}{Path-blind models} \\ \hline \multirow{2}{*}{Model} & \multicolumn{3}{c|}{Modality} \\ \cline{2-4} & Graph & Images & GPS \\ \hline AVG & - & - & - \\ LR & - & - & - \\ MURAT & + & - & - \\ DeepI2T & + & + & - \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|c|} \hline \multicolumn{4}{|c|}{Path-aware models} \\ \hline \multirow{2}{*}{Model} & \multicolumn{3}{c|}{Modality} \\ \cline{2-4} & Graph & Images & GPS \\ \hline WDR & + & - & - \\ DeepIST & - & + & - \\ DeepTTE & - & - & + \\ DeepI2T & + & + & - \\ \hline \end{tabular}
\end{table}
Table 1: Demonstration of utilized modalities in path-blind and path-aware models
representations. As well as DeppTTE, Deepl2T includes the segment-based prediction component implemented in the form of residual blocks on the top of the Bi-LSTM encoder.
In addition to extensively studied recurrent TTE methods, it is also important to mention recently emerged transformer models Liu et al. (2022); Semenova et al. (2022). Despite the limited comparison with classic LSTM-based methods, they have already demonstrated promising prediction quality, preserving the potential for further major improvements Shen et al. (2022); Fan et al. (2021). As most of the transformer models lack a comprehensive evaluation, we intend to explore GCT-TTE performance with respect to a sufficient number of state-of-the-art solutions to reveal its capabilities explicitly.
## Preliminaries
In this section, we introduce the main concepts required to operate with the proposed model.
**Rout**. A route \(r\) is defined as the set \(\{c^{r},a^{r},t^{r}\}\), where \(c^{r}\) is the sequence of GPS coordinates of a moving object, \(a^{r}\) is the vector of temporal and weather data, \(t^{r}\) is the travel time.
As the _image modality_\(p^{r}\) of a route \(r\), we utilize geographical map patches corresponding to each coordinate \(c_{i}^{r}\in c^{r}\). Each image has a fixed size \(W\times H\times 3\) across all of the GPS sequences in a specific dataset.
**Road network**. Road network is represented in the form of graph \(G=\{V,E,X\}\), where \(V=\{v_{1},\;...\;,v_{n}\}\) is the set of nodes corresponding to the segments of city roads, \(E=\{(v_{i},v_{j})\;|\;v_{i}\to v_{j}\}\) is the set of edges between connected nodes \(v_{i},v_{j}\in V\), \(X:n\times m\rightarrow\mathbf{R}\) is a feature matrix of nodes.
Description of a route \(r\) can be further extended by the _graph modality_\(g^{r}=\{v_{k}\;|\;k=argmin_{j}\,\rho(c_{i}^{r},v_{j})\}_{i=1}^{|c^{r}|}\), where \(\rho(c_{i}^{r},v_{j})\) is the minimum Euclidean distance between coordinates associated with \(v_{j}\) and \(c_{i}^{r}\).
**Travel time estimation**. For each entry \(r\), it is required to estimate the travel time \(t^{r}\) using the elements of feature description \(\{c^{r},p^{r},g^{r},a^{r}\}\).
## Data
We explored the predictive performance of the algorithm on two real-world datasets collected during the period from December 1, 2020 to December 31, 2020 in Abakan (112.4 square kilometers) and Omsk (577.9 square kilometers). Each dataset consists of a road graph and associated routes, Table 2. In the preprocessing stage, we excluded trips that lasted less than 30 seconds along with the ones that took more than 50 minutes.
In order to supply the image-based models with the relevant input data, we extended the road graphs with the map patches parsed via Open Street Map API2. Depending on the requirements of the considered learning model, image datasets had to be constructed regarding the fixed grid partitions or centered around the elements of GPS sequences. In the first case, a geographical map of a city was divided into equal disjoint patches, which were further mapped with the GPS data in accordance with the presence of coordinates in a specific partition. Trajectory-based approach to dataset construction does not require the disjoint property of images and relies on the extraction of patches with the center in the specified coordinate. The obtained grid-based image dataset consists of \(96\,101\) instances for Abakan and \(838\,865\) for Omsk while the trajectory-based dataset has \(544\,502\) and \(3\,376\,294\) images correspondingly.
Footnote 2: [https://www.openstreetmap.org](https://www.openstreetmap.org)
One of the crucial features of the considered datasets is the absence of traffic flow properties. The availability of such data is directly related to the specialized tracking systems (based on loop detectors or observation cameras), which are not presented in the majority of cities. In order to make the GCT-TTE suitable for the greatest number of urban environments, we decided not to limit the study by the rarely accessible data.
## Method
In this section, we provide an extensive description of the GCT-TTE main components: pointwise and sequence representation blocks, Figure 1.
#### Patches encoder
In order to extract features from the image modality, we utilized the RegNetY Radosavovic et al. (2020) architecture from the SEER model family. The key component of this architecture is the convolutional recurrent neural network (ConvRNN) which controls the spatio-temporal information flow between building blocks of the neural network.
Each RegNetY block consists of three operators. The initial convolution layer of \(t\)'th block processes the input tensor \(X_{1}^{t}\) and returns the feature map \(X_{2}^{t}\). Next, the obtained representation \(X_{2}^{t}\) is fed to ConvRNN:
\[H^{t}=\tanh(\mathrm{C_{x}}(X_{2}^{t})+\mathrm{C_{h}}(H^{t-1})+b_{h}), \tag{1}\]
where \(H^{t-1}\) is the hidden state of the previous RegNetY block, \(b_{h}\) is a bias tensor, \(\mathrm{C_{x}}\) and \(\mathrm{C_{h}}\) correspond to convolutional layers. In the following stage, \(X_{2}^{t}\) and \(H^{t}\) are utilized as input of the last convolution layer, which is further extended by residual connection.
As the SEER models are capable of producing robust features that are well-suited for out-of-distribution generalization Goyal et al. (2022), we pre-trained RegNetY with the following autoencoder loss:
\[\mathcal{L}(W\times RegNet(X),\,f(X))\to 0, \tag{2}\]
where \(\mathcal{L}\) is the binary cross-entropy function, \(f\) is an image flattening operator, and \(W\) is the projection matrix of learning parameters that maps model output to the flattened image.
#### Auxiliary encoder
Along with the map patches and graph elements, we apply additional features \(a^{r}\) corresponding to the temporal and weather data (e.g., trip hour, type of day, precipitation). The GCT-TTE model processes this part of the input with the help of a trivial linear layer:
\[A^{r}=Wa^{r}, \tag{3}\]
where \(W\) is a matrix of learning parameters.
#### Graph encoder
The graph data is handled with the help of the graph convolutional layers Kipf and Welling (2016) defined as follows:
\[h_{u}^{(k)}=\mathrm{ReLU}\left(W^{(k)}\underset{v\in\mathcal{N}(u)}{\mathrm{AG }}\left(\frac{h_{v}^{(k-1)}}{||\mathcal{N}_{uv}||}\right)\right), \tag{4}\]
where \(h_{u}^{(k)}\) is a \(k\)-hop embedding of \(u\in V\), \(h_{u}^{(0)}=x_{u}\), \(W^{(k)}\) is a matrix of learning parameters of \(k\)'th convolutional layer, \(\mathcal{N}(u)\) is a set of neighbour nodes of \(u\), \(\mathrm{AGG}_{v\in\mathcal{N}(u)}\) is a sum aggregarion function, and \(||\mathcal{N}_{uv}||=\sqrt{|\mathcal{N}(u)||\mathcal{N}(v)|}\).
To accelerate the convergence of the GCT-TTE model, we pre-trained the weights of the graph convolutions by the Deep Graph InfoMax algorithm Velickovic et al. (2019). This approach optimizes the loss function that allows learning the difference between initial and corrupted embeddings of nodes:
\[\mathcal{L}=\frac{1}{N+M}\Big{(}\sum_{i=1}^{N}E_{\mathcal{G}}\Big{[}log(D(h_{u },h_{\mathcal{G}}))\Big{]}+\sum_{j=1}^{M}\,E_{\mathcal{G}}\left[log(1-D(\tilde {h}_{u},h_{\mathcal{G}}))\right]\Big{)}, \tag{5}\]
\begin{table}
\begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{Road network} \\ \hline Property \(\backslash\) City & Abakan & Omsk \\ \hline Nodes & \(65\,524\) & \(231\,688\) \\ Edges & \(340\,012\) & \(1\,149\,492\) \\ Clustering & 0.5278 & 0.53 \\ Usage median & 12 & 8 \\ \hline \end{tabular}
\begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{Trips} \\ \hline Property \(\backslash\) City & Abakan & Omsk \\ \hline Trips number & \(121\,557\) & \(767\,343\) \\ Coverage & 53.3\% & 49.5\% \\ Average time & 427 sec & 608 sec \\ Average length & 3604 m & 4216 m \\ \hline \end{tabular}
\end{table}
Table 2: Description of the Abakan and Omsk datasets.
where \(h_{u}\) is an embedding of node \(u\) based on the initial graph \(\mathcal{G}\), \(\tilde{h}_{u}\) is an embedding of a node \(u\) from the corrupted version \(\tilde{\mathcal{G}}\) of the graph \(\mathcal{G}\), \(D\) corresponds to the discriminator function.
The final output of the pointwise block constitutes a concatenation of the weighted representations and auxiliary data for each route \(r\) with \(k\) segments:
\[P_{r}=\mathrm{CONCAT}(\alpha\cdot H^{r},(1-\alpha)\cdot I^{r},\ \beta\cdot A^{r}), \tag{6}\]
where \(H^{r}\) is the matrix of size \(k\times e_{g}\) of graph-based segment embeddings, \(I^{r}\) is the matrix of size \(k\times e_{i}\) obtained from a flattened RegNet output, \(\alpha\), \((1-\alpha)\), and \(\beta\) correspond to the weight coefficients of specific modalitites.
#### Sequence representation block
To extract sequential features from the output of the pointwise representation block, it is fed to transformer encoder Vaswani et al. (2017). The encoder consists of two attention layers with a residual connection followed by a normalization operator. The multi-head attention coefficients are defined as follows:
\[\alpha_{i,j}^{(h)}=\mathrm{softmax}_{w_{j}}\left(\frac{\langle W_{h,q}^{T}x_{ i},W_{h,k}^{T}x_{j}\rangle}{\sqrt{d_{k}}}\right), \tag{7}\]
where \(x_{i},x_{j}\in P_{r}\), \(h\) is an attention head, \(d_{k}\) is a scale coefficient, \(W_{h,q}^{T}\) and \(W_{h,k}^{T}\) are query and key weight matrices, \(w_{j}\) is a vector of softmax learning parameters. The output of the attention layer will be:
\[u_{i}=\mathrm{LayerNorm}\left(x_{i}+\sum_{h=1}^{H}W_{c,h}^{T}\sum_{j=1}^{n} \alpha_{i,j}^{(h)}W_{h,v}^{T}x_{j}\right), \tag{8}\]
where \(W_{h,v}^{T}\) is value weight matrix, \(H\) is a number of attention heads.
The final part of the sequence representation block corresponds to the flattening operator and several linear layers with the ReLU activation, which predict the travel time of a route.
### Results
In this section, we reveal the parameter dependencies of the model and compare the results of the considered baselines.
Figure 1: Demonstration of the GCT-TTE pipeline.
### Experimental setup
The experiments were conducted on 16 GPU Tesla V100. For the GCT-TTE training, Adam optimizer Kingma and Ba (2014) was chosen with a learning rate \(5\cdot 10^{-5}\) and batch size of 16. For better convergence, we apply the scheduler with patience equal to 10 epochs and 0.1 scaling factor. The training time for the final configuration of the GCT-TTE model is 6 hours in the case of Abakan and 30 for Omsk.
The established values of quality metrics were obtained from the 5-fold cross-validation procedure. As the measures of the model performance, we utilize mean absolute error (MAE), rooted mean squared error (RMSE), and 10\(\%\) satisfaction rate (SR). Additionally, we compute mean absolute percentage error (MAPE) as it is frequently used in related studies.
### Models comparison and evaluation
The results regarding path-blind evaluation are depicted in Table 3. Neighbor average (AVG) and linear regression (LR) demonstrated the worst results among the trivial baselines as long as gradient boosted decision trees (GBDT) explicitly outperformed more complex models in the case of the largest city. The MURAT model achieved the best score for Abakan in terms of MAE and RMSE, while GCT-TTE has the minimum MAPE among all of the considered architectures.
Demonstrated variability of metric values makes the identification of the best model rather a hard task for a path-blind setting. The simplest models are still capable to be competitive regarding such architectures as MURAT, which was expected to perform tangibly better on both considered datasets. The results regarding GCT-TTE can be partially explained by its structure as it was not initially designed for a path-blind evaluation.
As can be seen in Table 4, the proposed solution outperformed baselines in terms of the RMSE value, which proves the rigidity of GCT-TTE towards large errors prevention. The comparison of MAE and RMSE for considered methods has shown a minimal gap between these metrics in the case of GCT-TTE for both cities, signifying the efficiency of the technique with respect to dataset size. Overall, the results have confirmed that GCT-TTE appeared to be a more reliable approach than the LSTM-based models: while MAPE remains approximately the same across top-performing architectures, GCT-TTE achieves significantly better MAE and RMSE values. Conducted computational experiments also indicated that DeepI2T and WDR have intrinsic problems with the convergence, while GCT-TTE demonstrates smoother training dynamics.
### Performance analysis
In the case of both datasets, dependencies between the travelled distance and obtained MAE on the corresponding trips reveal similar dynamics: as the path length increases, the error rate continues to grow, Figure 2(b, d). The prediction variance is inversely proportional to the number of routes in a particular length interval except for the small percentage of the shortest routes. The main difference between the MAE curves is reflected in the higher magnitudes of performance fluctuations in Abakan compared to Omsk.
The temporal dynamics of GCT-TTE errors exhibit rich nonlinear properties during a 24-hour period. The shape of the error curves demonstrates that our model tends to accumulate a majority of errors in the period between 16:00 and 18:00, Figure 2(a, c). This time interval corresponds to the end of the working day, which has a crucial impact on the traffic flow foreseeability.
Despite the mentioned performance outlier, the general behaviour of temporal dependencies allows concluding that GCT-TTE successfully captures the factors influencing the target value in the daytime. With the growing sparsity of data during night hours, it is still capable of producing relevant predictions for Omsk. In the case of Abakan, the GCT-TTE performance drop can be associated with a substantial reduction in intercity trips number (which emerged to be an easier target for the model).
### Sensitivity analysis
In order to achieve better prediction quality, we extensively studied the dependencies between GCT-TTE parameters and model performance in the sense of the MAE metric. The best value for modality coefficient \(\alpha\) was 0.9, which reflects the significant contribution of graph data towards error reduction. For the final model, we utilized 2 graph convolutional layers with hidden size 192, Figure 3(a, b). The lack of aggregation depth can significantly reduce the performance of GCT-TTE, while the excessive number of layers has a less expressive negative impact on MAE. A similar situation can be observed in the case of the hidden size, which is getting close to a plateau after reaching a certain threshold value.
Along with the graph convolutions, we explored the configuration of the sequence representation part of GCT-TTE. Since the transformer block remains its main component, the computational experiments were focused on the influence of encoder depth on quality metrics, Figure 3(c). As it can be derived from the U-shaped dependency, the best number of attention layers is 3.
## Demonstration
In order to provide access to the inference of GCT-TTE, we deployed a demonstrational application2 in a website format, Figure 4. The application's interface consists of a user guide, navigation buttons, erase button, and a comparison button. A potential user can construct and evaluate an arbitrary route by clicking on the map at the desired start and end points: the system's response will contain the shortest path and the corresponding value of the estimated time of arrival.
Footnote 2: [http://gctte.online](http://gctte.online)
For additional evaluation of considered baselines, the limited number of predefined trajectories with known ground truth can also be requested. In this case, the response will contain three random trajectories from the datasets with associated predictions of WDR, Deepl2T, and GCT-TTE models along with the real travel time.
## Conclusion
In this paper, we introduced a multimodal transformer architecture for travel time estimation and performed an extensive comparison with the other existing approaches. Obtained results allow us to conclude that the transformer-based models can be efficiently utilized as sequence encoders in the path-aware setting. Our experiments with different data modalities revealed the superior importance of graphs compared to map patches. Such an outcome can be explained by the inheritance of main features between modalities where graph data represents the same properties more explicitly. In
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline & \multicolumn{4}{c|}{Abakan} & \multicolumn{4}{c|}{Omsk} \\ \hline _Baseline\textbackslash{}Metric_ & MAE & RMSE & MAPE & SR & MAE & RMSE & MAPE & SR \\ \hline DeepIST & 153.88 & 241.29 & 0.3905 & 18.08 & 256.50 & 415.16 & 0.6361 & 14.39 \\ \hline DeepTTE & 111.03 & 174.56 & 0.2165 & 31.45 & 179.07 & 296.98 & **0.1898** & 34.03 \\ \hline GridLSTM & 100.27 & 206.91 & 0.2202 & 30.74 & 135.74 & 257.18 & 0.2120 & 31.21 \\ \hline Deepl2T & 97.99 & 201.33 & **0.2128** & 31.34 & 136.66 & 260.90 & 0.2124 & 31.23 \\ \hline WDR & 97.22 & 190.09 & 0.2162 & **31.98** & 131.57 & 269.00 & 0.2039 & 33.34 \\ \hline \hline GCT-TTE & **92.26** & **147.89** & 0.2262 & 30.46 & **107.97** & **169.15** & 0.1961 & **35.17** \\ \hline \end{tabular}
\end{table}
Table 4: Path-aware models comparison
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline & \multicolumn{4}{c|}{Abakan} & \multicolumn{4}{c|}{Omsk} \\ \hline _Baseline \textbackslash{}Metric_ & MAE & RMSE & MAPE & SR & MAE & RMSE & MAPE & SR \\ \hline AVG & 322.77 & 477.61 & 0.761 & 0.018 & 439.05 & 628.75 & 0.741 & 0.012 \\ \hline LR & 262.33 & 456.63 & 1.169 & 9.527 & 416.81 & 593.01 & 1.399 & 7.187 \\ \hline GBDT & 245.77 & 433.91 & 1.106 & 10.28 & **209.99** & **372.11** & 0.656 & **17.72** \\ \hline \hline MURAT & **182.97** & **282.15** & 0.685 & 10.77 & 285.72 & 444.74 & 0.856 & 9.997 \\ \hline \hline GCT-TTE & 221.71 & 337.59 & **0.505** & **11.12** & 376.74 & 590.93 & **0.5486** & 8.99 \\ \hline \end{tabular}
\end{table}
Table 3: Path-blind models comparison
further studies, we intend to focus on the design of a more expressive image encoder as well as consider the task of path-blind travel time estimation, which currently remains challenging for the GCT-TTE model.
Figure 3: Parametric dependencies of GCT-TTE performance for Abakan: number of graph convolutions (a), hidden size of graph convolutions (b), and number of transformer encoder layers (c).
Figure 2: Spatial and temporal dependencies across the different groups of test entries for Abakan (a, b) and Omsk (c, d): blue and red lines depict mean and median values of MAE, borders of filled area correspond to Q1 and Q3 quartiles of a MAE distribution.
#### Declarations
Ethics approval and consent to participate
Not applicable.
### Consent for publication
Not applicable.
#### Availability of data and materials
Considered models and datasets are available in the project's GitHub repository.
### Competing interests
The authors declare that they have no competing interests.
## Funding
Not applicable.
## Authors contributions
V.M., V.C., and A.I.: Software, Data curation, Validation, Visualization; V.P.: Software, Visualization, Conceptualization, Methodology, Writing (original draft); N.S.: Conceptualization, Methodology, Supervision, Writing (review & editing).
Figure 4: An interface of the demonstrational application.
## Acknowledgements
The authors are grateful to Vladislav Zamkovy for the help with application deployment.
| この論文では、旅行時間予測問題に新たなトランスフォーマーベースモデルを導入します。提案された GCT-TTE アーキテクチャの重要な特徴は、入力経路の異なる特性を捉えるための異なるデータモダリティの利用です。モデル構成に関する広範な研究に加え、パス Aware とパス Blind 設定の基盤モデルを多数実装し評価しました。実施された計算実験により、私たちのパイプラインの可行性を確認しました。これは、考慮されたデータセットにおいて、最新モデルに上回るパフォーマンスを示しました。さらに、 GCT-TTE はユーザー定義ルートによる実験にアクセス可能なウェブサービスとしてデプロイされました。 |
2303.01482 | Modulation instability gain and localized waves by modified
Frenkel-Kontorova model of higher order nonlinearity | In this paper, modulation instability and nonlinear supratransmission are
investigated in a one-dimensional chain of atoms using cubic-quartic
nonlinearity coefficients. As a result, we establish the discrete nonlinear
evolution equation by using the multi-scale scheme. To calculate the modulation
instability gain, we use the linearizing scheme. Particular attention is given
to the impact of the higher nonlinear term on the modulation instability.
Following that, full numerical integration was performed to identify modulated
wave patterns, as well as the appearance of a rogue wave. Through the nonlinear
supratransmission phenomenon, one end of the discrete model is driven into the
forbidden bandgap. As a result, for driving amplitudes above the
supratransmission threshold, the solitonic bright soliton and modulated wave
patterns are satisfied. An important behavior is observed in the transient
range of time of propagation when the bright solitonic wave turns into a
chaotic solitonic wave. These results corroborate our analytical investigations
on the modulation instability and show that the one-dimensional chain of atoms
is a fruitful medium to generate long-lived modulated waves. | Alphonse Houwe, Souleymanou Abbagari, Lanre Akinyemi, Serge Yamigno Doka, Kofane Timoleon Crepin | 2023-02-25T17:29:35 | http://arxiv.org/abs/2303.01482v1 | Modulation instability gain and localized waves by modified Frenkel-Kontorova model of higher order nonlinearity
###### Abstract
In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic-quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To calculate the modulation instability gain, we use the linearizing scheme. Particular attention is given to the impact of the higher nonlinear term on the modulation instability. Following that, full numerical integration was performed to identify modulated wave patterns, as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a result, for driving amplitudes above the supratransmission threshold, the solitonic bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright solitonic wave turns into a chaotic solitonic wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves.
**Keywords:** Modified Frenkel-Kontorova model; Modulation instability; Modulated waves patterns; Rogue waves
## 1 Introduction
In recent years, investigation of the localized waves in nonlinear systems has grown. A wide class of nonlinear evolution equations have been employed in different fields such as optical fibers, Bose-Einstein condensates, optomechanical lattices, molecular chains, fluid mechanics, and ferromagnetic structures [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. More often, the phenomenon that exhibits the
behavior of the excited localized modes is the modulation instability (MI). MI is characterized by a rapidly growing plane wave amplitude subject to small perturbations where nonlinear and dispersion terms interplay [2, 6, 7, 13, 15, 17, 23]. Usually, MI is manifested by the unstable or stable zones generated by the perturbed wave vector or nonlinear strength, which leads to the formation of modulated wave (MW) patterns. The three types of localized excitations that can be obtained are bright solitons, rogue waves (RWs), and breathers. For example, Conrad and co-workers have recently pointed out the propagation of the RWs of A and B types in a nonlocal medium where a nonlinear saturation parameter is used [4]. The authors have shown that when the MI is developed, the MW patterns emerge. In [5], the authors have exhibited the localized modes in a nonlinear oscillator lattice through the MI. It is obvious that the MI is an appropriate mechanism for the generation of localized waves. If most of the models used before were in the continuum limit, it is evident today that the discrete MI of the continuous wave (CW) in the discrete nonlinear evolution equation (DNLEE) has gained a lot of interest. Houwe et al. have used the DNLEE, which describes the wave propagating in the ferromagnetic chains, to develop the MI under the effects of the nearest-neighbor coupling. The discrete MI has been the subject of theoretical and experimental research. The work of Mounouna et al. [7] is a well-known example of the MI growth rate, where the effects of the nonlinear cubic-quartic coupling of the modified Frenkel-Kontorova model were shown on the gain profile and the unstable zones that emerged during the long time of plane wave propagation.
If the MI is an important process for developing localized waves, nonlinear supratransmission also remains a powerful tool where energy can flow in the forbidden bandgap (FG). This phenomenon has been developed by F. Geniet and co-workers by using the Klein Gordon equation. They have shown that when the amplitude of the driving is considered above the threshold for supratransmission, energy can flow in the FG [11]. Khomeriki et al. used the same procedure to derive the static breather solution, which synchronizes and adjusts to the Fermi-Pasta-Ulam model [12]. Beside this, several other studies have been developed to show that nonlinear strength can also favor the formation of the MWs patterns in the FG [13, 22, 25].
Very recently, the supratransmission has been exhibited by driving one end of the chain where the on-site potential of a cubic form has been used [10]. It has been called the quartic nonlinear interaction potential. Nonlinear supratransmission has been used in other applications, such as three-wave sharp interaction and low-light pulses when a two-level medium produces solitary waves [20]. In the present study, we point out the MWs, RWs, and diverse other localized waves under the effects of the cubic and quartic nonlinear interaction potentials. Thereafter, we subject one end of the chain to an external periodic force to demonstrate the supratransmission phenomenon. It emerges that, above the threshold, supratransmission of a localized bright soliton is fulfilled. We equally observed that when the driven amplitude (DA) is strong enough, the transient regime manifests itself by the escape of the bright soliton to chaos-like motion.
The rest of the paper is sketched as follows: In Sect. 2, we present the proposed model and thereafter use the standard multi-scale scheme to derive the DNLEE. Sect. 3 gives the linear stability of the MI. An expression of the MI growth rate is deduced from the dispersion relation and used to show the unstable and stable zones. In particular, we focused on the dispersion coefficient and the impact of the cubic and quadratic nonlinear interactions. Sect. 4 uses the full numerical simulation to corroborate analytical predictions, MW patterns, and RWs. On the other hand, one end of the chain is driven by an external periodic force. In FG, the formation of excited localized modes and bright soliton is observed in equal measure. Sect. 5 is assigned to concluding the work.
## 2 Analytical model
Motivated by the work of Mounouna et al. [7], we consider in this work a chain of coupled atoms subjugated to a deformable external periodic potential where the total Hamiltonian is written as:
\[\mathbf{H}=\Gamma\sum_{n}\Bigg{[}\frac{1}{2}\left(\frac{d\theta_{n}}{dt}\right) ^{2}+\left(\frac{1}{2}G_{2}(\theta_{n}-\theta_{n-1})^{2}+\frac{1}{3}G_{3}( \theta_{n}-\theta_{n-1})^{3}+\frac{1}{4}G_{4}(\theta_{n}-\theta_{n-1})^{4} \right)+\omega_{0}^{2}\frac{\tan^{2}\left(\frac{\theta_{n}}{2}\right)}{\left( \sigma^{2}+\tan^{2}\left(\frac{\theta_{n}}{2}\right)\right)}\Bigg{]}, \tag{1}\]
where \(\Gamma\) denotes the energy scale, \(\theta_{n}\) the dimensionless movements of particles and \(\omega_{g}\) the angular frequency. The potential interaction coefficients are \(G_{j}(j=2,3,4)\) and the equation of the motion reads [7]:
\[\begin{split}\frac{d^{2}\theta_{n}}{dt^{2}}=G_{2}(\theta_{n+1}-2 \theta_{n}+\theta_{n-1})+G_{3}\left[(\theta_{n+1}-\theta_{n})^{2}-(\theta_{n}- \theta_{n-1})^{2}\right]+G_{4}\left[(\theta_{n+1}-\theta_{n})^{3}-(\theta_{n}- \theta_{n-1})^{3}\right]\\ -\omega_{g}^{2}(\theta_{n}+\alpha\theta_{n}^{2}+\beta\theta_{n}^ {3}).\end{split} \tag{2}\]
The frequency parameter is \(\omega_{g}\), the nonlinearities in the potential's shape are \(\alpha\) and \(\beta\). Eq. (2) is the discrete equation that describes the movement of the chains of particles in the presence of the nonlinear coupling terms, and it takes its origin from the modified Frenkel-Kontorova model. In [8], the authors have considered the model with \(G_{3}=0\) and \(G_{4}=0\) to exhibit the effects of the nonlinearity coefficients and the substrate's deformability on MI. The model was recently used to investigate the interaction between cubic-quartic nonlinearities and the substrate's deformable parameter on MI growth rates [7]. The authors have shown that the influence of the quartic nonlinearity has extended the MI bands and that the amplitude of the plane wave has risen exponentially. It is important to underline that Eq. (2) can take the form of the Klein Gordon equation when cubic-quartic interactions and coupling are omitted. In what follows, we aim to highlight the effect of the nonlinearity coefficients on the MI growth rates. Thereafter, we establish the threshold amplitude (TA) expression, from which we will consider the DA to drive one end of the model in the forbidden frequency band gap. A fascinating matter that has been studied in several nonlinear systems is driven at one end of the model. But, to our knowledge, this subject was not formulated in [7]. To do so, we consider the standard multi-scale analytical paths as follows:
\[\theta_{n}(t)=\epsilon\left(\psi_{n}(\epsilon^{2},t)e^{i\phi_{n}}+cc\right)+ \epsilon^{2}\Theta_{n}(\epsilon^{2},t)+\epsilon\left(\Gamma_{n}(\epsilon^{2}, t)e^{2i\phi_{n}}+cc\right). \tag{3}\]
Here, Eq. (3) is the slowly varying time and space of MW solution, which propagates at a carrier frequency \(\omega\) and wave vector \(q\), \(\epsilon\) counts for the small parameter and the phase is \(\phi_{n}=kn-\omega t.\) While \(\psi_{n}\) and \(\Gamma_{n}\) are the complex functions with cc denotes their complex conjugate, \(\Theta_{n}\) is a real function. Assuming that \(G_{2}\sim\epsilon^{2},\ G_{3}\sim 1,\ G_{4}\sim 1,\ \alpha\sim 1\) and \(\beta\sim 1\)[7, 10]. Inserting Eq. (3) into Eq. (2) and gathering terms in order \(\epsilon,\ \epsilon^{2}\) and \(\epsilon^{3}\) together with \(e^{2i\phi_{n}}\), we get the DNLEE:
\[\Theta_{n}=2\alpha|\psi_{n}|^{2},\ \ \Gamma_{n}=-\frac{\alpha\omega_{g}^{2} \psi_{n}^{2}}{4\omega^{2}-\omega_{g}^{2}}, \tag{4}\]
and
\[\begin{split}-2i\omega\dot{\psi}_{n}+(\omega_{g}^{2}-\omega^{2} )\psi_{n}-G_{2}(\psi_{n+1}e^{ik}-2\psi_{n}+\psi_{n-1}e^{-ik})+3G_{3}\left(\psi _{n-1}^{*}e^{ik}-\psi_{n}^{*}\right)\left(\psi_{n-1}e^{-ik}-\psi_{n}\right)^{2 }\\ -3G_{4}\left[\left(\psi_{n-1}^{*}e^{ik}-\psi_{n}^{*}\right) \left(\psi_{n-1}e^{-ik}-\psi_{n}\right)^{2}+\left(\psi_{n+1}^{*}e^{-ik}-\psi_{ n}^{*}\right)\left(\psi_{n+1}e^{ik}-\psi_{n}\right)^{2}\right]\\ +\omega_{g}^{2}(3\beta-4\alpha^{2}+\frac{2\omega_{g}^{2}\alpha^{2 }}{4\omega^{2}-\omega_{g}^{2}})|\psi_{n}|^{2}\psi_{n}=0.\end{split} \tag{5}\]
Thus, Eq. (5) is the DNLEE with cubic-quartic interaction potential. As we have mentioned above, the upcoming section will discuss the MI growth rates.
## 3 Modulation instability
MI is the phenomenon where nonlinearity and dispersion interplay. Some works has been reported in [2, 6, 13, 15, 19]. During the investigation of the MI, the DNLEEs are the models that are most widely involved. For example, in [2], the authors used the well-known discrete nonlinear Schrodinger equation with cubic-quintic nonlinearity to analyze the MI growth rates under the effects of the nonlinear term. Tanemura et al. investigate the modulational unstable or stable modes in loss-dispersion optical fibers. More recently, the effects of the nearest-neighbor coupling have been studied [13]. Without doubt, MI is the process where small perturbations are inserted into the CWs. One can also notice that the analytical investigation of the MI growth rate cannot say enough about the growing amplitude of the plane wave. As a result, numerical analysis is the most powerful tool for observing MW patterns over long periods of propagation. In what follows, we use small perturbations in the CW to establish the linearizing expression. Afterwards, we establish the
MI gain, where the effects of the nonlinear terms are highlighted. To confirm our analytical investigation, we use the numerical simulation to control the exponential growth rates of the plane wave. For this, we consider the plane wave with small perturbations as having a solution of Eq. (5) as:
\[\psi_{n}=(F_{0}+F_{n})\,e^{i(kn-\varpi)t}, \tag{6}\]
where \(F_{0}\) is the initial amplitude, \(k\) and \(\varpi\) are respectively the wave vector and angular frequency. Inserting Eq. (6) into Eq. (5), gives
\[i\frac{\partial}{\partial t}F_{n}+\Lambda_{1}F_{n+1}+\Lambda_{2}F_{n-1}+ \Lambda_{3}F_{n}+\Lambda_{4}F_{n+1}^{*}+\Lambda_{5}F_{n-1}^{*}+\Lambda_{6}F_{n }^{*}=0. \tag{7}\]
The parameters \(\Lambda_{j}(j=1,...,6)\) are in the Appendix. We consider the solution of Eq. (7) as follow:
\[F_{n}=f_{1}\cos(Qn+\Omega t)+if_{2}\sin(Qn+\Omega t), \tag{8}\]
where \(Q\) and \(\Omega\) are respectively the perturbed wave vector and angular frequency of the MI growth rate. Using Eq. (8) into Eq. (7), we obtain the matrix in the form
\[\left(\begin{array}{cc}i\Omega-(N_{1}-N_{2}+N_{4}-N_{5})\sin(Q)&i((N_{1}+N_ {2}-N_{4}-N_{5})\cos(Q)+N_{3}-N_{6})\\ (N_{1}+N_{2}+N_{4}+N_{5})\cos(Q)+N_{3}+N_{6}&\Omega+i(N_{1}-N_{2}-N_{4}+N_{5}) \sin(Q)\end{array}\right)\left(\begin{array}{c}f_{1}\\ f_{2}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right), \tag{9}\]
and Eq. (9) can vanish only for
\[\Omega^{2}+i(X_{1}-X_{2})\Omega+X_{1}X_{2}+\Delta=0, \tag{10}\]
with
\[X_{1}= (N_{1}-N_{2}+N_{4}-N_{5})\sin(Q), \tag{11}\] \[X_{2}= i(N_{1}-N_{2}-N_{4}+N_{5})\sin(Q),\] \[\Delta= i((N_{1}+N_{2}-N_{4}-N_{5})\cos(Q)+N_{3}-N_{6})((N_{1}+N_{2}+N_{ 4}+N_{5})\cos(Q)+N_{3}+N_{6}).\]
It is worth mentioning that the MI occurs when the frequency of the modulation is complex with a non-zero imaginary part. So, the corresponding MI growth rate takes the form of
\[Gain=|\Im(\Omega_{max})|. \tag{12}\]
In what follows, we highlight the impacts of the parameters of the cubic-quartic interaction potential on the MI. For this reason, the value of the cubic coupling is kept fixed along with that of \(G_{2}\). In Figure 1, we have depicted the variation of the MI growth rates under the effects of the quartic interaction potential. From Figure 1 a-b, we have shown the formation of the unstable zones (bright zones) for \(G_{4}=-0.01\). One can see that in panel (a), very slight stable zones emerge, while in panel (b), two side lobes appear with a large MI band. The higher amplitude of the plane waves is about 0.6. In Figure 1 c-d, we have increase negatively the quartic parameter to \(G_{4}=-0.1\). In contrast to panels (a-b), the unstable zones decrease to expand the stable modes (see panel (c)), and additional bands of the MI emerge. We equally observe that the amplitude of the plane wave increases to 2.6 and three side lobes emerge in panel (d). It results that when the quartic nonlinearity strength increases negatively, the amplitude of the perturbed plane wave and the stable modes increase together. In Figure 2, e-f, we increased the negative nonlinear strength interaction to \(G_{4}=-0.5\) once more. We have depicted the same scenario as in panels (c-d). The amplitude of the plane wave has increased strongly to reach 8.9 in panel (f), and the stable modes increase. We also notice that the additional bands amplitudes increase. Once more, we have exhibited in panels (g-h) the same behavior for \(G_{4}=-1\). From this analysis, it emerges that increasing negatively the quartic nonlinearity term reduces the unstable modes and increases the amplitude of the perturbed plane wave, which is a good argument for seeking numerically the evolution of the MI growth rates.
Following the same procedure as in Figure 1, we have depicted unstable modes of the MI in Figure 2 for \(G_{4}=0.01\), 0.1, 0.5, 1.5, and 2.5 in terms of \((Q,\,k)\). For \(G_{4}=0.01\) we have shown unstable zones in panel (a), indicated by the bright zones, and
three symmetric side lobes in panel (b). We increase the nonlinear term to \(0.1\), the unstable MI areas increase while additional bands emerge to shrink the MI bands in \(k\)-axis (see panel (c) and panel (d) respectively). One can observe that the amplitude of the plane wave has remained identical. To confirm the analytical predictions reported in [11], that the quartic nonlinear term induces unstable modes, we have increase its value to \(0.5\). It emerges with six peaks of the unstable modes, with small unstable lobes in the middle and a large enough amplitude of the plane wave in panels (e-f). Beside, in panel (g) and panel (h) respectively, we have depicted the same behavior, but the peak of the amplitude of the plane wave have increased strong enough to reach \(200\). Without doubt, positive values of the higher nonlinear term can generate instability in a chain of atoms and could probably induce the MWs patterns during long periods of propagation.
Since it is obvious that the quartic nonlinear term induces unstable or stable modes depending on its sign, we next turn to the effects of the cubic nonlinear strength. As a result, we consider \(G_{4}=0\). In Figure 3 a-d, we have depicted the variation of the unstable MI for \(G_{3}=-0.1\) panels (a-b) and \(G_{3}=0.5\) panels (c-d). One can observe that for negative values of the cubic term, the unstable mode is manifested by a set of two symmetrical lobes. By increasing this value to \(G_{3}=0.5\), the MI's stable modes increase along with the plane wave's amplitude. Another important effect of the cubic nonlinear term is observed through the unstable and stable modes in the atom's structure. Our analytical investigation has confirmed the previous predictions made by Nguetcho and co-workers. Last but not least, we used Figure 3, e to demonstrate the manifestation of the MI growth rate with the effects of the dispersion term \(G_{2}\). For \(G_{2}=-0.1\) and \(G_{2}=0.1\) respectively, the maximum amplitude of the plane wave have the same values. Meanwhile for \(G_{2}=-0.5\) and \(G_{2}=0.5\), one can observe that the plane wave gets large amplitude and the MI bands enlarge.
## 4 Numerical investigation
In the previous section, we underlined the interplay between dispersion and high-order nonlinear terms in the structure of the one-dimensional chain of atoms. We have shown that the cubic and quartic interactions can generate unstable or stable modes as well as wave patterns. In this section, we use the numerical integration of Eq. (5).
### Modulated waves patterns
To say the truth, the linear stability investigation can only say so much about the long-term propagation of the CWs. To answer this preoccupation, we use the numerical integration of Eq. (5). An initial condition in the form
\[\psi_{n}(t=0)=\psi_{0}\left(1+\xi\cos(Qn)\right)\cos(kn) \tag{13}\]
has been used with \(\psi_{0}=1\) and \(\xi=0.001\) to develop the MI growth rates. In what follows, attention is paid to the effects of the higher-order interaction coefficients on the development of the MI. The important aspect of this investigation is the fact that both cubic and quartic nonlinear terms of the nearest neighbor are used during the long period of the propagation of the MWs. On the other hand, the novelty of this present study lies in the inclusion of the higher-order nonlinearity that is manifested by the \(G_{4}\) parameter. The effects of the \(G_{4}\) nonlinear term have been revealed to be effective on the development of MI by increasing or reducing the unstable domains as well as the gain profiles [7]. Here, the effects of the higher nonlinear coupling coefficient are exhibited in Figure 4, where panel (a) shows the propagation of the trains of waves for \(G_{4}=0.001\). By increasing the value of the higher nonlinear strength to \(G_{4}=0.01\) in panel (b), one may observe the formation of the train of pulses with a different shape, despite the fact that the maximum amplitude of the plane wave remains constant as in panel (a). For more clarity, we have shown the same objects in panel (c) for large enough value of \(G_{4}\) parameter. Observing closely in panel (d), one realizes a similarity with RWs, where a 2D train depicted against the background displays an Akhmediev breather despite its small amplitude. Our results seem to be in accordance with the analytical predictions where the unstable MI is developed for positive values of the nonlinear term. Otherwise, the long time propagation has open new features in the dynamics of a one-dimensional chain made of atoms, harmonically coupled to their nearest-neighbors and subjected to an external on-site potential [7]. It is equally worth to mention that in the continuum limit Eq. (5) at \((k=0)\) and \((k=\pi)\) can turn to the nonlinear Schrodinger equation
which admits super RWs and Peregrine solitons. Beside, in Figure 5 we have shown in accordance with our analytical investigation that for negative values of the nonlinear parameter it is generated patterns waves. So, for \(G_{4}=-0.01,-0.1,\)\(-0.5,\) and \(-0.75,\) we have depicted the RWs that reinforce the argument that our structure can support Akhmediev breathers, which are related to the formation of the MI and were revealed in different studies as tools of regulation of the structures where nonlinear and dispersion terms are interplaying [9].
However, considering now the cubic nonlinear term gives the possibility to generate a train of pulse that comprise varied modes. In Figure 6, more precisely in panels (a,b) we have fixed \(G_{3}=-0.01\) and \(-0.5,\) one observes the MWs adopt different behaviors. Following the same procedure as in panels (a,b), we point out in panels (c,d) for \(G_{3}=0.01\) and \(0.5,\) the MWs patterns emerge. One results that the cubic coupling term can develop several modes when particles interaction is happened in the structure. So, in the next section we focuss on the MW bright soliton. Most of the studies carried out on the effects of the higher-order interaction have used a continuum limit which is only, our outcomes have been carried out by using a DNLEE. The obtained results have shown the robustness of this mechanism which reveals MWs with a particular properties.
### Nonlinear supratransmission
In this section, we aim to submit the left and right side of Eq. (5) to the external periodic force, which is different with regular mechanism of the supratransmission where only one end of the structure is driving in the FG. For this matter, we insert \(\psi_{n}\approx e^{i(kn-\omega t)}\) into Eq. (2) and the linear dispersion frequency is \(\omega=\sqrt{4G_{2}\sin^{2}(\frac{k}{2})+\omega_{g}^{2}}\). The lower and upper frequencies are respectively \(\omega_{0}=\omega_{g}\) and \(\omega_{max}=\sqrt{4G_{2}+\omega_{g}^{2}}.\) At the center (\(k=0\)) and in the limit (\(k=\pi\)) of the first Brillouin zone the DNLEE Eq. (5) are respectively
\[\begin{split}-2i\omega\dot{\psi}_{n}&+(\omega_{g}^ {2}-\omega^{2})\psi_{n}-G_{2}(\psi_{n+1}-2\psi_{n}+\psi_{n-1})+3G_{3}\left| \psi_{n-1}-\psi_{n}\right|^{2}(\psi_{n-1}-\psi)\\ &-3G_{4}\left[\left|\psi_{n-1}-\psi_{n}\right|^{2}(\psi_{n-1}- \psi_{n})+\left|\psi_{n+1}-\psi_{n}\right|^{2}(\psi_{n+1}-\psi_{n})\right]\\ &\hskip 142.26378pt+\omega_{g}^{2}(3\beta-4\alpha^{2}+\frac{2 \omega_{g}^{2}\alpha^{2}}{4\omega^{2}-\omega_{g}^{2}})|\psi_{n}|^{2}\psi_{n}=0.\end{split} \tag{14}\]
and
\[\begin{split}-2i\omega\dot{\psi}_{n}&+(\omega_{g}^ {2}-\omega^{2})\psi_{n}+G_{2}(\psi_{n+1}+2\psi_{n}+\psi_{n-1})-3G_{3}\left| \psi_{n-1}+\psi_{n}\right|^{2}(\psi_{n-1}+\psi_{n})\\ &+3G_{4}\left[\left|\psi_{n-1}+\psi_{n}\right|^{2}(\psi_{n-1}+ \psi_{n})+\left|\psi_{n+1}+\psi_{n}\right|^{2}(\psi_{n+1}+\psi_{n})\right]\\ &\hskip 142.26378pt+\omega_{g}^{2}(3\beta-4\alpha^{2}+\frac{2 \omega_{g}^{2}\alpha^{2}}{4\omega^{2}-\omega_{g}^{2}})|\psi_{n}|^{2}\psi_{n}=0.\end{split} \tag{15}\]
In the continuum limit the equations read at the lower FG
\[-2i\omega\dot{\psi}+(\omega_{g}^{2}-\omega^{2})\psi-G_{2}\frac{\partial^{2} \psi}{\partial x^{2}}+\omega_{g}^{2}(3\beta-4\alpha^{2}+\frac{2\omega_{g}^{2} \alpha^{2}}{4\omega^{2}-\omega_{g}^{2}})|\psi|^{2}\psi=0. \tag{16}\]
and the upper FG:
\[-2i\omega\dot{\psi}+(\omega_{g}^{2}-\omega^{2}+4G_{2})\psi+G_{2}\frac{ \partial^{2}\psi}{\partial x^{2}}+\left[-24G_{3}+48G_{4}+\omega_{g}^{2}\left( 3\beta-4\alpha^{2}+\frac{2\omega_{g}^{2}\alpha^{2}}{4\omega^{2}-\omega_{g}^{2 }}\right)\right]|\psi|^{2}\psi=0. \tag{17}\]
The static breather solutions of Eqs. (16) and (17) that synchronize and adjust to the driving in the end are respectively:
\[\begin{split}&\psi_{1}(x,t)=A_{1}e^{-i(\omega-\omega_{0})t} \operatorname{sech}\left(\sqrt{\frac{\omega^{2}+2\omega\left(\omega-\omega_{0} \right)-\omega_{g}{}^{2}}{G_{2}}}(x-x_{0})\right),\\ & A_{1}=\sqrt{-\frac{8\,\omega^{4}+16\omega^{3}\left(\omega-\omega _{0}\right)-10\omega^{2}\omega_{g}^{2}-4\,\omega\omega_{g}^{2}\left(\omega- \omega_{0}\right)+2\,\omega_{g}^{4}}{\omega_{g}^{2}\left(\frac{82\omega^{2}}{9} -\frac{19\omega_{g}^{2}}{6}\right)}},\end{split} \tag{18}\]
\[\psi_{2}(x,t)=A_{2}e^{-i\left(\omega-\omega_{max}\right)t}\operatorname{sech} \left(\sqrt{\frac{\omega^{2}+2\,\omega\,\left(\omega-\omega_{max}\right)-\omega _{g}^{2}-4G_{2}}{G_{2}}}(x-x_{0})\right), \tag{19}\]
\[A_{2}=\sqrt{-\frac{8\,\omega^{4}+16\,\omega^{3}\left(\omega-\omega_{max} \right)-10\,\omega^{2}\omega_{g}^{2}-4\,\omega\,\omega_{g}^{2}\left(\omega- \omega_{max}\right)+2\,\omega_{g}^{4}-32\,\omega^{2}G_{2}+8\,G_{2}\omega_{g}^ {2}}{16\,\omega^{2}\alpha^{2}\omega_{g}^{2}-6\,\alpha^{2}\omega_{g}^{4}-12\, \omega^{2}\beta\omega_{g}^{2}+3\beta\omega_{g}^{4}+96\,\omega^{2}G_{3}-192\, \omega^{2}G_{4}-24G_{3}\omega_{g}^{2}+48G_{4}\omega_{g}^{2}}}.\]
From there we derive the threshold boundary of the supratransmission in the lower and upper FGs respectively
\[A_{th_{1}}=2\sqrt{-\frac{8\,\omega^{4}+16\omega^{3}\left(\omega-\omega_{0} \right)-10\omega^{2}\omega_{g}^{2}-4\,\omega\omega_{g}^{2}\left(\omega-\omega _{0}\right)+2\,\omega_{g}^{4}}{\omega_{g}^{2}\left(\frac{82\omega^{2}}{9}- \frac{19\omega_{g}^{2}}{6}\right)}}, \tag{20}\]
and
\[A_{th_{2}}=2\sqrt{-\frac{8\,\omega^{4}+16\,\omega^{3}\left(\omega-\omega_{max }\right)-10\,\omega^{2}\omega_{g}^{2}-4\,\omega\,\omega_{g}^{2}\left(\omega- \omega_{max}\right)+2\,\omega_{g}^{4}-32\,\omega^{2}G_{2}+8\,G_{2}\omega_{g} ^{2}}{16\,\omega^{2}\alpha^{2}\omega_{g}^{2}-6\,\alpha^{2}\omega_{g}^{4}-12\, \omega^{2}\beta\omega_{g}^{2}+3\beta_{g}\omega_{g}^{4}+96\,\omega^{2}G_{3}-1 92\,\omega^{2}G_{4}-24G_{3}\omega_{g}^{2}+48G_{4}\omega_{g}^{2}}}. \tag{21}\]
For numerical simulation, we assume the boundary condition in the form of
\[\psi_{0}=A_{d}\cos(\omega t), \tag{22}\]
to drive Eq. (14). \(A_{d}\) is the DA and \(\omega\) the driven frequency (DF) with \(0<\omega<\omega_{0}\). In Figure 7 a, we have shown the propagation of the local energy for the DF belonging the lower FG \(\omega=0.25\) and the DA is \(A_{d_{1}}=1.8\). For specific cells index \(n=20\) and \(n=100\) in panels \((\mathrm{b},\mathrm{c})\) the spatiotemporal evolution of the train of traveling wave for the left boundary is fulfilled. Increasing the DA to \(1.85\), one can observe the propagation of the excited localized modes in the structure in panel (d). From the above assumption, it emerges that the model of Frenkel-Kontorova with cubic-quartic nonlinear on-site potential is opened to the nonlinear supratransmission phenomenon in the lower FG. We notice that a train of traveling wave occurs for the DA above the threshold supratransmission in the range of the propagation time \(t\,\epsilon\,[200,500]\). On the other hand, the energy goes to zero in the range of the time propagation \(t\,\epsilon\,[0,200]\) producing a transition phase in the structure.
Concerning the upper FG, we use the numerical integration of Eq. (15). From Figure 8 a-b, we have depicted the propagation of the localized waves in the upper FG for the DF \(\omega=1\). For DA \(A_{d_{2}}=0.8>A_{th_{2}}=0.78\), in panel (a) we show the evolution of the boundary driving of the coupled atoms with higher-order nonlinear term. In the bottom panel (c), the localized bright soliton is obtained in the range of time of propagation \(t\,\epsilon\,[200,400]\). However, we have increased the DA value to \(0.85\), one can observe that the energy flows in upper FG in panel (b). For specific range of propagation time, we have shown that the bright soliton turns to chaos-like motion in the structure in panel (d). This behavior corroborate our analytical prediction on the MI growth rates where a strong value of the higher-order nonlinear term is used. On the order hand, the propagation of the localized modes occurred in the upper FG despite the fact that the DA amplitude is considered below the threshold supratransmission of the lower FG.
Figure 1: Illustration of the MI growth rate with the variation of the quartic interaction potential. (a-b) \(G_{4}=-0.01\), (c-d) \(G_{4}=-0.1\), (e-f) \(G_{4}=-0.5\), and (g-h) \(G_{4}=-1\). The parameters used are \(G_{2}=0.01,\,G_{3}=0,\,\alpha=1.5,\,\beta=-\frac{1}{6},\,\omega_{g}=0.24,\, \omega=0.42\), and \(F_{0}=1\).
Figure 2: Variation of the MI growth rate under the positive value of the quartic nonlinearity strength. (a-b) \(G_{4}=0.01\), (c-d) \(G_{4}=0.1\), (e) \(G_{4}=0.5\), (f) \(G_{4}=1\), (g) \(G_{4}=1.5\), and (h) \(G_{4}=2.5\). The parameters used are \(G_{2}=0.01\), \(G_{3}=0\), \(\alpha=1.5\), \(\beta=-\frac{1}{6}\), \(\omega_{g}=0.24\), \(\omega=0.42\), and \(F_{0}=1\).
Figure 3: Top panel (a-d) MI growth rate with the variation of the cubic nonlinearity interaction. (a-b) \(G_{3}=-0.1\) and (c-d) \(G_{3}=0.5\) Bottom panel (f) is the illustration of the MI growth rate with the variation of the dispersion term. The parameters used are respectively \(G_{4}=0\), \(G_{3}=0.01\), \(\alpha=1.5\), \(\beta=-\frac{1}{6}\), \(\omega_{g}=0.24\), \(\omega=0.42\), and \(F_{0}=1\).
Figure 4: Numerical simulation of the intensity \(|\psi_{n}|^{2}\) with the variation of the quartic interaction coupling (\(G_{4}\)). (a) \(G_{4}=0.001\), (b) \(G_{4}=0.01\), (c) \(G_{4}=0.1\), and (d) \(G_{4}=0.5\). The parameters used are respectively \(G_{2}=0.01\), \(G_{3}=0\), \(\alpha=1.5\), \(\beta=-\frac{1}{6}\), and \(\omega_{g}=0.24\).
Figure 6: Numerical simulation of the intensity \(|\psi_{n}|^{2}\) with the variation of the cubic interaction coupling (\(G_{4}\)). (a) \(G_{3}=-0.01\), (b) \(G_{3}=-0.5\), (c) \(G_{3}=0.01\), and (d) \(G_{3}=0.5\). The parameters used are respectively \(G_{2}=0.01\), \(G_{4}=0\), \(\alpha=1.5\), \(\beta=-\frac{1}{6}\), and \(\omega_{g}=0.24\).
## 5 Conclusion
In this study, we investigated the variation of the modulation instability and the behavior of the wave propagating in the forbidden bandgap. We use the one-dimensional chain of atoms harmonically coupled to their nearest neighbors. A standard multi-scale method is used to derive the discrete nonlinear evolution equation. From the linear stability, the modulation instability gain is obtained, and the impact of the cubic-quartic nonlinearity on the modulation instability leads to unstable zones as well as modulated wave patterns for certain values of the higher nonlinear term. A numerical simulation of the derived discrete nonlinear evolution equation gives birth to rogue waves and diverse types of modulated waves. We derive static breather solutions that synchronize and adjust to the drive at the center and limit of the first Brillouin zone. Thereafter, we submit one end of the discrete model to an external periodic drive. The generation of modulated waves and bright soliton is observed for driven amplitudes above the threshold of supratransmission. When the driven amplitude is increased sufficiently, the bright soliton towers into chaos-like motion in the transient range of propagation time. These results shed light on the fact that at higher orders of complexity, the modified Frenkel-Kontorova model with cubic-quartic nonlinear coupling coefficients could be used to generate rogue waves, long-lived modulated wave patterns, and chaos-like motions that are very useful for data codification.
## Appendix
\(\Lambda_{1}=A_{2}a_{1}b_{1}+2\,{A_{5}{F_{0}}^{2}}a_{1}b_{1}\left(a_{1}b_{1}-1 \right)\left(a_{-1}b_{-1}-1\right),\)
\(\Lambda_{2}=A_{2}a_{-1}b_{-1}+2\,{A_{5}a_{-1}b_{-1}{F_{0}}^{2}}\left(a_{1}b_{1 }-1\right)\left(a_{-1}b_{-1}-1\right)\left(A_{7}+1\right),\)
\(\Lambda_{3}=\left(-a_{-1}b_{-1}-a_{1}b_{1}\right){A_{2}}-2\left(a_{1}b_{1}-1 \right)\left(a_{-1}b_{-1}-1\right)A_{5}{A_{7}}{F_{0}}^{2}-\left(a_{1}b_{1}-1 \right)\left(a_{-1}b_{-1}-1\right)\left(a_{-1}b_{-1}+a_{1}b_{1}+2\right){A_{5 }{F_{0}}^{2}}-{A_{7}}{F_{0}}^{2}\left(a_{1}b_{1}-1\right)\left(a_{-1}b_{-1}-1 \right)^{2}+{A_{3}}{F_{0}}^{2},\)
\(\Lambda_{4}=A_{5}a_{-1}b_{-1}{F_{0}}^{2}\left(a_{1}b_{1}-1\right)^{2},\)
\(\Lambda_{5}=A_{5}{F_{0}}^{2}a_{1}b_{1}\left(a_{-1}b_{-1}-1\right)^{2}\left(A_{7 }+1\right),\)
\(\Lambda_{6}=A_{3}F_{0}^{\ 2}+A_{5}\left(A_{7}\left(-{F_{0}}^{2}a_{-1}{{}^{2}b_{-1}}^{2}+2 \,{F_{0}}^{2}a_{-1}b_{-1}-F_{0}{}^{2}\right)-{F_{0}}^{2}\left(a_{-1}{{}^{2}b_{-1 }}^{2}+a_{1}{{}^{2}b_{1}}^{2}-2\,a_{-1}b_{-1}-2\,a_{1}b_{1}+2\right)\right),\)
\(N_{1}=A_{2}a_{1}b_{1}+2\,A_{5}{F_{0}}^{\ 2}a_{1}b_{1}\left(a_{1}b_{1}-1\right) \left(a_{-1}b_{-1}-1\right),\)
\(N_{2}=A_{2}a_{-1}b_{-1}+2\,A_{5}a_{-1}b_{-1}{F_{0}}^{\ 2}\left(a_{1}b_{1}-1 \right)\left(a_{-1}b_{-1}-1\right)\left(A_{7}+1\right).\)
\(N_{3}=\left(-a_{-1}b_{-1}-a_{1}b_{1}\right)A_{2}-2\left(a_{1}b_{1}-1\right) \left(a_{-1}b_{-1}-1\right)A_{5}A_{7}{F_{0}}^{\ 2}-\left(a_{1}b_{1}-1\right) \left(a_{-1}b_{-1}-1\right)\left(a_{-1}b_{-1}+a_{1}b_{1}+2\right)A_{5}{F_{0}} ^{\ 2}-\left(a_{-1}b_{-1}-1\right)^{2}\left(a_{1}b_{1}-1\right)A_{7}{F_{0}}^{\ 2}+A_{3}{F_{0}}^{\ 2},\)
\(N_{4}=A_{5}{F_{0}}^{\ 2}a_{-1}b_{-1}\left(a_{1}b_{1}-1\right)^{2},\)
\(N_{5}=A_{5}{F_{0}}^{\ 2}a_{1}b_{1}\left(a_{-1}b_{-1}-1\right)^{2}(A_{7}+1),\)
\(N_{6}=A_{3}{F_{0}}^{\ 2}+A_{5}\left(A_{7}\left(-{F_{0}}^{2}a_{-1}{{}^{2}b_{-1}}^{ 2}+2\,{F_{0}}^{2}a_{-1}b_{-1}-{F_{0}}^{2}\right)-{F_{0}}^{\ 2}\left(a_{-1}{{}^{2}b_{-1}}^{2}+a_{1}{{}^{2}b_{1}}^{2}-2\,a_{-1}b_{-1}-2a_{1 }b_{1}+2\right)\right),\)
\(A_{1}=\frac{\omega^{2}-\omega_{s}^{2}}{2\omega}\), \(A_{2}=\frac{C_{2}}{2\omega}\), \(A_{3}=-\frac{1}{2}\frac{\omega^{2}\left(3\beta-4\,\sigma^{2}+\frac{3\omega_{s }^{2}-\omega_{s}^{2}}{2\omega-\omega_{s}^{2}}\right)}{\omega}\), \(A_{5}=\frac{3}{2}\frac{C_{4}}{\omega}\), \(A_{7}=-\frac{3}{2}\frac{C_{3}}{\omega}\),
\(a_{1}=\cos(k)+i\sin(k);\ a_{-1}=\cos(k)+i\sin(k),\)
\(b_{1}=\cos(Q)+i\sin(Q);\ b_{-1}=\cos(Q)+i\sin(Q).\)
| この論文では、一次元の原子 chain に cubic-quartic non-linearity COEFFICIENTS を用いてモデレーション不安定性と非線形 supratransmission を調査します。その結果、マルチスケール Schemを用いてディスタンス非線形革命方程式を確立しました。モデレーション不安定性のゲインを計算するために、線形化スキームを使用します。特に、高非線形項の影響を調べる必要があります。その後、非線形 supratransmission Phenomenon を用いて、ディスタンスモデルの一端を禁断帯に駆動させました。これにより、 supratransmission Threshold を超える driving 係数では、ソリトニック Bright Soliton とモデレートされた波のパターンが達成されます。伝播時間の変遷領域では、Bright Soliton がchaotic Soliton に変わる重要な現象が見られます。これらの結果が、モデレーション不安定 |
2307.13571 | PT$\mathrm{L}^{p}$: Partial Transport $\mathrm{L}^{p}$ Distances | Optimal transport and its related problems, including optimal partial
transport, have proven to be valuable tools in machine learning for computing
meaningful distances between probability or positive measures. This success has
led to a growing interest in defining transport-based distances that allow for
comparing signed measures and, more generally, multi-channeled signals.
Transport $\mathrm{L}^{p}$ distances are notable extensions of the optimal
transport framework to signed and possibly multi-channeled signals. In this
paper, we introduce partial transport $\mathrm{L}^{p}$ distances as a new
family of metrics for comparing generic signals, benefiting from the robustness
of partial transport distances. We provide theoretical background such as the
existence of optimal plans and the behavior of the distance in various limits.
Furthermore, we introduce the sliced variation of these distances, which allows
for rapid comparison of generic signals. Finally, we demonstrate the
application of the proposed distances in signal class separability and nearest
neighbor classification. | Xinran Liu, Yikun Bai, Huy Tran, Zhanqi Zhu, Matthew Thorpe, Soheil Kolouri | 2023-07-25T15:23:15 | http://arxiv.org/abs/2307.13571v1 | # PTl\({}^{p}\): Partial Transport \(\mathrm{L}^{p}\) Distances
###### Abstract
Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a growing interest in defining transport-based distances that allow for comparing signed measures and, more generally, multi-channeled signals. Transport \(\mathrm{L}^{p}\) distances are notable extensions of the optimal transport framework to signed and possibly multi-channeled signals. In this paper, we introduce partial transport \(\mathrm{L}^{p}\) distances as a new family of metrics for comparing generic signals, benefiting from the robustness of partial transport distances. We provide theoretical background such as the existence of optimal plans and the behavior of the distance in various limits. Furthermore, we introduce the sliced variation of these distances, which allows for rapid comparison of generic signals. Finally, we demonstrate the application of the proposed distances in signal class separability and nearest neighbor classification.
## 1 Introduction
At the heart of Machine Learning (ML) lies the ability to measure similarities or differences between signals existing in different domains, such as temporal, spatial, spatiotemporal grids, or even graphs in a broader sense. The effectiveness of any ML model depends significantly on the discriminatory power of the metrics it employs. Several criteria are desired when quantifying dissimilarities among diverse multivariate signals, including: 1) the ability to compare signals with varying lengths, 2) adherence to the inherent structure and geometry of the signals' domain, 3) being invariant to local deformation and symmetries, 4) computational efficiency, and 5) differentiability. In recent literature, significant efforts have been dedicated to addressing these challenges. Prominent examples include
the Dynamic Time Warping (DTW) [1] technique and its numerous extensions [2, 3, 4, 5, 6], as well as more recent methods based on optimal transport principles [7, 8, 9, 10].
**Dynamic Time Warping (DTW).** DTW is a technique for comparing and aligning time series signals that may vary in lengths or exhibit temporal distortions. To compare two signals, DTW computes the minimal-cost alignment between the signals [1], enforcing the chronological order. The alignment problem in DTW is solved via dynamic programming (DP) using Bellman's recursion, with quadratic cost in lengths of the signals. A large body of work has studied extensions of the DTW approach. For instance, Ten Holt et al. [3] extend DTW to multivariate time series. Salvador and Chan [4] propose FastDTW, a linear time approximation of DTW with reasonable accuracy. To achieve robustness, Keogh and Pazzani [2] propose derivative DTW (DDTW), calculating the minimum-cost alignment based on derivatives of input signals, while Jeong et al. [5] consider the relative importance of alignments and propose weighted DTW (WDTW) providing robustness against outliers. Other notable extensions include Canonical Time Warping [11] and generalized time warping [12], which enable the application of DTW to multi-modal sequences whose instances may have different dimensions. More recently, Cuturi & Blondel [6] provide a differentiable variant of DTW, softDTW, allowing its seamless integration into end-to-end learning pipelines.
**Optimal Transport.** Optimal transport (OT) has gained recognition as a powerful tool for quantifying dissimilarities between probability measures, finding broad applications in data science, statistics, machine learning, signal processing, and computer vision [13, 14]. The dissimilarity metrics derived from OT theory define a robust geometric framework for comparing probability measures, exhibiting desirable properties such as a weak Riemannian structure [15], the concept of barycenters [16], and parameterized geodesics [17]. However, it is important to note that OT has limitations when it comes to comparing general multi-channel signals. OT is specifically applicable to non-negative measures with equal total mass, restricting its use to signals that meet specific criteria: 1) single-channel representation, 2) non-negativity, and 3) integration to a common constant, such as unity for probability measures. In cases where signals do not fulfill these criteria, normalization or alternative methods are required for meaningful comparison using OT.
**Unbalanced and Optimal Partial Transport.** Comparing non-negative measures with varying total amounts of mass is a common requirement in physical-world applications. In such scenarios, it is necessary to find partial correspondences or overlaps between two non-negative measures and compare them based on their respective corresponding and non-corresponding parts. Recent research has thus focused on extensions of the OT problem that enable the comparison of non-negative measures with unequal mass. The Hellinger-Kantorovich distance [18, 19], optimal partial transport (OPT) problem [20, 21, 22], Kantorovich-Rubinstein norm [23, 24] and unnormalized optimal transport [25, 26] are some of the variants that fall under the category of "unbalanced optimal transport" [18, 19]. These methods provide effective solutions for comparing non-negative measures in scenarios where the total amount of mass varies. It is important to note that although the unbalanced optimal transport methods have advanced the capabilities of comparing non-negative measures with unequal mass, they still cannot be used to compare multi-channel signals or signals with negative values.
**Transport-Based Comparison of Generic Signals.** Recent studies have proposed extensions of the Optimal Transport (OT) framework to compare multi-channel signals that may include negative values, while still harnessing the benefits of OT. For example, Su & Hua [8] introduced the Order-preserving Wasserstein distance, which computes the OT problem between elements of sequences while ensuring temporal consistency through regularization of the transportation plan. A more rigorous treatment of the problem was proposed in [7] that led to the so-called Transportation \(\mathrm{L}^{p}\) (\(\mathrm{TL}^{p}\)) distances. In short, to compare two signals \(f\) and \(g\), \(\mathrm{TL}^{p}\) uses the OT distance between their corresponding measures, e.g., the Lebesgue measure, raised onto the graphs of the signals (See Section 3). Later, Zhang et al. [10] utilized a similar approach to \(\mathrm{TL}^{p}\) while adding entropy regularization [27] and introduced Time Adaptive OT (TAOT). Lastly, in Spatio-Temporal Alignments, Janati et al. [9] combine OT with softDTW. They utilized regularized OT to capture spatial differences between time samples and employed softDTW for temporal alignment costs.
**Contributions.** In this paper, we tackle the problem of comparing multi-channel signals using transport-based methods and present a new family of metrics, denoted as \(\mathrm{P}\mathrm{TL}^{p}\), based on the optimal partial transport framework. Our approach is motivated by the realization that while \(\mathrm{TL}^{p}\) distances allow for the comparison of general signals, they require complete correspondences between input
signals, which limits their applicability to real-world signals that often exhibit partial correspondences. Our specific contributions are: 1) introducing a new family of metrics based on optimal partial transport for comparing multi-channel signals, 2) providing theoretical results on existence of the partial transport plan in the proposed metric, as well as the behavior of the distance in various limits, 3) providing the sliced variation of the proposed metric with significant computational benefits, and 4) demonstrating the robust performance of the proposed metric on nearest neighbor classification in comparison with various recent baselines.
**General Notations.** We provide an extensive list of our notations in the supplementary material. Here we provide a small subset used in the development of our proposed framework. We use \(\mathbb{R}_{+}\) for the set of postive real numbers, \(\mathbb{R}^{d}\) to denote the d-dimensional Euclidean space, and \(\mathbb{S}^{d-1}\subset\mathbb{R}^{d}\) to denote the unit hyper-sphere. Given \(\Omega\subseteq\mathbb{R}^{d},p\geq 1\), we use \(\mathcal{P}(\Omega)\) to denote the set of Borel probability measures and \(\mathcal{P}_{p}(\Omega)\) to denote the set of probability measures with finite \(p\)'th moment defined on a metric space \((\Omega,d)\). We use \(\mathcal{M}_{+}(\Omega)\) to denote the set of all positive Radon measures defined on \(\Omega\). For \(\mu\in\mathcal{P}_{p}(\Omega)\), we define \(\mathrm{L}^{p}(\mu;\mathbb{R}^{k}):=\{f:\Omega\to\mathbb{R}^{k}\mid\int_{ \Omega}\|f(x)\|^{p}\,\mathrm{d}\mu(x)<\infty\}\) to denote a Banach space with the usual norm. For \(f:\Omega\to\hat{\Omega}\) and measure \(\mu\) in \(\mathcal{M}_{+}(\Omega)\) we use \(f_{\#}\mu\) to denote the pushforward of measure \(\mu\) through \(f\), which is formally defined as \(f_{\#}\mu(A)=\mu(f^{-1}(A))\) for \(\forall A\subseteq\hat{\Omega}\).
## 2 Background - Optimal (Partial) Transport and Their Sliced Variations
**Optimal Transport**. The OT problem in the Kantorovich formulation [28] is defined for two probability measures \(\mu\) and \(\nu\) in \(\mathcal{P}(\Omega)\), and a lower semi-continuous cost function \(c:\Omega^{2}\to\mathbb{R}+\) by:
\[\mathrm{OT}_{c}(\mu,\nu):=\inf_{\gamma\in\Pi(\mu,\nu)}\int_{\Omega^{2}}c(x,y) \,\mathrm{d}\gamma(x,y), \tag{1}\]
Here, \(\Pi(\mu,\nu)\) is the set of all joint probability measures whose marginals are \(\mu\) and \(\nu\). We represent this by \(\pi_{1\#}\gamma=\mu\) and \(\pi_{2\#}\gamma=\nu\), where \(\pi_{1}\) and \(\pi_{2}\) are the canonical projection maps. If \(c(x,y)\) is a \(p\)-th power of a metric, then the \(p\)-th root of the resulting optimal value is known as the p-Wasserstein distance. This distance is a metric in \(\mathcal{P}_{p}(\Omega)\). We will ignore the subscript \(c\) if it is the default cost \(\|\cdot\|^{p}\). Please see the appendix for more details.
**Optimal Partial Transport.** The problem of Optimal Partial Transport (OPT) extends the concept of mass transportation to include mass destruction at the source and mass creation at the target, with corresponding penalties for such actions. More precisely, let \(\mu,\nu\in\mathcal{M}_{+}(\Omega)\), where \(\mathcal{M}_{+}(\Omega)\) is set of positive Radon measures defined on \(\Omega\). Let \(\lambda\geq 0\) denote the penalty for mass creation or destruction. Then the OPT problem is defined as:
\[\mathrm{OPT}_{\lambda,c}(\mu,\nu):=\inf_{\gamma\in\Pi_{\leq}(\mu,\nu)}\int_{ \Omega^{2}}c(x,y)\,\mathrm{d}\gamma(x,y)+\lambda(\|\mu\|_{\mathrm{TV}}+\|\nu \|_{\mathrm{TV}}-2\|\gamma\|_{\mathrm{TV}}) \tag{2}\]
where
\[\Pi_{\leq}(\mu,\nu):=\{\gamma\in\mathcal{M}_{+}(\Omega^{2}):\pi_{1\#}\gamma \leq\mu,\pi_{2\#}\gamma\leq\nu\},\]
\(\pi_{1\#}\gamma\leq\mu\) indicates that \(\pi_{1\#}\gamma\) is _dominated by_\(\mu\), i.e., for any Borel set \(A\subseteq\Omega\), \(\pi_{1\#}\gamma(A)\leq\mu(A)\), analogously for \(\pi_{2\#}\gamma\leq\nu\). The cost function \(c:\Omega^{2}\to\mathbb{R}\) is lower semi-continuous (generally, it is nonnegative), and \(\|\mu\|_{\mathrm{TV}}\) is the total variation (and the total mass) of \(\mu\), analogously for \(\|\nu\|_{\mathrm{TV}},\|\gamma\|_{\mathrm{TV}}\). When the transportation cost \(c(x,y)\) is a metric, \(\mathrm{OPT}_{\lambda,c}(\cdot,\cdot)\) defines a metric on \(\mathcal{M}_{+}(\Omega)\) (see [29, Proposition 2.10], [30, Proposition 5], [26, Section 2.1] and [31, Theorem 4]). For simplicity of notation, we drop the \(c\) in the subscript of \(\mathrm{OT}\) and \(\mathrm{OPT}\).
**Sliced Transport.** For one-dimensional measures, i.e., when \(\Omega\subseteq\mathbb{R}\), both OT and OPT problems have efficient solvers. In particular, the OT problem has a closed-form solution, and for discrete measures with \(M\) and \(N\geq M\) particles, it can be solved in \(\mathcal{O}(N\log(N))\). Moreover, a quadratic algorithm, \(\mathcal{O}(MN)\), was recently proposed in [32] for the one-dimensional OPT problem. To extend the computational benefits of one-dimensional OT and OPT problems to d-dimensional measures, recent works utilize the idea of slicing, which is rooted in the Cramer-Wold theorem [33] and the Radon Transform from the integral geometry [34, 35]. For \(\theta\in\mathbb{S}^{d-1}\), a one-dimensional slice of measure \(\mu\in\mathcal{M}_{+}(\Omega)\) can be obtained via \(\langle\theta,\cdot\rangle_{\#}\mu\) where \(\langle\cdot,\cdot\rangle:\Omega^{2}\to\mathbb{R}\) denotes the inner product.
Then for \(\mu,\nu\in\mathcal{P}_{p}(\Omega)\) we can define the Sliced-OT (SOT) as:
\[\mathrm{SOT}(\mu,\nu):=\int_{\mathbb{S}^{d-1}}\mathrm{OT}(\langle\theta,\cdot \rangle_{\#}\mu,\langle\theta,\cdot\rangle_{\#}\nu)\,\mathrm{d}\sigma(\theta), \tag{3}\]
where \(\sigma\in\mathcal{P}(\mathbb{S}^{d-1})\) is a probability measure such that \(\mathrm{supp}(\sigma)=\mathbb{S}^{d-1}\), e.g., the uniform distribution on the unit hyper-sphere. Similarly, for \(\mu,\nu\in\mathcal{M}_{+}(\Omega)\), Sliced-OPT (SOPT) [32] can be defined as:
\[\mathrm{SOPT}_{\lambda}(\mu,\nu):=\int_{\mathbb{S}^{d-1}}\mathrm{OPT}_{ \lambda(\theta)}(\langle\theta,\cdot\rangle_{\#}\mu,\langle\theta,\cdot\rangle_ {\#}\nu)\,\mathrm{d}\sigma(\theta), \tag{4}\]
where \(\lambda\in\mathrm{L}^{1}(\sigma;\mathbb{R}_{+})\) is generally a projection dependent function.
## 3 Partial Transport for Multi-Channel Signals
In the previous section, we discussed the suitability of OT and OPT problems (and similarly, SOT and SOPT problems) for comparing measures \(\mu\) and \(\nu\) in \(\mathcal{P}_{p}(\Omega)\) or \(\mathcal{M}_{+}(\Omega)\), respectively. In this section, we begin by defining a transport-based distance for multi-channel signals defined on a general class of measures, following the work of Thorpe et al. [7] on Transport \(\mathrm{L}^{p}\) distances. We then motivate the need for partial transportation when comparing such multi-channel signals and introduce our Partial-Transport \(\mathrm{L}^{p}\), \(\mathrm{PTL}^{p}\), distance.
**Transport \(\mathrm{L}^{p}\) Distances.** Following [7], a multi-channel signal with \(k\) channels can be defined as the pair \((f,\mu)\) for \(\mu\in\mathcal{P}_{p}(\Omega)\) and \(f\in L^{p}(\mu;\mathbb{R}^{k}):=\{f:\Omega\to A\subseteq\mathbb{R}^{k}\}\). We denote the set of all such signals as
\[\mathcal{Q}_{p}(\Omega;\mathbb{R}^{k}):=\{(f,\mu)|\mu\in\mathcal{P}_{p}( \Omega),f\in\mathrm{L}^{p}(\mu;\mathbb{R}^{k})\}.\]
We name it as the transport \(\mathrm{L}^{p}\) space. The \(\mathrm{TL}^{p}_{\beta}\) distance between two such k-dimensional signals \((f,\mu)\) and \((g,\nu)\) in \(\mathcal{Q}_{p}(\Omega;\mathbb{R}^{k})\) is defined as:
\[\mathrm{TL}^{p}_{\beta}((f,\mu),(g,\nu))=\inf_{\gamma\in\Pi(\mu,\nu)}\int_{ \Omega^{2}}\left(\frac{1}{\beta}\|x-y\|^{p}+\|f(x)-g(y)\|^{p}\right)\mathrm{d} \gamma(x,y). \tag{5}\]
For any \(p\in[1,\infty)\) and \(\beta>0\), the \(\mathrm{TL}^{p}_{\beta}\) distance defines a proper metric on \(\mathcal{Q}_{p}(\Omega;\mathbb{R}^{k})\), and \((\mathcal{Q}_{p}(\Omega;\mathbb{R}^{k}),\mathrm{TL}^{p}_{\beta})\) is a metric space. Intuitively, the \(\mathrm{TL}^{p}_{\beta}\) measures the OT between measures \(\mu\) and \(\nu\) raised onto the graphs of \(f\) and \(g\). Hence, \(\mathrm{TL}^{p}_{\beta}\) solves an OT problem in the \((d+k)\)-dimensional
Figure 1: Illustrating the fundamental idea of \(\mathrm{TL}^{p}\) distances. On the left, signals \(f\) and \(g\) are depicted along with their associated measures \(\mu\) and \(\nu\). In the middle, the measures \(\mu\) and \(\nu\) are lifted to the graphs of \(f\) and \(g\), respectively. On the right, the optimal transport plan is visualized, accompanied by the corresponding transportation cost.
space. Figure 1 shows the core concept behind \(\mathrm{TL}^{p}\) distances. Notably, the \(\mathrm{TL}^{p}_{\beta}\) distance satisfies the following properties:
\[\lim_{\beta\to 0}\mathrm{TL}^{p}_{\beta}((f,\mu),(g,\nu)) =\begin{cases}\|f-g\|^{p}_{\mathrm{L}^{p}(\mu)}&\text{if }\mu=\nu\\ \infty&\text{elsewhere}\end{cases} \tag{6}\] \[\lim_{\beta\to+\infty}\mathrm{TL}^{p}_{\beta}((f,\mu),(g,\nu)) =\mathrm{OT}(f_{\#}\mu,g_{\#}\nu) \tag{7}\]
Hence, the \(\mathrm{TL}^{p}_{\beta}\) distance interpolates between the \(\mathrm{L}^{p}\) distance between \(f,g\) and the p-Wasserstein distance between \(f_{\#}\mu\) and \(g_{\#}\nu\).
**Partial Transport \(\mathrm{L}^{p}\) Distances.** In many real-world scenarios, it is natural for two signals to only partially match each other. Figure 2 illustrates this phenomenon. However, because \(\mathrm{TL}^{p}\) distances are rooted in OT, they may sacrifice true correspondences in order to achieve a complete match between the two signals (as seen in Figure 2). To address this issue, we propose extending the definition of \(\mathrm{TL}^{p}\) distances to partial transport, allowing for partial matching for signal comparison.
To do so, we first expand the definition of \(k\)-dimensional signals to be defined on positive measures rather than probability measures. Specifically, we define a signal as the pair \((f,\mu)\) where \(\mu\in\mathcal{M}_{+}(\Omega)\) and \(f\in\mathrm{L}^{p}(f;\mathbb{R}^{k})\). We denote the set of all such signals as \(\mathcal{Q}^{+}_{p}(\Omega;\mathbb{R}^{k})\), that is,
\[\mathcal{Q}^{+}_{p}(\Omega;\mathbb{R}^{k}):=\{(f,\mu):\mu\in\mathcal{M}_{+}( \Omega),f\in\mathrm{L}^{p}(\mu;\mathbb{R}^{k})\}.\]
We now propose our Partial Transport \(\mathrm{L}^{p}\) (\(\mathrm{PTL}^{p}\)) distance between two signals \((f,\mu)\) and \((g,\nu)\) in \(\mathcal{Q}^{+}_{p}(\Omega;\mathbb{R}^{k})\) as:
\[\mathrm{PTL}^{p}_{\beta,\lambda}((f,\mu),(g,\nu)) =\inf_{\gamma\in\Pi_{\leq}(\mu,\nu)}\int_{\Omega^{2}}\left(\frac{ 1}{\beta}\|x-y\|^{p}+\|f(x)-g(y)\|^{p}\right)\mathrm{d}\gamma(x,y)\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\
**Theorem 3.1**.: _For any \(p\geq 1\), and \(\lambda,\beta>0\) there exists a minimizer for the \(\mathrm{PTL}^{p}\) problem (8). Furthermore, for the empirical \(\mathrm{PTL}^{p}\) problem (9), there exists a minimizer \(\gamma\in\Pi_{\leq}(1_{M},1_{N})\) that is induced by a 1-1 mapping. That is, the optimal \(\gamma\) satisfies \(\gamma_{ij}\in\{0,1\}\) for each \(\hat{i},j\), and each row and column of \(\gamma\) contains at most one nonzero element._
**Theorem 3.2**.: \((\mathcal{Q}_{+}(\Omega;\mathbb{R}^{k}),\mathrm{PTL}^{p}_{\beta,\lambda})\) _defines a metric space._
We refer to Section C in the appendix for the proofs of the above theorems and a detailed discussion of the \(\mathrm{PTL}^{p}\) space \(\mathcal{Q}_{+}(\Omega;\mathbb{R}^{k})\).
Similar to the \(\mathrm{TL}^{p}\) distance, we can also extend the definition for \(\beta=0\) and \(\beta=\infty\) by the following theorem:
**Theorem 3.3**.: _If \(\lambda>0\), we have_
\[\lim_{\beta\to 0}\mathrm{PTL}^{p}_{\beta,\lambda}((f,\mu),(g, \nu)) =\|f-g\|^{p}_{\mathrm{L}^{p}(\mu\wedge\nu),2\lambda}+\lambda(\|\mu-\nu\|_{ \mathrm{TV}}) \tag{10}\] \[\lim_{\beta\to\infty}\mathrm{PTL}^{p}_{\beta,\lambda}((f,\mu),(g,\nu)) =\mathrm{OPT}_{\lambda}(f_{\#}\mu,g_{\#}\nu), \tag{11}\]
_where \(\mu\wedge\nu\) is the minimum of measure \(\mu,\nu\),_
\[\|f-g\|^{p}_{\mathrm{L}^{p}(\mu\wedge\nu),2\lambda}:=\int_{\Omega}\|f-g\|^{p} \wedge 2\lambda\,\mathrm{d}(\mu\wedge\nu).\]
_and \(\|\mu-\nu\|_{\mathrm{TV}}\) is the total variation of the signed measure \(\mu-\nu\)._
See Section A in the appendix for the details of notations and Section D for the proof. Note, if we take \(\lambda\to\infty\), we can recover (6), (7) by the above limits. We note that \(\lambda\to 0\) is not an interesting case as it indicates zero cost for creation and destruction of mass, leading to an optimal \(\gamma\) of all zeros, i.e., \(\mathrm{PTL}^{p}_{\beta,0}((\mu,f),(\nu,g))=0\) for all \((\mu,f),(\nu,g)\in\mathcal{Q}^{p}_{+}(\Omega;\mathbb{R}^{k})\).
**Sliced Extensions of TLP and PTLP.** Using the connection between the \(\mathrm{TL}^{p}\) distance and OT distance [7], Eq. (5) can be rewritten as
\[\mathrm{TL}^{p}_{\beta}((f,\mu),(g,\nu))=\mathrm{OT}(\hat{\mu},\hat{\nu}) \tag{12}\]
where \(\hat{\mu}=(T_{\beta,f,p})_{\#}\mu\) is a push-forward measure of \(\mu\) by \(T_{\beta,f,p}(x)=\left[\begin{array}{c}x\beta^{-\frac{1}{p}}\\ f(x)\end{array}\right]\), and similarly \(\hat{\nu}=(T_{\beta,g,p})_{\#}\nu\). Eq. (12) allows us to apply SOT method to the \(\mathrm{TL}^{p}\) distance, and have the sliced-TLP distance as follows:
\[\mathrm{STL}^{p}_{\beta}((f,\mu),(g,\nu))=\int_{\mathbb{S}^{d+k-1}}\mathrm{OT} (\theta_{\#}\hat{\mu},\theta_{\#}\hat{\nu})d\sigma(\theta) \tag{13}\]
where \(\sigma(\theta)\) is a probability measure with non-zero density on \(\mathbb{S}^{d+k-1}\), for instance the uniform measure on the unit sphere. Similarly, by leveraging SOPT and the relation between \(\mathrm{PTL}^{p}\) and OPT (see proposition C.3), we can define Sliced \(\mathrm{PTL}^{p}\) as
\[\mathrm{SPTL}^{p}_{\beta,\lambda}((f,\mu),(g,\nu))=\int_{\mathbb{S}^{d+k-1}} \mathrm{OPT}_{\lambda(\theta)}(\theta_{\#}\hat{\mu},\theta_{\#}\hat{\nu})d \sigma(\theta) \tag{14}\]
where \(\lambda\) can be defined as an \(L^{1}(\sigma,\mathbb{R}_{++})\) function of \(\theta\). Note that \(\mathrm{STL}^{p}_{\beta}\) and \(\mathrm{SPTL}^{p}_{\beta,\lambda}\) are metrics on \(\mathcal{Q}(\Omega;\mathbb{R}^{k})\) and \(\mathcal{Q}_{+}(\Omega;\mathbb{R}^{k})\), respectively.
Equipped with the newly proposed distances, we now demonstrate their performance in separability and nearest neighbor classification.
## 4 Experiments
### Separability
A valid distance should be able to separate a mixture of different classes of signals. We aim to illustrate the separability of the \(\mathrm{PTL}^{p}\) distance on different classes of signals in this experiment.
**Synthetic Data**
We generate the following two classes of signals on the domain \([0,1]\):
\[\mathbf{S}_{0} =\{f(t)\mid f(t)=\varphi(t|x,\sigma_{0});\] \[\quad\quad\quad x=0.98z+0.01,z\sim\mathrm{Unif}[0,1]\}\] \[\mathbf{S}_{1} =\{g(t)\mid g(t)=\varphi(t|x+0.001,\sigma_{1})-\varphi(t|x-0.001, \sigma_{1});\] \[\quad\quad\quad x=0.98z+0.01,z\sim\mathrm{Unif}[0,1]\}\]
where \(\varphi\) denotes a Gaussian probability density function scaled within \([0,1]\), \(\sigma_{0}=0.01\) and \(\sigma_{1}=\frac{0.01}{\sqrt{2}}\); time \(t\sim\mathrm{Unif}[0,1]\). In short, \(\mathbf{S}_{0}\) is the class of signals with one positive Gaussian bump, whereas \(\mathbf{S}_{1}\) denotes the class of signals with both a positive and a negative Gaussian bumps. To further test the robustness, we add random blip noise \(\epsilon(t)\) to each signal as the second separability experiment:
\[\epsilon(t)=\alpha\varphi(t|x,\sigma_{e}=0.001\sqrt{5})+0.1\epsilon_{0}\]
where \(\alpha\) is randomly chosen from \(\{-0.5,0.5\}\); \(x=0.98z+0.01,z\sim\mathrm{Unif}[0,1]\); \(\epsilon_{0}\) is the Gaussian noise. \(\epsilon(t)\) can be considered as a tiny positive/negative bump with Gaussian oscillation.
**Results**
Figure 3 shows the 2D Multi-Dimensional Scaling (MDS) embeddings calculated from the precomputed pairwise \(\mathrm{L}^{p}\), \(\mathrm{TL}^{p}\) and \(\mathrm{PTL}^{p}\) distance matrices. We observe that \(\mathrm{PTL}^{p}\) not only achieves high performance in separating the two classes, but also exhibits robustness to noise. When adding blips, \(\mathrm{TL}^{p}\) tends to mistake the noise for the main trend and cluster signals based on the noise.
### 1 Nearest Neighbor Classification
**Experiment setup**
To demonstrate the effectiveness of our proposed \(\mathrm{PTL}^{p}\) metric and its sliced variant \(\mathrm{SPTL}^{p}\), we test these methods on the task of 1 Nearest Neighbor (1NN) classification, along with other baselines.
Figure 3: Visualizing manifold learning results for two classes of signals. For original signals (top row), both \(\mathrm{TL}^{p}\) and \(\mathrm{PTL}^{p}\) separates two classes well, but \(\mathrm{L}^{p}\) fails. However, for the noisy signals (bottom row), only \(\mathrm{PTL}^{p}\) shows a clear decision boundary.
Given a test signal, we seek the nearest training signal with respect to each metric/divergence, and predict the test label as that of the found nearest neighbor.
**Dataset**
We use three modified UCR datasets of varying lengths from [36]: Suffix, Prefix and Subsequence. The Suffix dataset is generated by simulating scenarios when sensors are activated at different times, thus may miss some observations from the start and record only suffix time series. Similarly, the Prefix dataset generator imitates the sensor behavior of stopping non-deterministically and produces only prefix time series. The Subsequence dataset contains time series that have variations on both starting and stopping time, i.e. the sensor may only capture subsequences.
**Baselines**
The \(\mathrm{L}^{p}\) distance between signals is known for its simplicity and efficiency, which fits signals in a fixed temporal grid. OT-based similarity metrics, p-Wasserstein distance (\(\mathrm{OT}\)), and \(\mathrm{TL}^{p}\) treat signals / the graph of signals as probability measures and solve the optimization problem of transporting one probability measure to the other in the most cost-efficient way. Moreover, \(\mathrm{STL}^{p}\) is included in the baselines as a fast approximation of \(\mathrm{TL}^{p}\).
Unlike the \(\mathrm{L}^{p}\) metric, Dynamic Time Warping (DTW) [1] applies an elastic (non-linear) warping to temporal sequences and finds an optimal matching between the warped time series. DTW is more robust to time distortions by its pathological alignment. An \((N,M)\)-warping path is a sequence \(p=(p_{1},p_{2},\cdots,p_{L})\) with \(p_{l}=(n_{l},m_{l})\in[1:N]\times[1:M]\), which defines an alignment between two sequences of length \(N\) and \(M\) that satisfies monotonicity, continuity and boundary conditions [37]. Given a pair of temporal sequences \(f=\{f_{i}\}_{i=0}^{N}\) and \(g=\{g_{j}\}_{j=0}^{M}\) on the domain \(\Omega\), DTW is calculated as
\[\mathrm{DTW}(f,g)=\min_{p}\{c_{p}(f,g)\mid p\text{ is an }(N,M)\text{-warping path}\}, \tag{15}\]
where \(c_{p}(f,g)=\sum_{(i,j)\in p}c(f_{i},g_{j})\) and \(c(f_{i},g_{j})\) is the cost of moving from \(f_{i}\) to \(g_{j}\). We also include variants of DTW, namely WDTW, DDTW, and Soft-DTW (SDTW) as baselines. For SDTW, we consider two cases for the smoothing parameter \(\gamma=0.01\) and \(\gamma=1\).
**Grid search for optimal \(\beta\) and \(\lambda\)**
To find the optimal \(\beta\) and \(\lambda\) for \(\mathrm{PTL}^{p}_{\beta,\lambda}\), we perform grid search based on the 5-fold cross validation. We use the scikit-learn built-in GridSearchCV tools for implementation. The search range for \(\beta\) is set to be \(\{10^{-3},10^{-2},10^{-1},1,10,100,10^{3},10^{4}\}\), and \(\lambda\) is chosen from a set of 10 evenly spaced values from \(0.1\) to the radius of the raised distribution on the graph of each signal.
In \(\mathrm{SPTL}^{p}_{\beta,\lambda}\), we also need to specify the slices, i.e. \(\theta\)'s for 1 dimensional projections. We obtain the optimal \(\beta\) from \(\mathrm{PTL}^{p}_{\beta,\lambda}\). As the amount of mass that should be transported may vary across slices, we adopt the strategy to search for the best \(\lambda\) for the most informative slice, and then set \(\lambda\)'s accordingly for other slices. We set \(\theta_{0}\) to be the first principle component of all signals. Note that \(\theta_{0}\) vanishes at dimensions corresponding to \(x\beta^{-\frac{1}{p}}\), but concentrates on \(f(x)\) in \(T_{\beta,f,p}(x)=\left[x\beta^{-\frac{1}{p}};f(x)\right]\) (refer to Eq. (12) and Eq. (13)). Similarly, we implement grid search for best \(\lambda_{\theta_{0}}\) corresponding to \(\theta_{0}\). Given \(\theta_{0}\) and \(\lambda_{\theta_{0}}\), for a specific slice \(\theta\), \(\lambda_{\theta}=\langle\theta,\theta_{0}\rangle\lambda_{\theta_{0}}\), where \(\langle\cdot,\cdot\rangle\) denotes inner product.
**Results**
Table 4.2 presents the results of nearest neighbor classification using different metrics/divergences on three subsets of the modified UCR dataset: Prefix, Subsequence, and Suffix. The table indicates that no single metric/divergence exhibits a significant advantage over others on a single dataset. However, \(\mathrm{SPTL}^{p}\) achieves the best performance on two out of three datasets and performs nearly as well as the top performers on the remaining dataset, resulting in an overall win. It is worth noting that although the improvement margins are small, the computational advantage of \(\mathrm{SPTL}^{p}\) and \(\mathrm{STL}^{p}\) compared to other competitors (see Figure 2), make them more favorable choices in terms of efficiency.
### Computation efficiency using Sliced \(\mathrm{PTL}^{p}\)
We summarize the time complexities of all methods considered in Table 2.
In implementation, DTW-based methods are solved by a dynamic programming algorithm. For DTW, soft-DTW, we use the solvers from tislearn, which are accelerated by numba. \(\mathrm{TL}^{p}\) and \(\mathrm{PTL}^{p}\) are solved by linear programming solvers in PythonOT, whose time complexity is cubic with respect to the length of signals in the worst case, and quadratic in practice when the measures are
empirical. \(\mathrm{STL}^{p}\), \(\mathrm{SPTL}^{p}\) can be accelerated by numba. For \(\mathrm{STL}^{p}\) and \(\mathrm{SPTL}^{p}\), we set the number of projections to be 50. Note, the computation of \(\mathrm{STL}^{p}\) and \(\mathrm{SPTL}^{p}\) can be further accelerated by parallel computation with respect to slices.
## 5 Conclusion
In this paper, we propose partial transport \(\mathrm{L}^{p}\) (\(\mathrm{PTL}^{p}\)) distance as a similarity measure for generic signals. We have shown that \(\mathrm{PTL}^{p}\) defines a metric that comes with an optimal transport plan. We further characterize the behaviors of \(\mathrm{PTL}^{p}_{\beta,\lambda}\) as \(\beta\) goes to various limits. We extend \(\mathrm{PTL}^{p}\) to sliced partial transport \(\mathrm{L}^{p}\) (\(\mathrm{SPTL}^{p}\)), which is more computationally efficient. In the experimental section, we have demonstrated that the proposed metric is superior to other baselines in separability, and shown promising results on 1 nearest neighbor classification.
\begin{table}
\begin{tabular}{l|l} \hline Method & Worst-case Complexity \\ \hline \(\mathrm{PTL}^{p}\) & \(\mathcal{O}(N^{3}(d+k))\) \\ \(\mathrm{SPTL}^{p}\) & \(\mathcal{O}(LN((d+k)+N+\log(N)))\) \\ \(\mathrm{TL}^{p}\) & \(\mathcal{O}(N^{3}(d+k))\) \\ \(\mathrm{STL}^{p}\) & \(\mathcal{O}(LN((d+k)+\log(N)))\) \\ OT & \(\mathcal{O}(N^{3}k)\) \\ *DTW & \(\mathcal{O}(N^{2}k)\) \\ \(\mathrm{L}^{p}\) & \(\mathcal{O}(Nk)\) \\ \hline \end{tabular}
\end{table}
Table 2: Worst case time complexities for our proposed methods and baselines. Here \(N\) denotes the length of the signals, \(d\) and \(k\) are the signal dimension and number of channels respectively. \(L\) is the number of slices for sliced methods. Note that DTW and its variants used in this paper share the same complexity, which is denoted by *DTW in the table. | 最適な輸送とそれに関連する問題、特に最適な部分輸送は、確率または正の測度間の有意義な距離を計算するための機械学習における有益なツールとなっています。この成功は、符号付き測度と比較可能で、より一般的に、多チャネル信号を扱うための輸送基盤の距離の定義に注目を集めていることを意味します。輸送 $\mathrm{L}^{p}$ 距離は、最適輸送の枠組みを符号付きおよび多チャネル信号に拡張するものです。この論文では、部分輸送 $\mathrm{L}^{p}$ 距離を、一般的な信号を比較するための新しい metri ックとして導入します。これは、部分輸送距離のRobustnessの利点を生み出しています。この距離の理論的背景には、最適なプランの存在と、さまざまな制限における距離の挙動が含まれています。さらに、これらの距離の切断変動を導入し、一般的な信号間の迅速な比較を可能 |
2304.14996 | Maximizing Reachability Probabilities in Rectangular Automata with
Random Clocks | This paper proposes an algorithm to maximize reachability probabilities for
rectangular automata with random clocks via a history-dependent prophetic
scheduler. This model class incorporates time-induced nondeterminism on
discrete behavior and nondeterminism in the dynamic behavior. After computing
reachable state sets via a forward flowpipe construction, we use backward
refinement to compute maximum reachability probabilities. The feasibility of
the presented approach is illustrated on a scalable model. | Joanna Delicaris, Stefan Schupp, Erika Ábrahám, Anne Remke | 2023-04-28T17:32:57 | http://arxiv.org/abs/2304.14996v3 | # Maximizing Reachability Probabilities in Rectangular Automata with Random Clocks
###### Abstract
This paper proposes an algorithm to maximize reachability probabilities for rectangular automata with random clocks via a history-dependent prophetic scheduler. This model class incorporates time-induced nondeterminism on discrete behavior and nondeterminism in the dynamic behavior. After computing reachable state sets via a forward flowpipe construction, we use backward refinement to compute maximum reachability probabilities. The feasibility of the presented approach is illustrated on a scalable model.
## 1 Introduction
Hybrid automata [2] are a modeling formalism for systems whose evolution combines continuous dynamics interrupted by discrete steps. This work considers a subclass of rectangular automata [13], which we equip with stochasticity via random delays. The duration of a random delay in a hybrid automaton can be measured either (i) implicitly, via the semantics, or (ii) explicitly via a stopwatch and constraints on the syntax. While the first is user-friendly and intuitive, the latter makes restrictions on the modeling formalism explicit.
We follow the syntactical modeling variant and explicitly define the corresponding modeling restrictions. Similar to [7, 20], we use stopwatches to model random delays on jumps. We propose an algorithm to optimize reachability probabilities in _rectangular automata with random clocks_ (_RAR_).
Nondeterminism, which arises naturally, e.g., in concurrent systems, is often resolved probabilistically in stochastic models [17, 1, 4], and is usually not explicitly resolved in non-stochastic hybrid systems. Recently, _history-dependent_ schedulers have been proposed to resolve _discrete_ nondeterminism in hybrid Petri nets [19] and in singular automata with random clocks and urgent transitions either prophetically, or nonprophetically [20]. The prophetic scheduler knows the expiration times of random variables and is considered more powerful than the nonprophetic scheduler, who does not know these. Prophetic schedulers model the _worst/best case scenario_ and induce maximal bounds on probabilities.
When adding random clocks to rectangular automata, the challenge lies in the correct handling of continuous nondeterminism to compute correct probabilities. We propose a measure-driven backward computation which partitions the infinite set of schedulers according to their ability to reach the goal set. Prophetic scheduling hence computes a symbolic refinement of schedulers when performing a backward analysis through the precomputed reachability tree. Maximizing reachability probabilities requires taking the union over the reachable states leading to the goal, whose transition delays have been refined by backward analysis. To compute the optimal reachability probabilities, that union is projected onto the integration domain of the corresponding probability densities, before performing a multidimensional integration over the joint probability density.
For a bounded number of jumps reachability is decidable for non-initialized rectangular automata [2, 9]. Hence, flowpipe construction computes the exact reachable state-set for this model class, e.g. using a state representation of polytopes [9]. Backward refinement is then used to resolve the inherent continuous nondeterminism such that reachability probabilities are maximized. Consequently, the analysis approach presented here is exact up to numerical integration for the considered model formalism. To the best of our knowledge, the only approach able to compute reachability for this model class without resolving nondeterminism probabilistically computes a safe overapproximation via the tool ProHVer[12]. The feasibility of our approach is illustrated on a scalable model and validated by results computed in ProHVer.
Related work.The application of model checking methods for probabilistic HA was enabled by CEGAR-style abstractions [26]. Extending decidable subclasses of HA by discrete probability distributions on jumps ([24],[25]) preserves decidability of reachability. An extension of probabilistic timed automata is presented in [16], where continuously distributed resets are used and randomized schedulers resolve _discrete_ nondeterminism for a discretized state-space. Further approaches for (networks of) stochastic timed automata ([3]) maintain the probabilistic approach of resolving nondeterminism. Approaches for more general classes either rely on stochastic approximation (also for nondeterminism) [21] or on a combination of discretization and randomness [15]. Several approaches that abstract to finite Markov decision processes have been proposed: In [23] abstractions for uncountable-state discrete-time stochastic processes are proposed for SHA, where all nondeterminism is resolved probabilistically. In [5] an abstraction to interval Markov decision processes is proposed. Both approaches feature a stochastic kernel, and can hence not be compared to our work.
The analysis proposed in [12] resolves _discrete_ nondeterminism prophetically and _continuous_ nondeterminism via a safe overapproximation (c.f. [26]). The same approach has been specified for stochastic timed automata in [11].
In [6], _discrete_ nondeterminism is resolved (non-)prophetically for stochastic automata, where all continuous variables are random clocks. Similarly, scheduling of _discrete_ nondeterminism is introduced for hybrid Petri nets in [19] and for singular automata with random clocks in [20], where a forward analysis without refinement is sufficient to compute maximal reachability probabilities.
Contribution.We propose both (i) the _modeling formalism_ of rectangular automata with random clocks, which combines discrete and continuous nondeterminism with stochastic delays; and (ii) an _analytical approach_ which computes maximum reachability probabilities induced by a prophetic scheduler. We provide a feasibility study that shows that our computations are highly accurate.
Organization of the paper.Section 2 introduces the considered model class. The computation of the maximum probabilities is explained in Section 3. The feasibility study is shown in Section 4 and the paper is concluded in Section 5.
## 2 Rectangular Automata with Random Clocks
Let \(\mathbb{I}\) denote the set of all closed intervals \([a,b],[a,\infty),(-\infty,b],(-\infty,\infty)\subseteq\mathbb{R}\) with infinite or rational endpoints \(a,b\in\mathbb{Q}\), with the standard set semantics.
For \(\mathit{distr}:\mathbb{R}_{\geq 0}\to[0,1]\subseteq\mathbb{R}\) let \(\mathit{supp}(\mathit{distr})=\{v\in\mathbb{R}_{\geq 0}\,|\,\mathit{distr}(v)>0\}\). We call \(\mathit{distr}\) a _continuous distribution_ if it is absolute continuous with \(\int_{0}^{\infty}\mathit{distr}(v)dv=1\). We call \(\mathit{distr}\) a _discrete distribution_ if \(\mathit{supp}(\mathit{distr})\) is countable and \(\sum_{v\in\mathit{supp}(\mathit{distr})}f(v)=1\). We use \(\mathbb{F}_{c}\) and \(\mathbb{F}_{d}\) to denote the set of all continuous resp. discrete distributions, and let \(\mathbb{F}=\mathbb{F}_{c}\cup\mathbb{F}_{d}\) contain all _distributions_.
_Hybrid automata_[2] are a modeling formalism for systems whose evolution combines continuous dynamics (flow) interrupted by discrete steps (jumps). In this work we restrict ourselves to the subclass of _rectangular automata_[13].
Definition 1: A _rectangular automaton (RA)_ is a tuple \(R=(\mathit{Loc},\mathit{Var}_{C},\mathit{Inv},\)\(\mathit{Init},\mathit{Flow}_{C},\mathit{Edge}_{C})\) with
* a finite set \(\mathit{Loc}\) of _locations_;_
* a finite ordered set \(\mathit{Var}_{C}=\{x_{1},\ldots,x_{d_{C}}\}\) of _variables_; for \(\nu=(\nu_{1},\ldots,\nu_{d_{C}})\in\mathbb{R}^{d_{C}}\) we define \(\nu_{x_{i}}=\nu_{i}\), and for \(I=I_{1}\times\ldots\times I_{d_{C}}\in\mathbb{I}^{d_{C}}\) we set \(I_{x_{i}}=I_{i}\);_
* functions \(\mathit{Inv}\)_,_ \(\mathit{Init}\) and \(\mathit{Flow}_{C}\)_, all of type \(\mathit{Loc}\to\mathbb{I}^{d_{C}}\), assigning an _invariant_, _initial states_ resp. _flow_ to each location; we call \(\mathit{Flow}_{C}(\ell)_{x}\) the _rate_ of \(x\in\mathit{Var}_{C}\) in location \(\ell\in\mathit{Loc}\);_
* a finite set \(\mathit{Edge}_{C}\subseteq\mathit{Loc}\times\mathbb{I}^{d_{C}}\times 2^{ \mathit{Var}_{C}}\times\mathit{Loc}\) of _non-stochastic jumps_ \(e=(\ell,\mathit{pre},\mathit{post},\mathit{reset},\ell^{\prime})\in\mathit{Edge }_{C}\) with _source (target) location_\(\ell\) (\(\ell^{\prime}\)), such that \(\mathit{pre}_{x}=\mathit{post}_{x}\) for all \(x\in\mathit{Var}_{C}\setminus\mathit{reset}\) and \(\mathit{post}\subseteq\mathit{Inv}(\ell^{\prime})\);
\(R\) is _non-blocking_ if for each location \(\ell\in\mathit{Loc}\) and each variable \(x\in\mathit{Var}_{C}\):
* if \(\mathit{Inv}(\ell)_{x}\) is lower-bounded by \(a\in\mathbb{Q}\) (i.e. it has the form either \([a,b]\) or \([a,\infty)\)) and \(\mathit{Flow}_{C}(\ell)_{x}\cap\mathbb{R}_{<0}\neq\emptyset\) then there exists a non-stochastic jump \(e=(\ell,\mathit{pre},\mathit{post},\mathit{reset},\ell^{\prime})\in\mathit{Edge }_{C}\) such that \(\{\nu\in\mathit{Inv}(\ell)\mid\nu_{x}=a\}\subseteq\mathit{pre}\),
* if \(\mathit{Inv}(\ell)_{x}\) is upper-bounded by \(b\in\mathbb{Q}\) (i.e. it has the form either \([a,b]\) or \((-\infty,b]\)) and \(\mathit{Flow}_{C}(\ell)_{x}\cap\mathbb{R}_{>0}\neq\emptyset\) then there exists a non-stochastic jump \(e=(\ell,\mathit{pre},\mathit{post},\mathit{reset},\ell^{\prime})\in\mathit{Edge }_{C}\) such that \(\{\nu\in\mathit{Inv}(\ell)\mid\nu_{x}=b\}\subseteq\mathit{pre}\).
next event, yielding a sequence of random variables \(S_{r,0},S_{r,1},S_{r,2},\dots\) for each random clock \(r\). An illustrative example is depicted in Figure 1.
Definition 2: A _rectangular automaton with random clocks (RAR)_ is a tuple \(\mathcal{A}=(\mathit{Loc},\mathit{Var}_{C},\mathit{Var}_{R},\mathit{Distr}, \mathit{Inv},\mathit{Init},\mathit{Flow}_{C},\mathit{Flow}_{R},\mathit{Edge}_{C },\mathit{Edge}_{R})\) with \((\mathit{Loc},\mathit{Var}_{C},\mathit{Inv},\mathit{Init},\mathit{Flow}_{C}, \mathit{Edge}_{C})\) a non-blocking RA and:
* a finite ordered set \(\mathit{Var}_{R}=\{r_{1},\dots,r_{d_{R}}\}\) of _random clocks_; we use analogously \(\mu_{r_{i}}=\mu_{i}\) for \(\mu\in\mathbb{R}^{d_{R}}\) and \(I_{r_{i}}=I_{i}\) for \(I=I_{1}\times\dots\times I_{d_{R}}\in\mathbb{I}^{d_{R}}\);
* a function \(\mathit{Distr}:\mathit{Var}_{R}\to\mathbb{F}_{c}\);
* a function \(\mathit{Flow}_{R}:\mathit{Loc}\to\{0,1\}^{|\mathit{Var}_{R}|}\);
* a finite set \(\mathit{Edge}_{R}\subseteq\mathit{Loc}\times\mathit{Var}_{R}\times\mathit{Loc}\) of _stochastic jumps_\(e=(\ell,r,\ell^{\prime})\) with _source (target) location_\(\ell\) (\(\ell^{\prime}\)) and random clock \(r\), where (i) each two stochastic jumps \(e,e^{\prime}\in\mathit{Edge}_{R}\), \(e\neq e^{\prime}\), with the same source location \(\ell\) have different random clocks, (ii) for all locations \(\ell\in\mathit{Loc}\) and all random clocks \(r\in\mathit{Var}_{R}\), if \(\mathit{Flow}_{R}(\ell)_{r}=1\) then \((\ell,r,\ell^{\prime})\in\mathit{Edge}_{R}\) for some \(\ell^{\prime}\in\mathit{Loc}\), and (iii) for each stochastic jump \((\ell,r,\ell^{\prime})\in\mathit{Edge}_{R}\) it holds that \(\mathit{Inv}(\ell)\subseteq\mathit{Inv}(\ell^{\prime})\).
Let \(\mathcal{A}=(\mathit{Loc},\mathit{Var}_{C},\mathit{Var}_{R},\mathit{distr}, \mathit{Inv},\mathit{Init},\mathit{Flow}_{C},\mathit{Edge}_{C},\mathit{Edge}_{R})\) with \(\mathit{Var}_{C}=\{x_{1},\dots,x_{d_{C}}\}\) and \(\mathit{Var}_{R}=\{r_{1},\dots,r_{d_{R}}\}\) be a RAR. A _state_\(\sigma=(\ell,\nu,\mu,s)\in\mathcal{S}=\mathit{Loc}\times\mathbb{R}^{d_{C}} \times\mathbb{R}^{d_{R}}\times\mathbb{R}^{d_{R}}\) of \(\mathcal{A}\) specifies the current location \(\ell\), the values \(\nu\) of the continuous variables, and the values \(\mu\) and expiration times \(s\) of the random clocks. The operational semantics [14] specifies the evolution of a state of \(\mathcal{A}\), by letting time elapse or by taking a discrete jump (non-stochastic or stochastic):
\[\begin{array}{c}\begin{array}{c}t\in\mathbb{R}_{\geq 0}\quad\mathit{rate}\in \mathit{Flow}_{C}(\ell)\quad\nu^{\prime}=\nu+t\cdot\mathit{rate}\quad\nu^{ \prime}\in\mathit{Inv}(\ell)\\ \qquad\qquad\qquad\qquad\mu^{\prime}=\mu+t\cdot\mathit{Flow}_{R}(\ell)\quad \forall r\in\mathit{Var}_{R}.\,\mu^{\prime}_{r}\leq s_{r}\end{array}\\ \end{array}\]
\[\begin{array}{c}\begin{array}{c}e=(\ell,\mathit{pre},\mathit{post}, \mathit{reset},\ell^{\prime})\in\mathit{Edge}_{C}\quad\nu\in\mathit{pre}\quad \nu^{\prime}\in\mathit{post}\\ \qquad\qquad\qquad\forall x\in\mathit{Var}_{C}\setminus\mathit{reset}.\,\,\nu^{ \prime}_{x}=\nu_{x}\quad\nu^{\prime}\in\mathit{Inv}(\ell^{\prime})\\ \qquad\qquad\qquad(\ell,\nu,\mu,s)\stackrel{{ e}}{{\to}}(\ell^{ \prime},\nu^{\prime},\mu,s)\end{array}\\ \end{array}\]
\[\begin{array}{c}e=(\ell,r,\ell^{\prime})\in\mathit{Edge}_{R}\quad\mu_{r}=s_{ r}\quad\mu^{\prime}_{r}=0\quad s^{\prime}_{r}\in\mathit{supp}(\mathit{Distr}(r))\\ \qquad\qquad\qquad\forall r^{\prime}\in\mathit{Var}_{R}\setminus\{r\}.\mu^{ \prime}_{r^{\prime}}=\mu_{r^{\prime}}\wedge s^{\prime}_{r^{\prime}}=s_{r^{ \prime}}\end{array}\\ \end{array}\]
For \(\sigma\in\mathcal{S}\), let \(\mathit{EnabledJumps}(\sigma)=\{e\in\mathit{Edge}_{C}\cup\mathit{Edge}_{R}\,|\,\exists \sigma^{\prime}\in\mathcal{S}.\,\sigma\xrightarrow{e}\sigma^{\prime}\}\) be the set of jumps _enabled_ in \(\sigma\in\mathcal{S}\). We will also use \(\mathit{EnabledTime}(\sigma):=\{(t,\mathit{rate})\in\mathbb{R}_{\geq 0}\times \mathbb{R}^{d_{C}}\,|\,\exists\sigma^{\prime}\in\mathcal{S}.\,\sigma \xrightarrow{t,\mathit{rate}}\sigma^{\prime}\}\).
We set \(\rightarrow=(\bigcup_{t\in\mathbb{R}_{\geq 0},\mathit{rate}\in\mathbb{R}^{d_{C}}} \xrightarrow{t,\mathit{rate}})\cup(\bigcup_{e\in\mathit{Edge}_{C}\cup\mathit{ Edge}_{R}}\xrightarrow{e})\). A _path_ of \(\mathcal{A}\) is a finite sequence \(\pi=\sigma_{0}\xrightarrow{a_{0}}\sigma_{1}\xrightarrow{a_{1}}\ldots\) of states with \(\sigma_{0}=(\ell_{0},\nu_{0},\mu_{0},s_{0})\), \(\nu_{0}\in\mathit{Inv}(\ell_{0})\), and \(\sigma_{i}\xrightarrow{a_{i}}\sigma_{i+1}\) for all \(i\in\mathbb{N}\); we call \(\pi\)_initial_ if \(\nu_{0}\in\mathit{Init}(\ell_{0})\), \(\mu_{0}=0\in\mathbb{R}^{d_{R}}\) and \(s_{0_{r}}\in\mathit{supp}(\mathit{Distr}(r))\) for all \(r\in\mathit{Var}_{R}\). A state is _reachable_ if there is an initial path leading to it. Let _Paths_ denote the set of all paths.
For every reachable state, there is a path with alternating delays and jumps, as delays may have duration zero and consecutive delays can be combined. This holds even for consecutive delays with different rates, as flow sets are convex4:
Footnote 4: For a proof of Lemma 1, we refer to Appendix 0.A.
Lemma 1: _Let \(\sigma_{0}\xrightarrow{t_{1},\mathit{rate}_{1}}\sigma_{1}\xrightarrow{t_{2}, \mathit{rate}_{2}}\sigma_{2}\) for \(\sigma_{0},\sigma_{1},\sigma_{2}\in\mathcal{S}\) with location \(\ell\), \(t_{1},t_{2}\in\mathbb{R}_{\geq 0}\) and \(\mathit{rate}_{1},\mathit{rate}_{2}\in\mathit{Flow}_{C}(\ell)\). Then there is \(\mathit{rate}\in\mathit{Flow}_{C}(\ell)\) s.t. \(\sigma_{0}\xrightarrow{t_{1}+t_{2},\mathit{rate}}\sigma_{2}\)._
The _duration_ of a path is defined as the sum of the durations of its steps, where jumps are considered instantaneous: \(\mathit{dur}(\sigma)=0\), \(\mathit{dur}(\pi\xrightarrow{e}\sigma)=\mathit{dur}(\pi)\) and \(\mathit{dur}(\pi\xrightarrow{t,\mathit{rate}}\sigma^{\prime})=\mathit{dur}( \pi)+t\). Let the _jump-depth_ of a path be the number of jumps in it. We call a RAR _Zeno-free_ iff for all \(t\in\mathbb{R}_{\geq 0}\) there exists a \(k\in\mathbb{N}\) such that all initial paths of jump-depth at least \(k\) have duration at least \(t\). We deviate from the standard definition, which requires that only finitely many jumps are possible in finite time. Our definition assures a _concrete bound on the number of jumps_ per path for each time bound. In the following, we assume all considered models to be Zeno-free.
RAR allow for (i) _initial nondeterminism_ in the choice of the initial state, (ii) _time nondeterminism_ when time can elapse but also jumps are enabled during the whole flow, and (iii) _rate nondeterminism_ when continuous variables can evolve with different rates. In addition to these continuous types of nondeterminism, we also consider (iv) _discrete nondeterminism_ when different jumps are enabled simultaneously. We use prophetic schedulers to resolve nondeterminism, which have full information not only on the history but also on the future expiration times of all random clocks, as introduced in [20]. While prophetic scheduling may seem unrealistic, they are well-suited to perform a _worst-case_ analysis, especially when uncontrollable uncertainties are modeled nondeterministically.
Definition 3: A _(prophetic history-dependent) scheduler_ is a function \(\mathfrak{s}:\mathit{Paths}\rightarrow\mathbb{F}\) which assigns to every path \(\pi=\sigma_{0}\xrightarrow{a_{1}}\ldots\xrightarrow{a_{n}}\sigma_{n}\in \mathit{Paths}\) a distribution \(\mathit{distr}=\mathfrak{s}(\pi)\), such that if \(n\geq 1\) and \(a_{n}\in\mathit{Edge}_{C}\cup\mathit{Edge}_{R}\) is an edge then \(\mathit{distr}\) is continuous with \(\mathit{supp}(\mathit{distr})\subseteq\mathit{EnabledTime}(\sigma_{n})\) and otherwise \(\mathit{distr}\) is discrete with \(\mathit{supp}(\mathit{distr})\subseteq\mathit{EnabledJumps}(\sigma_{n})\). The set of schedulers is denoted \(\mathfrak{S}\).
We maximize the probability of _time-bounded_ reachability, i.e., of reaching a set of goal states \(\mathcal{S}^{\mathit{goal}}\subseteq\mathcal{S}\) along initial paths \(\pi\) with \(\mathit{dur}(\pi)\leq t_{\max}\).
Resolving nondeterminism in a RAR \(\mathcal{A}\) via the set of schedulers \(\mathfrak{S}\) induces an interval of _reachability probabilities_\([p_{\text{min}}^{\mathfrak{S}}(\mathcal{S}^{\text{goal}},t_{\text{max}}),p_{ \text{max}}^{\mathfrak{S}}(\mathcal{S}^{\text{goal}},t_{\text{max}})]\), where the bounds are referred to as minimum and maximum. We define \(\mathfrak{S}_{\text{goal}}\subseteq\mathfrak{S}\) as the set of schedulers that reach \(\mathcal{S}^{\text{goal}}\) along initial paths \(\pi\) with \(\mathit{dur}(\pi)\leq t_{\text{max}}\) and induce \(p_{\text{max}}^{\mathfrak{S}}\). Let \(\mathcal{V}^{\mathfrak{s}}\subseteq\mathbb{R}^{d_{R}}\) denote the sample values for all random variables that allow scheduler \(\mathfrak{s}\) to reach \(\mathcal{S}^{\text{goal}}\). This yields the following definition:
Definition 4: The _prophetic maximum reachability probability is:_
\[p_{\text{max}}^{\mathfrak{S}}(\mathcal{S}^{\text{goal}},t_{\text{max}})=\int_ {\bigcup_{s\in\mathfrak{S}_{\text{goal}}}\mathcal{V}^{\mathfrak{s}}}G(\mathbf{ s})\ d\mathbf{s}, \tag{1}\]
_where \(G(\mathbf{s})=\prod_{r_{n}}\mathit{Distr}(r)\) is the joint probability density function for all random delays \(r_{n}\) and random clocks \(r\)._
Note that due to the independence of the random variables, \(G(\mathbf{s})\) equals the product over the probability density functions \(\mathit{Distr}(r)\).
## 3 Computation of Maximum Reachability Probabilities
The inherent continuous nondeterminism in RAR leads to an uncountable number of choices and hence schedulers. We propose a measure-driven state space construction, which partitions the infinite set of continuous schedulers w.r.t. their ability to reach the set of goal states. We remark that our model class allows resets on continuous variables, but our method does not yet support this.
Section 3.1 explains the forward flowpipe construction and Section 3.2 introduces the backward refinement. Sample domains are extracted in Section 3.3 and maximum prophetic reachability probabilities are computed in Section 3.4.
### Forward flowpipe construction
To compute reachability for rectangular automata, flowpipe construction has been proposed in [2], which results in a geometric representation of all states reachable up to a predefined time bound. To apply this method to RAR, we disregard the stochasticity by removing all constraints on the sample values \(s\) from the operational semantics. The resulting automaton is a regular rectangular automaton, where the \(n\)-th random delay induced by random clock \(r\) in the original automaton is treated as a continuous variable \(r_{n}\). Replacing every random clock \(r\) with one continuous variable for each possible delay corresponds to an enrollment of the automaton (c.f. Figure 2). The set \(\mathit{Var}_{R}\) of the enrolled rectangular automaton then contains continuous variables \(r_{0},r_{1},\dots\) for each random clock \(r\) in the original automaton and \(d_{R}\) is the number of all random delays.
In the following, we omit the sampled values \(s\) from states \(\sigma=(\ell,\nu,\mu,s)\). To simplify, we also combine the valuations of continuous variables \(\nu\) and random clocks \(\mu\), such that we call \((\ell,\mathcal{V})\)_state set_, where \(\mathcal{V}\) is a set of valuations and \(\nu_{x}\) refers to the valuation of a variable \(x\in\mathit{Var}_{C}\cup\mathit{Var}_{R}\). Even though formally we
work on state sets \((\ell,\mathcal{V})\), for readability our notation restricts to valuation sets \(\mathcal{V}\). We refer to valuation sets as _segments_ and to the respective location of the state set as _corresponding_ location.
We execute a forward flowpipe construction, i.e., starting with the initial states we alternate between computing the forward time closure and the jump successors until a predefined \(t_{\text{max}}\) is reached. The forward time closure \(T_{\ell}^{+}(\mathcal{V})\) of \(\mathcal{V}\) in \(\ell\) (c.f. [2]), represents the set of states reachable from the set of entry states, i.e., computes states reachable via a time delay in location \(\ell\). Jumps are represented by the one-step relation \(D_{e}^{+}(\mathcal{V})\), which defines the set of valuations reachable by choosing transition \(e\in\mathit{Edge}_{C}\cup\mathit{Edge}_{R}\) from valuations \(\mathcal{V}\)[2].
Our computation relies on a state representation via convex polytopes, usually in _H-representation_, defined as an intersection of multiple halfspaces. Some computations require the _V-representation_, defined as convex hull of a set of vertices (convex hull combined with convex cone for unbounded polytopes) [27].
The forward flowpipe construction then computes all reachable segments and stores them together with their corresponding location in the _reach tree_\(\mathcal{R}\) according to their occurrence. We define a _reach tree_:
Definition 5: For a rectangular automaton \(R\) and a time bound \(t_{\text{max}}\), a _reach tree_ is a tuple \(\mathcal{R}=(N,E)\) with symbolic state sets \((\ell,\mathcal{V})\in N\) as nodes and edges \(e\in E=N\times(\mathit{Edge}_{C}\cup\mathit{Edge}_{R})\times N\) annotated with jumps, where for each state \(\sigma=(\ell,\nu)\in\mathcal{S}\) it holds that, if and only if \(\sigma\) is reachable in \(R\) before time \(t_{\text{max}}\), there exists a node \(n=(\ell,\mathcal{V})\in N\), such that \(\sigma\in\mathcal{V}\).
When performing qualitative analysis in the purely hybrid case, flowpipe construction ends as soon as the intersection of a computed segment with the set of goal states is non-empty. As we perform quantitative reachability analysis, we have to complete the flowpipe construction until the time bound \(t_{\text{max}}\) and collect all trajectories leading to goal states in order to fully resolve the nondeterminism present in the system. After computing the flowpipe up to \(t_{\text{max}}\), the resulting segments are intersected with the goal set to determine reachability. We define
goal states as \(\mathcal{S}^{\text{goal}}=(L^{\text{goal}},\mathcal{V}^{\text{goal}})\), where \(L^{\text{goal}}\) is a set of locations and \(\mathcal{V}^{\text{goal}}\) is a set of valuations defined as a convex polytope, with constraints only for continuous variables \(x\in\mathit{Var}_{C}\). Hence, \(\mathcal{V}^{\text{goal}}\) is unbounded in the dimensions of random clocks \(r\in\mathit{Var}_{R}\).
We define the set \(\mathcal{S}^{\text{goal}}_{\text{reach}}\) of _reachable goal states_, such that it contains all subsets of the goal state set that can be reached via a trace in the reach tree:
\[(\ell,\mathcal{V})\in\mathcal{S}^{\text{goal}}_{\text{reach}}\Leftrightarrow \ell\in L^{\text{goal}}\wedge\exists(\ell^{\prime},\mathcal{V}^{\prime})\in \mathcal{R}.\mathcal{V}^{\prime}\cap\mathcal{V}^{\text{goal}}=\mathcal{V}( \neq\emptyset)\wedge\ell^{\prime}=\ell.\]
We use \(\mathcal{V}^{i}\) and \(\mathcal{S}^{i}=(\ell,\mathcal{V}^{i})\) to refer to these subsets of goal states and index \(i\) to refer to the trace of the reach tree leading to \(\mathcal{S}^{i}\) (c.f. Figure 3(c)). We define \(I_{\mathcal{S}}\) as the set collecting all trace indices \(i\). All segments \(\mathcal{V}^{i}\), then serve as input for the backward refinement (c.f. Section 3.2).
Running example.The forward time closure of the initial set in location \(\ell_{0}\) (c.f. Figure 1) corresponds to the segment indicated by a gray solid line in Figure 2(a). Moving from location \(\ell_{0}\) to \(\ell_{3}\) corresponds to the expiration of the random clock \(r\), which models the only random delay present in the automaton. In location \(\ell_{2}\), only \(y\) evolves (green dotted border). For states with \(x\geq 2\) and \(y\geq 10\), taking the transition to location \(\ell_{4}\) is possible (solid dark green border). Here, \(y\) is stopped and only \(x\) evolves. Alternatively, moving from \(\ell_{0}\) to \(\ell_{1}\) (blue dotted border) is possible for \(x=4\). All states with \(y\geq 5\) can reach location \(\ell_{2}\), as long as \(y\leq 7\) holds. This leads to overlapping segments for \(\ell_{1}\) and \(\ell_{2}\) (solid dark blue border). The goal states (yellow border) are defined by \(\mathcal{S}^{\text{goal}}=\{(\ell,\nu)\in\mathcal{S}\mid\ell\in Loc\ \wedge\ \nu_{x}\in[8,10]\wedge\nu_{y}\in[8,11]\}\). Figure 2(b) illustrates the reach tree \(\mathcal{R}\).
The goal can be reached both, via locations \(\ell_{1},\ell_{2}\) and via locations \(\ell_{3},\ell_{4}\). Hence, intersecting the forward flowpipe segments with the goal set results in two traces \(i=0,1\), leading to state sets \(\mathcal{S}^{0}=(\ell_{2},\mathcal{V}^{0})\) and \(\mathcal{S}^{1}=(\ell_{4},\mathcal{V}^{1})\), where:
\[\mathcal{V}^{0} =\{\nu\in\mathbb{R}^{3}\ |\ \nu_{x}\leq 10\wedge\nu_{y}\geq 8 \wedge\nu_{y}\leq 3.5+\nicefrac{{1}}{{2}}\cdot\nu_{x}\wedge\nu_{r}=3\}\] \[\mathcal{V}^{1} =\{\nu\in\mathbb{R}^{3}\ |\ \nu_{x}\in[8,10]\wedge\nu_{y}\in[10,11] \wedge\nu_{r}\in[0,3]\}.\]
### Backward refinement
Starting from each \(\mathcal{V}^{i}\subseteq\mathcal{V}^{\text{goal}}\), we perform a backward computation along the reach tree to refine state sets according to prophetic maximum schedulers until the root of the tree is reached. We refine backward by computing _refined segments_ and _intermediate goal segments_ for all forward flowpipe segments on trace \(i\). The result of the backward refinement then is given by all refined segments and \(\mathcal{V}^{i}\) in traces \(i\) along the reach tree, containing exactly the fragment of the reach tree that allows reaching the goal.
Backward propagation relies on the definitions of backward time closure \(T_{\ell}^{-}(\mathcal{V})\) and backward one-step relation \(D_{e}^{-}(\mathcal{V})\) (c.f. [2]). Similar to the forward time closure, \(T_{\ell}^{-}(\mathcal{V})\) computes all states valid in location \(\ell\) by regressing the time delay in that location, that are able to reach valuations in \(\mathcal{V}\). \(D_{e}^{-}(\mathcal{V})\) reverts the effects of transition \(e\) leading to \(\mathcal{V}\), defined through guards and resets.
Starting from each segment \(\mathcal{V}^{i}\), the backward time closure \(T^{-}_{\ell}(\mathcal{V})\) is computed according to the activity of the corresponding location \(\ell\). This yields an unbounded cone, containing all states which can reach \(\mathcal{V}^{\text{goal}}\) from an arbitrary state in location \(\ell\). We define the corresponding precomputed forward segment \(\mathcal{V}^{i}_{k}\) as the flowpipe segment on trace \(i\) in the reach tree, which is encountered in step \(k\) going backward from \(\mathcal{V}^{i}\) for \(\mathcal{V}^{i}\subset\mathcal{V}^{i}_{0}\) (c.f. Figure 3(b)). It is then intersected with the unbounded backward time closure to restrict the state set to states that are actually reachable in the hybrid automaton via a maximizing scheduler (c.f. Figure 4(a)). This results in a so-called _refined segment_\(\hat{\mathcal{V}}^{i}_{k}\), containing all states which can reach the goal set from location \(\ell\) (in step \(k\)) on trace \(i\):
Definition 6: Given an intermediate goal segment \(\mathcal{Q}^{i}_{k}\) within a segment \(\mathcal{V}^{i}_{k}\) and its corresponding location \(\ell\), the _refined segment_\(\hat{\mathcal{V}}^{i}_{k}\) on trace \(i\) is \(\hat{\mathcal{V}}^{i}_{k}=T^{-}_{\ell}(\mathcal{Q}^{i}_{k})\cap\mathcal{V}^{i}_ {k}\). The first intermediate goal segment \(\mathcal{Q}^{i}_{0}\) on trace \(i\) is \(\mathcal{V}^{i}\) itself.
Figure 5 illustrates refined segments computed from intermediate goal segments, corresponding to the running example. Given a transition \(e\) from \(\ell_{p}\) to \(\ell\) connecting segments \(\mathcal{V}^{i}_{k+1}\) and \(\mathcal{V}^{i}_{k}\), we compute the backward one-step relation of a refined segment \(\hat{\mathcal{V}}^{i}_{k}\). By intersecting with the next forward segment \(\mathcal{V}^{i}_{k+1}\) it is ensured, that the resulting segment \(\mathcal{Q}^{i}_{k+1}\) is a subset of \(\mathcal{V}^{i}_{k+1}\). This will be used as intermediate goal segment for the next computation step.
Definition 7: For a refined segment \(\hat{\mathcal{V}}^{i}_{k}\) and its corresponding location \(\ell\), the next _intermediate goal segment_\(\mathcal{Q}^{i}_{k+1}\) on trace \(i\) is \(\mathcal{Q}^{i}_{k+1}=D^{-}_{e}(\hat{\mathcal{V}}^{i}_{k})\cap\mathcal{V}^{i}_ {k+1}\). If \(\hat{\mathcal{V}}^{i}_{k}\) corresponds to an initial location \(\ell_{0}\) (i.e., \(\text{Init}(\ell_{0})\neq\emptyset\)), we use the initial set of \(\ell_{0}\) instead of \(\mathcal{V}^{i}_{k+1}\) for the intersection, i.e., \(\mathcal{Q}^{i}_{k+1}=D^{-}_{e}(\hat{\mathcal{V}}^{i}_{k})\cap\text{Init}( \ell_{0})\).
Running example: Backward refinement for the running example is illustrated in Figure 3(a) and Figure 3(b). Figure 3(c) illustrates the two traces \(i=0,1\). Starting from segments \(\mathcal{V}^{0}\) and \(\mathcal{V}^{1}\), the refined segments \(\hat{\mathcal{V}}^{0}_{k}\) and \(\hat{\mathcal{V}}^{1}_{k}\) can be computed iteratively. Figure 5 illustrates one refined segment on each trace. The refined segment \(\hat{\mathcal{V}}^{0}_{2}\) and the next intermediate goal segment \(\mathcal{Q}^{0}_{3}\) (c.f. Figure 4(a)) can be computed from \(\mathcal{Q}^{0}_{2}\). In addition to the constraints visible in Figure 4(a),
Figure 4: Backward refinement.
then contains the constraint \(\nu_{r}=\nu_{x}-1\), stemming from the initial valuation of \(\nu_{x}=1\). Note that \(\mathcal{Q}_{3}^{0}\) is computed via intersection with \(\mathit{Init}(\ell_{0})\).
The refined segment \(\hat{\mathcal{V}}_{1}^{1}\) and the next intermediate goal segment \(\mathcal{Q}_{2}^{1}\) (c.f. Figure 4(b)) are computed from \(\mathcal{Q}_{1}^{1}\). Again, \(\hat{\mathcal{V}}_{1}^{1}\) and \(\mathcal{Q}_{2}^{1}\) contain the constraint \(\nu_{r}=\nu_{x}-1\). Note that, since also \(\nu_{x}\in[2,4]\), \(\nu_{r}\) is implicitely restricted to be contained by \([1,3]\). These computation steps can also be found in Appendix 0.B.
### Extracting the sample domain
Backward refinement results in refined segments \(\hat{\mathcal{V}}_{k}^{i}\), from which the sample domain for all random variables is extracted as follows. Refined segments contain information on interdependencies between all variables and valuations leading to the goal in \(\mathcal{V}^{i}\). The sample domain for each random variable \(S_{r,n}\) is derived from the segment which corresponds to the \(n\)-th expiration of random clock \(r\).
Polytopes \(\mathcal{P}^{i}\subset\mathbb{R}^{d_{R}}\) collect the sample values which lead to the goal via trace \(i\). Traversing the traces in a forward manner, for each segment we derive information on the sample domains from the constraints on the valuations of the random clocks which allow taking the next step on trace \(i\) (in the reach tree). Thus, we collect all valid values for each random delay on trace \(i\).
For each segment, constraints on the samples are derived as follows. First, we project the segment onto the stochastic dimensions. Second, we collect information about all random delays, which are either already expired, about to expire in the next step on the trace, or not expired. (i) Random delays that are already expired cannot provide any new information, and hence do not have to be considered again. (ii) A step in the trace that corresponds to the expiration of a random delay, induces (upper) bounds on the sample domain. Each edge in the reach tree maps to exactly one jump in the automaton. In case of a stochastic jump, the step results in the upper bounds of exactly one random clock. (iii) To account for information on future expirations, the upper bounds of all other random delays have to be _lifted_.
Figure 5: Detailed backward refinement steps.
Definition 8: We define the _lifting_ for a variable \(r\) in a polytope \(P\subseteq\mathbb{R}^{d_{R}}\) as \(\Lambda_{r}(P)\)\(:=\{(s_{1},\ldots,s_{r}+c,\ldots,s_{d_{R}})\in\mathbb{R}^{d_{R}}\,|\,(s_{1},\ldots,s_{r}, \ldots,s_{d_{R}})\in P\wedge c\in[0,\infty)\}\).
This iterative collection of constraints on the sample domain leads to a polytope \(\mathcal{P}^{i}\) that contains all sample values which allow to follow trace \(i\). To compute maximum reachability probabilities, all possibilities of reaching the goal have to be included in the integration. This corresponds to taking the union over all \(\mathcal{P}^{i}\) for \(i\in I_{\mathcal{S}}\).
Summarizing, this leads to a polytope \(\mathcal{P}_{\text{max}}=\bigcup_{i\in I_{\mathcal{S}}}\mathcal{P}^{i}\).
Running example: The random delay \(r\) does not expire on trace \(0\), hence the lower constraints on the sample domain for \(s_{r}\) are iteratively collected, leading to the strongest lower constraint in \(\mathcal{V}^{0}\). By projecting \(\mathcal{V}^{0}\) onto \(\mathbb{R}^{d_{R}}\) and lifting the resulting polytope in dimension \(r\), the constraint \(s_{r}\geq 3\) can be derived.
On trace \(1\), the expiration of the random clock corresponds to the transition from \(\ell_{0}\) to \(\ell_{3}\), i.e., \(r\) is about to expire in segment \(\hat{\mathcal{V}}^{1}_{2}\). Hence, from \(\hat{\mathcal{V}}^{1}_{2}\), the constraints \(s_{r}\geq 1\) and \(s_{r}\leq 3\) are derived, i.e. \(\mathcal{P}^{1}=[1,3]\). This leads to \(\mathcal{P}_{\text{max}}=[3,\infty)\cup[1,3]=[1,\infty)\), which contains all values for random variable \(S_{r}\) for which the goal is reachable: for \(s_{r}\geq 3\) via trace \(0\), and for \(s_{r}\in[1,3]\) via trace \(1\).
### Maximum prophetic reachability probabilities
To maximize continuous nondeterminism prophetically, we partition the potentially infinite set of prophetic schedulers \(\mathfrak{S}_{\text{goal}}\) with respect to their ability to reach the goal. The backward refinement returns the fragment of the reach tree that leads to goal states \(\mathcal{S}^{\text{goal}}\) (and specifically state sets \(\mathcal{S}^{i}\in\mathcal{S}^{\text{goal}}_{\text{reach}}\)), that allows extracting the sample domain. This process incorporates knowledge of future expiration times of random variables, leading to prophetic schedulers.
Reachability defines an equivalence relation on the set of schedulers \(\mathfrak{S}_{\text{goal}}\) with respect to the state sets \(\mathcal{S}^{i}\), reachable via trace \(i\). This results in equivalence classes \([\mathfrak{s}]_{i}\), which contain all schedulers \(\mathfrak{s}\) able to reach \(\mathcal{S}^{i}\). Hence, \(\bigcup_{i\in I_{\mathcal{S}}}[\mathfrak{s}]_{i}=\mathfrak{S}_{\text{goal}}\). Via this equivalence relation, we are able to resolve different types of (continuous) nondeterminism. This is explained in the following for nondeterministic time delays on transitions, rectangular flow sets and conflicting transitions.
Initial and time nondeterminism.Taking a transition (from \(\ell_{p}\) to \(\ell\)) at different points in time leads to different states in the set, with which target location \(\ell\) is entered. This set \(\mathcal{Q}^{i}_{k+1}\subset\hat{\mathcal{V}}^{i}_{k}\) then contains all states corresponding to time delays which enable reaching \(\mathcal{Q}^{i}_{k}\). Recall, that at the end of the trace, \(\mathcal{Q}^{i}_{0}\) equals \(\mathcal{V}^{i}\). This corresponds to a maximizing scheduler choosing (in a forward way) a time delay for that transition, such that from each state in \(\mathcal{Q}^{i}_{k+1}\), entering location \(\ell_{p}\), it is possible to reach the intermediate goal segment \(\mathcal{Q}^{i}_{k}\). Similarly, for a corresponding initial location, \(\mathcal{Q}^{i}_{k+1}\) restricts the initial set. The scheduler then chooses an initial state, such that \(\mathcal{Q}^{i}_{k}\) can be reached via \(\hat{\mathcal{V}}^{i}_{k}\).
Figure (b)b illustrates overlapping segments caused by a nondeterministic guard. The backward time closure from \(\mathcal{Q}^{1}_{1}\) in segment \(\mathcal{V}^{1}_{1}\) restricts the range of possible time delays: the (intermediate) goal segment can solely be reached from the pink fragment, i.e., from states with \(\nu_{y}\leq 11\).
Rate nondeterminismThe backward refinement results in restricted segments, which implicitly define a partitioning of the schedulers. For every state in \(\mathcal{Q}^{i}_{k+1}\), a maximizing prophetic scheduler can pick at least one slope, which leads to the intermediate goal set \(\mathcal{Q}^{i}_{k}\). In contrast, for all states outside of \(\mathcal{Q}^{i}_{k+1}\), such a slope does not exist. Figure (a)a illustrates that only initial states in \(\mathcal{V}^{0}_{2}\) (and hence in \(\mathcal{Q}^{0}_{3}\)) can reach the goal. The choice of the initial state restricts the possible rates. In this example, for initial state \(\nu_{x}=1,\nu_{y}=2\), only the largest possible rate (\(\dot{y}=\nicefrac{{1}}{{3}}\)) leads to the intermediate goal segment \(\mathcal{Q}^{0}_{2}\) and enables reaching \(\mathcal{V}^{0}\).
Discrete nondeterminismEvery discrete choice leads to a different trace in the forward flowpipe. The backward analysis starts from \(\mathcal{S}^{\text{goal}}_{\text{reach}}\), i.e., only considers traces leading to goal states. Hence, the union of all \(\mathcal{P}^{i}\) over all traces \(i\), obtained from backward refinement, represents all discrete choices which reach the goal. Consequently, this maximizes discrete nondeterminism.
Summarizing, the valuations induced by all maximizing schedulers match the valuation sets computed by the backward refinement.
Lemma 2: _Given the set of valuations \(\mathcal{V}^{\mathfrak{s}}\) over the space of the random variables that allow scheduler \(\mathfrak{s}\) to reach the goal set \(\mathcal{S}^{\text{goal}}\), and an equivalence relation over \(\mathfrak{S}_{\text{goal}}\) defining equivalence classes \([\mathfrak{s}]_{i}\), \(\bigcup_{\mathfrak{s}\in\mathfrak{S}_{\text{goal}}}\mathcal{V}^{\mathfrak{s}}\) can be computed as follows:_
\[\bigcup_{\mathfrak{s}\in\mathfrak{S}_{\text{goal}}}\mathcal{V}^{\mathfrak{s}} =\bigcup_{i\in I_{S}}\bigcup_{\mathfrak{s}\in[\mathfrak{s}]_{i}} \mathcal{V}^{\mathfrak{s}}=\bigcup_{i\in I_{S}}\mathcal{V}^{[\mathfrak{s}]_{ i}}=\bigcup_{i\in I_{S}}\mathcal{P}^{i}=\mathcal{P}_{\text{max}}, \tag{2}\]
_where \(\mathcal{V}^{[\mathfrak{s}]_{i}}\) is the union of valuations induced by schedulers \(\mathfrak{s}\in[\mathfrak{s}]_{i}\)._
For a proof of Lemma 2, we refer to Appendix 0.A. We can now compute the maximum reachability probability by integration over \(\mathcal{P}_{\text{max}}\):
\[p^{\mathfrak{S}}_{\text{max}}(\mathcal{S}^{\text{goal}},t_{\text{max}})=\int _{\bigcup_{\mathfrak{s}\in\mathfrak{S}_{\text{goal}}}\mathcal{V}^{\mathfrak{s} }}G(\mathbf{s})\ d\mathbf{s}=\int_{\mathcal{P}_{\text{max}}}G(\mathbf{s})d \mathbf{s}, \tag{3}\]
where \(G(\mathbf{s})=\prod_{r_{n}\in\mathit{Var}_{R}}\mathit{Distr}(r_{n})\) is the joint probability density function.
The maximum reachability probability stems from all schedulers which can reach the goal. Hence, taking the union over all sample values \(\mathcal{V}^{\mathfrak{s}}\) reached by \(\mathfrak{s}\in\mathfrak{S}_{\text{goal}}\) results in the integration domain for maximum reachability probabilities. By construction, \(\mathfrak{S}_{\text{goal}}\) contains all maximizing prophetic schedulers:
1. All schedulers \(\mathfrak{s}\) able to reach the goal \(\mathcal{S}^{\text{goal}}\), reach a state set \(\mathcal{S}^{i}\) and are hence represented by an equivalence class \([\mathfrak{s}]_{i}\), collectively reaching \(\mathcal{S}^{i}\).
2. Schedulers \(\mathfrak{s}\in\mathfrak{S}\setminus\mathfrak{S}_{\text{goal}}\) cannot reach the goal states \(\mathcal{S}^{\text{goal}}\). In case a scheduler \(\mathfrak{s}\in\mathfrak{S}\setminus\mathfrak{S}_{\text{goal}}\) could reach a goal state, the state(s) reachable by \(\mathfrak{s}\) would belong to a flowpipe segment, and hence belong to a state set \(\mathcal{S}^{i}\).
3. The combination of forward analysis and backward refinement results in prophetic schedulers \(\mathfrak{S}_{\text{goal}}\). The former ensures that all sample values leading to goal states are known. The latter partitions the infinite set of schedulers w.r.t the reachability of different state sets \(\mathcal{S}^{i}\) using this knowledge.
Integration error.Multidimensional integration over unbounded polytopes is in practice not possible. Hence, we lift to a predefined _integration bound_\(t_{\text{int}}\geq t_{\text{max}}\) and not to infinity, as stated in Section 3.3. This results in an overapproximation error:
\[e_{\infty}=1-\int_{[0,t_{\text{int}}]^{d_{R}}}G(\mathbf{s})d\mathbf{s}.\]
This error is exact if: (i) no random clock has ever expired upon reaching the goal set on all traces, or (ii) the support of all random clocks that have not expired upon reaching the goal is finite. Clearly, increasing \(t_{\text{int}}\) decreases \(e_{\infty}\). Integration is done statistically with Monte Carlo Vegas [18], which introduces an additional statistical error \(e_{\text{stat}}\), depending on the number of integration samples.
Running example.Maximizing schedulers form two equivalence classes, where \([\mathfrak{s}]_{0}\) contains all schedulers reaching \(\mathcal{S}^{0}\) and \([\mathfrak{s}]_{1}\) contains all schedulers reaching \(\mathcal{S}^{1}\). Schedulers in \([\mathfrak{s}]_{0}\) start with \(\nu_{y}\geq 2\) in \(\ell_{0}\), then pick a rate of \(y\) that is at least \(1-\nicefrac{{\nu_{y}}}{{3}}\) in \(\ell_{1}\). Finally, the time delay in \(\ell_{1}\) has to ensure \(\nu_{y}\geq 3+\nicefrac{{1}}{{2}}\cdot\nu_{x}\). Schedulers in \([\mathfrak{s}]_{1}\) choose to leave \(\ell_{3}\) such that \(\nu_{y}\leq 11\).
Assuming a folded normal distribution \(\mathcal{N}(2,1)\) for the stochastic delay, the maximum reachability probability to reach \(\mathcal{S}^{\text{goal}}\) before \(t_{\text{max}}=10\) is computed by integration over \([1,\infty]\): \(p_{\text{max}}^{\mathfrak{S}}(\mathcal{S}^{\text{goal}},10)=\int_{[1,\infty]} G(\mathbf{s})d\mathbf{s}=1-0.2464=0.7536\).
Computational complexity.The complexity of forward reachability analysis _fwra_ is exponential in the state-space dimension and depends on the automaton, the set of initial states, and the number of jumps _jmp_ depending on \(t_{\text{max}}\). The complexity of computing one refined segment \(\tilde{\mathcal{V}}_{k}^{i}\) from its predecessor \(\tilde{\mathcal{V}}_{k-1}^{i}\) is denoted _bwra_ with worst-case length \(\mathcal{O}(\textit{jmp})\) of traces \(i\in I_{\mathcal{S}}\). We can then bound the complexity of the proposed analysis by \(\mathcal{O}(\textit{fwra}+|I_{\mathcal{S}}|\cdot\textit{jmp}\cdot\textit{bwra})\), where the number of sets used for numerical integration is in \(\mathcal{O}(|I_{\mathcal{S}}|\cdot\textit{jmp})\).
## 4 Feasibility study
Figure 6 illustrates our model of an electric car with different charging modes, which choose the charging rate nondeterministically from different intervals. Charging stops whenever the battery is full. The driving distances are sampled from random variables. See [8] for the automaton with all dynamics.
Locations \(\textit{charging}_{A}\), \(\textit{charging}_{B}\), \(\textit{charging}_{C}\) and _full_ model decreasing charging rates, depending on the state of charge of the battery. The charging time is modeled by random clock \(c\) which follows a folded normal distribution, i.e. \(\mathcal{N}(2,2)\). In location _driving_, the battery decreases, where random clock \(d\) models the time spent driving. The expiration of \(d\) leads to location _arrival_, while draining the battery leads to location _empty_. The driving time follows a folded normal distribution (\(\mathcal{N}(4,1)\)).
The model is scalable, as it includes the possibility of taking \(0\) or more detours. In location _driving_, the random delay until the next detour (\(r\)) competes
with the end of the drive (\(d\)). Random clock \(r\) follows an exponential distribution with \(\lambda=2\). In the _detour_ location, \(d\) is still active and the battery still decreases. The expiration of \(d\) corresponds to the end of the detour and the transition to location _charge_ is taken, which marks the start of the next charging cycle. Depending on the current charge of the battery, the transition to the matching charging location is chosen immediately. By explicitly restricting the number of detours, we ensure that the model is Zeno-free. Performing a worst case analysis, we compute the maximum probability of reaching an _empty_ battery.
Implementation and reproducibilityWe rely on a prototypical implementation of the tool RealySt5 and use the library HyPro[22] to compute flowpipe segments; particularly HyPro is used to compute the forward flowpipe, the reach tree, and the backward time closure and one-step relation as used in our backward refinement. Additionally, we use GNU Scientific Library[10] to perform multi-dimensional integration. All experiments were run on a machine equipped with an Intel(r) Core(r) i7 with 6\(\times\)3.30 GHz and 32 GB RAM.
Footnote 5: Tool and models: [https://zivgitlab.uni-muenster.de/ag-sks/tools/realyst/](https://zivgitlab.uni-muenster.de/ag-sks/tools/realyst/).
ResultsWe consider three different model variants, namely ABC, AB and A, that include exactly those charging locations indicated in the model name. If locations _charging\({}_{B}\)_ and/or _charging\({}_{C}\)_ are removed, the invariant in the last charging location is extended to \(x\leq 10\) and the in- and outgoing transitions of that location are adapted accordingly. To further reduce model complexity, a singular version is created, where the rate of the continuous variables equals the lower bound of the corresponding rectangular interval. Continuous nondeterminism is maintained via the initial set, hence results cannot be compared with [20].
Table 1 contains maximum reachability probabilities and the corresponding computation time for the full model presented in Figure 6 (indicated by ABC)
Figure 6: Car model with detours. Random clock \(c\) is active (\(\dot{c}=1\)) in the charging locations, \(d\) is active in locations _driving_ and _detour_ and \(r\) is active in location _driving_. The state of charge \(x\) is restricted to \([0,10]\) in all locations unless stated otherwise. No time is spent in location _charge_ due to invariants not shown.
and for reduced model versions (AB and A). We scale \(t_{\text{max}}=(\#\text{detours}+1)\cdot 21\), for \(\#\text{detours}\in\{0,1,2\}\) and use \(1\cdot 10^{5}\), \(2\cdot 10^{6}\), \(1\cdot 10^{7}\) integration samples for \(0,1,2\) detours. We validate results for different parameter settings computed by our prototype RealySt with results from ProHVer[12], which computes a safe overapproximation of the reachability probabilities via discretization.
Table 1 indicates for every considered number of detours the resulting size of the model as tuple \(K=(|\mathit{Var}_{R}|,|\mathcal{R}|,|\mathcal{S}^{\text{goal}}_{\text{reach}}|)\), where \(|\mathit{Var}_{R}|\) is the number of random variables, \(|\mathcal{R}|\) the number of nodes in the reach tree and \(|I_{S}|=|\mathcal{S}^{\text{goal}}_{\text{reach}}|\) the number of traces leading to the goal set. The dimension of the polytopes constructed by forward and backward analysis is \(|\mathit{Var}_{R}|+3\), and the dimension of integration equals \(|\mathit{Var}_{R}|\).
RealySt computes maximum reachability probabilities for all model variants with \(0\) and \(1\) detour in at most \(30\) minutes. For \(2\) detours, the complexity of the model increases considerably. Computations in the singular model A take up to \(3.5\) hours, and in the rectangular variant for \(2\) detours just below \(10\) hours. The number of dimensions in variants AB and ABC with \(2\) detours becomes too large, such that flowpipe construction does not terminate or takes very long. RealySt is able to complete the singular variant of AB in slightly less than \(83\,\mathrm{h}\) and results in probability \(0.431\,565\) with \(e_{\text{stat}}=7.064\cdot 10^{-3}\) statistical error.
The probability to reach an empty battery increases considerably with additional detours, as they introduce uncertainty to the state of charge of the battery. The scheduler can exploit this to maximize the probability of an empty battery.
Maximizing the reachability probability for an undesired goal yields a _worst case_ probability. Reaching an empty battery is undesirable, hence, the computed probability to reach the goal provides an upper bound when everything goes wrong. The results indicate that modeling the charging process in more detail has a relatively low impact on the computed probability. This is expected, as the influence of charging on the state of charge of the battery (rates between \(1\) and
6) is in any case higher than the influence of driving on the state of charge of the battery (rate \(-1\)). Rectangular behavior gives a scheduler plenty opportunities to impact model evolution, which may increase the reachability probability.
As of 1 detour, the results for rectangular and singular models are close. This is due to the singular rates being equal to the lower bounds of the rectangular rate intervals. The scheduler aims to reduce the state of charge of the battery, hence, in most cases it will choose the lowest possible rate.
ProHVer computes safe overapproximations of maximum reachability probabilities. However, its precision highly depends on the chosen number of discretization intervals. A recent release of ProHVer automates the interval generation as well as a refinement thereof, w.r.t. to given parameters. For 0 detours, ProHVer computes a substantial overapproximation of the reachability probability obtained by RealySt. Computation times of ProHVer are between 28 and 140 times larger. For 1 detour, ProHVer takes up to 8 h and results in a better approximation of the reachability probabilities. For 2 detours, ProHVer is not able to perform a refinement on its discretization, yielding quick computation times with a substantial overapproximation. Running ProHVer with alternative parameters which enforce more discretization intervals does not terminate in less than 15 h. We refer to Appendix 0.C.2 for details on the parameter setting of ProHVer; and to Appendix 0.C.3 for details on the computation times of RealySt.
RealySt indicates an error between \(10^{-5}\) up to \(10^{-3}\), which due to Lemma 2, solely stems from integration. For the choice of \(t_{\text{int}}=100\) and the distributions \(\mathcal{N}(4,1)\), \(\mathcal{N}(2,2)\) and \(Exp(2)\), the computed error \(e_{\infty}=0\), using IEEE 754 double precision. Hence, the probabilities computed by RealySt plus the indicated error \(e_{\text{stat}}\) agree with the overapproximations provided by ProHVer.
## 5 Conclusions and Future Work
We propose rectangular automata with random clocks as a new modeling formalism that combines discrete and continuous behavior with random delays. Nondeterminism is usually resolved probabilistically in stochastic hybrid systems. In contrast, this paper presents the first approach to compute maximum reachability probabilities for rectangular automata with random clocks, fully resolving all kinds of discrete and continuous nondeterminism prophetically. The computation requires a combination of forward flowpipe construction with a backward refinement to partition the potentially infinite set of schedulers. The resulting error solely stems from the multidimensional integration. The results of the feasibility study show that RealySt performs very well for up to five random variables. Reachability probabilities are highly accurate and obtained fast in comparison to ProHVer. Future work aims to improve scalability via other state set representations, and compute prophetic minimum reachability probabilities. We will provide an equivalent notion of RAR where restrictions on random delays are placed implicitly via the semantics to ease modeling; a transformation between both will maintain analyzability via RealySt. | この論文は、ランダムクロックを持つrectangular automataにおける可及可能性の最大化を目的としたアルゴリズムを提案します。このモデルクラスには、時間的な非決定性と、動的な非決定性があります。reachable state setsを前向き流体構造を用いて計算した後、後退 refinement を用いて最大可及可能性を計算します。この提案手法の可行性をスケーラブルなモデルを用いて示します。
Please let me know if you need anything else. |
2306.13658 | Synthetic Dimensions | Novel geometries can be created by coupling internal states of atoms or
molecules to mimic movement in real-space | Kaden R. A. Hazzard, Bryce Gadway | 2023-06-05T20:44:21 | http://arxiv.org/abs/2306.13658v1 | ## Synthetic Dimensions
###### Abstract
One of the most basic laws of nature is that spatial motion is restricted to three dimensions, but a wide range of experiments have recently manipulated atoms, molecules, and light to engineer artificial matter that is so configurable that even this fundamental rule can be broken. This matter can behave as if it were in four or more spatial dimensions, or restricted to two or one dimension, as determined by the experimental design. These techniques can control not only dimensionality, but spatial geometries and potential energy landscapes. Frequently, the synthetic dimensions are created in quantum mechanical systems, so these experiments provide powerful windows into the hard-to-understand world of interacting quantum matter, which underpins many fields from quantum gravity to solid-state physics.
To understand synthetic dimensions, it helps to abstract away many details and note that physical theories - whether classical or quantum mechanical - have two ingredients: first, a set of states the system can occupy, and, second, rules for how to move between these states. For example, in classical mechanics, the state is the positions (and velocities) of particles and the rules are Newton's laws. Dimensionality is defined by these rules. Particles in one dimensional systems can step forward or backward, much like a walker on a tightrope. In three dimensions, they can also move up or down, and left or right.
The idea of synthetic dimensions is simply to allow some set of states to play the role of spatial positions, and apply controls, usually electromagnetic fields, that implement the desired rules of motion. This creates a system that is mathematically equivalent to a particle moving in a new spatial dimension, and provides novel capabilities to control geometry and other important aspects of motion in the synthetic dimension. We now look at a concrete example.
## Highly Excited (Rydberg) atoms as a platform for synthetic dimensions
Synthetic dimensions have been created in numerous systems. To begin concretely, we concentrate on how synthetic dimensions have been realized using highly excited atoms known as Rydberg atoms.
This platform has been experimentally demonstrated in recent work by Profs. Tom Killian and Barry Dunning that one of us (K.H.) collaborated on as a theorist, and is being actively explored in B.G.'s lab, as well as in others around the world. Synthetic "positions" are realized by excited electronic states, while microwave-frequency electromagnetic radiation drives transitions between the electronic states, providing the "rules of the road" for what motion is allowed. These rules are mathematically equivalent to those obeyed by particles in a real lattice, for example electrons in a crystal or a molecule.
Figure 1 shows an example of how motion in a real molecule - here polyacetylene, (C\({}_{2}\)H\({}_{2}\))\({}_{n}\) - can be equivalent to motion in a synthetic dimension. In the portion of the molecule shown, an electron can occupy one of six carbon atoms (dark gray balls) along the backbone, labeled as \(r\) = 0, 1,..., 5. The equivalent synthetic positions are six highly excited electronic states, known as Rydberg states. You can think of these states essentially
Figure 1: Rydberg atom synthetic dimensions. Left: Highly excited electronic states of *Sr atoms that are cooled to nanoKelvin temperatures act as position states in a synthetic dimension, while microwave electromagnetic radiation drives transitions that act as quantum tunnelings between sites in the synthetic dimension. In the illustrated scenario, states of the Rydberg atoms are connected in a one-dimensional geometry, and a staggered pattern of tunneling strengths mimics the lattice structure found in the organic conductor polyacetylene, which gives rise to interesting topological phenomena. | The sentence you want to translate is: Novel geometries can be created by coupling internal states of atoms or molecules to mimic movement in real-space.
Here is the translation:
**原子や分子の内部状態を連携させることで、実空間の動きを模倣した新しい幾何学模様を作ることができる。** |
2305.04674 | Entangled coherent states and violations of Bell-CHSH inequalities | Three classes of entangled coherent states are employed to study the
Bell-CHSH inequality. By using pseudospin operators in infinite dimensional
Hilbert spaces, four dichotomic operators $(A,A',B,B')$ entering the inequality
are constructed. For each class of coherent states, we compute the correlator
$\langle \psi \vert A B + A' B + A B' - A' B' \vert \psi \rangle$, analyzing
the set of parameters that leads to a Bell-CHSH inequality violation and,
particularly, to the saturation of Tsirelson's bound. | Philipe De Fabritiis, Fillipe M. Guedes, Giovani Peruzzo, Silvio P. Sorella | 2023-05-08T12:51:01 | http://arxiv.org/abs/2305.04674v2 | # Entangled coherent states and violations of Bell-CHSH inequalities
###### Abstract
Three classes of entangled coherent states are employed to study the Bell-CHSH inequality. By using pseudospin operators in infinite dimensional Hilbert spaces, four dichotomic operators \((A,A^{\prime},B,B^{\prime})\) entering the inequality are constructed. For each class of coherent states, we compute the correlator \(\langle\psi|AB+A^{\prime}B+AB^{\prime}-A^{\prime}B^{\prime}|\psi\rangle\), analyzing the set of parameters that leads to a Bell-CHSH inequality violation and, particularly, to the saturation of Tsirelson's bound.
## I Introduction
The advent of Quantum Mechanics has changed our way of seeing the world, showing that Nature can be subtle and does not always work in the way our intuition suggests. A striking example of a non-intuitive feature present in Nature is the phenomenon of entanglement [1], that is, the existence of states of a composite system that cannot be written as a product of the states of individual subsystems [2]. An entangled system has quantum correlations between its constituents that cannot be accounted for by any local realistic theory, as shown by Bell in 1964 through the formulation of his renowned inequality [3; 4]. There is a particularly interesting and popular version of Bell's inequality that is more suitable for experimental verifications, the so-called Clauser-Horne-Shimony-Holt (CHSH) inequality [5; 6; 7]. The CHSH inequality is violated by entangled states in Quantum Mechanics [8; 9; 10; 11; 12; 13; 14; 15; 16; 17], with its maximum violation being given by Tsirelson's bound [18; 19]: \(2\sqrt{2}\). The existence of entanglement can be considered the deepest departure from classical physics [20], having far-reaching consequences, both from the theoretical and technological sides [21].
Among the large set of quantum states which exists in a given Hilbert space, the so-called coherent states [22; 23; 24; 25] exhibit properties which, in the large occupation number, allow us to regard them as the most classical objects that one can devise in a quantum system. Yet, systems described by these states can exhibit entanglement, leading to relevant developments in many areas such as quantum information and quantum optics [26]. There is a great interest in superpositions of coherent states, a subject that first appeared in [27; 28], and was analyzed in more detail later in [29; 30]. The production of such states in the case of a single mode of the electromagnetic field was studied in [31], and their main properties and extensions in [32; 33; 34; 35; 36; 37; 38; 39]. For a review on this subject, see [40].
Entangled coherent states first appeared in 1967 [41] and had to wait almost twenty years for their next appearance, in [29]. The so-called pair coherent state [42; 43; 44] appeared in the same year, being a special case of the already known state considered in [39]. Although these states were somehow present in the literature since 1967, the first time entangled coherent states were directly studied was in [45; 46], where the authors generalized to multimode coherent states what was done in [27; 28; 29]. The term _entangled coherent state_ was introduced only in 1992 [47], in a paper that analyzed the Bell-CHSH inequality violation for such states and also how to produce them. In that work, the few-photon limit was considered, but later on, it was shown that entangled coherent states violate the Bell inequality in the large photon number limit as well [48].
These states can be generalized to superpositions of multimode coherent states [49; 50; 51]. There are important examples of multipartite states such as the Greenberger-Horne-Zeilinger (GHZ) and W-types of states [52; 53] as well as the cluster states [54; 55; 56]. One can also consider more general entangled coherent states upon considering abstract generalizations of coherent states [35; 36; 37; 38]. It is worth underlining that entangled coherent states are currently employed in quantum teleportation [57; 58; 59; 60; 61; 62; 63], in quantum information processing [64; 65; 66], in quantum networks [67; 68] and in quantum metrology [69; 70; 71; 72; 73].
The aim of this paper is to present a study of the Bell-CHSH inequality violation by considering a variety of entangled coherent states. We shall check out when
\[|\langle\psi|\,\mathcal{C}_{CHSH}\,|\psi\rangle|>2\;, \tag{1}\]
where \(\mathcal{C}_{CHSH}=(A+A^{\prime})B+(A-A^{\prime})B^{\prime}\) is the Bell-CHSH operator and \(|\psi\rangle\) stands for an entangled coherent state. Here, the dichotomic Bell operators are defined such that
\[A^{2}=B^{2}=1;\quad A^{\dagger}=A,\,B^{\dagger}=B;\quad[A,B]=0, \tag{2}\]
with similar equations holding for the primed operators. These operators will be obtained by relying on the so-called pseudospin operators [74; 75; 76], which reproduce the algebra of the Pauli matrices in the case of infinite dimensional Hilbert spaces needed for defining coherent states.
This work is organized as follows. In Section II we give a brief account of entangled coherent states. Section III is devoted to the construction of Bell operators in the infinite dimensional Hilbert space by means of the pseudospin operators. In Section IV, we perform a detailed investigation of the Bell-CHSH inequality violation for the so-called symmetric entangled coherent states. Section V deals with the asymmetric case. The Schrodinger cat states are studied in Section VI. Finally, in Section VII, we state our conclusions.
## II Entangled coherent states
Bosonic coherent states can be defined as the annihilation operator eigenstates and can be represented in the number state basis as
\[|\alpha\rangle=e^{-\frac{\alpha^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{ \sqrt{n!}}|n\rangle\,, \tag{3}\]
where \(|n\rangle=\frac{(a^{\dagger})^{n}}{\sqrt{n!}}|0\rangle\) are the Fock basis states, and we have by construction \(a|\alpha\rangle=\alpha|\alpha\rangle\). Here, \(\alpha\) is taken as a real number. An equivalent formulation can be given by employing the unitary displacement operator \(\mathcal{D}(\alpha)\):
\[\mathcal{D}(\alpha)=e^{(\alpha a^{\dagger}-\alpha a)}=e^{\frac{ \alpha^{2}}{2}}e^{\alpha a^{\dagger}}e^{-\alpha a},\] \[\mathcal{D}^{\dagger}(\alpha)\mathcal{D}(\alpha)=\mathcal{D}( \alpha)\mathcal{D}^{\dagger}(\alpha)=1\;,\] \[\mathcal{D}^{\dagger}(\alpha)\,a\,\mathcal{D}(\alpha)=a+\alpha \tag{4}\]
The coherent state \(|\alpha\rangle\) is obtained by acting on the vacuum state with the operator \(\mathcal{D}(\alpha)\), namely,
\[|\alpha\rangle=\mathcal{D}(\alpha)|0\rangle. \tag{5}\]
Entangled coherent states can be built by means of the superposition of multimode coherent states and have a huge number of applications, as highlighted in the Introduction. For instance, in optics, there is much interest in the use of two-mode maximally entangled number states, the so-called \(N00N\) states
\[|\psi\rangle_{N00N}=\frac{1}{\sqrt{2}}\left[|N\rangle_{a}|0\rangle_{b}+e^{i \phi}|0\rangle_{a}|N\rangle_{b}\right], \tag{6}\]
that can find applications in quantum metrology, quantum sensing, and quantum interferometric photolithography [77; 78]. In [79], the authors investigated possible non-local correlation experiments using \(N00N\) states with a relative phase \(\phi=\pi\), showing the violation of some Bell-type inequalities for a total number of photons \(N\)1. In [81], the authors extended the analysis done in [79] considering maximally entangled coherent states
Footnote 1: The Bell-CHSH inequality for \(N=1\) has been studied in [80].
\[|\psi_{\alpha}\rangle=N_{\alpha}e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty} \frac{\alpha^{n}}{\sqrt{n!}}\big{[}|n\rangle_{a}|0\rangle_{b}+e^{i(\phi+n \theta)}|0\rangle_{a}|n\rangle_{b}\big{]}, \tag{7}\]
where \(N_{\alpha}=\frac{1}{\sqrt{2}}\left(1+\cos\phi\,e^{-|\alpha|^{2}}\right)^{- \frac{1}{2}}\). These states can be considered as superpositions of \(N00N\) states, and can be useful in the context of interferometry [82; 83]. For the cases considered by them, it was found a greater degree of violation for the Bell-type inequalities considered, in comparison with the \(N00N\) states results [79].
More elaborated states can be obtained by considering the symmetric and asymmetric entangled coherent states, a slight generalization of the cases considered in [81; 84]. Here we define the symmetric case as
\[|\psi\rangle_{S}=N_{S}\left[|\alpha\rangle_{a}|\beta\rangle_{b}+e^{i\phi}|\beta \rangle_{a}|\alpha\rangle_{b}\right], \tag{8}\]
and the asymmetric case as
\[|\psi\rangle_{A}=N_{A}\left[|\alpha\rangle_{a}|\beta\rangle_{b}+e^{i\phi}|- \alpha\rangle_{a}|-\beta\rangle_{b}\right], \tag{9}\]
where \((N_{S},N_{A})\) stand for normalization factors which will be given below and \(\phi\) is a relative phase. Finally, we shall also consider entangled coherent states built out from Schrodinger cat states [85], _i.e._,
\[|\psi\rangle_{\pm}=C_{\pm}\left[|\alpha\rangle_{\pm}|\beta\rangle_{\pm}+e^{i \phi}|\beta\rangle_{\pm}|\alpha\rangle_{\pm}\right], \tag{10}\]
where \(C_{\pm}\) is a normalization factor and we defined
\[|\alpha\rangle_{\pm}=N_{\pm}\left[|\alpha\rangle_{\pm}|-\alpha\rangle\right]. \tag{11}\]
In what follows, we shall present a detailed investigation of the Bell-CHSH inequality violation, Eq. (1), for both symmetric and asymmetric entangled coherent states, Eqs. (8),(9), as well as for the cat states (10). We would like to emphasize that, although we used the specific states cited above to illustrate our method, we could apply it in principle to any entangled coherent state.
## III Bell's operators construction
The first goal to introduce the Bell-CHSH correlator,
\[\mathcal{C}_{CHSH}=AB+A^{\prime}B+AB^{\prime}-A^{\prime}B^{\prime}, \tag{12}\]
is to construct Bell's operators \((A,A^{\prime},B,B^{\prime})\) fulfilling:
\[A^{2}=B^{2}=1;\quad A^{\dagger}=A,\,B^{\dagger}=B;\quad[A,B]=0, \tag{13}\]
with similar equations holding for the primed operators. As we are working in Hilbert spaces which have infinite dimensions, this task can be accomplished by means of the pseudospin operators [74; 75; 76] defined as
\[s_{x}=\sum_{n=0}^{\infty}s_{x}^{(n)}\;,\qquad s_{y}=\sum_{n=0}^{\infty}s_{y}^{( n)}\;,\qquad s_{z}=\sum_{n=0}^{\infty}s_{z}^{(n)}\;, \tag{14}\]
where
\[s_{x}^{(n)} = |2n+1\rangle\langle 2n|+|2n\rangle\langle 2n+1|,\] \[s_{y}^{(n)} = i\left(|2n\rangle\langle 2n+1|-|2n+1\rangle\langle 2n|\right),\] \[s_{z}^{(n)} = |2n+1\rangle\langle 2n+1|-|2n\rangle\langle 2n|. \tag{15}\]
An easy calculation shows that
\[\left[s^{(n)}_{x},s^{(n)}_{y}\right] = 2is^{(n)}_{z}\;,\] \[\left[s^{(n)}_{y},s^{(n)}_{z}\right] = 2is^{(n)}_{x}\;,\] \[\left[s^{(n)}_{z},s^{(n)}_{x}\right] = 2is^{(n)}_{y}\;. \tag{16}\]
As a consequence, it follows that these operators obey the same algebraic relations of the spin \(1/2\) Pauli matrices
\[[s_{x},s_{y}]=2is_{z},\quad[s_{y},s_{z}]=2is_{x},\quad[s_{z},s_{x}]=2is_{y}, \tag{17}\]
from which the name _pseudospin_ follows.
In particular, from expressions (15) one observes that the introduction of the pseudospin operators can be related to a pairing mechanism in Hilbert space, a pair being given by two states, namely \((|2n\rangle,|2n+1\rangle)\), with \(n=0,1,2,...\) Each pair of states gives raise to a set of operators, \((s^{(n)}_{x},s^{(n)}_{y},s^{(n)}_{z})\), which obey the same spin \(1/2\) algebra of Pauli matrices.
The observation of the pairing mechanism goes back to [86]. More recently, its applications to the study of the Bell-CHSH inequality has been discussed in [87; 88; 89], where it has been shown that each single pair might be employed for a test of the Bell-CHSH inequality. This is the setup which we shall adopt in the following.
More precisely, we shall analyze the Bell-CHSH inequality by considering two cases, namely:
* The Bell operators act non-trivially on a single pair, identified, for example, with the states \((|0\rangle,|1\rangle)\). Let \(|x,y\rangle\) stand for a generic basis element of the Hilbert space \(\mathcal{H}\otimes\mathcal{H}\) to which the entangled coherent states, Eqs. (8),(9),(10), belong. Then, for the operators \((A,B)\) we shall set \[A|0,y\rangle=e^{ia}|1,y\rangle;\;A|1,y\rangle=e^{-ia}|0,y\rangle;\; \forall y,\] \[B|x,0\rangle=e^{ib}|x,1\rangle;\;B|x,1\rangle=e^{-ib}|x,0\rangle; \;\forall x,\] (18) and acting as the identity on all the other states, _i.e._, \[A|x,y\rangle=|x,y\rangle,\quad\forall x\geq 2,\] \[B|x,y\rangle=|x,y\rangle,\quad\forall y\geq 2.\] (19) The quantities \((a,b)\) are arbitrary real parameters. One sees that the operator \(A\) acts only on the first entry of \(|x,y\rangle\), while the operator \(B\) only on the second one. In terms of the pseudospin operators, it turns out that the operator \(A\) can be written as \[A=\left(\tilde{u}\cdot\tilde{s}^{(0)}+\mathcal{R}\right)\otimes I,\] (20) where \(\tilde{u}\) denotes the unit vector \[\tilde{u}=\left(\cos(a),-\sin(a),0\right),\qquad\tilde{u}\cdot\tilde{u}=1,\] (21) and \(\mathcal{R}\) is the identity operator for \(x\geq 2\): \[\mathcal{R}=\sum_{n=2}^{\infty}|n\rangle\langle n|.\] (22) Analogous expressions can be written down for \(B\), \(A^{\prime}\) and \(B^{\prime}\). For the primed operators, the parameters \(a\) and \(b\) are simply replaced by \(a^{\prime}\) and \(b^{\prime}\).
* In the same vein, we can define a second Bell setup: \[A|2n,y\rangle = e^{ia}|2n+1,y\rangle;\] \[A|2n+1,y\rangle = e^{-ia}|2n,y\rangle;\;\forall y,\] \[B|x,2n\rangle = e^{ib}|x,2n+1\rangle;\] \[B|x,2n+1\rangle = e^{-ib}|x,2n\rangle;\;\forall x.\] (23) with similar expressions for the primed operators \((A^{\prime},B^{\prime})\). One sees that, in the second setup, all states of the Hilbert space have been grouped in pairs. In terms of pesusdospin operators, we have \[A=\tilde{u}\cdot\tilde{s}\otimes I,\] (24) where \(\tilde{u}\) is the unit vector of expression (21).
It is immediate to check that in both setups the required properties (13) for the Bell-type operators are satisfied. We remark that in order to investigate the Bell-CHSH inequality violation, one needs only to compute the expected value of the product \(AB\) on the state \(|\psi\rangle\) of interest, since all the other combinations, that is, \(A^{\prime}B\), \(AB^{\prime}\), and \(A^{\prime}B^{\prime}\), can be achieved by putting primes on the respective parameters. Therefore, in the following we will restrict ourselves to explicitly state only the result for \(\langle\psi|AB|\psi\rangle\), being the complete expression for \((\psi|\mathcal{C}_{CHSH}|\psi\rangle)\) left understood to avoid cluttering the text.
We are ready now to investigate the Bell-CHSH inequality violation for the three entangled coherent states (8),(9) and (10), starting with the symmetric state.
## IV Symmetric coherent states
Let us consider here the symmetric entangled coherent states, a generalization of the states considered in [81], obtained by the entanglement of two different coherent states with an arbitrary relative phase \(\phi\):
\[|\psi\rangle_{S}=N_{S}\!\left[|\alpha\rangle_{a}|\beta\rangle_{b}+e^{i\phi}| \beta\rangle_{a}|\alpha\rangle_{b}\right], \tag{25}\]
where the normalization factor is given by
\[N_{S}=\frac{1}{\sqrt{2}}\left(1+\cos\phi\exp\left[-\left(\alpha-\beta\right)^ {2}\right]\right)^{\!\!-1\!/2}. \tag{26}\]
In the sequel, we analyze the Bell-CHSH violation for the symmetric states (25) considering the two different Bell setups defined in Eqs. (18),(19),(23).
### First setup
Adopting the first setup, defined by Eqs. (18),(19), we can compute the expectation value \(\langle\psi|AB|\psi\rangle\) in the symmetric state defined by Eq. (25). Thus, we find:
\[\langle AB\rangle_{S_{1}} = \Omega_{S}\Big{\{}\left(\cos a+\cos b\right)\left[\beta\left(e^{ \alpha^{2}}-1-\alpha^{2}\right)\right. \tag{27}\] \[+ \alpha\left(e^{\beta^{2}}-1-\beta^{2}\right)\Big{]}+\left(e^{ \alpha\beta}-1-\alpha\beta\right)\] \[\times \left[\alpha\left(\cos(a+\phi)+\cos(b-\phi)\right)\right.\] \[+ \beta\left(\cos(a-\phi)+\cos(b+\phi)\right)\Big{]}\] \[+ \left[4\alpha\beta\cos a\cos b+2\alpha\beta\cos\phi\cos(a+b)\right.\] \[+ \beta^{2}\cos(a-b-\phi)+\alpha^{2}\cos(a-b+\phi)\Big{]}\] \[+ \left[\left(e^{\alpha^{2}}-1-\alpha^{2}\right)\left(e^{\beta^{2}} -1-\beta^{2}\right)\right.\] \[\left.+\cos\phi\left(e^{\alpha\beta}-1-\alpha\beta\right)^{2} \right]\right\},\]
where the overall factor is given by
\[\Omega_{S}=\frac{\exp\left[-\left(\alpha^{2}+\beta^{2}\right)\right]}{1+\cos \phi\,\exp\left[-\left(\alpha-\beta\right)^{2}\right]}. \tag{28}\]
Considering a relative phase \(\phi=\pi\), we will adopt the following choice of parameters to find a Bell-CHSH inequality violation: \(a=0\), \(a^{\prime}=\pi/2\), \(b=+\pi/4\), \(b^{\prime}=-\pi/4\). In this case, the expression for \(\langle\mathcal{C}_{CHSH}\rangle\) simplifies considerably, giving us:
\[\langle\mathcal{C}_{CHSH}\rangle = \frac{e^{-(\alpha^{2}+\beta^{2})}}{1-e^{-(\alpha-\beta)^{2}}} \Big{\{}2e^{\alpha^{2}+\beta^{2}}\left(1-e^{-(\alpha-\beta)^{2}}\right) \tag{29}\] \[- 2(\sqrt{2}-1)(\alpha-\beta)^{2}-e^{\beta^{2}}\left[2-(2+\sqrt{2} )\alpha+2\alpha^{2}\right]\] \[- e^{\alpha^{2}}\left[2-(2+\sqrt{2})\beta+2\beta^{2}\right]\] \[+ e^{\alpha\beta}\left[4+4\alpha\beta-(2+\sqrt{2})(\alpha+\beta) \right]\Big{\}}.\]
From the above expression, taking \(\alpha=0.1\) and \(\beta=0.2\) we already obtain \(|\langle\mathcal{C}_{CHSH}\rangle|=2.6939\). Interestingly, if we keep one of them small, we can take the other large and still find a violation. For instance, keeping \(\beta=0.01\), we can take \(\alpha=0.70\) and still find \(|\langle\mathcal{C}\rangle|=2.1699\). Considering smaller \(\alpha\) and \(\beta\), for instance, \((\alpha,\beta)=(0.0001,0.0002)\), we can saturate the Tsirelson's bound finding \(|\langle\mathcal{C}\rangle|=2.8284\approx 2\sqrt{2}\). It is important to remark that with the parameters adopted here, the result is symmetric under the exchange \(\alpha\leftrightarrow\beta\). Finally, considering fixed values for \(\alpha\) and \(\beta\), we can find a small range for the relative phase around \(\phi=\pi\) that still exhibits a violation. For instance, taking \((\alpha,\beta)=(0.5,0.1)\), we still observe a violation for phases \(\phi\) in the range \((\pi-0.2,\pi+2)\). Without a relative phase \((\phi=0)\), we were not able to find any violation.
In order to simplify the visualization of these features, we show \(\langle\mathcal{C}\rangle\) as a function of \(\alpha\), considering \(\beta=\alpha+0.001\) in Fig. 1. Notice that there is violation for small values of \(\alpha\), a saturation of the Tsirelson's bound for very small \(\alpha\), and the result asymptotes to \(2\) for large values of \(\alpha\). One can see explicitly the region of parameters \((\alpha,\beta)\) leading to Bell-CHSH inequality violation in Fig. 2.
### Second setup
Now, continuing our analysis, we consider the more general setup given by Eq. (23). In this case, we find for the expectation value of \(AB\) in the symmetric state:
\[\langle AB\rangle_{S_{2}} = \Omega_{S}\sum_{n,m=0}^{\infty}\frac{1}{\sqrt{(2n)!(2n+1)!(2m)!(2m+ 1)!}}\] (30) \[\times \Big{\{}4\cos a\cos b\,\alpha^{4n+1}\beta^{4m+1}
\[+(\alpha\beta)^{2n+2m}\big{[}2\alpha\beta\cos\phi\,\cos(a+b)\] \[+\alpha^{2}\cos(a-b+\phi)+\beta^{2}\cos(a-b-\phi)\big{]}\Big{]}, \tag{30}\]
where the overall factor here is the same as before, given by Eq. (28). This time, unfortunately, we were not able to find a closed analytical expression for the above sum. Even tough, we can do the analysis by considering the appropriate number of terms in the series in order to stabilize the result for a given choice of parameters.
Once again, we consider \(\phi=\pi\) and the following choice of parameters: \(a=0\), \(a^{\prime}=\pi/2\), \(b=+\pi/4\), \(b^{\prime}=-\pi/4\). First of all, we take the first contribution in the series, that is, the \(n=m=0\) term. It can be written as:
\[\langle AB\rangle|_{n=m=0}=\frac{2\sqrt{2}(\alpha-\beta)^{2}}{e^{2\alpha\beta }-e^{\alpha^{2}+\beta^{2}}}. \tag{31}\]
For small values of \(\alpha\) and \(\beta\), the first order term is already \(-2\sqrt{2}\) plus small corrections, encouraging us to proceed and consider the complete expression. Upon considering \((\alpha,\beta)=(0.1,0.2)\), we already find \(|\langle C_{CHSH}\rangle|=2.7018\). As before, we can saturate the Tsirelson's bound if we take smaller values of \(\alpha\) and \(\beta\). Once again, keeping one of them small, we have freedom to enlarge the other and still observe a violation. For instance, taking \((\alpha,\beta)=(0.70,0.001)\), we obtain \(|\langle\mathcal{C}_{CHSH}\rangle|=2.1732\). Upon choosing \((\alpha,\beta)=(0.5,0.1)\) as before, we still observe a violation for phases \(\phi\) in the range \((\pi-0.2,\pi+2)\), and we were not able to find any violation for \(\phi=0\).
As before, we show \((\mathcal{C})\) as a function of \(\alpha\), considering \(\beta=\alpha+0.001\) in Fig. 3. The region of parameters \((\alpha,\beta)\) leading to Bell-CHSH inequality violation can be seen in Fig. 4. We remark that the qualitative behavior of these graphs are the same as in the last subsection, with only small quantitative differences between them.
## V Asymmetric coherent states
Now, we consider the asymmetric entangled coherent states, similar to the case analyzed in [84], but with a different Bell setup, and here considering a general relative phase \(\phi\). Thus, we define these states as
\[|\psi\rangle_{A}=N_{A}\big{[}|\alpha\rangle_{a}|\beta\rangle_{b}+e^{i\phi}|- \alpha\rangle_{a}|-\beta\rangle_{b}\big{]}, \tag{32}\]
where the normalization factor here is given by
\[N_{A}=\frac{1}{\sqrt{2}}\big{(}1+\cos\phi\exp\big{[}-2\left(\alpha^{2}+\beta ^{2}\right)\big{]}\big{)}^{-1/2}. \tag{33}\]
In the following, we analyze the Bell-CHSH violation for the asymmetric states (32) considering the two Bell setups defined before, as was done for the symmetric states.
### First setup
To begin with, we consider the first setup defined by Eqs. (18),(19). Computing the expectation value \((\psi|AB|\psi)\) on the asymmetric state defined in Eq. (32):
\[\langle AB\rangle_{A_{1}} =\Omega_{A}\Big{\{}4\alpha\beta\left(\cos a\cos b-\cos\phi\sin a \sin b\right)\] \[-2\sin\phi\big{[}\alpha\sin a\left(-1+\beta^{2}+e^{-\beta^{2}} \right)\] \[+\beta\sin b\left(-1+\alpha^{2}+e^{-\alpha^{2}}\right)\big{]}\] \[+\big{[}\left(e^{\alpha^{2}}-1-\alpha^{2}\right)\left(e^{\beta^{ 2}}-1-\beta^{2}\right)\] \[+\cos\phi\left(e^{-\alpha^{2}}-1+\alpha^{2}\right)\left(e^{-\beta ^{2}}-1+\beta^{2}\right)\big{]}\Big{\}}, \tag{34}\]
Figure 3: \((\mathcal{C})\) correlator in blue, as a function of \(\alpha\) in the symmetric case (second setup), with \(\beta=\alpha+0.001\). Here we considered \(\phi=\pi,\,a=0,\,a^{\prime}=\pi/2,\,b=+\pi/4,\,b^{\prime}=-\pi/4\). The red line represents the Tsirelson’s bound.
where the overall factor here is given by
\[\Omega_{A}=\frac{\exp\left[-\left(\alpha^{2}+\beta^{2}\right)\right]}{1+\cos\phi \exp\left[-2\left(\alpha^{2}+\beta^{2}\right)\right]}. \tag{35}\]
Here we also consider the relative phase \(\phi=\pi\) and the same set of parameters used in the last section, that is: \(a=0,\,a^{\prime}=\pi/2,\,b=+\pi/4,\,b^{\prime}=-\pi/4\). With this choice, the expression for \(\langle\mathcal{C}\rangle\) significantly simplifies:
\[\langle\mathcal{C}_{CHSH}\rangle=2-\frac{2}{\sinh(\alpha^{2}+ \beta^{2})}\Big{[}-2\sqrt{2}\alpha\beta-\alpha^{2}-\beta^{2}\] \[+\sinh\alpha^{2}+\sinh\beta^{2}+\alpha^{2}\cosh\beta^{2}+\beta^{ 2}\cosh\alpha^{2}\Big{]}. \tag{36}\]
From the above expression one can see that there is a considerable range of parameters \(\alpha\) and \(\beta\) providing a violation. In fact, we observe that the Tsirelson's bound is saturated already with \(\alpha=\beta=0.06\). Moreover, upon keeping \(\alpha=\beta\), we can find violations from \(\alpha=\beta=0.1\) (with \(\langle\mathcal{C}_{CHSH}\rangle=2.8282\)) to \(\alpha=\beta=0.8\) (with \(\langle\mathcal{C}\rangle=2.0218\)). But if we do not take them close to each other, we do not find any violation. For instance, considering \((\alpha,\beta)=(0.1,0.3)\) we find \(\langle\mathcal{C}_{CHSH}\rangle=1.6942\) while for \((\alpha,\beta)=(0.3,0.3)\) we find \(\langle\mathcal{C}_{CHSH}\rangle=2.8132\). Therefore, in this case it is important to keep \(\alpha\) and \(\beta\) close to find violations. As before, fixing values for \(\alpha\) and \(\beta\), we can search for violation with relative phases around \(\phi=\pi\). In the asymmetric case, we have more space for violation in the relative phase than in the previous case. In fact, adopting \(\alpha=\beta=0.5\), we find violation for \(\phi\in(\pi-0.78,\pi+0.78)\), a considerably larger range in comparison with the symmetric case studied before.
In order to visualize better the results, we show \(\langle\mathcal{C}\rangle\) as a function of \(\alpha\) in Fig. 5, considering \(\alpha=\beta\). Notice that the region of parameters \(\alpha\) leading to a violation is considerably larger than the one obtained in the symmetric case studied before. Moreover, we saturate the Tsirelson's bound for small \(\alpha\), and the result asymptotes to \(2\) for large values of \(\alpha\), as before. The region of parameters \((\alpha,\beta)\) leading to a violation is shown in Fig. 6. We also remark that there is a more pronounced violation in the asymmetric case, comparing with the symmetric one.
### Second setup
Giving sequence to our analysis, we now consider the more general setup defined by Eq. (23). The expectation value \(\langle\psi|AB|\psi\rangle\) in the asymmetric state defined in Eq. (32) is given by:
\[\langle AB\rangle_{A}=4\Omega_{A_{2}}\,\left(\cos a\cos b-\cos \phi\sin a\sin b\right)\\ \times\sum_{n,m=0}^{\infty}\Bigg{[}\frac{\alpha^{4n+1}\beta^{4m+1 }}{\sqrt{(2n)!(2n+1)!(2m)!(2m+1)!}}\Bigg{]}, \tag{37}\]
where the overall factor here is the same as before, given by Eq. (35). Although the expression obtained is pretty simple, this infinite sum cannot be written in a closed analytic form. The simple angular structure of this expression will make this example very interesting.
Also here, we choose to work with the same parameters \(a=0,\,a^{\prime}=\pi/2,\,b=+\pi/4,\,b^{\prime}=-\pi/4\). Firstly, we consider the first contribution of this sum, taking the \(n=m=0\) term, that can be written as:
\[\langle AB\rangle|_{n=m=0}=\frac{4\sqrt{2}\,\alpha\beta}{\sinh \left(\alpha^{2}+\beta^{2}\right)}. \tag{38}\]
For small values of \(\alpha\) and \(\beta\) we have a result close to \(2\sqrt{2}\), encouraging us to proceed with the analysis using the
complete expression. Taking \(\alpha=\beta=0.1\), we already find \((\mathcal{C}_{CHSH})=2.8284\simeq 2\sqrt{2}\). Once again, for small values of \(\alpha\) and \(\beta\), it is important to keep these parameters close to each other in order to find a violation. Considering \(\alpha=\beta\), we have found a Bell-CHSH inequality violation for all the considered parameters \(\alpha\). For instance, taking \(\alpha=\beta=0.01\) we find \((\mathcal{C}_{CHSH})=2.8284\); for \(\alpha=\beta=1\), we have \((\mathcal{C})=2.6678\); considering \(\alpha=\beta=5\), we have \((\mathcal{C})=2.7997\). We remark that here we also have a considerable freedom in the phase. For example, adopting \(\alpha=\beta=0.5\), we find violation for any \(\phi\in(\pi-0.81,\pi+0.81)\).
Once more, we show \((\mathcal{C})\) as a function of \(\alpha\) in Fig. 7, considering \(\alpha=\beta\). In this case, we are finding a large violation for all the considered values of \(\alpha\). The region of parameters \((\alpha,\beta)\) leading to a violation is also large, as one can immediately see in Fig. 8.
Finally, it is noteworthy that in this case, we managed to find a Bell-CHSH inequality violation even in the case without a relative phase, that is, with \(\phi=0\). In fact, taking \(\phi=0\) we can consider now a slightly different set of parameters: \(a=0\), \(a^{\prime}=\pi/2\), \(b=-\pi/4\), \(b^{\prime}=+\pi/4\). Thus, for \(\alpha=\beta=0.7\) we find already \((\mathcal{C})=2.0895\). There is violation in this case also for larger values of \(\alpha\), as one can see in Fig. 9. For different values of \(\alpha\) and \(\beta\), see Fig. 10. We remark that a Bell-CHSH inequality violation for the states defined by Eq. (32) for \(\phi=\pi\) and \(\phi=0\) was also obtained in [84], although with different Bell setups.
## VI Schrodinger's cat states
Let us proceed now with the so-called entangled Schrodinger cat states. One can define both the even (+) and odd (-) cases simultaneously by the equation:
\[|\alpha\rangle_{\pm}=N_{\pm}\left[|\alpha\rangle\pm|-\alpha\rangle\right], \tag{39}\]
with the normalizations respectively given by
\[N_{+}=\frac{e^{\alpha^{2}/2}}{2\sqrt{\cosh(\alpha^{2})}},\quad N_{-}=\frac{e^ {\alpha^{2}/2}}{2\sqrt{\sinh(\alpha^{2})}}. \tag{40}\]
Therefore, we can consider the two entangled states, considering the even and odd cases separately:
\[|\psi\rangle_{\pm}=C_{\pm}\left[|\alpha\rangle_{\pm}|\beta\rangle_{\pm}+e^{i \phi}|\beta\rangle_{\pm}|\alpha\rangle_{\pm}\right], \tag{41}\]
with normalizations in this case given respectively by
\[C_{+} =\frac{1}{\sqrt{2}}\left[1+\cos\phi\frac{\cosh^{2}(\alpha\beta) }{\cosh(\alpha^{2})\cosh(\beta^{2})}\right]^{-1/2}, \tag{42}\] \[C_{-} =\frac{1}{\sqrt{2}}\left[1+\cos\phi\frac{\sinh^{2}(\alpha\beta) }{\sinh(\alpha^{2})\sinh(\beta^{2})}\right]^{-1/2}. \tag{43}\]
In this case, we need to adapt a bit our Bell setup, since for the even cat state there are only even modes \(|2n\rangle\) and for the odd cat states there are only odd modes \(|2n+1\rangle\). Thus, in this section, we slightly modify our first setup for the even cat states, using
\[A|0,y\rangle =e^{ia}|2,y\rangle;\ A|2,y\rangle=e^{-ia}|0,y\rangle;\ \forall y,\] \[B|x,0\rangle =e^{ib}|x,2\rangle;\ B|x,2\rangle=e^{-ib}|x,0\rangle;\ \forall x, \tag{44}\]
and also for the odd cat states, by defining
\[A|1,y\rangle =e^{ia}|3,y\rangle;\ A|3,y\rangle=e^{-ia}|1,y\rangle;\ \forall y,\] \[B|x,1\rangle =e^{ib}|x,3\rangle;\ B|x,3\rangle=e^{-ib}|x,1\rangle;\ \forall x, \tag{45}\]
and acting as the identity in all other states in both setups. Furthermore, since the cat states \(|\alpha\rangle_{+}\) and \(|\alpha\rangle_{-}\)
have only even and odd modes, respectively, the second Bell setup defined earlier would give vanishing expectation value in this case. Therefore, in the following, we will not consider it, limiting ourselves to the Bell-CHSH inequality violation analysis for the setup described above in this section, for both even and odd entangled cat states defined by Eq. (41).
### Even Cat States
First, we consider the even cat state \(|\psi\rangle_{+}\) as defined in Eq. (41). For this state, one can compute the expectation value of the product \(AB\) and find:
\[\langle AB\rangle_{+}=\Omega_{+}\Bigg{\{}4\alpha^{2}\beta^{2} \cos a\cos b+2\alpha^{2}\beta^{2}\cos\phi\cos(a+b)\] \[+\alpha^{4}\cos(a-b+\phi)+\beta^{4}\cos(a-b-\phi)\] \[+\sqrt{2}\left(\cos a+\cos b\right)\left[\alpha^{2}\left(\cosh( \beta^{2})-1-\frac{\beta^{4}}{2}\right)\right.\] \[+\beta^{2}\left(\cosh(\alpha^{2})-1-\frac{\alpha^{4}}{2}\right) \Big{]}\] \[+\sqrt{2}\bigg{[}\alpha^{2}\left(\cos(a+\phi)+\cos(b-\phi)\right)+\] \[+\beta^{2}\left(\cos(a-\phi)+\cos(b+\phi)\right)\bigg{]}\left( \cosh(\alpha\beta)-1-\frac{(\alpha\beta)^{2}}{2}\right)\] \[+2\left(\cosh(\alpha^{2})-1-\frac{\alpha^{4}}{2}\right)\left( \cosh(\beta^{2})-1-\frac{\beta^{4}}{2}\right)\] \[+2\cos\phi\left(\cosh(\alpha\beta)-1-\frac{\alpha^{2}\beta^{2}}{ 2}\right)^{2}\Bigg{\}}, \tag{46}\]
where the overall factor can be written as
\[\Omega_{+}=\frac{1}{2}\left[\cosh(\alpha^{2})\cosh(\beta^{2})+\cos\phi\,\cosh^ {2}(\alpha\beta)\right]^{-1}. \tag{47}\]
Once more, we adopt the phase \(\phi=\pi\) and the usual set of parameters \(a=0\), \(a^{\prime}=\pi/2\), \(b=+\pi/4\), \(b^{\prime}=-\pi/4\). In this case, the expression for \((\mathcal{C}_{CHSH})\) significantly simplifies:
\[\langle\mathcal{C}_{CHSH}\rangle =\frac{1}{\kappa_{+}}\Big{\{}1+\left(\sqrt{2}-1\right)\left( \alpha^{2}-\beta^{2}\right)^{2}\] \[+\cosh(2\alpha\beta)-2\cosh(\alpha^{2})\cosh(\beta^{2})\] \[-\cosh(\alpha\beta)\left[4+2\alpha^{2}\beta^{2}-(1+\sqrt{2})( \alpha^{2}+\beta^{2})\right]\] \[+\cosh(\alpha^{2})\Big{[}2-(1+\sqrt{2})\beta^{2}+\beta^{4}\Big{]}\] \[+\cosh(\beta^{2})\Big{[}2-(1+\sqrt{2})\alpha^{2}+\alpha^{4}\Big{]} \Big{\}}, \tag{48}\]
where \(\kappa_{+}=\cosh^{2}(\alpha\beta)-\cosh(\alpha^{2})\cosh(\beta^{2})\). Considering \((\alpha,\beta)=(0.1,0.2)\) we find \(|\langle\mathcal{C}\rangle|=2.8278\) and the Tsirelson's bound is saturated already at \((\alpha,\beta)=(0.07,0.08)\). We remark that with the parameters adopted, the result is also symmetric under the exchange \(\alpha\leftrightarrow\beta\) and that one finds a Bell-CHSH violation for almost the whole interval of \(\alpha\) and \(\beta\) between \(0\) and \(1\). Considering the parameters \((\alpha,\beta)=(0.1,0.8)\), we still find violation for relative phases around \(\phi\in(\pi-0.3,\pi+0.3)\).
We exhibit \(\langle\mathcal{C}\rangle\) as a function of \(\alpha\) in Fig. 11, taking \(\beta=\alpha+0.001\). Notice that there is violation for almost all values of \(\alpha\) between \(0\) and \(1\), and that for large values of \(\alpha\) the result asymptotes to \(2\). For the range of parameters \((\alpha,\beta)\) leading to a violation, see Fig. 12.
### Odd Cat States
Continuing our analysis, we consider now the odd cat state \(|\psi\rangle_{-}\) defined in Eq. (41). For this state, the expectation value of the product \(AB\) is given by:
\[\langle AB\rangle_{-}=\Omega_{-}\Bigg{\{}\frac{4}{3}\alpha^{4} \beta^{4}\cos a\cos b+\frac{2}{3}\cos\phi\,\alpha^{4}\beta^{4}\cos(a+b)\] \[+\frac{1}{3}\alpha^{2}\beta^{2}\Big{[}\alpha^{4}\cos(a-b+\phi)+ \beta^{4}\cos(a-b-\phi)\Big{]}\] \[+\frac{2}{\sqrt{6}}\left(\cos a+\cos b\right)\Big{[}\alpha^{4} \left(\sinh(\beta^{2})-\beta^{2}-\frac{\beta^{6}}{6}\right)\] \[+\beta^{4}\left(\sinh(\alpha^{2})-\alpha^{2}-\frac{\alpha^{6}}{6 }\right)\Big{]}\] \[+\frac{2}{\sqrt{6}}\alpha\beta\Big{[}\alpha^{2}\left(\cos(a+\phi )+\cos(b-\phi)\right)\] \[+\beta^{2}\left(\cos(a-\phi)+\cos(b+\phi)\right)\Big{]}\left( \sinh(\alpha\beta)-\alpha\beta-\frac{(\alpha\beta)^{3}}{6}\right)\] \[+2\left(\sinh(\alpha^{2})-\alpha^{2}-\frac{\alpha^{6}}{6}\right) \left(\sinh(\beta^{2})-\beta^{2}-\frac{\beta^{6}}{6}\right)\] \[+2\cos\phi\left(\sinh(\alpha\beta)-\alpha\beta-\frac{(\alpha \beta)^{3}}{6}\right)^{2}\Bigg{\}}, \tag{49}\]
where the overall factor is given by
\[\Omega_{-}=\frac{1}{2}\left[\sinh(\alpha^{2})\sinh(\beta^{2})+\cos\phi\, \sinh^{2}(\alpha\beta)\right]^{-1}. \tag{50}\]
In the same way as before, we adopt \(\phi=\pi\) and the choice of parameters \(a=0\), \(a^{\prime}=\pi/2\), \(b=+\pi/4\), \(b^{\prime}=-\pi/4\). In this case, we immediately find for \(\langle\mathcal{C}\rangle\):
\[\langle\mathcal{C}_{CHSH}\rangle =\frac{1}{3\kappa_{-}}\Big{[}\left(\sqrt{2}-1\right)\alpha^{2} \beta^{2}(\alpha^{2}-\beta^{2})^{2}\] \[+6\sinh^{2}(\alpha\beta)-6\sinh(\alpha^{2})\sinh(\beta^{2})\] \[-\alpha\beta\sinh(\alpha\beta)\left[12+2\alpha^{2}\beta^{2}-( \sqrt{3}+\sqrt{6})(\alpha^{2}+\beta^{2})\right]\] \[+\alpha^{2}\sinh(\beta^{2})\Big{[}6-(\sqrt{3}+\sqrt{6})\alpha^{2} +\alpha^{4}\Big{]}\] \[+\beta^{2}\sinh(\alpha^{2})\Big{[}6-(\sqrt{3}+\sqrt{6})\beta^{2} +\beta^{4}\Big{]}\Big{]}, \tag{51}\]
where \(\kappa_{-}=\sinh^{2}(\alpha\beta)-\sinh(\alpha^{2})\sinh(\beta^{2})\). Considering \((\alpha,\beta)=(0.1,0.2)\) we find \(\langle\mathcal{C}_{CHSH}\rangle=2.8280\) and the Tsirelson's bound is saturated already at \((\alpha,\beta)=(0.08,0.09)\). Furthermore, here we observe a violation for the whole range of parameters \(\alpha\) and \(\beta\) between \(0\) and \(1\). Adopting the parameters \((\alpha,\beta)=(0.1,0.8)\), we still find violation for relative phases around \(\phi\in(\pi-0.2,\pi+0.2)\).
The results for the odd cat states are very similar to the ones obtained for even cat states, as one can immediately see in Fig. 13, where we show \(\langle\mathcal{C}\rangle\) as a function of \(\alpha\) with \(\beta=\alpha+0.001\), and also in Fig. 14, where the range of parameters \((\alpha,\beta)\) leading to a violation is shown.
## VII Conclusions
In this work, we investigated the violation of Bell-CHSH inequalities considering some interesting examples of entangled coherent states. The Bell's operators construction by using the pseudospin operators plays a prominent role, allowing a detailed analysis of the entangled coherent states considered here. We have studied in two different Bell setups three types of entangled coherent states: the symmetric, the asymmetric, and the
cat states. In each of them, we computed the Bell-CHSH correlator and presented the set of parameters that leads to the explicit violation of Bell-CHSH inequalities. Furthermore, we highlighted the particular values of \(\alpha\) and \(\beta\) leading to the saturation of Tsirelson's bound for each case, collecting them in Table 1.
It is worth pointing out that in the symmetric case, we need to choose \(\alpha\) and \(\beta\) small in order to find a violation, and very small to saturate the Tsirelson's bound. On the other hand, in the asymmetric case, we can find a Bell-CHSH inequality violation even with \(\alpha\) and \(\beta\) near to \(1\), but it is important to keep them close to each other, finding a larger violation if we consider the \(\alpha=\beta\) situation. Finally, the entangled cat states exhibit Bell-CHSH inequality violation for almost all the values of \(\alpha\) and \(\beta\) in the interval \((0,1)\). In all the above statements, we are tacitly considering a relative phase \(\phi=\pi\) and the set of parameters \(a=0,a^{\prime}=\pi/2,b=+\pi/4,b^{\prime}=-\pi/4\), but the computations were done for a general set of parameters. In almost all the considered cases, the Bell-CHSH correlator \(|(\mathcal{C})|\) asymptotes to \(2\) for large values of \(\alpha\), and we didn't find any violation without a relative phase (\(\phi=0\)).
The asymmetric case in the second Bell setup studied in Sec. V.2 stands as an exception, exhibiting Bell-CHSH inequality violation in the case \(\phi=\pi\) for all the considered values of \(\alpha\). Furthermore, in this case we were able to find a Bell-CHSH violation even in the situation without a relative phase. However, we emphasize that these particular features of asymmetric coherent states (32) were also observed in [84], although using a different Bell setup.
It would be extremely interesting if one could devise a realistic experimental setup corresponding to the theoretical framework presented here, in order to measure the Bell-CHSH inequality violation for these entangled coherent states and confirm our predictions. Furthermore, the extension of this investigation to the realm of multipartite systems could bring quite interesting developments. This is work in progress and will be reported soon.
###### Acknowledgements.
The authors thank the Brazilian agencies CNPq, CAPES and FAPERJ for financial support. S. P. Sorella is a level 1 CNPq researcher under the contract 301030/2019-7. G. Peruzzo is a FAPERJ postdoctoral fellow in the _Pos-Doutorado Nota 10_ program under the contracts E-26/205.924/2022 and E-26/205.925/2022. PDF is grateful to Fernando de Melo for the very interesting discussion and also for the useful suggestions. | Entangled Coherent State の３つのクラスを用いて、Bell-CHSH不等式を研究しています。擬素スピン演算子を無限次元Hilbert空間を用いて、不等式の４つの二値演算子（A,A',B,B'）を構築しています。各クラスの相関状態に対して、演算子$\langle \psi \vert AB + A' B + AB' - A' B' \vert \psi \rangle$ を計算し、Bell-CHSH不等式の違反をもたらすパラメータセットと、Tsirelsonの境界の飽和に焦点を当てています。 |
2301.07852 | On an inverse problem for the plate equation with passive measurement | This paper focuses on an inverse problem associated with the plate equation
which is derived from models in fluid mechanics and elasticity. We establish
the unique identifying results in simultaneously determining both the unknown
density and internal sources from passive boundary measurement. The proof
mainly relies on the asymptotic analysis and harmonic analysis on integral
transforms | Yixian Gao, Hongyu Liu, Yang Liu | 2023-01-19T02:40:03 | http://arxiv.org/abs/2301.07852v1 | # On an inverse problem for the plate equation with passive measurement
###### Abstract.
This paper focuses on an inverse problem associated with the plate equation which is derived from models in fluid mechanics and elasticity. We establish the unique identifying results in simultaneously determining both the unknown density and internal sources from passive boundary measurement. The proof mainly relies on the asymptotic analysis and harmonic analysis on integral transforms.
Key words and phrases:the plate equation, density, internal sources, passive boundary measurement 2010 Mathematics Subject Classification: 35R30, 31B10, 74K20 The research of YG was supported by NSFC grants 11871140, 12071065 and National Key R&D Program of China 2020YFA0714102. The research of HL was supported by Hong Kong RGC General Research Funds (project numbers, 11300821, 12301218 and 12302919) and the NSFC-RGC Joint Research Grant (project number, N_CityU101/21).
biharmonic equation is not as extensive as the results of second order differential equations. The increase of the order leads to the failure of the methods which work for the second order equations. A detailed description of the properties of the solution can be found in [17].
Global uniqueness results of recovering potential function or medium parameters associated with bi-harmonic or poly-harmonic operators by active measurements can be found in [1, 11, 15, 20]. To our best knowledge, no uniqueness identifiability result is known in the literature in determining unknown medium parameters associated with the bi-harmonic operator by the passive measurement. Indeed, in our study of (1.3), we aim at determining both the medium parameter \(\rho\) and the unknown sources \(f,g\) from the associated passive measurement. In recent years, simultaneously determining both an unknown source and its surrounding medium by the associated passive measurement has received considerable interest in the literature due to its practical importance in emerging applications. In [10, 16], the authors proved the uniqueness in determining both an acoustic density and an internal source for the scalar wave equation by the passive measurement, which arises in thermo- and photo-acoustic tomography. In [7], the authors established unique recovery results in simultaneously determining an unknown internal current and its surrounding medium by the passive measurement associated with a Maxwell system, which arises in brain imaging. In [5, 6], similar inverse problems were considered associated with the geo-dynamical system which arises in geomagnetic anomaly detection. We also refer to [2, 13, 14] for more related studies in different physical and mathematical setups. These results show that it is possible to prove the uniqueness of two or more unknowns simultaneously by the passive measurement. Motivated by [2, 7, 10, 14, 16], we are interested in recovering both the density \(\rho\) and the sources \(f,g\) by the passive boundary measurement for the biharmonic plate equation (1.1).
In this article, we shall make use of the temporal Fourier transform, converting the time-dependent problem (1.1) into the frequency domain. To that end, we need to require that the plate equation of (1.1) is exponentially decaying in time, which can guarantee the well-posedness of the temporal Fourier transform. In order to appeal for a general study, we shall always assume the exponentially decaying in time for the plate equation. Nevertheless, we would like to emphasise that such a property is satisfied by generic mediums and sources. In what follows, we refer to \((f,g,\rho)\) as admissible if the aforementioned time decaying property is fulfilled. In fact, one can refer to the case of the acoustic wave equation (cf. Theorem 6.1, page 113, [12]), and derive general admissibility conditions by following similar arguments. However, this is not the focus of the current study and our main purpose is to study the related inverse problem. The main methods to obtain the uniqueness results are by performing certain asymptotic analysis in the low frequency regime. We derive some integral identities involving source functions and density coupling. Combining those integral identities and using harmonic analysis techniques, the uniqueness results are obtained. The uniqueness relies on assuming that the density \(\rho\) and internal source functions \(f,g\) along an arbitrary direction vector \(\iota\) satisfy \(\nabla\rho\cdot\iota=\nabla f\cdot\iota=\nabla g\cdot\iota=0\). Additionally, the above assumption can be replaced by setting the size of the corresponding parameters to obtain a more general uniqueness results.
The paper is organized as follows. In Section 2, we introduce the temporal Fourier transform, converting the time domain problem (1.1) into frequency domain and give some notations. In Section 3, we derive the asymptotic expansions of the solutions with respect to the frequency \(\kappa\), and get some integral identities which the source functions and the density are coupled together. Then the uniqueness results established by a natural admissibility assumption on the parameters. The more general uniqueness results are given in Section 4.
## 2. Problem Formulation
In this section, we introduce the temporal Fourier transform to convert the time domain problem (1.1) into the frequency domain and introduce some notations.
Our argument depends on the temporal Fourier transform of the function \(u(t,\boldsymbol{x})\) defined by
\[\hat{u}(\omega,\boldsymbol{x}):=\frac{1}{2\pi}\int_{0}^{\infty}u(t, \boldsymbol{x})e^{\mathrm{i}wt}\,\,\mathrm{d}t,\ \ \ \ (\omega,\boldsymbol{x})\in\mathbb{R}_{+}\times \mathbb{R}^{3}.\]
Applying the temporal Fourier transform to equation (1.1) and assume that \(\kappa=\omega^{1/2}\), it yields that \(\hat{u}(\kappa,\boldsymbol{x})\) satisfies
\[\Delta^{2}\hat{u}(\kappa,\boldsymbol{x})-\kappa^{4}\rho( \boldsymbol{x})\hat{u}(\kappa,\boldsymbol{x})=-\frac{\mathrm{i}\kappa^{2}}{2 \pi}\rho(\boldsymbol{x})f(\boldsymbol{x})+\frac{1}{2\pi}\rho(\boldsymbol{x})g (\boldsymbol{x}),\ \ \ (\kappa,\boldsymbol{x})\in\mathbb{R}_{+}\times\mathbb{R}^{3}, \tag{2.1}\]
and the boundary measurement (1.2) in the frequency domain
\[\hat{\Lambda}_{\rho,f,g}(\kappa,\boldsymbol{x})=(\hat{u}(\kappa, \boldsymbol{x}),\ \Delta\hat{u}(\kappa,\boldsymbol{x}))\,,\ \ \ \ (\kappa,\boldsymbol{x})\in\mathbb{R}_{+}\times \partial\Omega. \tag{2.2}\]
To ensure the well-posedness of the equation (2.1), we impose an analogue of the Sommerfeld radiation conditions
\[\underset{r\to\infty}{\lim}r\left(\partial_{r}\hat{u}(\kappa, \boldsymbol{x})-\mathrm{i}\kappa\hat{u}(\kappa,\boldsymbol{x})\right)=0,\ \ \ \ \underset{r\to\infty}{\lim}r\left(\partial_{r}(\Delta\hat{u}(\kappa, \boldsymbol{x}))-\mathrm{i}\kappa(\Delta\hat{u}(\kappa,\boldsymbol{x}))\right) =0, \tag{2.3}\]
uniformly in all directions \(\hat{\boldsymbol{x}}=\boldsymbol{x}/|\boldsymbol{x}|\) with \(r=|\boldsymbol{x}|\) (cf. [19]).
One of the key technical ingredients to establish the unique recovery results is first by performing certain asymptotic analysis in the low frequency regime to derive certain integral identities involving the source function and density, which are coupled together.
We introduce some notations. The fundamental solution for biharmonic operator \(\Delta^{2}-\kappa^{4}\) in \(\mathbb{R}^{3}\) is
\[G_{\kappa}(|\boldsymbol{x}-\boldsymbol{y}|)=\frac{e^{\mathrm{i }\kappa|\boldsymbol{x}-\boldsymbol{y}|}-e^{-\kappa|\boldsymbol{x}-\boldsymbol {y}|}}{8\pi\kappa^{2}|\boldsymbol{x}-\boldsymbol{y}|}\ \ \ \ \text{for}\ \boldsymbol{x}\neq \boldsymbol{y}. \tag{2.4}\]
Notice that when \(\kappa=0,\) the fundamental solution to \(\Delta^{2}\) is
\[G_{0}(|\boldsymbol{x}-\boldsymbol{y}|)=-\frac{|\boldsymbol{x}- \boldsymbol{y}|}{8\pi}\ \ \ \ \text{for}\ \boldsymbol{x}\neq\boldsymbol{y},\]
and the fundamental solution for \(-\Delta\) is
\[g_{0}(|\boldsymbol{x}-\boldsymbol{y}|)=\frac{1}{4\pi|\boldsymbol{x}- \boldsymbol{y}|}\ \ \ \ \text{for}\ \boldsymbol{x}\neq\boldsymbol{y}.\]
## 3. The uniqueness results for density and internal sources
In this section, we will prove the uniqueness for both the unknown density \(\rho\) and the internal sources \(f\) and \(g\).
### Auxiliary results
Before proving the uniqueness results, we first derive several auxiliary results.
**Lemma 3.1**.: _Let \(\hat{u}(\kappa,\mathbf{x})\in H^{2}_{loc}(\mathbb{R}^{3})\) be the solution of (2.1) and (2.3). Then \(\hat{u}(\kappa,\mathbf{x})\) is uniquely given by the following integral equation_
\[\hat{u}(\kappa,\mathbf{x})= \kappa^{4}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\hat{u}(\kappa,\mathbf{ y})G_{\kappa}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}-\frac{\mathrm{i}\kappa^{2}}{2 \pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y})G_{\kappa}(|\mathbf{x}-\mathbf{y}|)\; \mathrm{d}\mathbf{y}\] \[+\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})G_{ \kappa}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y},\hskip 14.226378pt\mathbf{x}\in\mathbb{R}^{3}. \tag{3.1}\]
_In addition, if taking the expansion of \(e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}\) as \(\kappa\to+0\), we have_
\[\hat{u}(\kappa,\mathbf{x})= \frac{1}{2\pi\kappa}\int_{\mathbb{R}^{3}}\frac{(\mathrm{i}+1)}{8 \pi}\rho(\mathbf{y})g(\mathbf{y})\;\mathrm{d}\mathbf{y}-\frac{1}{2\pi}\int_{\mathbb{R}^{3} }\frac{1}{8\pi}\rho(\mathbf{y})g(\mathbf{y})|\mathbf{x}-\mathbf{y}|\;\mathrm{d}\mathbf{y}\] \[+\frac{\kappa}{2\pi}\int_{\mathbb{R}^{3}}\frac{(\mathrm{1}- \mathrm{i})}{8\pi}\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{2}}{3!}\;\mathrm{ d}\mathbf{y}\] \[-\frac{\mathrm{i}\kappa}{2\pi}\int_{\mathbb{R}^{3}}\frac{( \mathrm{i}+1)}{8\pi}\rho(\mathbf{y})f(\mathbf{y})\;\mathrm{d}\mathbf{y}+\mathcal{O}(\kappa ^{2}),\hskip 14.226378pt\mathbf{x}\in B_{R}, \tag{3.2}\]
_where \(B_{R}:=B(0,R)\) is a central ball of radius \(R\in\mathbb{R}_{+}\) and satisfies \(\Omega\subset B_{R}\)._
Proof.: From the regularity theorem in [8], it is easy to verify that the solution \(\hat{u}\in H^{4}_{loc}(\mathbb{R}^{3})\). By applying the fundamental solution to (2.1), we obtain a Lippmann-Schwinger integral equation
\[\hat{u}(\kappa,\mathbf{x})= \kappa^{4}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\hat{u}(\kappa,\bm {y})G_{\kappa}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y })f(\mathbf{y})G_{k}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[+\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})G_{k}(| \mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}. \tag{3.3}\]
Since \(\rho(x)-1=0\) in \(\mathbb{R}^{3}\backslash\bar{\Omega}\) and \(\rho\in L^{\infty}(\mathbb{R}^{3})\), we assume that \(\Omega\subset B_{R}\) and \(\mathcal{K}_{\rho,\kappa}:C(\overline{B_{R}})\longrightarrow C(\overline{B_{R}})\) satisfies
\[\mathcal{K}_{\rho,\kappa}(\hat{u})=\kappa^{4}\int_{B_{R}}(\rho(\mathbf{y})-1)\hat{ u}(\kappa,\mathbf{y})G_{\kappa}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}.\]
Suppose that \(M:=\sup\limits_{|\mathbf{x}|\leq R}|\rho(\mathbf{x})-1|\) and \(\kappa^{2}<\frac{2}{MR^{2}}\), we have \(\|\mathcal{K}_{\rho,\kappa}\|_{L^{\infty}(B_{R})}\leq 1\). Therefore, there exsits a Neumann series
\[(I-\mathcal{K}_{\rho,\kappa})^{-1}=I+\mathcal{K}_{\rho,\kappa}+\mathcal{K}_{ \rho,\kappa}^{2}+\cdots.\]
If taking \(\kappa\longrightarrow+0\) and replacing \(e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}\) by the series expansion
\[e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}=\kappa( \mathrm{i}+1)|\mathbf{x}-\mathbf{y}|-\kappa^{2}|\mathbf{x}-\mathbf{y}|^{2}+\frac{\kappa^{3}( 1-\mathrm{i})}{3!}|\mathbf{x}-\mathbf{y}|^{3}+\mathcal{O}(\kappa^{5}),\]
we calculate that
\[(I-\mathcal{K}_{\rho,\kappa})^{-1}=I+\mathcal{O}(\kappa^{3}). \tag{3.4}\]
Additionally, substituting (2.4) into (3.3), implies
\[\hat{u}(\kappa,\mathbf{x})= (I-\mathcal{K}_{\rho,\kappa})^{-1}\big{(}-\frac{\mathrm{i}\kappa^{2 }}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y})\frac{e^{\mathrm{i}\kappa|\mathbf{ x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}}{8\pi\kappa^{2}|\mathbf{x}-\mathbf{y}|}\,\mathrm{d}\mathbf{y}\] \[+\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{e^{ \mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}}{8\pi\kappa^{2}| \mathbf{x}-\mathbf{y}|}\,\,\mathrm{d}\mathbf{y}\big{)}\] \[= \frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{( \mathrm{i}+1)}{8\pi\kappa}-\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|}{8\pi}+ \rho(\mathbf{y})g(\mathbf{y})\frac{\kappa(1-\mathrm{i})}{3!}\frac{|\mathbf{x}-\mathbf{y}|^{2}} {8\pi}\,\,\mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}\kappa}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f (\mathbf{y})\frac{(\mathrm{i}+1)|\mathbf{x}-\mathbf{y}|}{8\pi|\mathbf{x}-\mathbf{y}|}\,\,\mathrm{d }\mathbf{y}+\mathcal{O}(\kappa^{2}),\quad\mathbf{x}\in B_{R}.\]
The proof is completed.
**Lemma 3.2**.: _Let \(\hat{u}(\kappa,\mathbf{x})\in H^{2}_{loc}(\mathbb{R}^{3})\) be the solution of (2.1) and (2.3). Then the integral equation (3.1) can be rewritten as_
\[\hat{u}(\kappa,\mathbf{x})=\sum_{m=-1}^{3}M_{m}(\mathbf{x})\kappa^{m}+\mathcal{O}( \kappa^{4})\quad\text{ as }\kappa\to+0,\quad\mathbf{x}\in B_{R}, \tag{3.5}\]
_where_
\[M_{-1}:= \frac{1}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{R}^{3}} \rho(\mathbf{y})g(\mathbf{y})\,\mathrm{d}\mathbf{y},\] \[M_{0}:= \frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})G_{0}(| \mathbf{x}-\mathbf{y}|)\,\,\mathrm{d}\mathbf{y},\] \[M_{1}:= -\frac{\mathrm{i}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{ R}^{3}}\rho(\mathbf{y})f(\mathbf{y})\,\,\mathrm{d}\mathbf{y}+\frac{1}{2\pi}\frac{(1- \mathrm{i})}{8\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y }|^{2}}{3!}\,\,\mathrm{d}\mathbf{y},\] \[M_{2}:= (\frac{\mathrm{i}+1}{8\pi})\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)M _{-1}\,\,\mathrm{d}\mathbf{y}-\frac{\mathrm{i}}{2\pi}\int_{\mathbb{R}^{3}}\rho( \mathbf{y})f(\mathbf{y})G_{0}(|\mathbf{x}-\mathbf{y}|)\,\,\mathrm{d}\mathbf{y},\] \[M_{3}:= \int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)M_{-1}\,\,G_{0}(|\mathbf{x}-\bm {y}|)\,\,\mathrm{d}\mathbf{y}+\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{R}^{3}}( \rho(\mathbf{y})-1)M_{0}(\mathbf{y})\,\,\mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}}{2\pi}\frac{(1-\mathrm{i})}{8\pi}\int_{\mathbb{R }^{3}}\rho(\mathbf{y})f(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{2}}{3!}\,\,\mathrm{d}\mathbf{y} +\frac{1}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g( \mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{4}}{5!}\,\,\mathrm{d}\mathbf{y}.\]
_Taking the laplacian for (3.5) with respect to \(\mathbf{x}\), imply_
\[\Delta_{\mathbf{x}}\hat{u}(\kappa,\mathbf{x})= -\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})g_{0}(| \mathbf{x}-\mathbf{y}|)\,\,\mathrm{d}\mathbf{y}+\frac{\kappa}{2\pi}\frac{(1-\mathrm{i})}{8 \pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\,\,\mathrm{d}\mathbf{y}\] \[+\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y} )f(\mathbf{y})g_{0}(|\mathbf{x}-\mathbf{y}|)\,\,\mathrm{d}\mathbf{y}-\kappa^{3}\int_{\mathbb{R }^{3}}(\rho(\mathbf{y})-1)M_{-1}\,\,g_{0}(|\mathbf{x}-\mathbf{y}|)\,\,\mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}\kappa^{3}}{2\pi}\frac{(1-\mathrm{i})}{8\pi}\int_ {\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y})\,\,\mathrm{d}\mathbf{y}\] \[+\frac{\kappa^{3}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{R }^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{2}}{3!}\,\,\mathrm{d}\mathbf{y}+ \mathcal{O}(\kappa^{4})\quad\text{ as }\kappa\to+0,\quad\mathbf{x}\in B_{R}. \tag{3.6}\]
Proof.: Plugging (3.2) into (3.1) and using the series expansion
\[e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}= \kappa(\mathrm{i}+1)|\mathbf{x}-\mathbf{y}|-\kappa^{2}|\mathbf{x}-\mathbf{y}|^{2}+ \frac{\kappa^{3}(1-\mathrm{i})}{3!}|\mathbf{x}-\mathbf{y}|^{3}\] \[+\frac{\kappa^{5}(\mathrm{i}+1)}{5!}|\mathbf{x}-\mathbf{y}|^{5}+\mathcal{ O}(\kappa^{6})\quad\text{ as }\kappa\to+0,\]
uniformly for \(\mathbf{x}\in B_{R}\), we get
\[\hat{u}(\kappa,\mathbf{x})= \kappa^{4}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\big{(}\frac{1}{2 \pi\kappa}\int_{\mathbb{R}^{3}}\frac{(\mathrm{i}+1)}{8\pi}\rho(\mathbf{z})g(\mathbf{z })\;\mathrm{d}\mathbf{z}-\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{z})g(\mathbf{z}) \frac{|\mathbf{y}-\mathbf{z}|}{8\pi}\;\mathrm{d}\mathbf{z}\] \[+\frac{\kappa}{2\pi}\int_{\mathbb{R}^{3}}\frac{(1-\mathrm{i})}{8 \pi}\rho(\mathbf{z})g(\mathbf{z})\frac{|\mathbf{y}-\mathbf{z}|^{2}}{3!}\;\mathrm{d}\mathbf{z}\] \[-\frac{\mathrm{i}\kappa}{2\pi}\int_{\mathbb{R}^{3}}\frac{( \mathrm{i}+1)}{8\pi}\rho(\mathbf{z})f(\mathbf{z})\;\mathrm{d}\mathbf{z}+\mathcal{O}( \kappa^{2})\big{)}\frac{e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}- \mathbf{y}|}}{8\pi\kappa^{2}|\mathbf{x}-\mathbf{y}|}\;\mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{ y})f(\mathbf{y})\frac{e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}}{8 \pi\kappa^{2}|\mathbf{x}-\mathbf{y}|}\;\mathrm{d}\mathbf{y}\] \[+\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{e^ {\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}}{8\pi\kappa^{2}| \mathbf{x}-\mathbf{y}|}\;\mathrm{d}\mathbf{y}\] \[:= I_{1}+I_{2}+I_{3}+\mathcal{O}(\kappa^{4}). \tag{3.7}\]
By direct calculation, one has
\[I_{1}= \kappa^{4}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\big{(}\frac{1}{2 \pi\kappa}\int_{\mathbb{R}^{3}}\frac{(\mathrm{i}+1)}{8\pi}\rho(\mathbf{z})g(\mathbf{z} )\;\mathrm{d}\mathbf{z}-\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{z})g(\mathbf{z}) \frac{|\mathbf{y}-\mathbf{z}|}{8\pi}\;\mathrm{d}\mathbf{z}\] \[+\frac{\kappa}{2\pi}\int_{\mathbb{R}^{3}}\frac{(1-\mathrm{i})}{8 \pi}\rho(\mathbf{z})g(\mathbf{z})\frac{|\mathbf{y}-\mathbf{z}|^{2}}{3!}\;\mathrm{d}\mathbf{z}\] \[-\frac{\mathrm{i}\kappa}{2\pi}\int_{\mathbb{R}^{3}}\frac{( \mathrm{i}+1)}{8\pi}\rho(\mathbf{z})f(\mathbf{z})\;\mathrm{d}\mathbf{z}+\mathcal{O}( \kappa^{2})\big{)}\frac{e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}- \mathbf{y}|}}{8\pi\kappa^{2}|\mathbf{x}-\mathbf{y}|}\;\mathrm{d}\mathbf{y}\] \[= \kappa^{2}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\frac{1}{2\pi}\int _{\mathbb{R}^{3}}\frac{(\mathrm{i}+1)}{8\pi}\rho(\mathbf{z})g(\mathbf{z})\;\mathrm{d} \mathbf{z}\;\frac{(\mathrm{i}+1)}{8\pi}\;\mathrm{d}\mathbf{y}\] \[+\kappa^{3}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\frac{1}{2\pi} \int_{\mathbb{R}^{3}}\frac{(\mathrm{i}+1)}{8\pi}\rho(\mathbf{z})g(\mathbf{z})\; \mathrm{d}\mathbf{z}\;G_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[+\kappa^{3}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)\frac{1}{2\pi} \int_{\mathbb{R}^{3}}\rho(\mathbf{z})g(\mathbf{z})G_{0}(|\mathbf{y}-\mathbf{z}|)\;\mathrm{d} \mathbf{z}\;\frac{(\mathrm{i}+1)}{8\pi}\;\mathrm{d}\mathbf{y}+\mathcal{O}(\kappa^{4}), \quad\mathbf{x}\in B_{R},\]
\[I_{2}= -\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y} )f(\mathbf{y})\frac{e^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}} {8\pi\kappa^{2}|\mathbf{x}-\mathbf{y}|}\;\mathrm{d}\mathbf{y}\] \[= -\frac{\mathrm{i}\kappa}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{ \mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y})\;\mathrm{d}\mathbf{y}-\frac{\mathrm{i}\kappa^ {2}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y})G_{0}(|\mathbf{x}-\mathbf{y}|)\; \mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}\kappa^{3}}{2\pi}\frac{(1-\mathrm{i})}{8\pi}\int _{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{2}}{3!}\;\mathrm{d} \mathbf{y}+\mathcal{O}(\kappa^{4}),\quad\mathbf{x}\in B_{R},\]
and
\[I_{3}= \frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{ \mathrm{e}^{\mathrm{i}\kappa|\mathbf{x}-\mathbf{y}|}-e^{-\kappa|\mathbf{x}-\mathbf{y}|}}{8\pi \kappa^{2}|\mathbf{x}-\mathbf{y}|}\;\mathrm{d}\mathbf{y}\] \[= \frac{1}{2\pi\kappa}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{R}^ {3}}\rho(\mathbf{y})g(\mathbf{y})\;\mathrm{d}\mathbf{y}+\frac{1}{2\pi}\int_{\mathbb{R}^{3} }\rho(\mathbf{y})g(\mathbf{y})G_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[+\frac{\kappa}{2\pi}\frac{(1-\mathrm{i})}{8\pi}\int_{\mathbb{R}^ {3}}\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{2}}{3!}\;\mathrm{d}\mathbf{y}\] \[+\frac{\kappa^{3}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{ R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{4}}{5!}\;\mathrm{d}\mathbf{y}+ \mathcal{O}(\kappa^{4}),\;\;\;\;\;\mathbf{x}\in B_{R}.\]
Taking the Laplacian on both side of equality (3.7) with respect to \(\mathbf{x}\), we obtain
\[\Delta_{\mathbf{x}}\hat{u}(\kappa,\mathbf{x})= -\frac{\kappa^{3}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{ R}^{3}}(\rho(\mathbf{y})-1)\int_{\mathbb{R}^{3}}\rho(\mathbf{z})g(\mathbf{z})\;\mathrm{d} \mathbf{z}\;g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[+\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y })f(\mathbf{y})g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}-\frac{\mathrm{i}\kappa^{ 3}}{2\pi}\frac{(1-\mathrm{i})}{8\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y}) \;\mathrm{d}\mathbf{y}\] \[-\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})g_{0}(| \mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}+\frac{\kappa}{2\pi}\frac{(1-\mathrm{i})}{8 \pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\;\mathrm{d}\mathbf{y}\] \[+\frac{\kappa^{3}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\int_{\mathbb{ R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}|^{2}}{3!}\;\mathrm{d}\mathbf{y}+ \mathcal{O}(\kappa^{4}),\;\;\;\;\;\mathbf{x}\in B_{R}\]
as \(\kappa\to+0\). The proof is completed.
**Remark 3.1**.: _Let \(n\in\mathbb{N}\cup\{0\}\), then the solution \(\hat{u}(\kappa,\mathbf{x})\) can be represented as_
\[\hat{u}(\kappa,\mathbf{x})=\sum_{m=-1}^{n+1}\!M_{m}(\mathbf{x})\kappa^{m}+M_{n+2}(\bm {x})\kappa^{n+2}+\mathcal{O}(\kappa^{n+3}),\]
_and_
\[\Delta_{\mathbf{x}}\hat{u}(\kappa,\mathbf{x})=\sum_{m=0}^{n+2}\!N_{m}(\mathbf{x})\kappa^{ m}+N_{n+3}(\mathbf{x})\kappa^{n+3}+\mathcal{O}(\kappa^{n+4})\;\;\;\;\text{as}\; \kappa\to+0,\]
_where_
\[M_{n+2}(\mathbf{x}):= \sum_{m=0}^{n}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)M_{n-m-1}(\bm {y})\frac{\mathrm{i}^{m+1}-(-1)^{m+1}}{8\pi}\frac{|\mathbf{x}-\mathbf{y}|^{m}}{(m+1)!} \;\mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y}) \frac{\mathrm{i}^{n+2}-(-1)^{n+2}}{8\pi}\frac{|\mathbf{x}-\mathbf{y}|^{n+1}}{(n+2)!} \;\mathrm{d}\mathbf{y}\] \[+\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{ \mathrm{i}^{n+4}-(-1)^{n+4}}{8\pi}\frac{|\mathbf{x}-\mathbf{y}|^{n+3}}{(n+4)!}\; \mathrm{d}\mathbf{y},\]
_and_
\[N_{n+3}:= \sum_{m=1}^{n+1}\int_{\mathbb{R}^{3}}(\rho(\mathbf{y})-1)M_{n-m}(\mathbf{y}) \frac{\mathrm{i}^{m+1}-(-1)^{m+1}}{8\pi}\frac{|\mathbf{x}-\mathbf{y}|^{m-2}}{(m-1)!}\; \mathrm{d}\mathbf{y}\] \[-\frac{\mathrm{i}}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})f(\mathbf{y} )\frac{\mathrm{i}^{n+3}-(-1)^{n+3}}{8\pi}\frac{|\mathbf{x}-\mathbf{y}|^{n}}{(n+1)!}\; \mathrm{d}\mathbf{y}\] \[+\frac{1}{2\pi}\int_{\mathbb{R}^{3}}\rho(\mathbf{y})g(\mathbf{y})\frac{ \mathrm{i}^{n+5}-(-1)^{n+5}}{8\pi}\frac{|\mathbf{x}-\mathbf{y}|^{n+2}}{(n+3)!}\; \mathrm{d}\mathbf{y},\;\;\mathbf{x}\in B_{R}.\]
_In addition, \(N_{m}(\mathbf{x})\kappa^{m},m=0,1,2\) corresponds to the first three terms in (3.6), respectively._
**Theorem 3.1**.: _Assume that \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) are two sets of admissible configurations and supported in \(\Omega\). If_
\[\Lambda_{\rho_{1},f_{1},g_{1}}(t,\mathbf{x})=\Lambda_{\rho_{2},f_{2},g_{2}}(t,\bm {x}),\;\;\;\;\;(t,\mathbf{x})\in\mathbb{R}_{+}\times\partial\Omega. \tag{3.8}\]
_Then for any harmonic function \(h(\mathbf{x})\), we have_
\[\int_{\mathbb{R}^{3}}(\rho_{1}f_{1}-\rho_{2}f_{2})(\mathbf{x})h(\mathbf{x })\;\mathrm{d}\mathbf{x}=0, \tag{3.9}\] \[\int_{\mathbb{R}^{3}}(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{x})h(\mathbf{x })\;\mathrm{d}\mathbf{x}=0. \tag{3.10}\]
_Furthermore, for any \(\mathbf{x}\in\partial B_{R}\), the following holds_
\[\int_{\mathbb{R}^{3}}(\rho_{1}(\mathbf{y})-1)\int_{\mathbb{R}^{3}} \rho_{1}(\mathbf{z})g_{1}(\mathbf{z})\;\mathrm{d}\mathbf{z}\;g_{0}(|\mathbf{x}-\mathbf{y}|)\; \mathrm{d}\mathbf{y}\] \[+\int_{\mathbb{R}^{3}}\rho_{1}(\mathbf{y})f_{1}(\mathbf{y})\;\mathrm{d} \mathbf{y}-\int_{\mathbb{R}^{3}}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}| ^{2}}{3!}\;\mathrm{d}\mathbf{y}\] \[= \int_{\mathbb{R}^{3}}(\rho_{2}(\mathbf{y})-1)\int_{\mathbb{R}^{3}} \rho_{2}(\mathbf{z})g_{2}(\mathbf{z})\;\mathrm{d}\mathbf{z}\;g_{0}(|\mathbf{x}-\mathbf{y}|)\; \mathrm{d}\mathbf{y}\] \[+\int_{\mathbb{R}^{3}}\rho_{2}(\mathbf{y})f_{2}(\mathbf{y})\;\mathrm{d} \mathbf{y}-\int_{\mathbb{R}^{3}}\rho_{2}(\mathbf{y})g_{2}(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y}| ^{2}}{3!}\;\mathrm{d}\mathbf{y}. \tag{3.11}\]
Proof.: Using the temporal Fourier transform, let \(\hat{u}_{1}(\kappa,\mathbf{x})\) and \(\hat{u}_{2}(\kappa,\mathbf{x})\) denote the solution of (2.1), corresponding to \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) respectively. It follows from (2.2) and (3.8) that
\[(\hat{u}_{1}(\mathbf{x}),\;\Delta\hat{u}_{1}(\mathbf{x}))=(\hat{u}_{2}(\mathbf{x}),\; \Delta\hat{u}_{2}(\mathbf{x})),\;\;\;\;\;\mathbf{x}\in\partial\Omega.\]
Since the priori information for density and internal sources, it is easy to verify that both \(\hat{u}_{1}\) and \(\hat{u}_{2}\) satisfy the same equation
\[\Delta^{2}u-\kappa^{4}u=0\;\;\;\;\;\text{in }\mathbb{R}^{3}\backslash\overline{\Omega}.\]
Additionally, from the uniqueness of exterior boundary value problem in Theorem A.1, we have
\[(\hat{u}_{1}(\mathbf{x}),\;\Delta\hat{u}_{1}(\mathbf{x}))=(\hat{u}_{2}(\mathbf{x}),\; \Delta\hat{u}_{2}(\mathbf{x})),\;\;\;\;\;\mathbf{x}\in\mathbb{R}^{3}\backslash\Omega.\]
Due to \(\partial B_{R}\subset\mathbb{R}^{3}\backslash\overline{\Omega}\), we can obtain
\[(\hat{u}_{1}(\mathbf{x}),\;\Delta\hat{u}_{1}(\mathbf{x}))=(\hat{u}_{2}(\mathbf{x}),\; \Delta\hat{u}_{2}(\mathbf{x})),\;\;\;\;\;\mathbf{x}\in\partial B_{R}. \tag{3.12}\]
Combining (3.6) and (3.12), we imply the following integral identities
\[\frac{1}{2\pi}\int_{B_{R}}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})g_{0}(|\mathbf{x }-\mathbf{y}|)\;\mathrm{d}\mathbf{y}= \frac{1}{2\pi}\int_{B_{R}}\rho_{2}(\mathbf{y})g_{2}(\mathbf{y})g_{0}(|\mathbf{x }-\mathbf{y}|)\;\mathrm{d}\mathbf{y}, \tag{3.13}\] \[\frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{B_{R}}\rho_{1}(\mathbf{y})f_{1 }(\mathbf{y})g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}= \frac{\mathrm{i}\kappa^{2}}{2\pi}\int_{B_{R}}\rho_{2}(\mathbf{y})f_{2}(\mathbf{y})g _{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}, \tag{3.14}\]
and
\[\frac{\kappa^{3}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\big{(}\int_{B _{R}}(\rho_{1}(\mathbf{y})-1)\int_{B_{R}}\rho_{1}(\mathbf{z})g_{1}(\mathbf{z})\;\mathrm{d }\mathbf{z}\;g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[\qquad\qquad\qquad\qquad+\int_{B_{R}}\rho_{1}(\mathbf{y})f_{1}(\mathbf{y} )\;\mathrm{d}\mathbf{y}-\int_{B_{R}}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})\frac{|\mathbf{x}- \mathbf{y}|^{2}}{3!}\;\mathrm{d}\mathbf{y}\big{)}\] \[= \frac{\kappa^{3}}{2\pi}\frac{(\mathrm{i}+1)}{8\pi}\big{(}\int_{B _{R}}(\rho_{2}(\mathbf{y})-1)\int_{B_{R}}\rho_{2}(\mathbf{z})g_{2}(\mathbf{z})\;\mathrm{d }\mathbf{z}\;g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[\qquad\qquad\qquad+\int_{B_{R}}\rho_{2}(\mathbf{y})f_{2}(\mathbf{y})\; \mathrm{d}\mathbf{y}-\int_{B_{R}}\rho_{2}(\mathbf{y})g_{2}(\mathbf{y})\frac{|\mathbf{x}-\mathbf{y} |^{2}}{3!}\;\mathrm{d}\mathbf{y}\big{)}\]
for \(\mathbf{x}\in\partial B_{R}\).
Note that the fundamental solution of \(-\Delta\) can be written as
\[\frac{1}{4\pi|\mathbf{x}-\mathbf{y}|}=\sum_{m=0}^{\infty}\sum_{n=-m}^{m}\!\frac{1}{2m +1}\frac{|\mathbf{y}|^{m}}{|\mathbf{x}|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|}) \overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\quad\text{ for }|\mathbf{x}|>|\mathbf{y}|, \tag{3.15}\]
where \(Y_{m}^{n}(\cdot)\) denotes the spherical harmonics of order \(m\in\mathbb{N}\cup\{0\}\) for \(n=-m,\cdots,m\).
Substituting (3.15) into (3.13) and (3.14), we calculate that
\[\int_{B_{R}}\sum_{m=0}^{\infty}\!\!\sum_{n=-m}^{m}\!\frac{1}{2m+ 1}\frac{1}{|\mathbf{x}|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\rho_{1}(\mathbf{y})f _{1}(\mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{ d}\mathbf{y}\] \[= \int_{B_{R}}\sum_{m=0}^{\infty}\!\!\sum_{n=-m}^{m}\!\frac{1}{2m+ 1}\frac{1}{|\mathbf{x}|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\rho_{2}(\mathbf{y})f _{2}(\mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{ d}\mathbf{y}\quad\text{for }\mathbf{x}\in\partial B_{R},\]
and
\[\int_{B_{R}}\sum_{m=0}^{\infty}\!\!\sum_{n=-m}^{m}\!\frac{1}{2m+ 1}\frac{1}{|\mathbf{x}|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\rho_{1}(\mathbf{y})g _{1}(\mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{ d}\mathbf{y}\] \[= \int_{B_{R}}\sum_{m=0}^{\infty}\!\!\sum_{n=-m}^{m}\!\frac{1}{2m+ 1}\frac{1}{|\mathbf{x}|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\rho_{2}(\mathbf{y})g _{2}(\mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{ d}\mathbf{y}\quad\text{for }\mathbf{x}\in\partial B_{R}.\]
That is,
\[\sum_{m=0}^{\infty}\!\!\sum_{n=-m}^{m}\!\!\frac{1}{2m+1}\frac{1}{ |R|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\int_{B_{R}}\rho_{1}(\mathbf{y})f_{1}( \mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{d}\mathbf{y}\] \[= \sum_{m=0}^{\infty}\!\!\sum_{n=-m}^{m}\!\!\frac{1}{2m+1}\frac{1}{ |R|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\int_{B_{R}}\rho_{2}(\mathbf{y})f_{2}( \mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{d}\mathbf{y} \quad\text{for }\mathbf{x}\in\partial B_{R},\]
and
\[\sum_{m=0}^{\infty}\!\!\!\sum_{n=-m}^{m}\!\!\!\frac{1}{2m+1}\frac{1}{| R|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\int_{B_{R}}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})| \mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{d}\mathbf{y}\] \[= \!\!\!\sum_{m=0}^{\infty}\!\!\!\sum_{n=-m}^{m}\!\!\!\frac{1}{2m+1} \frac{1}{|R|^{m+1}}Y_{m}^{n}(\frac{\mathbf{x}}{|\mathbf{x}|})\int_{B_{R}}\rho_{2}(\mathbf{ y})g_{2}(\mathbf{y})|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\; \mathrm{d}\mathbf{y}\quad\text{for}\;\mathbf{x}\in\partial B_{R}.\]
Indeed, \(\{Y_{m}^{n}(\cdot)\}_{m=0,1,2,...,n=-m,...,m}\) is a complete orthonormal basis of \(L^{2}(\mathbb{S}^{2})\) (note that \(\mathbb{S}^{2}\) means the unit sphere), we get
\[\int_{B_{R}}\big{(}\rho_{1}(\mathbf{y})f_{1}(\mathbf{y})-\rho_{2}(\mathbf{y})f_{2}(\mathbf{y} )\big{)}|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{d} \mathbf{y}=0,\]
and
\[\int_{B_{R}}\big{(}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})-\rho_{2}(\mathbf{y})g_{2}(\mathbf{y} )\big{)}|\mathbf{y}|^{m}\overline{Y}_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\;\mathrm{d} \mathbf{y}=0,\]
where \(m\in\mathbb{N}\cup\{0\}\) for \(n=-m,...,m\).
For any homogeneous harmonic function \(h(\cdot)\) can be represent by \(|\mathbf{y}|^{m}Y_{m}^{n}(\frac{\mathbf{y}}{|\mathbf{y}|})\) for \(m=0,1,2,...\), and \(n=-m,...,m\), which yields
\[\int_{B_{R}}(\rho_{1}f_{1}-\rho_{2}f_{2})(\mathbf{y})h(\mathbf{y})\; \mathrm{d}\mathbf{y}= 0,\quad\mathbf{x}\in\partial B_{R}, \tag{3.16}\] \[\int_{B_{R}}(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{y})h(\mathbf{y})\; \mathrm{d}\mathbf{y}= 0,\quad\mathbf{x}\in\partial B_{R}.\]
**Remark 3.2**.: _Since \(f_{i},g_{i}\) are supported in \(\Omega\) and \(\text{supp}(\rho_{i}-1)\subset\Omega,i=1,2\), we note that the integral domains in (3.16) can be replaced by \(\Omega\)._
### The uniqueness results
**Theorem 3.2**.: _Assume that \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) are two sets of admissible configurations and supported in \(\Omega\), respectively. Furthermore, suppose that_
\[F(\mathbf{x}):= (\rho_{1}f_{1}-\rho_{2}f_{2})(\mathbf{x}),\quad\ G(\mathbf{x}):=(\rho_{1}g _{1}-\rho_{2}g_{2})(\mathbf{x}),\quad\ \mathbf{x}\in\Omega,\]
_satisfy the either of the following conditions:_
1. \(F(\mathbf{x})=h_{1}(\mathbf{x})\) _and_ \(G(\mathbf{x})=h_{2}(\mathbf{x})\) _for_ \(\mathbf{x}\in\Omega\)_, where_ \(h_{1}(\mathbf{x})\) _and_ \(h_{2}(\mathbf{x})\) _are harmonic functions in_ \(\mathbb{R}^{3}\)_;_
2. \(\nabla F(\mathbf{x})\cdot\mathbf{\iota}=0\) _and_ \(\nabla G(\mathbf{x})\cdot\mathbf{\iota}=0\)_, where_ \(\mathbf{\iota}\) _is an arbitrary direction vector in_ \(\mathbb{R}^{3}\)_._
_Then_
\[F(\mathbf{x})=G(\mathbf{x})=0\quad\text{for}\;a.e.\;\mathbf{x}\in\Omega.\]
Proof.: For the first case, taking \(h(\mathbf{x})=h_{1}(\mathbf{x})\) and \(h(\mathbf{x})=h_{2}(\mathbf{x})\) into (3.9) and (3.10), respectively, we get
\[\int_{\Omega}h_{1}^{2}(\mathbf{x})\;\mathrm{d}\mathbf{x}=\int_{\Omega}h_{2}^{2}(\mathbf{x} )\;\mathrm{d}\mathbf{x}=0.\]
It shows that \(F(\mathbf{x})=G(\mathbf{x})=0\).
For the second case, because of the rotation invariance of the biharmonic operator \(\Delta^{2}\), the vector \(\iota\) can be rotated appropriately to any coordinate axis. Without loss of generality, we assume that \(\iota\) rotates to the \(x_{3}\)-axis, then we have
\[\partial_{x_{3}}F(\mathbf{x})=\partial_{x_{3}}G(\mathbf{x})=0,\]
which means \(F(\mathbf{x}),G(\mathbf{x})\) only depend on the variables \(x_{1},x_{2}\) for \((x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\).
Denote by
\[h(\mathbf{x})=e^{\mathrm{i}\tilde{\mathbf{\xi}}\cdot\mathbf{x}},\quad\mathbf{x}\in\mathbb{R}^{3} \tag{3.17}\]
be a harmonic function, where
\[\tilde{\mathbf{\xi}}=\mathbf{\xi}_{1}+\mathrm{i}\mathbf{\xi}_{2},\quad\ \mathbf{\xi}_{1}=(\xi_{1},\xi_{2},0)^{\top}\in\mathbb{R}^{3},\quad\ \mathbf{\xi}_{2}=(0,0,\xi_{3})^{\top}\in\mathbb{R}^{3},\]
and satisfies \(\xi_{1}^{2}+\xi_{2}^{2}=\xi_{3}^{2}\).
Plugging (3.17) into (3.9), we compute
\[\int_{\mathbb{R}^{3}}F(x_{1},x_{2})e^{\mathrm{i}\tilde{\mathbf{\xi}} \cdot\mathbf{x}}\ \mathrm{d}\mathbf{x} =\int_{B_{R}}F(x_{1},x_{2})e^{\mathrm{i}\tilde{\mathbf{\xi}}\cdot\mathbf{ x}}\ \mathrm{d}\mathbf{x}\] \[=\int_{\mathbb{R}^{2}}F(x_{1},x_{2})e^{\mathrm{i}\xi_{1}\cdot x_{ 1}+\mathrm{i}\xi_{2}\cdot x_{2}}\ \mathrm{d}x_{1}\mathrm{d}x_{2}\int_{\{x_{3};(x_{1},x_{2},x_{3})\in B_{R}\}}e ^{-\xi_{3}x_{3}}\mathrm{d}x_{3}=0.\]
It follows from the priori information of \(\rho_{i},f_{i}\) and \(g_{i},i=1,2\) that
\[0=\int_{\mathbb{R}^{2}}F(x_{1},x_{2})e^{\mathrm{i}\xi_{1}\cdot x_{1}+\mathrm{ i}\xi_{2}\cdot x_{2}}\ \mathrm{d}x_{1}\mathrm{d}x_{2}=(\mathcal{F}F)(\mathbf{\xi}_{1}),\]
which holds for any \((\xi_{1},\xi_{2})\in\mathbb{R}^{2}\). Since \((\mathcal{F}F)(\mathbf{\xi}_{1})\) means the Fourier transform of \(F(x_{1},x_{2})\), it clearly showed that \(F(\mathbf{x})=0\). We can state \(G(\mathbf{x})=0\) by using the same methods. The proof is completed.
Now, we discuss the uniqueness for density and internal sources by using above orthogonality results.
**Theorem 3.3**.: _Assume that \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) are two sets of configurations and supported in \(\Omega\), which satisfy both conditions:_
1. \(\rho_{1}\) _and_ \(\rho_{2}\) _are positive constants;_
2. \(\nabla f_{i}(\mathbf{x})\cdot\mathbf{\iota}=0\) _and_ \(\nabla g_{i}(\mathbf{x})\cdot\mathbf{\iota}=0,i=1,2\)_, where_ \(\mathbf{\iota}\) _is an arbitrary direction vector in_ \(\mathbb{R}^{3}\)_._
_If_
\[\Lambda_{\rho_{1},f_{1},g_{1}}(t,\mathbf{x})=\Lambda_{\rho_{2},f_{2},g_{2}}(t,\bm {x}),\quad\ (t,\mathbf{x})\in\mathbb{R}_{+}\times\partial\Omega, \tag{3.18}\]
_and suppose that_
\[\int_{\Omega}g_{i}(\mathbf{x})\ \mathrm{d}\mathbf{x}\neq 0,\quad\ i=1,2,\quad\ \mathbf{x}\in\Omega. \tag{3.19}\]
_Then_
\[\rho_{1}=\rho_{2},\quad\ f_{1}(\mathbf{x})=f_{2}(\mathbf{x}),\quad\ g_{1}(\mathbf{x})=g_{ 2}(\mathbf{x}).\]
Proof.: By rotation invariance, without loss of generality, we still assume that \(\iota\) rotates to the \(x_{3}\)-axis. Then \(\rho_{1}f_{1}(\mathbf{x})-\rho_{2}f_{2}(\mathbf{x})\) and \(\rho_{1}g_{1}(\mathbf{x})-\rho_{2}g_{2}(\mathbf{x})\) only depend on the variables \(x_{1},x_{2}\). By Theorem 3.2 and (3.18), we deduce
\[\rho_{1}f_{1}(\mathbf{x})=\rho_{2}f_{2}(\mathbf{x}),\quad\ \rho_{1}g_{1}(\mathbf{x})=\rho_{2}g_{2}(\mathbf{x}), \quad\ \mathbf{x}\in\Omega.\]
It can be seen that
\[\int_{B_{R}}\rho_{1}f_{1}(\mathbf{y})\;\mathrm{d}\mathbf{y}= \int_{B_{R}}\rho_{2}f_{2}(\mathbf{y})\;\mathrm{d}\mathbf{y}, \tag{3.20}\] \[\int_{B_{R}}\rho_{1}g_{1}(\mathbf{y})\;\mathrm{d}\mathbf{y}= \int_{B_{R}}\rho_{2}g_{2}(\mathbf{y})\;\mathrm{d}\mathbf{y}, \tag{3.21}\]
and
\[\int_{B_{R}}\rho_{1}g_{1}(\mathbf{y})|\mathbf{x}-\mathbf{y}|^{2}\;\mathrm{d}\mathbf{y}= \int_{B_{R}}\rho_{2}g_{2}(\mathbf{y})|\mathbf{x}-\mathbf{y}|^{2}\;\mathrm{d}\mathbf{y}. \tag{3.22}\]
Substituting (3.20)-(3.22) into (3.11), imply
\[(\rho_{1}-\rho_{2})\int_{B_{R}}\big{(}\int_{B_{R}}\rho_{1}g_{1}( \mathbf{z})\;\mathrm{d}\mathbf{z}\big{)}g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}=0,\; \mathbf{x}\in\partial B_{R}.\]
From the assumption (3.19), we have
\[\rho_{1}\int_{B_{R}}g_{1}(\mathbf{z})\;\mathrm{d}\mathbf{z}\neq 0.\]
It is easy to verify that
\[\int_{B_{R}}\big{(}\int_{B_{R}}\rho_{1}g_{1}(\mathbf{z})\;\mathrm{d} \mathbf{z}\big{)}g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\neq 0\quad\text{ for }\mathbf{x}\in \partial B_{R}.\]
Therefore, we have \(\rho_{1}=\rho_{2}\) and imply \(f_{1}(\mathbf{x})=f_{2}(\mathbf{x})\) and \(g_{1}(\mathbf{x})=g_{2}(\mathbf{x})\), respectively.
Next, we prove the uniqueness for density and internal sources in a domain with an anomalous inclusion. Let \(\rho_{0}(\mathbf{x})\) be a positive background density which is known in advance and \(\varrho_{i},i=1,2\) be a positive constant denoting different anomalous inclusion supported in \(\Omega_{0}\subset\Omega\).
**Corollary 3.1**.: _Assume that \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) are two sets of configurations and supported in \(\Omega\), which satisfy the conditions_
1. \(\rho_{i}(\mathbf{x})=\rho_{0}(\mathbf{x})+\varrho_{i}\chi_{\Omega_{0}}\) _with_ \(\varrho_{i},i=1,2\) _is a constant;_
2. \(\nabla\rho_{0}(\mathbf{x})\cdot\mathbf{\iota}=0\)_,_ \(\nabla f_{i}(\mathbf{x})\cdot\mathbf{\iota}=0\) _and_ \(\nabla g_{i}(\mathbf{x})\cdot\mathbf{\iota}=0,i=1,2\)_, where_ \(\mathbf{\iota}\) _is an arbitrary direction vector in_ \(\mathbb{R}^{3}\)_._
_If_
\[\Lambda_{\rho_{1},f_{1},g_{1}}(t,\mathbf{x})=\Lambda_{\rho_{2},f_{2},g_ {2}}(t,\mathbf{x}),\quad\;(t,\mathbf{x})\in\mathbb{R}_{+}\times\partial\Omega,\]
_and suppose that_
\[\int_{\Omega_{0}}\big{(}\int_{B_{R}}\rho_{1}(\mathbf{z})g_{1}(\mathbf{z} )\;\mathrm{d}\mathbf{z}\big{)}h(\mathbf{x})\;\mathrm{d}\mathbf{x}\neq 0,\quad\;\mathbf{x}\in \Omega_{0} \tag{3.23}\]
_for any harmonic function \(h(\mathbf{x})\). Then_
\[\varrho_{1}=\varrho_{2},\quad f_{1}(\mathbf{x})=f_{2}(\mathbf{x}),\quad \;g_{1}(\mathbf{x})=g_{2}(\mathbf{x}).\]
Proof.: Using analogue analysis as Theorem 3.3. We assume that \(\rho_{0}\), \(f_{i}\) and \(g_{i}\) only depend on the variables \(x_{1},x_{2}\), and write them as \(\rho_{0}(\mathbf{x})=\rho_{0}(x_{1},x_{2}),f_{i}(\mathbf{x})=f_{i}(x_{1},x_{2}),g_{i}( \mathbf{x})=g_{i}(x_{1},x_{2})\) for \((x_{1},x_{2})\in\mathbb{R}^{2},i=1,2\), then we deduce
\[\rho_{1}(\mathbf{x})f_{1}(\mathbf{x})= \rho_{2}(\mathbf{x})f_{2}(\mathbf{x}),\quad\;\rho_{1}(\mathbf{x})g_{1}(\mathbf{x} )=\rho_{2}(\mathbf{x})g_{2}(\mathbf{x}),\quad\;\mathbf{x}\in\Omega.\]
It is easy to see that
\[\int_{B_{R}}\rho_{1}(\mathbf{y})f_{1}(\mathbf{y})\;\mathrm{d}\mathbf{y}= \int_{B_{R}}\rho_{2}(\mathbf{y})f_{2}(\mathbf{y})\;\mathrm{d}\mathbf{y},\] \[\int_{B_{R}}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})\;\mathrm{d}\mathbf{y}= \int_{B_{R}}\rho_{2}(\mathbf{y})g_{2}(\mathbf{y})\;\mathrm{d}\mathbf{y},\]
and
\[\int_{B_{R}}\rho_{1}(\mathbf{y})g_{1}(\mathbf{y})|\mathbf{x}-\mathbf{y}|^{2}\;\mathrm{d}\mathbf{y}= \int_{B_{R}}\rho_{2}(\mathbf{y})g_{2}(\mathbf{y})|\mathbf{x}-\mathbf{y}|^{2}\;\mathrm{d}\mathbf{y} \quad\text{ for }\mathbf{x}\in\partial B_{R}.\]
Taking above identities into (3.11), imply
\[\int_{B_{R}}(\rho_{1}(\mathbf{y})-1)\big{(}\int_{B_{R}}\rho_{1}(\mathbf{z })g_{1}(\mathbf{z})\;\mathrm{d}\mathbf{z}\big{)}g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{y}\] \[= \int_{B_{R}}(\rho_{2}(\mathbf{y})-1)\big{(}\int_{B_{R}}\rho_{2}(\mathbf{z })g_{2}(\mathbf{z})\;\mathrm{d}\mathbf{z}\big{)}g_{0}(|\mathbf{x}-\mathbf{y}|)\;\mathrm{d}\mathbf{ y}\quad\text{ for }\mathbf{x}\in\partial B_{R}.\]
It follows from the proof of Theorem 3.1 that
\[\int_{B_{R}}(\rho_{1}-\rho_{2})(\mathbf{y})\big{(}\int_{B_{R}}\rho_{1}(\mathbf{z})g_{ 1}(\mathbf{z})\;\mathrm{d}\mathbf{z}\big{)}h(\mathbf{y})\;\mathrm{d}\mathbf{y}=0\quad\text{ for }\mathbf{x}\in\partial B_{R}, \tag{3.24}\]
where \(h(\cdot)\) is any harmonic function.
Substituting \(\rho_{i}=\rho_{0}(\mathbf{x})+\varrho_{i}\chi_{\Omega_{0}}\) into (3.24), we have
\[(\varrho_{1}-\varrho_{2})\int_{\Omega_{0}}\big{(}\int_{B_{R}}\rho_{1}(\mathbf{z}) g_{1}(\mathbf{z})\;\mathrm{d}\mathbf{z}\big{)}h(\mathbf{y})\;\mathrm{d}\mathbf{y}=0.\]
Because of the condition (3.23), we get
\[\varrho_{1}=\varrho_{2},\]
which yields \(g_{1}(\mathbf{x})=g_{2}(\mathbf{x})\) and \(f_{1}(\mathbf{x})=f_{2}(\mathbf{x})\).
**Remark 3.3**.: _In fact, the condition (3.19) is a special form for (3.23). There are other ways to achieve the non-zero condition for (3.23)._
Besides the assumptions of density and internal sources in Theorem 3.3 and Corollary 3.1, we also consider whether there are other circumstances in which a more general uniqueness result can be derived. The following example illustrates a more general result.
**Example 3.1**.: _Let \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) be two sets of configurations and supported in \(\Omega\), which satisfy_
\[(\rho_{2},f_{2},g_{2})=(\rho_{1}+a,f_{1}+b,g_{1}+c)\]
_with \(a(\mathbf{x}),b(\mathbf{x}),c(\mathbf{x})\in L^{\infty}(\mathbb{R}^{3})\) are nonnegative and supported in \(\Omega\). Furthermore, suppose that \(f_{1}(\mathbf{x}),g_{1}(\mathbf{x})>0\). If_
\[\Lambda_{\rho_{1},f_{1},g_{1}}(t,\mathbf{x})=\Lambda_{\rho_{2},f_{2},g_{2}}(t,\bm {x}),\quad\;(t,\mathbf{x})\in\mathbb{R}_{+}\times\partial\Omega,\]
_then_
\[a(\mathbf{x})=b(\mathbf{x})=c(\mathbf{x})=0.\]
Proof.: Assume that at least one of \(a(\mathbf{x}),b(\mathbf{x})\) and \(c(\mathbf{x})\) is not zero. Without losing of generality, we set \(a\neq 0\), then we have
\[(\rho_{1}f_{1}-\rho_{2}f_{2})(\mathbf{x})= (\rho_{1}f_{1}-(\rho_{1}f_{1}+af_{1}+b\rho_{1}+ab))(\mathbf{x})=-(af_{1 }+b\rho_{1}+ab)(\mathbf{x})<0,\] \[(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{x})= (\rho_{1}g_{1}-(\rho_{1}g_{1}+ag_{1}+c\rho_{1}+ac))(\mathbf{x})=-(ag_{1 }+c\rho_{1}+ac)(\mathbf{x})<0,\]
which yields
\[\int_{\Omega}(\rho_{1}f_{1}-\rho_{2}f_{2})(\mathbf{x})\;\mathrm{d}\mathbf{ x}<0, \tag{3.25}\] \[\int_{\Omega}(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{x})\;\mathrm{d}\bm {x}<0. \tag{3.26}\]
It follows from (3.9) and (3.10) that
\[\int_{\Omega}(\rho_{1}f_{1}-\rho_{2}f_{2})(\mathbf{x})\;\mathrm{d} \mathbf{x}= 0,\] \[\int_{\Omega}(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{x})\;\mathrm{d}\bm {x}= 0,\]
when taking \(h(\mathbf{x})=1\). Therefore, the inequalities (3.25) and (3.26) are contradiction. The proof is completed.
It can be seen from the above example that a more general uniqueness result can be obtained if additional assumptions of density and sources are considered. The detail process will be shown in the following section.
## 4. Extension to general results
In this section, the previous assumptions of density and internal sources will be replaced by assuming some size relationships of density, internal sources and their coupling term, which implies more general unique results.
**Lemma 4.1**.: _Assume that \((\rho_{1},f_{1},g_{1})\) and \((\rho_{2},f_{2},g_{2})\) are two sets of configurations and supported in \(\Omega\). If_
\[\Lambda_{\rho_{1},f_{1},g_{1}}(t,\mathbf{x})=\Lambda_{\rho_{2},f_{2},g_{2}}(t,\bm {x}),\ \ \ \ (t,\mathbf{x})\in\mathbb{R}_{+}\times\partial\Omega,\]
_and satisfies_
\[(\rho_{1}g_{1})(\mathbf{x})\leq(\rho_{2}g_{2})(\mathbf{x})\ \ \ \ \text{or}\ \ \ \ (\rho_{1}g_{1})(\mathbf{x})\geq(\rho_{2}g_{2})(\mathbf{x}),\ \ \ \ \mathbf{x}\in\Omega. \tag{4.1}\]
_Then_
\[\rho_{1}(\mathbf{x})g_{1}(\mathbf{x})=\rho_{2}(\mathbf{x})g_{2}(\mathbf{x}). \tag{4.2}\]
_In addition, if_
\[\int_{\mathbb{R}^{3}}(\rho_{i}g_{i})(\mathbf{x})\;\mathrm{d}\mathbf{x}\neq 0,\ \ \ \ i=1,2,\]
_we have_
\[\int_{\mathbb{R}^{3}}(\rho_{1}-\rho_{2})(\mathbf{x})h(\mathbf{x})\;\mathrm{d}\mathbf{x}=0, \tag{4.3}\]
_where \(h(\mathbf{x})\) is any harmonic function in \(\mathbb{R}^{3}\)._
Proof.: Let \(h(\mathbf{x})=1\), then it follows from (3.9) and (3.10) that
\[\int_{B_{R}}(\rho_{1}f_{1})(\mathbf{y})\;\mathrm{d}\mathbf{y}= \int_{B_{R}}(\rho_{2}f_{2})(\mathbf{y})\;\mathrm{d}\mathbf{y}, \tag{4.4}\] \[\int_{B_{R}}(\rho_{1}g_{1})(\mathbf{y})\;\mathrm{d}\mathbf{y}= \int_{B_{R}}(\rho_{2}g_{2})(\mathbf{y})\;\mathrm{d}\mathbf{y}. \tag{4.5}\]
Given by the conditions (4.1), if
\[(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{x})>0\quad\text{ or }\quad(\rho_{1}g_{1}-\rho_{ 2}g_{2})(\mathbf{x})<0,\]
we have
\[\int_{B_{R}}(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{y})\;\mathrm{d}\mathbf{y}>0\quad \text{ or }\quad\int_{B_{R}}(\rho_{1}g_{1}-\rho_{2}g_{2})(\mathbf{y})\;\mathrm{d}\mathbf{y}<0.\]
This contradiction with (4.5). Therefore, we get
\[(\rho_{1}g_{1})(\mathbf{x})=(\rho_{2}g_{2})(\mathbf{x}),\]
and imply
\[\int_{B_{R}}(\rho_{1}g_{1})(\mathbf{y})|\mathbf{x}-\mathbf{y}|^{2}\;\mathrm{d}\mathbf{y}=\int_ {B_{R}}(\rho_{2}g_{2})(\mathbf{y})|\mathbf{x}-\mathbf{y}|^{2}\;\mathrm{d}\mathbf{y}. \tag{4.6}\]
Substituting (4.4)-(4.6) into (3.11), we obtain
\[\int_{B_{R}}(\rho_{1}-\rho_{2})(\mathbf{y})\;g_{0}(|\mathbf{x}-\mathbf{y}|)\; \mathrm{d}\mathbf{y}=0.\]
Repeating the process of (3.9) and (3.10) in Theorem 3.1, we derive an orthogonal relation
\[\int_{\mathbb{R}^{3}}(\rho_{1}-\rho_{2})(\mathbf{x})\;h(\mathbf{x})\; \mathrm{d}\mathbf{x}=0\]
for any harmonic function \(h(\mathbf{x})\) in \(\mathbb{R}^{3}\).
**Corollary 4.1**.: _With the same assumptions of Lemma 4.1. If \(\rho_{i}(\mathbf{x}),i=1,2\) satisfies the either of the following conditions:_
1. \((\rho_{1}-\rho_{2})(\mathbf{x})\) _is a harmonic function in_ \(\mathbb{R}^{3}\)_;_
2. \(\rho_{1}(\mathbf{x})\leq\rho_{2}(\mathbf{x})\) _or_ \(\rho_{1}(\mathbf{x})\geq\rho_{2}(\mathbf{x}),\quad\mathbf{x}\in\Omega\)_._
_Then_
\[\rho_{1}(\mathbf{x})=\rho_{2}(\mathbf{x})\quad\text{ and }\quad g_{1}(\mathbf{x})=g_{2}(\bm {x}).\]
_Furthermore, suppose that_
\[f_{1}(\mathbf{x})\leq f_{2}(\mathbf{x})\quad\text{ or }\quad f_{1}(\mathbf{x})\geq f_{2}(\mathbf{x }),\quad\mathbf{x}\in\Omega. \tag{4.7}\]
_Then_
\[f_{1}(\mathbf{x})=f_{2}(\mathbf{x}).\]
Proof.: For the first case, taking \((\rho_{1}-\rho_{2})(\mathbf{x})=h(\mathbf{x})\) into (4.3), which implies
\[\int_{\Omega}h^{2}(\mathbf{x})\;\mathrm{d}\mathbf{x}=0.\]
Thus we can obtain \(\rho_{1}(\mathbf{x})=\rho_{2}(\mathbf{x})\).
For the second case, substituting \(h(\mathbf{x})=1\) into (4.3), we get
\[\int_{\Omega}(\rho_{1}-\rho_{2})(\mathbf{x})\;\mathrm{d}\mathbf{x}=0.\]
By using the conditions of \(\rho_{1}(\mathbf{x})\) and \(\rho_{2}(\mathbf{x})\), we deduce \(\rho_{1}(\mathbf{x})=\rho_{2}(\mathbf{x})\). It follows from (4.2) that \(g_{1}(\mathbf{x})=g_{2}(\mathbf{x})\).
Let \(\rho_{1}(\mathbf{x})=\rho_{2}(\mathbf{x})=\rho(\mathbf{x})\) and \(h(\mathbf{x})=1\), plugging them into (3.9), we have
\[\int_{\Omega}\rho(\mathbf{y})(f_{1}-f_{2})(\mathbf{y})\;\mathrm{d}\mathbf{y}=0.\]
Therefore, given by (4.7), \(f_{1}(\mathbf{x})=f_{2}(\mathbf{x})\) is proved.
| この論文は、流体力学と弾性力学のモデルから導出されるプレート方程式に関連する逆問題に焦点を当てています。私たちは、 pasive boundary measurement から未知の密度と内部の源を同時に決定することに焦点を当てています。証明は主に、積分変換における漸近的分析と調和分析に依存しています。 |
2304.13021 | Face Feature Visualisation of Single Morphing Attack Detection | This paper proposes an explainable visualisation of different face feature
extraction algorithms that enable the detection of bona fide and morphing
images for single morphing attack detection. The feature extraction is based on
raw image, shape, texture, frequency and compression. This visualisation may
help to develop a Graphical User Interface for border policies and specifically
for border guard personnel that have to investigate details of suspect images.
A Random forest classifier was trained in a leave-one-out protocol on three
landmarks-based face morphing methods and a StyleGAN-based morphing method for
which morphed images are available in the FRLL database. For morphing attack
detection, the Discrete Cosine-Transformation-based method obtained the best
results for synthetic images and BSIF for landmark-based image features. | Juan Tapia, Christoph Busch | 2023-04-25T17:51:23 | http://arxiv.org/abs/2304.13021v1 | # Face Feature Visualisation of Single Morphing Attack Detection
###### Abstract
This paper proposes an explainable visualisation of different face feature extraction algorithms that enable the detection of bona fide and morphing images for single morphing attack detection. The feature extraction is based on raw image, shape, texture, frequency and compression. This visualisation may help to develop a Graphical User Interface for border policies and specifically for border guard personnel that have to investigate details of suspect images. A Random forest classifier was trained in a leave-one-out protocol on three landmarks-based face morphing methods and a StyleGAN-based morphing method for which morphed images are available in the FRLL database. For morphing attack detection, the Discrete Cosine-Transformation-based method obtained the best results for synthetic images and BSIF for landmark-based image features.
Morphing Attack Detection, Explainability, Visualisation +
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Some studies suggest applying image forensics techniques to detect the origin of image manipulation. They focus on noise patterns by analysing pixel discontinuities that may be impacted by morphing algorithms - like Photo Response Non-Uniformity (PRNU) [9] and Sensor Pattern Noise (SPN) [10], or on image quality by quantifying image degradation of artefacts in morphed faces [11].
Tapia et al. [3] proposed adding an extra Feature Selection (FS) stage after feature extraction of LBP, HOG and Raw images based on Mutual Information. Since high redundancy between features confuses the classifier, they identify the most relevant features, and remove the most redundant ones from the feature vector, to better separate bona fide and morphed images in an S-MAD scenario. The authors also conclude that the eyes and nose are the most relevant facial areas.
Very recently, Dargaud et al. [12] proposed a visualisation approach based on 50 Principal Component Analyses (PCA) and explored several colour channels, such as RGB, HSV and others, to determine that the blue channel is one of the most relevant to visualise the difference between bona fide and morphed images.
## III Databases
In this study, four different databases of frontal faces images are used: the Facial Recognition Technology (FERET) [13], the Face Recognition Grand Challenge (FRGCv2) [14], the Face Research London Lab (FRLL) [15] and AMSL database [16]. The morphed images in these datasets have been created using a morphing factor of 0.5, meaning both parent images contribute equally to the morphed image. FERET and FRGCv2 morphed images have been used to complement bona fide images. The FRLL and AMSL morphed images have been generated with the following morphing tools: FaceMorpher, FaceFusion and Webmorpher based on landmarks and StyleGAN from FRLL without landmarks. Table I provides a summary of the datasets.
FERET dataset is a subset of the Colour FERET Database, generated in the context of the Facial Recognition Technology program technically handled by the National Institute of Standards and Technology (NIST). It contains 569 faces bona fide.
The FRGCv2 dataset used in this work is a constrained subset of the second version of the Face Recognition Grand Challenge dataset. It contains 979 bona fide face images.
The FRLL dataset is a subset of the publicly available Face Research London Lab dataset. It contains 102 bona fide neutral and 102 smiling images. Three morphing algorithms were applied to obtain 1,222 morphs from the FaceMorpher algorithm, 1,222 morphs from the StyleGAN algorithm, and 1,222 morphs from the WebMorph algorithm [17].
The AMSL Face Morph Image Data Set is a collection of bona fide and morphed face images that can be used to evaluate the detection performance of MAD algorithms. The images are organised as follows: genuine-neutral with 102 genuine neutral face images, genuine-smiling with 102 genuine smiling face images and 2,175 morphing face images. The following morphing tool has been used:
* **FaceFusion**[18]: this proprietary mobile application developed by MOMENT generates realistic faces since morphing artefacts are almost invisible.
* **FaceMorpher**[19]: this open-source Python implementation relies on STASM, a facial feature finding the package, for landmark detection, but generated morphs show many artefacts which make them more recognisable.
* **OpenCV**[20]: this open-source morphing algorithm is similar to the FaceMorpher method, but it uses Dlib to detect face landmarks. Again, some artefacts remain in generated morphs.
* **Webmorpher**[17]: this open-source morphing algorithm is a web-based version of Psychomorph with several additional functions. While WebMorph is optimized for averaging and transforming faces, you can delineate and average any image.
* **StyleGAN2**[21]: this open-source morphing algorithm by NVIDIA, No landmarks are used.
## IV Metrics
The detection performance of the investigated S-MAD algorithms was measured according to ISO/IEC 30107-3 1 using the Equal Error Rate (EER), Bona fide Presentation Classification Error Rate (BPCER), and Attack Presentation Classification Error Rate (APCER) metric defined as (1) and (2).
Footnote 1: [https://www.iso.org/standard/79520.html](https://www.iso.org/standard/79520.html)
\[BPCER=\frac{\sum_{i=1}^{N_{BF}}RES_{i}}{N_{BF}} \tag{1}\]
\[APCER=\frac{1}{N_{PAIS}}\sum_{i=1}^{N_{PAIS}}{(1-RES_{i})} \tag{2}\]
where \(N_{BF}\) is the number of bona fide presentations, \(N_{PAIS}\) is the number of morphing attacks for a given attack instrument species and \(RES_{i}\) is \(1\) if the system's response to the \(i-th\) attack is classified as an attack and \(0\) if classified as bona fide. In this work, S-MAD performance is reported using the Equal Error Rate (EER), which is the point where the APCER is equal to BPCER. Also, two operational points are reported BPCER10 and BPCER20. The BPCER20 is the BPCER value obtained when the APCER is fixed at 5%, and BPCER10 (APCER at 10%).
\begin{table}
\begin{tabular}{|c|c|c|c|} \hline Dataset & Bona fide & Morphing & Notes \\ \hline AMSL & 204 & 2,175 & * Bona fide subjects are the same of FRLL \\ \hline \multirow{3}{*}{FRLL} & \multirow{3}{*}{0*} & 1,222 & MT: Opcevr \\ \cline{3-3} & & 1,222 & MT: FaceMorpher \\ \cline{3-3} & & 1,222 & MT: StyleGAN2 \\ \cline{3-3} & & 1,222 & MT: WebMorpher \\ \hline FERET & 529 & 2,116 & N/A \\ \hline FRGC & 979 & 3,904 & N/A \\ \hline Total & 1.718 & 13.083 & 14,801 \\ \hline \end{tabular}
\end{table} TABLE I: Summary databases. MT: Morphing Tools.
## V Method
### _Feature extraction methods_
Our intention is to determine which feature would be the most useful and can deliver the most specific information to separate both classes. In order to obtain and leverage different features for the bona fide and morphing images, eight different feature extraction methods and several combinations are utilised: RAW images (Intensity levels), Discrete Fourier Transform (DFT) [22], Steganalysis Rich Model (SRM) [23], Error Level Analysis (ELA) [24], Single Value Decomposition (SVD), Local Binary Patterns (LBP), Binary Statistical Image Feature (BISF). These methods are used separately as input for the Random Forest Classifier and tested against each other.
The purpose of the DFT [22] is to transform the image into its frequency domain representation. The intuition behind this is that differences between the frequencies of multiple face capture devices, which were used to generate the parent images.
SRM has been used successfully in related works [23] in order to detect MA by utilising the local noise features in an image.
ELA is used to detect differences of compression in distinct regions of a JPEG image, which may be a residual effect caused by tampering with an image in JPEG format and resaving it in that same format.
These feature extraction methods are applied to the original \(1280\times 720\) resolution image, which is then resized and cropped to the input shape of the network. This specific order of preprocessing contributes to a better separation of the classes, whereas resizing the image and extracting the features resulted in worse classification performance in all tests.
#### V-A1 Intensity
For raw data, the intensity of the values in grayscale was used and normalised between 0 and 1.
#### V-A2 Discrete Fourier transform
The discrete Fourier transform (DFT) decomposes a discrete time-domain signal into its frequency components.
For training purposes, only the magnitude (real) and not the phase (complex) is used. The magnitude image is then transformed from a linear scale to a logarithmic scale to compress the range of values. Furthermore, the quadrants of the matrix are shifted so that zero-value frequencies are placed at the centre of the image.
#### V-A3 Uniform Local Binary Pattern
The histogram of uniform LBP and BSIF [25] were used for texture. LBP is a grey-scale texture operator which characterises the spatial structure of the local image texture. Given a central pixel in the image, a binary pattern number is computed by comparing its value with those of its neighbours.
#### V-A4 The Binary Statistical Image Feature
was also explored as a texture method. BSIF is a local descriptor designed by binarising the responses to linear filters. The filters learn from 13 natural images. The code value of pixels is considered a local descriptor of the image intensity pattern in the pixels' surroundings. The value of each element (i.e bit) in the binary code string is computed by binarising the response of a linear filter with a zero threshold. Each bit is associated with a different filter, and the length of the bit string determines the number of filters used. A grid search from the 60 filters available in BSIF implementation was explored. The filter \(5\times 5\) and 9 bits obtained the best results estimated from the baseline approach. The resulting BSIF images were used as input for the classifiers.
#### V-A5 Inverse Histogram Oriented Gradient
For the purpose of describing the shape, the inverse Histogram of oriented gradients [26] was used. The distribution directions of gradients are used as features. Gradients, \(x\), and \(y\) derivatives of an image are helpful because the magnitude of gradients is large around edges and corners (regions of abrupt intensity changes). We know edges and corners contain more information about object shapes than flat regions. We used the visualisation proposed by Vondrik et al. [26] to select the best parameters that allow us to visualise the artefacts contained in morphed images. This implementation used \(10\times 12\) blocks and \(3\times 3\) filter sizes.
#### V-A6 Steganalysis Rich Model
SRM filters yield noise features from neighbouring pixels, which can be applied to detect discrepancies between real and tampered images. The input and output are 3-channel images. As used by Zhou et al. [23], the kernels shown in Figure 1 are applied to the images, which are then directly used as the input for training the networks.
#### V-A7 Error level Analysis
ELA [24, 27] is a forensic method to identify portions of an image saved in a JPEG format with a different level of compression. ELA is based on characteristics of image formats that are based on a lossy image compression technique that could be used to determine if an image has been digitally modified.
JPEG is a method of lossy compression for digital images. The compression level is chosen as a trade-off between image size and quality. A JPEG compression scale is usually 70%. The compression of data discards or loses information. The JPEG algorithm works on image grids, compressed independently, having a size of \(8\times 8\) pixels. The \(8\times 8\) dimension was chosen using a grid search. Meanwhile, any matrices of size less than \(8\times 8\) do not have enough information. They result in poor-quality compressed images.
ELA highlights differences in the JPEG compression rate. Regions with uniform colourings, like solid blue or white pixels, will likely have a lower ELA result (darker colour) than high-contrast edges. Highlighted regions can be, potentially, tampered regions in the image that suffered a second JPEG compression after the user saves the tampered image.
Figure 2 presents visualisation examples for ELA, DFT, DCT, SVD and SRM morphed images. The last row, show visualisation examples of a random bona fide image.
Fig. 1: SRM filter kernels.
## VI Experiments and Results
In this paper, a LOO protocol was defined in order to evaluate the influence of the feature extracted. According to each experiment, we explored five cross-dataset tests. In the end, we performed 125 evaluations in total. In order to train S-MAD, all the datasets were divided into 70.0% for train and 30,0% for testing.
A LOO protocol was applied to train and test the S-MAD system, which means in the first round, FaceFusion was used to compute the morphing images used for the test, and training was carried out with FaceMorpher, OpenCV-Morpher, UBO-Morpher and FRLL. In the second round, FaceMorpher was used for testing and training was done with morphed images created with FaceFusion, OpenCV-Morpher, UBO-Morpher and FRLL, and so on. All the images were aligned, cropped and resized to \(180\times 240\).
Different kinds of features were extracted from faces based on SRM, ELA, DFT, SVD, LBP and BSIF filters for all experiments. We used the intensity values of the pixels from raw images normalised between 0 and 1. The histogram of the uniform LBP and BSIF was used for texture. For the uLBP, all radii values were explored from uLBP81 to uLBP88. The fusion of LBPs was also investigated, concatenating the LBP81 to LBP88. The image's vertical (uLBP_VERT) and horizontal (uLBP_HOR) concatenation divided into eight patches was also explored.
After feature extraction, we fused that information at the feature level by concatenating the feature vectors from different sources into a single feature vector that becomes the input to the classifier. From BSIF, the resulting images of filter \(3\times 3-5bits\) were used considering BISF images (BSIF-IM), BSIF Histogram (BSIF-H) and BSIF normalised histogram (BSIF-NH). All the features were extracted after applying our proposed transfer texture method.
The S-MAD system was trained with a Random Forest classifier with each feature extraction method described above as input. All the individual and average results through all features extracted are presented in Table II. The first column of Table II identifies each evaluation's LOO dataset test.
Figure 3 show a bar plot with all the feature methods used on X-axis and the EER on the Y-axis. A dot-plot line was added in order to show the average results of all the methods for each dataset. For the AMSL dataset reaches the lower EER. The higher EER is obtained for MA developed with FaceMorpher images. The results obtained in LBP show that this texture feature is not suitable for identifying synthetic images such as StyleGAN2 images. Conversely, DCT shows impressive results across the datasets, including the StyleGANs MA.
Figure 4 shows two DET plots illustrating the error trade-off for the four S-MAD methods with the EER on percentages in parentheses for the best case. The left images show the results of the DET curve for the BSIF feature on the AMSL database. Where FaceMorpher reached an EER 11.90%, OpenCV 8.38%, StyleGAN2 3,30%, and WebMorph obtained a 3,23%. The BPCER10/20 obtained is 13.5% and 16.3%. The right DET shows the DCT feature's results in the normalised histogram. Where FaceMorpher reached an EER 0.73%, OpenCV 1.41%, StyleGAN2 4.06%, and WebMorph obtained an 0.29%. The BPCER10/20 obtained is 0.46% and 0.83%.
## VII Conclusions
This work shows that different feature extractors can deliver relevant information to guide the analysis of MAD. FaceMorpher has been identifying, on average, the morphing tool with the highest EER. Textures and frequencies are more effective in visualising the details of bona fide and morphed images without compression. ELA has been identified as a very good tool for detecting changes in JPEG compression. After this work, the main challenge is identifying common parameters to tune all the filters. As we can declare now, different datasets need proper parameters. The synthetics images based on GANs are not difficult to identify using the DCT feature compared to Landmark-based such as FaceMorpher. As a future work, fusion-specific features may be extended to Deep learning methods to identify specific morphing tools.
| |
2307.05313 | Programmable and arbitrary-trajectory ultrafast flying focus pulses | "Flying focus" techniques produce laser pulses with dynamic focal points that
travels distances much greater than a Rayleigh length. The implementation of
these techniques in laser-based applications requires the design of optical
configurations that can both extend the focal range and structure the radial
group delay. This article describes a method for designing optical
configurations that produce ultrashort flying focus pulses with
arbitrary-trajectory focal points. The method is illustrated by several
examples that employ an axiparabola for extending the focal range and either a
reflective echelon or a deformable mirror-spatial light modulator pair for
structuring the radial group delay. The latter configuration enables rapid
exploration and optimization of flying foci, which could be ideal for
experiments. | M. V. Ambat, J. L. Shaw, J. J. Pigeon, K. G. Miller, T. T. Simpson, D. H. Froula, J. P. Palastro | 2023-07-11T15:00:07 | http://arxiv.org/abs/2307.05313v1 | # Programmable and arbitrary-trajectory ultrafast flying focus pulses
###### Abstract
"Flying focus" techniques produce laser pulses with dynamic focal points that travels distances much greater than a Rayleigh length. The implementation of these techniques in laser-based applications requires the design of optical configurations that can both extend the focal range and structure the radial group delay. This article describes a method for designing optical configurations that produce ultrashort flying focus pulses with arbitrary-trajectory focal points. The method is illustrated by several examples that employ an axiparabola for extending the focal range and either a reflective echelon or a deformable mirror-spatial light modulator pair for structuring the radial group delay. The latter configuration enables rapid exploration and optimization of flying foci, which could be ideal for experiments.
## 1 Introduction
The intensity peak of a flying focus pulse can travel at any velocity, independent of the group velocity, over distances much longer than a Rayleigh range [1, 2, 3, 4, 5]. These properties offer a new approach to optimizing the wide range of laser-based applications that require velocity matching or extended interaction lengths. For instance, recent experiments have used a flying focus to create long, contiguous plasma channels [6, 7] and to synchronize the pump and probe pulses in soft x-ray lasers [8]. The potential uses of flying focus pulses extend beyond these demonstrations to enhancing laser wakefield acceleration [3, 9, 10], nonlinear Thomson scattering [11], or THz generation [12] and to facilitating observations of fundamental processes, such as radiation reaction [13] and Compton scattering [14]. The ultimate success of these applications relies on the design of practical, and preferably adaptive, optical configurations for preparing flying focus pulses.
The first experimental realization of a flying focus used a highly chromatic diffractive optic to focus a chirped laser pulse [2]. The diffractive optic focuses each wavelength of the pulse to a different longitudinal location, while the chirp controls the arrival time of each wavelength at its focus. The resulting intensity peak traverses the focal range, i.e., the distance between the focal points of the minimum and maximum wavelengths, with a constant velocity that can be adjusted by changing the chirp. More complex spectral phases allow for more complex focal trajectories [1, 15]. Despite its tunability, this "chromatic flying focus" has several limitations. First, because the extended focal range is produced by a static diffractive optic, it cannot be modified from shot to shot. Second and more importantly, the bandwidth of the pulse is spread across the focal region. This precludes the formation of an ultrashort (<100 fs) intensity peak, which is a requirement for many applications.
The need for ultrashort intensity peaks has motivated the development of flying focus techniques that preserve the entire bandwidth of the laser pulse at every location within the focal range [3, 5, 9]. In contrast to the chromatic flying focus, which uses radial group delay to extend the focal range, these "ultrafast flying focus" schemes employ separate optics to independently extend the focal range and structure the radial group delay. As an example, a recent demonstration of an ultrafast, constant-velocity flying focus [5] used the geometric aberration of an axiparabola [16, 17, 18] to focus different annuli in the near field to different longitudinal locations in the far field and the
radial group delay imparted by an echelon [3] to control the relative timing of the annuli. Despite the success of these experiments, the configuration relies on the use of a static echelon designed for a specific focal trajectory. An alternative configuration that replaces the echelon with adaptive optics, such as a deformable mirror-spatial light modulator pair [19, 20], would allow for on-shot programmability of the radial group delay and, as a result, the focal trajectory.
This work describes a method for designing optical configurations that produce ultrashort flying focus pulses with arbitrary focal trajectories at velocities close to the speed of light (Section II). The general method is independent of the optical configuration but is illustrated for specific examples of an axiparabola combined with either an echelon or a deformable mirror-spatial light modulator pair (Section III). The method is applied to create flying focus pulses exhibiting constant velocity, constant acceleration, and oscillating focal trajectories (Section IV). In each case, the intensity peak of the flying focus maintains an ultrashort duration as it traverses the extended focal range. The flexibility afforded by this method and the deformable mirror-spatial light modulator pair (DM-SLM) enable rapid and automated control over the focal trajectory, which can facilitate the use of the ultrafast flying focus in laser-based applications.
## 2 The focal trajectory of an ultrafast flying focus
Figure 1 compares the trajectories of focal points produced by a focusing optic alone (a) and a focusing optic used in combination with optics that structure the radial group delay (b) and (c).
Figure 1: The effect of optics on the focal trajectory. (a) A laser pulse with a flat pulse front (red) and flat phase front (grey) is focused by an optic that extends the focal range \(L\) (blue). The trajectory of the focus is completely determined by the focal geometry. (b) and (c) The pulse front, or radial group delay \(\tau_{D}(r)\), is structured by a preliminary optic (purple). The structure of the pulse front can be used to create a constant-velocity focus (b), an oscillating focus (c), or otherwise dynamic trajectories.
In Fig. 1(a), a laser pulse with a flat phase front and a flat pulse front is incident at \(z=0\) on a focusing optic with a surface defined by the sag function \(s_{f}(r)\). The focusing optic extends the range of high intensity by using geometric aberration to focus different radial locations \(r\) in the near field to different longitudinal locations in the far field \(z=f(r)\). The resulting focal point travels a distance \(L=\max(f)-\min(f)\) along a trajectory that is fully determined by the sag function. In Figs. 1(b) and (c), additional optics are used to structure the pulse front, or radial group delay \(\tau_{D}(r)\), before focusing. Structuring the delay provides control over the trajectory of the focus and can produce a constant-velocity (b), oscillating (c), or otherwise dynamic focal point.
Each optical element in Fig. 1 applies a spatio-spectral phase to the laser pulse. The phase imparted by the entire optical assembly \(\phi(\omega,r)\) can be written as the sum of contributions from the focusing optic and the elements that structure the radial group delay (RGD). In the paraxial approximation (see Appendix A),
\[\phi(\omega,r)=-\frac{2\omega}{c}s_{f}(r)+\phi_{D}(\omega,r). \tag{1}\]
The first term provides the initial phase front curvature required to focus each radius to the location \(z=f(r)\). With \(f(r)\) specified, the sag function \(s_{f}(r)\) can be found by solving
\[\frac{ds_{f}}{dr}=\frac{r}{2f(r)}. \tag{2}\]
The second term in Eq. (1) modifies the relative timing of the near-field radii,
\[\tau_{D}(r)=\frac{\partial\phi_{D}(\omega,r)}{\partial\omega}. \tag{3}\]
To preserve the desired focusing, the elements that structure the RGD cannot significantly distort the phase fronts. The constraint \(\partial_{r}\phi_{D}(\omega,r)|_{\omega=\omega_{0}}=0\) ensures that \(\phi_{D}\) only modifies the RGD and, equivalently, that the central frequency of the laser pulse \(\omega_{0}\) focuses to the locations described by \(f(r)\).
For applications, one would like to specify a focal trajectory, i.e., the time-dependent velocity of the focus \(v_{f}(t)\), and use this trajectory to determine the required \(\tau_{D}(r)\). To calculate the required \(\tau_{D}(r)\), first note that each near-field radius of the laser pulse can arrive at its focal location \(z=f(r)\) at a different time. The focal time \(t_{f}(r)\) for each radius has contributions from the structured RGD and the focal geometry:
\[t_{f}(r)\approx\tau_{D}(r)+\frac{1}{c}\left[f(r)+\frac{r^{2}}{2f(r)}-2s_{f}(r )\right]. \tag{4}\]
The variation in the focal time and location with radius results in a moving focal point with a velocity
\[\tilde{v}_{f}(r)=\frac{df}{dr}\left(\frac{dt_{f}}{dr}\right)^{-1}\approx c \left[1+\frac{r^{2}}{2f^{2}(r)}-c\left(\frac{df}{dr}\right)^{-1}\frac{d\tau_{ D}(r)}{dr}\right]. \tag{5}\]
Equation 5 demonstrates that the structured RGD can be used to control the trajectory of the focus independently of the focal geometry. If \(\tau_{D}(r)=0\), \(\tilde{v}_{f}(r)=c\left[1+r^{2}/2f^{2}(r)\right]\), which is dictated soley by \(f(r)\). Rearranging Eq. (5) provides a differential equation for the \(\tau_{D}(r)\) needed to produce a specified trajectory \(v_{f}(t)\):
\[c\frac{d\tau_{D}}{dr}=\left[1-\frac{v_{f}\big{(}t_{f}(r)\big{)}}{c}+\frac{r^{ 2}}{2f^{2}(r)}\right]\frac{df}{dr}, \tag{6}\]
where \(v_{f}(t_{f}(r))=\tilde{v}_{f}(r)\) depends on \(\tau_{D}\) through Eq. (4) and a one-to-one mapping between near-field radius and time has been assumed. The solutions to Eqs. (2) and (6) form the basis for designing the optical elements necessary to create an ultrafast flying focus.
In order to preserve the ultrashort duration of the intensity peak at every point within the focal range, the focal velocity must be close to the speed of light, \(v_{f}(t)\approx c\). Even if a \(\phi_{D}\) satisfies the constraint \(\partial_{r}\phi_{D}|_{\omega=\omega_{0}}=0\) and maintains the focal locations of the central frequency, it will modify the focal locations of every other frequency. This spreads the frequency content of the laser pulse across the focal region, which reduces the bandwidth available at each location and places a lower bound on the minimum duration. Noting that the transverse wavenumber is the radial derivative of the phase and using similar triangles, one can show that the RGD modifies the focal locations by a distance \(\Delta f(\omega,r)\approx-cf^{2}(\partial_{r}\phi_{D})/(r\omega)\). This longitudinal chromatism will have a negligible effect on the duration of the intensity peak when \(\Delta f\) is much smaller than the focal range \(L\), i.e., when
\[\frac{\Delta\omega}{\omega_{0}}\frac{f^{2}}{rL}\left|\frac{df}{dr}\left(1- \frac{v_{f}}{c}\right)\right|\ll 1, \tag{7}\]
where \(\Delta\omega\) is the bandwidth of the laser pulse and Eq. (6) has been used with a simple form of \(\phi_{D}(\omega,r)=(\omega-\omega_{0})\tau_{D}(r)\).
## 3 Optical elements to create an ultrafast flying focus
### Optics to extend the focal range
The optics that extend the focal range use geometric aberration to focus different radial locations \(r\) in the near field to different longitudinal locations in the far field \(z=f(r)\). In principle, this can be accomplished using refractive optics like lenses. However, for broadband, ultrashort pulses, the B-integral, group velocity dispersion, and higher-order dispersion of these optics can broaden or distort the temporal profile. In addition, the damage threshold of refractive optics typically prohibits their use as final focusing elements for high-intensity pulses. Thus, reflective optics are often preferable for extending the focal range of high-intensity, ultrashort flying focus pulses.
One such optic, the axiparabola [16, 17], produces an near-constant, on-axis intensity maximum over the entire focal range, making it ideal for many applications. The focal length as a function of near-field radius \(f(r)\) is designed so that a flattop transverse intensity profile incident on the optic results in a uniform on-axis intensity maximum in the far field. Specifically,
\[f(r) =f_{0}+L\left(\frac{r}{R}\right)^{2}, \tag{8}\] \[s_{f}(r) =\frac{R^{2}}{4L}\ln\left[1+\frac{L}{f_{0}}\left(\frac{r}{R} \right)^{2}\right], \tag{9}\]
where \(f_{0}\) is the nominal focal length, \(R\) is the maximum radius of the axiparabola, and \(L\) determines the length of the focal range. Expanding Eq. (9) in powers of \(q\equiv L/f_{0}\) shows that the axiparabola is primarily a parabolic mirror \(\mathcal{O}(q^{0})\) with spherical aberration \(\mathcal{O}(q^{1})\). For \(L>0\) (\(<0\)), rays incident at larger radii are focused farther from (closer to) the optic than rays incident at smaller radii. With this choice of \(f(r)\), Eq. (7) simplifies to \(2(\Delta\omega/\omega_{0})(f_{0}/R)^{2}|1-v_{f}/c|\ll 1\), which is independent of \(L\).
Figure 2 displays the results of propagation simulations (see Appendix B) for a laser pulse focused by an axiparabola with \(f_{0}=50\) cm, \(R=5\) cm, and \(L=1\) cm. The laser pulse had a central wavelength \(\lambda_{0}=2\pi c/\omega_{0}=920\) nm and \(\Delta\lambda=78\) nm of bandwidth in a Gaussian power spectrum, corresponding to a 27 fs full-width at half-maximum (FWHM) duration. The transverse profile was initialized as a flattop with a 5 cm radius that filled the aperture of the axiparabola. The maximum on-axis intensity is nearly uniform over the entire focal range \(L\), which is \(\sim\)340\(\times\) longer than the Rayleigh range of the full-aperture focal spot \(Z_{R}=\lambda_{0}f_{0}^{2}/\pi R^{2}\)
[Fig. 2(b)]. The modulations in the on-axis intensity result from diffraction of the spherically aberrated phase fronts (see Appendix C). The near-uniform on-axis intensity comes at the cost of a spot size \(w\) that narrows over the focal range [Fig. 2(c)]. More specifically, the effective \(f/\#\) at the beginning of the focal range is larger than that at the end, such that within the focal region
\[w(z)\approx\frac{\lambda_{0}f_{0}}{\pi R}\left|\frac{L}{z-f_{0}}\right|^{1/2}. \tag{10}\]
The ring-like structures visible in the fluence [Fig. 2(c)] are the natural diffraction pattern created by the axiparabola.
Figure 2(d) illustrates the focal trajectory produced by the axiparabola. Here, the on-axis intensity is plotted as a function of propagation distance \(z-f_{0}\) and the moving frame coordinate \(\xi=t-z/c\). In these coordinates, a vertical line indicates a signal travelling at the vacuum speed of light. The intensity peak accelerates from its initial focal point at \(z-f_{0}=0\) and \(\xi=0\) to its final focal point at \(z-f_{0}=L\) and \(\xi\approx-75\) fs, following a trajectory consistent with \(\tilde{v}_{f}(r)=c\left[1+r^{2}/2f^{2}(r)\right]\). The pulse maintains its ultrashort duration over the entire focal range as shown by the white lineouts taken at the start (right) and end (left) of the focal region.
### Optics to structure the radial group delay
The trajectory of the focus can be programmed by structuring the radial group delay of the laser pulse. Ideal, achromatic focusing optics impart the exact amount of RGD needed to ensure that all frequency components within a pulse arrive at their focus at the same time. More generally, optics can impart unwanted RGD, resulting in asynchronous focusing and a reduction in the maximum focused intensity. For instance, with refractive optics, the combination of group velocity dispersion and the radially dependent thickness of the optic produce unfavorable RGD [21]. Below, optical elements are discussed that can impart favorable RGD, thereby enabling control over the trajectory of the focal point and the peak laser intensity.
The recently proposed and demonstrated radial echelon provides a reflective approach to structuring the radial group delay [3, 5]. The mirrored surface of the echelon consists of concentric rings with variable widths determined by the desired RGD and depths \(d\) equal to a half-integer multiple of the central wavelength \(d=(\ell/2)\lambda_{0}=\pi\ell/\omega_{0}\), where \(\ell\) is a positive integer. For a
Figure 2: The focal properties of an axiparabola alone. (a) The sag function of an axiparabola with \(f_{0}=50\) cm, \(R=5\) cm, and \(L=1\) cm. The axiparabola focuses a laser pulse with a central wavelength of \(\lambda_{0}=920\) nm and a \(27\) fs FWHM duration. (b) The maximum on-axis intensity of the pulse as a function of distance from the nominal focal point \(z=f_{0}\). (c) The fluence profile. (d) The focal trajectory as a function of propagation distance and moving frame coordinate \(\xi=t-z/c\). The peak intensity travels at a superluminal velocity and accelerates. The white lineouts show the temporal profile of the pulse at the beginning (right) and end (left) of the focal region.
given \(\tau_{D}(r)\) and \(\ell=1\), the phase imparted by the echelon is given by
\[\phi_{D}^{\rm ech}(\omega,r)=-\frac{2\omega}{c}\left\{\frac{1}{4}\lambda_{0} \left[{\rm ceil}\left(\frac{c\tau_{D}(r)}{\lambda_{0}}\right)+{\rm floor}\left( \frac{c\tau_{D}(r)}{\lambda_{0}}\right)\right]\right\}. \tag{11}\]
By discretizing the continuous delay \(c\tau_{D}(r)\) in steps of the central wavelength, the echelon satisfies the constraint \(\partial_{r}\phi_{D}^{\rm ech}(\omega,r)|_{\omega=\omega_{0}}=0\) and thus does not affect the focusing of the frequency component \(\omega_{0}\). Said differently, the phase fronts of the central wavelength maintain their transverse coherence upon reflection from the echelon. For any other wavelength, the echelon introduces a shear in the phase front between each ring. This shear smooths out as higher-spatial orders diffract, leaving the desired radial group delay. The widths of the echelon rings can also lead to diffractive losses. These losses are negligible when \(\Delta R\gg\lambda_{0}f_{0}/2R\), which is easily satisfied for a large range of designs. Importantly, for \(v_{f}(t)\approx c\), the combined axiparabola-echelon system preserves an ultrashort pulse duration.
Despite its advantage as a reflective optic with a higher damage threshold, each echelon is a static optical element that can only impart a single, pre-designed RGD. Adaptive optics, such as deformable mirrors and spatial light modulators, offer dynamic programmability of the radial group delay and, as a result, the focal trajectory. A deformable mirror (DM) consists of pistons or piezoelectric segments that shape a flexible, reflective membrane [22, 23]. A DM can be programmed to apply the continuous phase
\[\Phi_{\rm dm}(\omega,r)=-\frac{2\omega}{c}s_{\rm dm}(r)=\omega\tau_{D}(r), \tag{12}\]
where \(s_{\rm dm}(r)=-c\tau_{D}(r)/2\) is the sag function of the membrane. However, the phase \(\Phi_{\rm dm}(\omega,r)\) does not satisfy the constraint \(\partial_{r}\Phi_{\rm dm}(\omega,r)|_{\omega=\omega_{0}}=0\). Thus a second optical element must be introduced to eliminate the phase distortion at the central frequency.
A spatial light modulator (SLM) can partially correct the phase front distortion at the central frequency [20]. An SLM consists of a pixelated, two-dimensional array of liquid crystals that possess electrical and optical anisotropy. The voltage delivered to each pixel can be adjusted to change the optical path length of an incident laser pulse as a function of transverse location [24, 25]. By appropriately programming the SLM voltages, the phase front of the central frequency can be
Figure 3: (a) The radial group delays, i.e., the \(\tau_{D}(r)\), required to produce constant-velocity focal trajectories with \(v_{f}=1.001c\), (blue, solid), \(v_{f}=c\) (green, dashed), and \(v_{f}=0.999c\) (red, dotted) with the axiparabola described in Fig. 2. (b) The echelon profile for \(v_{f}=c\). (c) The deformable mirror sag function (green) and spatial light modulator phase (black) for \(v_{f}=c\).
flattened to an extent allowed by the discreteness of the pixels. Specifically, for the DM phase in Eq. (12),
\[\Phi_{\rm slm}(\omega,r)=-\frac{\omega}{c}\lambda_{0}{\rm mod}\left[\frac{c\tau_{D }(r_{p})}{\lambda_{0}},1\right], \tag{13}\]
where \(r_{p}=\frac{1}{2}[{\rm floor}(\frac{r}{p})+{\rm ceil}(\frac{r}{p})]p\) and \(p\) is the SLM pixel size. The total phase of the DM-SLM pair is then
\[\phi_{D}^{\rm dm\text{-}{\rm slm}}(\omega,r)=\Phi_{\rm dm}(\omega,r)+\Phi_{\rm slm }(\omega,r). \tag{14}\]
In the limit of infinitesimal pixels, \(p\to 0\) and \(\phi_{D}^{\rm dm\text{-}{\rm slm}}(\omega,r)\rightarrow\phi_{D}^{\rm ech}( \omega,r)\). Note that Eq. (13) was discretized into radial zones; for Cartesian zones, one can instead use \(\tau_{D}(x_{p},y_{p})\).
Figures 3 and 4 illustrate how these optics modify the electric field profile of a laser pulse in the near field to produce a constant-velocity focus. Figure 3(a) shows the \(\tau_{D}(r)\) required for subluminal (\(v_{f}<c\)), luminal (\(v_{f}=c\)), and superluminal (\(v_{f}>c\)) focal velocities when using the axiparabola described in Fig. 2. Because the axiparabola naturally produces a superluminal and accelerating focus, the subluminal (superluminal) velocity requires a larger (smaller) delay than the luminal velocity at larger radii. The echelon and DM-SLM designs for \(v_{f}=c\) are displayed in Figs. 3(b) and (c). In this configuration, the incident laser pulse propagates from right to left, so that the center of the pulse encounters the optics first. Figure 4 shows the effect that each optic
Figure 4: Modification to the electric field in the near field for \(v_{f}=c\). (a) The input field has flat phase fronts, a flat pulse front, and an ultrashort duration (\(\sim 27\) fs). (b) The echelon imparts the desired radial group delay to the pulse while maintaining the flat phase fronts. (c) The DM imparts the desired radial group delay to the pulse. However, as shown in the inset, the phase fronts are now curved with respect to the propagation direction. (d) The SLM corrects the undesired phase front curvature. The inset shows that the phase fronts are now globally flat, but retain a residual tilt within each pixel. Each inset is a \(500~{}\mu\)m \(\times~{}15\) fs window, and the SLM had a \(p=50~{}\mu\)m pixel size. The pulse propagates from left to right.
has on the electric field profile. After the echelon [Fig. 4(b)], the field has flat phase fronts and a radially dependent delay consistent with \(\tau_{D}(r)\). After the DM [Fig. 4(c)], the field has the correct delay, but also has curved phase fronts. The SLM undoes this curvature [Fig. 4(d)]. The combined DM-SLM system reproduces the field profile created by the echelon to within the resolution limits of the SLM.
A DM-SLM pair with sufficiently small pixels can create a flying focus that is virtually indistinguishable from a flying focus created by an echelon [Fig. 5]. While an echelon flattens the phase fronts globally and locally, an SLM can only flatten the phase fronts globally. Within each pixel, the phase fronts remain curved [Fig. 4(d) inset]. As a result, the constraint \(\partial_{r}\phi_{D}^{\text{dm-slim}}(\omega,r)|_{\omega=\omega_{0}}=0\) is only approximately satisfied. When the SLM pixel size is too large, the local curvature of the phase fronts affects the structure of the flying focus pulse in the far field. The inequality \(\max(\partial_{r}\phi_{D}^{\text{dm-slim}})p\ll 1\) provides a rough condition for the SLM pixel size required to reproduce the flying focus created with an echelon. Failing to meet this condition in the near field results in a decreased intensity at corresponding locations in the far field [cf. Figs. 5(b) and (c)]. As the pixel size is reduced, the intensity profile converges to the profile produced using an echelon [cf. Figs. 5(a) and (d)].
## 4 Examples of ultrashort flying focus trajectories
This section presents examples that demonstrate the flexibility and far-field properties of the ultrafast flying focus. The examples, i.e., constant-velocity, accelerating, and oscillating focal trajectories, are motivated by applications in plasma physics and nonlinear optics. The propagation of pulses that exhibit these trajectories was simulated in the near and far fields using a combination of the Fresnel diffraction integral and the modified paraxial wave equation (see Appendix B for details) [15, 26]. In all cases, an axiparabola with \(f_{0}=50\) cm, \(R=5\) cm, and \(L=1\) cm, a deformable mirror with a 5 cm radius, and a spatial light modulator with a pixel size of \(p=50\)\(\mu\)m were used to extend the focal range and structure the RGD. The parameters were chosen based on the capabilities of current technology.
### Constant-velocity focal trajectories
A constant-velocity flying focus can enhance applications that rely on velocity matching over long distances, such as laser wakefield acceleration [3, 9, 10, 27], THz generation [12], and photon acceleration [28, 29]. Figure 6 shows the on-axis intensity for the (a) superluminal, (b) luminal, and (c) subluminal velocities described in Fig. 3. In each case, the intensity peak travels along
Figure 5: The maximum on-axis intensity of flying focus pulses with \(v_{f}=c\) created using (a) an echelon or a DM-SLM pair with an SLM pixel size of (b) \(p=200\)\(\mu\)m, (c) \(p=100\)\(\mu\)m, and (d) \(p=50\)\(\mu\)m.
the designed constant-velocity trajectory. The images also reveal that the combination of the DM-SLM and axiparabola produce features similar to those of the axiparabola alone. Namely, the on-axis intensity is modulated, and the ultrashort pulse duration is preserved over the entire focal region [cf. Fig. 2].
### Exotic focal trajectories
An accelerating focus can be used to control the trapping and acceleration of electrons in a laser wakefield accelerator. Initializing the intensity peak, and therefore the wakefield, with a subluminal velocity would facilitate the trapping of background plasma electrons in the plasma wave [3, 30]. After sufficient trapping has occurred, the intensity peak can be accelerated to a luminal or superluminal velocity. This change in velocity has the dual benefit of preventing electrons from outrunning the accelerating phase of the wakefield, i.e., dephasing, and of improving the quality of the electron bunch by eliminating unwanted trapping [31].
Figure 7 illustrates an ultrafast flying focus that accelerates from an initial subluminal velocity to a superluminal velocity over the focal range. The design trajectory was specified as
\[v_{f}(t)=v_{0}+\Delta v\left(\frac{ct-f_{0}}{L}\right), \tag{15}\]
with an initial velocity \(v_{0}=0.99c\) and a velocity increment \(\Delta v=0.02c\). Over the first half of the focal range, the on-axis intensity falls back in a frame moving at the vacuum speed of light [Fig. 7(a)]. At the half-way point the velocity has increased to \(c\), and thereafter the intensity peak advances in the speed of light frame. Interestingly, the radial group delay required for this trajectory [Figs. 7(b) and (c)] smooths the intensity modulations that were observed with both the axiparabola alone and with the DM-SLM constant-velocity trajectories [cf. Figs. 2 and 6].
A pulse with an oscillating focal point could provide a novel method for quasi-phase-matching nonlinear optical processes, a wiggler for generating radiation from relativistic electrons, or an additional degree of freedom for accessing new parametric resonances in direct laser acceleration [32]. An example of such a focus is shown in Fig. 8. In this case, the design focal
Figure 6: Ultrafast flying foci with constant velocities. The maximum on-axis intensity of the pulse as a function of distance from the nominal focal point \(z=f_{0}\) for (a) \(v_{f}=1.001c\), (b) \(v_{f}=c\), and (c) \(v_{f}=0.999c\).
trajectory was specified as
\[v_{f}(t)=v_{0}+\Delta v\sin\left(\frac{2\pi N(ct-f_{0})}{L}\right), \tag{16}\]
with a nominal velocity \(v_{0}=c\), an oscillation magnitude \(\Delta v=0.002c\), and \(N=3\) periods. As shown in Fig. 8(a), the on-axis intensity peak oscillates between the expected velocities. While the pulse maintains its ultrashort duration, the maximum value of the intensity exhibits modulations, as it did in the case of the axiparabola alone. In general, the oscillation period of the velocity should be much greater than the Rayleigh range of the full-aperture focal spot, so that the intensity modulations do not obscure the velocity oscillations, i.e., \(N\ll\pi R^{2}L/\lambda_{0}f_{0}^{2}\).
## 5 Conclusions and outlook
This work has described a method for structuring ultrashort laser pulses with dynamic focal points. The moving focal point, or "flying focus," can follow a near-arbitrary trajectory over distances much greater than a Rayleigh range, while maintaining an ultrashort duration. The method employs separate optics to extend the focal range and structure the radial group delay (RGD). This overcomes a disadvantage of previous flying focus techniques, which place a lower bound on the duration of the moving intensity peak. Two specific optical configurations were considered: an axiparabola, which uses geometric aberration to extend the focal range, combined with either an echelon or a deformable mirror-spatial light modulator (DM-SLM) pair to structure the RGD. While an echelon can apply the exact RGD required for a particular focal trajectory, it is a static optic that cannot be modified on a shot-to-shot basis. The DM-SLM pair, on the other hand, has constraints imposed by the resolution of the SLM, but allows for dynamic programmability and optimization of the focal trajectory. This capability could enable rapid exploration of exotic flying foci that benefit laser-based applications in plasma physics and nonlinear optics.
Figure 7: An ultrafast flying focus that accelerates from an initial subluminal velocity to a superluminal velocity over the focal range. (a) The maximum on-axis intensity of the pulse as a function of distance from the nominal focal point \(z=f_{0}\). (b) The radial group delay, i.e., the \(\tau_{D}(r)\), required to produce this trajectory. (c) The corresponding deformable mirror sag function (green) and spatial light modulator phase (black). The pulse propagates from right to left.
## Appendix A Focal trajectory produced by an extended focal range optic
Consider a laser pulse with an initially flat phase front and flat pulse front propagating in the negative \(\hat{\mathbf{z}}\)-direction. Assuming cylindrical symmetry, the rays composing the phase and pulse front can be identified by their radial distance \(r=(x^{2}+y^{2})^{1/2}\) from the propagation axis and their frequency \(\omega\). The rays travel parallel to the axis and are incident on a reflective optic defined by the sag function \(s_{f}(r)\). At the point of reflection, each ray acquires a transverse wavenumber \(k_{r}(\omega,r)=(\omega/c)\sin[2\theta(r)]\), where \(\theta(r)=\arccos[\hat{\mathbf{z}}\cdot\hat{\mathbf{n}}(r)]\) defines the angle between the \(+\hat{\mathbf{z}}\)-direction and the normal vector to the surface of the optic \(\hat{\mathbf{n}}(r)=[D(r)\hat{\mathbf{r}}-\hat{\mathbf{z}}]/\sqrt{1+D^{2}(r)}\) with \(D(r)\equiv ds_{f}/dr\). After some algebra, one finds
\[k_{r}(\omega,r)=-\frac{2\omega}{c}\frac{D(r)}{1+D^{2}(r)}. \tag{17}\]
The perpendicular wavenumber is simply the radial derivative of the phase, such that
\[\phi_{f}(\omega,r)=-\frac{2\omega}{c}\int\,\frac{D(r)}{1+D^{2}(r)}dr. \tag{18}\]
In the paraxial approximation, Eq. (18) simplifies to \(\phi_{f}(\omega,r)=-2\omega s_{f}(r)/c\), which is the first term on the right-hand side of Eq. (1).
The trajectory of the rays as they travel to the far field can be found by integrating the ray equations \(\dot{\mathbf{x}}^{\prime}=c^{2}\mathbf{k}/\omega\), where the overdot denotes a total time derivative and the prime denotes the instantaneous location of the ray. The radial and longitudinal locations of the rays evolve according to
\[r^{\prime}(t) =r+\frac{ck_{r}(\omega,r)}{\omega}[ct+s_{f}(r)] \tag{19}\] \[z^{\prime}(t) =s_{f}(r)+\frac{ck_{z}(\omega,r)}{\omega}[ct+s_{f}(r)], \tag{20}\]
where \(ct\geq-s_{f}(r)\), \(t=0\) corresponds to the time at which the ray with \(r=0\) reflects from the optic, and \(k_{z}(\omega,r)=[\omega^{2}/c^{2}-k_{z}(\omega,r)]^{1/2}\). The focal time \(t_{f}(r)\) and location \(f(r)\) of each ray
Figure 8: An ultrafast flying focus that oscillates between subluminal and superluminal velocities. (a) The maximum on-axis intensity of the pulse as a function of distance from the nominal focal point \(z=f_{0}\). (b) The radial group delay, i.e., the \(\tau_{D}(r)\), required to produce this trajectory. (c) The corresponding deformable mirror sag function (green) and spatial light modulator phase (black). The pulse propagates from right to left.
are defined as the values of \(t\) and \(z^{\prime}\) where \(r^{\prime}=0\). Solving for the value of \(t\) where Eq. (19) equals zero and using this in Eq. (20) yields
\[ct_{f}(r)=-s_{f}(r)+\frac{1+D^{2}(r)}{2D(r)}r \tag{21}\]
\[f(r)=s_{f}(r)+\frac{1-D^{2}(r)}{2D(r)}r, \tag{22}\]
where Eq. (17) has been used. The focal time and location are both independent of frequency.
The focal location depends implicitly on the focal time through their shared dependence on \(r\). This dependence results in a focal point that moves in time. The velocity of the focal point \(\tilde{v}_{f}(r)\) is given by
\[\frac{\tilde{v}_{f}(r)}{c}=\frac{df}{dr}\left(\frac{dct_{f}}{dr}\right)^{-1}= \frac{1+D^{2}(r)}{1-D^{2}(r)}, \tag{23}\]
which is constrained by the focal geometry \(D(r)\) and is always superluminal (\(D^{2}\) is positive definite).
When each ray is delayed by a time \(\tau_{D}(r)\) before reflecting from the optic, the focal time \(t_{f}(r)\to t_{f}(r)+\tau_{D}(r)\), and Eq. (23) can be rewritten as a differential equation for the delay needed to produce a specified focal trajectory \(v_{f}(t)\):
\[\frac{d\tau_{D}}{dr}=\left[\frac{c}{v_{f}\big{(}t_{f}(r)\big{)}}-\left(\frac {1-D^{2}(r)}{1+D^{2}(r)}\right)\right]\frac{df}{dr}, \tag{24}\]
where \(v_{f}\big{(}t_{f}(r)\big{)}=\tilde{v}_{f}(r)\). The paraxial limits of these equations are presented in the main text for simplicity.
### Simulation details
The evolution of the flying focus pulse was simulated in two steps. The first step used the frequency-domain Fresnel integral to propagate the laser pulse from the flying focus optical configuration to the far field. The second step used the modified paraxial wave equation to propagate the pulse through the far field [15, 26]. The results shown in the figures were obtained from this second step.
To solve for the evolution of the flying focus pulse, the transverse electric field was written as a carrier modulating an envelope: \(\mathrm{E}(\xi,r,z)=\frac{1}{2}e^{-i\omega_{0}\xi}E(\xi,r,z)+\mathrm{c.c.}\), where \(\xi=t-z/c\) is the moving frame coordinate. The carrier frequency \(\omega_{0}\) was chosen so that the central wavelength \(\lambda_{0}=2\pi c/\omega_{0}=920\) nm. The envelope \(E\) was initialized just before the optical configuration in the frequency domain with the profile
\[\tilde{E}_{0}(\delta\omega,r)=\tilde{E}_{i}\Theta(r-R)\exp{(-\frac{1}{4}\tau^ {2}\delta\omega^{2})}, \tag{25}\]
where \(\sim\) denotes a frequency domain field, \(\delta\omega=\omega-\omega_{0}\), \(\Theta\) is the Heaviside function, \(\tilde{E}_{i}\) is the initial amplitude, \(R=5\) cm, and \(\tau=23\) fs, corresponding to a full width at half maximum duration and bandwidth of \(27\) fs and \(\Delta\lambda=78\) nm, respectively.
The phase imparted by the optical configuration, i.e., an axiparabola combined with either an echelon or a deformable mirror-spatial light modulator pair, was applied to the initial envelope. Just after the optical configuration at \(z=0\), the envelope can be expressed as \(\tilde{E}_{0}(\delta\omega,r)e^{i\phi(\omega,r)}\), where \(\phi(\omega,r)\) is the phase applied by the optical configuration [Eq. (1)]. The envelope was propagated in vacuum from \(z=0\) to the far-field location \(z=z_{i}\) using the frequency-domain Fresnel integral:
\[\tilde{E}(\delta\omega,r,z=z_{i})=\frac{\omega}{icz_{i}}\int J_{0}\left(\frac {\omega rr^{\prime}}{cz_{i}}\right)\exp{\left[\frac{i\omega(r^{2}+r^{\prime 2})}{ 2cz_{i}}+i\phi(\omega,r^{\prime})\right]}\tilde{E}_{0}(\delta\omega,r^{\prime}) r^{\prime}dr^{\prime}, \tag{26}\]
where \(J_{0}\) is the zeroth-order Bessel function of the first kind. The electric field from the Fresnel integral \(\tilde{E}(\omega,r,z=z_{i})\) provided the initial condition for the modified paraxial wave equation [26]:
\[[2(i\omega_{0}-\partial_{\xi})\partial_{z}+c\nabla_{\perp}^{2}]E(r,z,\xi)=0. \tag{27}\]
The mixed space-time derivative in Eq. (27) ensures that effects such as radial group delay and angular dispersion are modelled correctly--a requirement for accurately modeling an ultrafast flying focus. Note that Eqs. (26) and (27) are fully consistent with one another: Eq. (26) is the integral solution to Eq. (27). The use of the Fresnel integral decouples the radial grids in the near field and far field, reducing computational expense compared to using Eq. (27) over the entire domain, especially when considering smaller \(f/\#\)'s [15].
The simulation parameters were motivated by the MTW-OPAL laser system at the Laboratory for Laser Energetics [33], where future ultrafast flying focus experiments are being planned. The longitudinal step size \(\Delta z=2.83\)\(\mu\)m, temporal resolution \(\Delta\xi=0.74\) fs, and radial resolution \(\Delta r=0.60\)\(\mu\)m, were chosen to resolve the Rayleigh range, transform-limited pulse duration, and spot size, respectively.
## Appendix C On-axis intensity modulation from an axiparabola
The Fresnel diffraction integral can be used to derive an approximate expression for the far-field, on-axis intensity profile of a laser pulse focused by an axiparabola. The expression reveals that the on-axis intensity modulations result from the spherical aberration imparted by the axiparabola and provides a condition for mitigating these modulations. The derivation begins by substituting Eq. (25) into Eq. (26) and approximating the axiparabola phase as
\[\phi(\omega,r^{\prime})=-\frac{\omega{r^{\prime}}^{2}}{2cf_{0}}\left(1-\frac{ L}{2f_{0}}\frac{{r^{\prime}}^{2}}{R^{2}}\right), \tag{28}\]
which includes the parabolic and spherical contributions and is accurate to second order in \(L/f_{0}\). Evaluating Eq. (26) on-axis, i.e., at \(r=0\), provides
\[\tilde{E}(\delta\omega,0,z)=\frac{\omega}{icz}\int_{0}^{R}\exp\left[\frac{i \omega{r^{\prime}}^{2}}{2c}\left(\frac{1}{z}-\frac{1}{f_{0}}\right)+\frac{i \omega{Lr^{\prime}}^{4}}{4cf_{0}^{2}R^{2}}\right]\tilde{E}_{0}(\delta\omega){ r^{\prime}}{dr^{\prime}}, \tag{29}\]
where \(\tilde{E}_{0}(\delta\omega)=\tilde{E}_{i}\exp\left(-\frac{1}{4}\tau^{2}\delta \omega^{2}\right)\). Upon integrating, one finds
\[\frac{|\tilde{E}(\delta\omega,0,z)|^{2}}{|\tilde{E}_{0}(\delta\omega)|^{2}} \approx\frac{\pi\omega R^{2}}{4cL}\left|\text{erfi}\left[\left(\frac{i\omega R ^{2}}{4cLf_{0}^{2}}\right)^{1/2}(f_{0}-z)\right]-\text{erfi}\left[\left(\frac {i\omega R^{2}}{4cLf_{0}^{2}}\right)^{1/2}(f_{0}+L-z)\right]\right|^{2}, \tag{30}\]
where erfi is the imaginary error function and \(z\approx f_{0}\) has been assumed. Equation (30) oscillates with a period that varies throughout the focal region. The scale length apparent in Eq. (30) provides a rough estimate for the modulation period: \(L_{M}\sim(4Lf_{0}^{2}\lambda_{0}/R^{2})^{1/2}\). The modulations can be mitigated when \(L\gg L_{M}\) or \(L\gg 4\pi Z_{R}\), where \(Z_{R}=\lambda_{0}f_{0}^{2}/\pi R^{2}\) is the Rayleigh range of the full-aperture focal spot.
Funding.U.S. Department of Energy Office of Fusion Energy Award Number DE-SC00215057, U.S. Department of Energy National Nuclear Security Administration Award Number DE-NA0003856. The authors would like to thank D. Ramsey, J. Bromage, C. Dorrer, S.-W. Bahk, C. Jeon, B. Webb, and I. Begishev for productive discussions.
This material is based upon work supported by the Department of Energy Office of Fusion Energy under Award Number DE-SC00215057 and by the Department of Energy National Nuclear Security Administration under Award Number DE-NA0003856. This report was prepared as an account of work sponsored by an | 飛ぶ焦点技術によるレーザパルスは、動的な焦点ポイントを持ち、レイリー長よりも長い距離を移動します。レーザ技術へのこの技術の適用には、焦点範囲を延ばすことができる光学構成の設計が必要です。また、Radialグループ遅延の構造化も必要です。この論文では、この方法を使って、任意の軌道を示す飛ぶ焦点パルスを生成する光学構成の設計方法について説明しています。この方法を、軸形Parabolaを用いて焦点範囲を延ばす方法と、反射的なエchelonsまたは可変ミラー-空間光の変動装置の組み合わせを用いてRadialグループ遅延を構造化する方法で説明しています。後者配置は、飛ぶ焦点の迅速な探索と最適化を可能にします。これらは、実験に最適です。 |
2305.13308 | If at First You Don't Succeed, Try, Try Again: Faithful Diffusion-based
Text-to-Image Generation by Selection | Despite their impressive capabilities, diffusion-based text-to-image (T2I)
models can lack faithfulness to the text prompt, where generated images may not
contain all the mentioned objects, attributes or relations. To alleviate these
issues, recent works proposed post-hoc methods to improve model faithfulness
without costly retraining, by modifying how the model utilizes the input
prompt. In this work, we take a step back and show that large T2I diffusion
models are more faithful than usually assumed, and can generate images faithful
to even complex prompts without the need to manipulate the generative process.
Based on that, we show how faithfulness can be simply treated as a candidate
selection problem instead, and introduce a straightforward pipeline that
generates candidate images for a text prompt and picks the best one according
to an automatic scoring system that can leverage already existing T2I
evaluation metrics. Quantitative comparisons alongside user studies on diverse
benchmarks show consistently improved faithfulness over post-hoc enhancement
methods, with comparable or lower computational cost. Code is available at
\url{https://github.com/ExplainableML/ImageSelect}. | Shyamgopal Karthik, Karsten Roth, Massimiliano Mancini, Zeynep Akata | 2023-05-22T17:59:41 | http://arxiv.org/abs/2305.13308v1 | If at First You Don't Succeed, Try, Try Again: Faithful Diffusion-based Text-to-Image Generation by Selection
###### Abstract
Despite their impressive capabilities, diffusion-based text-to-image (T2I) models can lack faithfulness to the text prompt, where generated images may not contain all the mentioned objects, attributes or relations. To alleviate these issues, recent works proposed post-hoc methods to improve model faithfulness without costly retraining, by modifying how the model utilizes the input prompt. In this work, we take a step back and show that large T2I diffusion models _are more faithful than usually assumed_, and can generate images faithful to even complex prompts without the need to manipulate the generative process. Based on that, we show how faithfulness can be simply treated as a candidate selection problem instead, and introduce a straightforward pipeline that generates candidate images for a text prompt and picks the best one according to an automatic scoring system that can leverage already existing T2I evaluation metrics. Quantitative comparisons alongside user studies on diverse benchmarks show consistently improved faithfulness over post-hoc enhancement methods, with comparable or lower computational cost. Code is available at [https://github.com/ExplainableML/ImageSelect](https://github.com/ExplainableML/ImageSelect).
## 1 Introduction
Text-to-Image (T2I) Generation [42; 55; 73] has seen drastic progress in recent times with the advent of modern generative models. Starting from GAN-based [22] approaches [55; 73], this process was supercharged and popularized with the release of Stable Diffusion [57] and other large-scale pretrained generative models [7; 61; 53; 20; 70; 30]. However, even these large models appear to exhibit shortcomings, particularly when it comes to faithfully generating the input prompt, failing to correctly reflect attributes, counts, semantic object relations or even entire objects [39; 19; 10]. Consequently, recent works such as Composable Diffusion [39], Structure Diffusion [19], Space-Time Attention [66] or Attend-and-Excite [10] propose to improve faithfulness in these baseline models by modifying the inference procedure. While resulting in a more expensive generation process (e.g. Attend-and-Excite [10] being around six times slower, and [66] over a hundred times), qualitative demonstrations showcase superior faithfulness compared to the baselines. However, these methods are often tailored to special prompt types. Paired with the mostly qualitative support, it remains unclear if they can work in general-purpose settings with a larger and more diverse set of prompts.
As such, in this work, we take a step back and investigate how unfaithful these diffusion models really are. Upon closer inspection, we observe that the faithfulness of Stable Diffusion is affected heavily by the random seed that determines the initial latent noise, suggesting that within the explorable latent space, faithful image generations are possible (c.f. for example image candidates in Fig. 1). Motivated by this observation, we thus propose to improve the faithfulness in diffusion models not through an explicit change in the baseline model, but instead by simply querying it multiple
times and finding ways to automatically select the most suitable output. We denote this simple pipeline as ImageSelect. We utilize metrics from recently proposed text-to-image faithfulness benchmarks, TIFA [28] and ImageReward [68], to evaluate the faithfulness of our image generation. TIFA simplifies the text-to-image matching process into a set of Visual Question Answering tasks, which can be more easily solved with existing pretrained models than the complex input prompts used in direct matching. ImageReward proposes a matching model trained on human preferences, where candidates assign preference scores to generated images. In both cases, the matching qualities are significantly better than those of previous approaches that use global image-text matching with a vision-language model, such as CLIPScore [26] or CLIP-R-Precision [48]. Our results with these metrics provide evidence that candidate selection can improve faithfulness, and improvements in faithfulness measures can directly translate to better generation faithfulness using ImageSelect.
To understand the efficacy of ImageSelect, we first study each selection mechanism against all reference methods evaluated with opposing metrics - TIFA as the selection mechanism evaluated on the ImageReward metric, and vice versa. To ensure sufficient generality of our results, we generate a diverse collection of over 1000 prompts, diverse-1k, aggregated from multiple datasets (HRS [4], TIFA [28]/MSCOO [37], Structure Diffusion [19]), spanning different textual aspects such as counting, spatial relations and attribute binding. Doing so also mitigates overfitting to a particular prompt generation approach from a specific dataset. Results on diverse-1k in both cases indicate significant performance improvements against reference methods, with gains in faithfulness through automatic candidate selection consistently higher than that even achieved by changed model version generations (going for example from Stable Diffusion 1.4 to 2.1). This improvement in faithfulness holds even when investigating faithfulness for specific prompt types. In addition, we perform an extensive human evaluation in which ImageSelect is compared against baseline methods on human
Figure 1: Our ImageSelect introduces automatic candidate selection to increase the faithfulness of a T2I generative model. We show that existing models are more faithful than assumed, and by simply querying them multiple times and selecting the most suitable image, we achieve significant improvements in T2I faithfulness, without requiring to explicitly adapt the generative process.
evaluated faithfulness. Results produced by over 5000 image comparisons covering 68 voluntary participants strongly support our observations made on the quantitative tests, with ImageSelect outputs preferred in parts over three times as often as baseline method outputs. The results showcase a simple, but large step forward for text-to-image faithfulness, and highlight our insights as a crucial sanity check for future work tackling the task of post-hoc enhancement of text-to-image generation.
To summarize, we make the following contributions: (1) We highlight that, given a prompt, the faithfulness (and quality) of images generated by diffusion-based text-to-image generative approaches varies significantly across multiple generations with different seeds. (2) From this insight, we propose ImageSelect, a simple pipeline which generates multiple candidate images and selects the most faithful one via an automatic scoring mechanism. (3) Quantitative studies and extensive user studies on diverse benchmarks show that ImageSelect significantly outperforms existing methods in text-to-image faithfulness while matching or even improving their inference speeds.
## 2 Related Work
**Faithful Text-to-Image Generation.** T2I generation was first introduced with GAN [22] models generalizing to unseen concepts [55; 56; 72]. Later works explored other generative architectures such as VQ-VAE/VQ-GANs [18; 54; 15; 20; 32; 24; 1] and diffusion models [62; 27; 16; 44; 57; 53; 60]. The latter dominate the current state-of-the-art, with text conditioning coming from either a language [52] or a vision-language [51] model. However, even these advanced methods struggle to capture detailed prompt semantics, such as composing arbitrary concepts, counting [46], spelling [40], and handling biases [43; 6]. Recent works address these shortcomings post-hoc by changing the latent diffusion process in models s.a. Stable Diffusion [57] or DALL-E 2 [53]. Composable Diffusion [39] handles conjunction and negation operations by recomposing diffusion outputs at every timestep. Structure Diffusion [19] performs multi-guidance via CLIP [51] text embeddings of different noun phrases in a prompt. Attend-and-Excite [10] optimizes cross-attention maps [25], ensuring they attend to manually selected prompt parts. Space-Time Attention [66] improves faithfulness with a separate layout predictor and temporal attention control. Unlike these approaches, we found that T2I diffusion models s.a. Stable Diffusion already exhibit a large degree of faithfulness that a simple and automatic candidate selection process can capture without altering the generative process.
**Evaluating Image-Text Alignment.** Large vision-language models [51; 29; 64] offer direct tools to evaluate and leverage image-text alignment (e.g. [26; 59; 69; 17]), but lack compositional understanding [71]. Other approaches [47; 2; 5; 65] propose to caption the generated image and measure the textual similarity between the prompt and caption. However, these metrics are not well correlated with human preferences [48; 28; 45], and may miss fine-grained details of the prompt. Inspired by the success of reinforcement learning from human feedback [23; 14; 63], several works [68; 33; 67] trained models to predict human preferences instead. However, this requires expensive annotations, while not disentangling preferences regarding the quality of the generation and faithfulness to the prompt. Instead, TIFA [28] measures faithfulness by answering questions about the prompt using a VQA model (s.a. BLIP [36; 35]), producing a fine-grained and interpretable rating. These metrics are part of ongoing efforts to provide quantitative benchmarks for T2I models, s.a. MS-COCO [37; 12], CompT2i [48], DALL-E-Eval [13], HRS [4], VSR [38], TIFA [28], CC [19], ABC [19], PaintSkill [13], DrawBench [60], PartiPrompts [70] or VISOR [21]. To ensure the generality of our results beyond the prompt generation process of a single dataset, we also leverage an aggregate prompt collection using TIFA, MS-COCO, HRS, and Structure Diffusion to test general-purpose T2I faithfulness across a wide range of categories.
## 3 Achieving Faithfulness through Selection
We first provide an overview of Latent Diffusion Models and a motivation for faithfulness through candidate selection. From these findings, we describe measures for text-to-image alignment and how they can be used to improve T2I faithfulness via selection. Finally, we provide details for our diverse benchmark, diverse-1k, which we use in the experiments to validate our findings.
### Background: Latent Diffusion Models
Latent Diffusion Models (LDMs) [57] extend Denoising Diffusion Probabilistic Models (DDPM) [27] into the latent space of pretrained encoder-decoder models s.a. VAEs [31], where the compression allows for improved scalability. Unlike generic DDPMs which model the generation of an image \(x_{0}\) as an iterative denoising process with \(T\) steps starting from noise \(x_{T}\) (sampled from a Normal prior), LDMs deploy the denoising process over spatial latents \(z_{T}\to z_{0}\) of the pretrained model. Starting from \(z_{T}\), these LDMs (often parametrized as a UNet [58] with parameters \(\theta\)) provide a perturbation \(\epsilon_{\theta}(z_{t},t)\) for every timestep \(t\in[1,...,T]\), which is subtracted from \(z_{t}\) to generate subsequent latents
\[z_{t-1}=z_{t}-\epsilon_{\theta}(z_{t},t)+\mathcal{N}(0,\sigma_{t}^{2}I) \tag{1}\]
with learned covariances \(\sigma_{t}^{2}I\). When \(z_{0}\) is reached, the decoder projects the latent back into the image space. The favorable scaling properties of operating in latent spaces allow LDMs to produce large-scale pretrained, high-quality generative models such as Stable Diffusion [57]. Additional text-conditioning can then be performed during the denoising process. For Stable Diffusion, this condition is simply a text embedding produced by CLIP [51], \(c(y)\), corresponding to associated prompts \(y\). By extending the standard UNet with cross-attention layers (e.g. [25, 10, 19, 11]) to connect these embeddings with the latent features, the text-conditioned LDM can then simply be trained in the same manner as standard LDMs. While these LDMs can generate high-quality images when trained at scale, recent works [39, 19, 10, 66] strongly emphasize that they lack faithfulness to the text prompt, as shown in a qualitative fashion on specific input prompts and seeds.
### ImageSelect: Faithfulness through Selection
Indeed, our first qualitative study on various prompts over multiple seeds using vanilla Stable Diffusion indicates that faithful images _can be_ generated, but are simply hidden behind a suitable selection of the starting latent noise (see Fig. 1). Based on this insight, we thus introduce a simple, efficient and effective mechanism to provide more faithful outputs for a given prompt by simply looking at candidates from multiple seeds and automatically selecting the most suitable image.
**Measuring Faithfulness in Text-to-Image Alignment.** For our automatic selection, we show that one can simply leverage already existing advanced T2I evaluation methods. As _proof-of-concept_, we simply select two - TIFA and ImageReward - which we explain in the following in more detail.
TIFA Scores [28] evaluate T2I alignment using the auxiliary task of Visual-Question Answering (VQA) [3]. Specifically, given a text prompt \(y\), and a generated image \(I\), a Large Language Model (LLM) such as GPT3.5 [8] is used to generate question-answer pairs \(\mathcal{Q}(y):=\{(Q_{i},A_{i})\}_{i}\) related to the prompt or caption \(y\)[9]. An off-the-shelf VQA model \(\Psi_{\text{VQA}}\) such as BLIP [36, 35] or mPLUG [34] is then used to answer these generated questions using the generated image \(I\), providing respective
Figure 2: Given a text prompt and a set of latent starting points \(\epsilon_{i}\), we generate corresponding candidate images with off-the-shelf T2I models s.a. Stable Diffusion. A scoring mechanism then assigns faithfulness scores per image, with the highest scoring one simply selected as the final output.
answers \(A_{i}^{\text{VQA}}\) for given questions \(Q_{i}\). Doing so breaks down the matching process into many easier-to-solve, small-scale matching problems. The resulting faithfulness score \(\mathcal{F}\) of the generated image \(I\) is simply defined as the ratio of questions that the VQA model answered correctly,
\[\mathcal{F}_{\text{TIFFA}}(I,y)=\frac{1}{|\mathcal{Q}(y)|}\sum_{(Q_{i},A_{i}) \sim\mathcal{Q}(y)}\mathbb{I}\left[\Psi_{\text{VQA}}(I,Q_{i})=A_{i}\right]. \tag{2}\]
where \(\mathbb{I}\left[\Psi_{\text{VQA}}(I,Q_{i})=A_{i}\right]\) is 1 if the answer is correct. This evaluation strategy has the benefits of being interpretable, fine-grained, and avoiding any manual annotations for text-image alignment.
ImageReward Scores [68] are produced from a completely different direction, following more closely the trend of just end-to-end training on suitable data. In particular, [68] simply train a Multi-Layer Perception (MLP) on top of image and text features produced by BLIP to regress 137k expert human preference scores on image-text pairs, with higher scores denoting higher levels of faithfulness. The resulting rating model \(\Psi_{\text{ImageReward}}\), while not normalized, is well-correlated with human ratings even on samples outside the training dataset, and gives the faithfulness score simply as
\[\mathcal{F}_{\text{ImageReward}}(I,y)=\Psi_{\text{ImageReward}}(I,y). \tag{3}\]
Faithfulness through Selection.Both TIFA and ImageReward are only utilized as a benchmarking mechanism to evaluate current and future T2I methods on faithfulness. Instead, we showcase that these metrics can be easily utilized to supercharge the faithfulness of existing models without any additional retraining, by simply re-using them in a contrastive framework as a candidate selection metric. In particular, given a budget of \(N\) initialization starting points and a text prompt \(y\), our associated generated output image \(I\) is thus simply given as
\[I_{\text{ImageSelect}}(y)=\operatorname*{arg\,max}_{n\in N}\mathcal{F}_{\text{ ImageSelect}}\left(\mathcal{D}(\epsilon_{\theta}(\epsilon_{n},T,y)),y\right) \tag{4}\]
where \(\epsilon_{\theta}\) denotes the text-conditioned denoising diffusion model in the latent space of the encoder-decoder model with decoder \(\mathcal{D}\), total number of denoising iterations \(T\), and initial latent noise \(\epsilon_{n}\sim\mathcal{N}(0,1)\) sampled anew for each \(n\). We note that we use ImageSelect to refer to the use of any faithfulness measure s.a. \(\mathcal{F}_{\text{TIFA}}\), \(\mathcal{F}_{\text{ImageReward}}\), and highlight that this can be extended to any other scoring mechanism or combinations thereof. For a given selection method, we denote the respective ImageSelect operation as TIFASelect or RewardSelect.
### The Diverse Prompts Dataset
While multiple benchmarks have recently been proposed to study text-to-image faithfulness, most benchmarks introduce their unique sets of prompts. These are grouped under different fine- or coarse-grained categories like _shape_, _attribute_ or _color_ in TIFA, which are shared in e.g. HRS [4], or more general prompt types such as _emotions_ or _long prompts_ specifically introduced in HRS. To ensure that our results are as representative as possible and do not overfit to a particular type of prompt generation mechanism introduced in a benchmark, we aggregate prompts from HRS, TIFA (containing also captions from MS-COCO), and prompts utilized in [19]. Given the higher diversity and count of prompts in HRS and TIFA, we oversample from both. For HRS, we cover each sub-category. We avoid duplicates or semantic equivalents by first filtering based on language similarity (using a CLIP text encoder) before manual removal. We plan to release the prompt collection to aid future research on faithful text-to-image generation.
## 4 Experiments
**Implementation Details.** We take off-the-shelf Stable Diffusion 1.4 and 2.1 and evaluate them on the TIFAv1.0 [28] benchmark - consisting of prompts from MS-COCO and other sources that benchmark
\begin{table}
\begin{tabular}{l|l|l} \hline \hline
**Sources\(\downarrow\)** & Subsets & Count \\ \hline \multirow{3}{*}{HRS [4]} & Bias, Spatial, Counting, & \multirow{3}{*}{38} \\ & Emotion, Size, Fairness, & Length, Color, Synthetic \\ & Writing & \\ \hline StrD [19] & ABC & 127 \\ & CC & 125 \\ \hline TIFA [28] & N/A & 381 \\ \hline \multicolumn{3}{c}{**Total:** 1011} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary statistics in our diverse-1k dataset. For further details, see supplementary.
T2I generation for more creative tasks - and our diverse-1k prompts list. We consider the Structure Diffusion (StrD) [19] & Composable Diffusion (CD) [39] (both available only with Stable Diffusion 1.4) and the Attend-and-Excite (A&E) [10] methods as our baselines. While StrD can be applied directly, CD requires us to split the prompts and join them together using the "AND" operator.
**Extending Attend & Excite for automatic usage.** A&E requires a user to manually select tokens the model should attend to. We modify this to work automatically by selecting categories from MS-COCO, as well as utilizing NLTK [41] to determine nouns which cannot be treated as either a verb or adjective. For any prompt for which the above protocol provides no target tokens, we continuously relax the constraints over the nouns. In limit cases where nothing suitable is selected, A&E defaults back to the original Stable Diffusion it extends. We denote A&E equipped with this formalism as _Attend-and-Excite++_ (A&E++). We find that on normal prompts or those qualitatively studied in the original paper [10], our protocol comes very close to the generations reported in [10].
### Quantitative comparison between Stable Diffusion variants
**Faithfulness on diverse-1k.** We begin by evaluating the faithfulness of baselines on top of Stable Diffusion Version 1.4 (SD1.4) and Version 2.1 (SD2.1, where possible) on diverse-1k, which we evaluate using both TIFA (eq. 2) and ImageReward score (eq. 3). We use RewardSelect for TIFA scores, and vice versa TIFASelect for the ImageReward score evaluation, over a pool of 10 randomly generated images per prompt to evaluate the quantitative impact of ImageSelect. Results in Fig. 3 highlight a **clear** increase in faithfulness of ImageSelect over all baseline methods across both evaluation metrics. We also find that across diverse, non-cherry-picked prompts, both Composable and Structure Diffusion can actually have an overall detrimental effect, with standard SD1.4 scoring \(71.6\%\) on TIFA and \(-0.22\) on ImageReward, and Structure Diffusion only \(70.6\%\) on TIFA and \(-0.35\) on ImageReward. For Composable Diffusion, performance also falls below on ImageReward (\(-0.35\)). On the opposite end, we find our extension of [10], Attend-and-Excite++, to offer faithfulness benefits (e.g. \(75.2\%\) TIFA score) across SD1.4 and SD2.1. However, this change in performance is overshadowed by ImageSelect, which e.g. on SD1.4 achieves an impressive \(80.4\%\) - over \(4pp\)_higher than the change from SD1.4 to SD2.1_ gives in terms of text-to-image faithfulness. This fact is only exacerbated on the ImageReward score (\(-0.22\) SD1.4, \(0.18\) SD2.1 and \(0.32\) for TIFASelect). Together, these results provide a first clear quantitative indicator that suitable candidate selection can have a much higher impact on faithfulness than current explicit changes to the generative process. For completeness, we test simple CLIPScore selection (in the same fashion as Eq.4) against
Figure 4: RewardSelect offers improved faithfulness across faithfulness categories as used in [28]
Figure 3: Quantitative results for baselines and ImageSelect on diverse-1k. For Stable Diffusion 1.4 and 2.1, ImageSelect outperforms all, irrespective of the selection and evaluation metric.
RewardSelect on TIFA (\(72.9\%\) versus \(80.8\%\) and \(71.6\%\) for SD V1.4), and against TIFASelect on ImageReward (\(-0.129\) vs \(0.316\) and \(-0.22\) for standard Stable Diffusion V1.4). As can be seen, while faithfulness over the Stable Diffusion baseline is increased, the overall performance falls short compared to more suitable selection mechanisms. We believe these insights hint towards the potential impact of further research into selection approaches to improve faithfulness.
**Breakdown by Categories.** We repeat our previous experiments on the original TIFAv1.0 benchmark [28] (where parts were integrated into diverse-1k), as the benchmark offers easy category-level grouping such as "counting", "spatial (relations)", "shape" etc. While diverse-1k also offers subset breakdowns (c.f. Table 1), the grouping in TIFAv1.0 provides a simple, straightforward attribute-style separation. For all methods and RewardSelect on SD1.4, we showcase results in Fig. 4. When breaking down the overall improvement in faithfulness into respective categories, the benefits of ImageSelect become even clearer. ImageSelect improves over every baseline across every single category, with especially significant changes in categories such as "counting" (over \(10pp\)) - a well-known shortcoming of T2I diffusion models [46]. While not a complete remedy, the change in performance is remarkable. Similarly, we see other scenarios such as "spatial (relations)" or "object (inclusion)" improving from \(0.71\) to \(0.78\) and \(0.77\) to \(0.85\), respectively. Again, it is important to highlight that these improvements are not a result of potential overfitting to the evaluation metric, as the scoring approaches are entirely different (VQA versus modeling human preferences).
**Comparison to Ground Truth Faithfulness.** To provide a better reference for the quantitative change in performance, we also evaluate on the MS-COCO captions used in [28], for which ground truth images exist. Using RewardSelect and the TIFAScore for evaluation, we report results in Tab. 2. While clearly outperforming baseline methods, we also see RewardSelect matching ground truth TIFA faithfulness scores of true MS-COCO image-caption pairs (\(89.85\%\) versus \(89.09\%\)). While attributable to increases in measurable faithfulness through ImageSelect, it is important to note both the noise in ground truth captions on MS-COCO [37] and a focus on a particular prompt-style (descriptive natural image captions - hence also our use of diverse-1k for most of this work). Still, these ground truth scores provide strong support for the benefits of candidate selection as a means to increase overall faithfulness.
**Relation between Faithfulness and Number of Candidate Images.** We further visualize the relation between text-to-image faithfulness and the number of candidate images taken into consideration in Fig. 5, as measured by the ImageReward score on diverse-1k. Our experiments show a drastic improvement with already two candidates, raising the faithfulness of SD1.4 to that of SD2.1. Going further, we find monotonic improvements, but with diminishing returns becoming more evident for larger candidate counts. This also means that a small number of candidate images (e.g. 4) is already sufficient to beat all baselines. We highlight that this is not caused by any single seed being more effective [50], as we find all seeds to behave similarly (\(77.9\%\) to \(78.5\%\) for 10 seeds on TIFAv1.0), _but rather the per-prompt candidate selection_.
**Computational Efficiency.** While Stable Diffusion takes 5 seconds to generate a single image (NVIDIA 2080Ti), Attend-and-Excite requires 30 with double the memory requirements. Other recent methods such as Space-Time-Attention [66] can require nearly five times the VRAM and over 10 minutes. Thus even from a computational perspective, there is a clear benefit of leveraging simple candidate selection through ImageSelect, and generating as many candidates as possible
\begin{table}
\begin{tabular}{c|c c c} \hline \hline V1.4 & SD & A\&E++ & RS \\ & \(82.69\%\) & \(82.04\%\) & \(88.69\%\) \\ \hline V2.1 & SD & A\&E++ & RS \\ & \(85.28\%\) & \(85.87\%\) & \(\mathbf{89.85\%}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Faithfulness comparison with our RewardSelect (RS) using the TIFA-score on the ground-truth MS-COCO image-caption pairs. Our RS closes the gap with GT=\(89.09\%\) in faithfulness.
Figure 5: Faithfulness increases with number of candidate images per prompt to select from.
within a computational budget. Finally, the process of producing respective images for a prompt is parallelizable, and directly benefits from extended GPU counts even on a single-prompt level.
### User Study
Since quantitative metrics alone can be inadequate for tasks which have subjective choices such as image generation, we expand our quantitative studies with extensive human evaluations. For every diverse-1k prompt, we generate images using all baselines (Composable Diffusion [39], Structure Diffusion [19] and Attend-and-Excite++) as well as RewardSelect and TIFASelect on SD1.4. For all ImageSelect variants and Attend-and-Excite++, we also utilize SD2.1. Using the generated images, we set up a comparative study following the layout shown in supplementary. Voluntary users interact with the study through a webpage, and are tasked to select the most faithful generation between the output of either a baseline method or an ImageSelect variant. We ensure that the underlying Stable Diffusion model is shared, and the relative positioning on the interface is randomly shuffled for each selection. Baseline and ImageSelect method are sampled anew after each choice. In total, we collect 5093 human preference selections, distributed over 68 unique users and each comparative study. The number of selections performed for a comparative study is between 456 and 538. Results are shown in Fig. 6, where we also compare RewardSelect and TIFASelect directly.
Looking at the results, we find a clear preference in faithfulness for images generated by ImageSelect, particularly RewardSelect. Indeed, when looking at the relative improvements w.r.t. each baseline in Table 3, we find ImageSelect to be chosen in parts twice (e.g. \(+126.3\%\) for TIFASelect vs Comp. Diffusion on SD1.4) or even three times more often (e.g. \(+207.9\%\) on RewardSelect vs. Structure Diffusion on SD1.4). Even against our adaptation of [10] (Attend-and-Excite++) and on the improved Stable Diffusion V2.1, RewardSelect still has a \(84.4\%\) higher chance to be chosen as more faithful. In general, we found RewardSelect to be better aligned with human insights on text-to-image faithfulness, and better suited as a candidate selector. This is further supported when looking at the direct comparisons with TIFASelect in Fig. 6i-j, and Tab 3, where RewardSelect is preferred with a \(53.6\%\) higher chance on SD V1.4 and \(46.5\%\) on SD V2.1. This indicates that a model trained to mimic human preferences might work better as a selection metric than one that looks for faithfulness as a numerical metric, weighing every semantic aspect equally.
Regardless of the variations in ImageSelect, our user study provides compelling evidence that automatic candidate selection is a highly promising approach for post-hoc text-to-image faithfulness in large-scale pretrained text-to-image diffusion models, especially when compared to existing approaches that explicitly adapt the generative process in a costly manner. We intend to publicly release all user preferences collected during the study to facilitate further exploration in this direction.
\begin{table}
\begin{tabular}{l|l|c c c|c} \hline \hline
**Versus**\(\rightarrow\) & CD [39] & SD [19] & A\&E & TIFASelect \\ \hline
**V1.4** & TIFASelect & \(126.3\) & \(101.24\) & \(58.7\) & \(\times\) \\ & RewardSelect & \(207.9\) & \(201.5\) & \(125.7\) & \(53.6\) \\ \hline
**V2.1** & TIFASelect & \(\times\) & \(\times\) & \(22.5\) & \(\times\) \\ & RewardSelect & \(\times\) & \(\times\) & \(84.4\) & \(46.5\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relative improvements of ImageSelect approaches over faithfulness baselines. Human participants are in parts \(\times 2\) or even \(\times 3\) as likely to find RewardSelect images more faithful to the prompt. Even against our updated, automatic variation of A\&E, selection preference are in parts \(>\times 2\). Finally, comparing selection methods, we find the learned RewardSelect approaches to generally outperform TIFASelect which decomposes the matching tasks.
Figure 6: Performing human faithfulness comparisons between baselines and ImageSelect shows ImageSelect being preferred in the majority of cases for prompts from diverse-1k.
### Qualitative Examples and Limitations
We also show additional qualitative examples to illustrate the successes of ImageSelect in Fig. 7, which captures both simple and complex prompts well, particularly compared to other methods that struggle with the issues of catastrophic neglect [10], attribute binding [19], and incorrect spatial arrangement. For instance, ImageSelect is able to capture the objects and spatial relations in prompts like "three small yellow boxes on a large blue box" or "Two men in yellow jackets near water and a black plane.", while also faithfully rendering creative prompts like "an oil painting of a cat playing checkers.". Other methods perform worse in comparison, often missing objects entirely or generating objects with an incorrect spatial arrangement or false association of attributes (c.f. "A green chair and a red horse").
**Limitations.** We illustrate failures in Fig. 8. While ImageSelect significantly improves faithfulness, it can still struggle with challenges inherent to the underlying model such as rendering text, exact spatial relations, counting or very long prompts. However, due to its applicability to any T2I model, these shortcomings can be addressed by jointly tackling fundamental issues in vision-language models [71] and leveraging orthogonal extensions such as e.g. [40] for character generation.
## 5 Conclusion
In this work, we both highlight and leverage the dependence of faithfulness on initial latent noises in diffusion-based text-to-image models to introduce ImageSelect. By viewing the problem of post-hoc faithfulness improvements as a candidate selection problem, we propose a simple pipeline,
Figure 8: _Qualitative Failure Cases._ Despite significantly improving faithfulness, ImageSelect can not fully account for fundamental shortcomings. Details on faithfulness categories, see e.g. Fig. 4.
Figure 7: _Additional Examples highlighting favorable faithfulness of ImageSelect (rightmost) compared to Attend-and-Excite++, Composable Diffusion [39] and Structure Diffusion [19]._ | Despite their impressive capabilities, diffusion-based text-to-image (T2I) models can lack faithfulness to the text prompt, where generated images may not contain all the mentioned objects, attributes or relations. 改善するこれらの問題を解決するために、最近の研究では、コストがかかる再訓練を避けて、入力プロンプトをどのように利用するかを修正することでモデルの信憑性を向上させるための後処理方法を提案しています。この研究では、大規模なT2I拡散モデルは通常想定されていたよりも信憑性に優れていることを示し、複雑なプロンプトに対しても生成画像を信憑性の高い画像に生成できることを示しています。その上で、信憑性を候補選択問題として扱うことができ、既存のT2I評価メトリクスを活用した自動スコアシステムを用いて候補画像を生成し、最適な画像を選択する直截なパイプラインを提案しました。多様なベンチマークにおける定量的比較とユーザー |
2306.12546 | Highly depleted alkali metals in Jupiter's deep atmosphere | Water and ammonia vapors are known to be the major sources of spectral
absorption at pressure levels observed by the microwave radiometer (MWR) on
Juno. However, the brightness temperatures and limb darkening observed by the
MWR at its longest wavelength channel of 50 cm (600 MHz) in the first 9
perijove passes indicate the existence of an additional source of opacity in
the deep atmosphere of Jupiter (pressures beyond 100 bar). The absorption
properties of ammonia and water vapor, and their relative abundances in
Jupiter's atmosphere do not provide sufficient opacity in deep atmosphere to
explain the 600 MHz channel observation. Here we show that free electrons due
to the ionization of alkali metals, i.e. sodium, and potassium, with sub-solar
metallicity [M/H] (log based 10 relative concentration to solar) in the range
of [M/H] = -2 to [M/H] = -5 can provide the missing source of opacity in the
deep atmosphere. If the alkali metals are not the source of additional opacity
in the MWR data, then their metallicity at 1000 bars can only be even lower.
The upper bound of -2 on the metallicity of the alkali metals contrasts with
the other heavy elements -- C, N, S, Ar, Kr, and Xe -- which are all enriched
relative to their solar abundances having a metallicity of approximately +0.5. | Ananyo Bhattacharya, Cheng Li, Sushil K. Atreya, Paul G. Steffes, Steven M. Levin, Scott J. Bolton, Tristan Guillot, Pranika Gupta, Andrew P. Ingersoll, Jonathan I. Lunine, Glenn S. Orton, Fabiano A. Oyafuso, J. Hunter Waite, Amadeo Belloti, Michael H. Wong | 2023-06-21T20:20:24 | http://arxiv.org/abs/2306.12546v1 | # Highly depleted alkali metals in Jupiter's deep Atmosphere
###### Abstract
Water and ammonia vapors are known to be the major sources of spectral absorption at pressure levels observed by the microwave radiometer (MWR) on Juno. However, the brightness temperatures and limb darkening observed by the MWR at its longest wavelength channel of 50 cm (600 MHz) in the first 9 perrijove passes indicate the existence of an additional source of opacity in the deep atmosphere of Jupiter (pressures beyond 100 bar). The absorption properties of ammonia and water vapor, and their relative abundances in Jupiter's atmosphere do not provide sufficient opacity in the deep atmosphere to explain the 600 MHz channel observation. Here we show that free electrons due to the ionization of alkali metals, i.e. sodium, and potassium, with sub-solar metallicity, [M/H] (log based 10 relative concentration to solar) in the range of [M/H] = -2 to [M/H] = -5 can provide the missing source of opacity in the deep atmosphere. If the alkali metals are not the source of additional opacity in the MWR data, then their metallicity at 1000 bars can only be even lower. This upper bound of -2 on the metallicity of the alkali metals contrasts with the other heavy elements - C, N, S, Ar, Kr, and Xe - which are all enriched relative to their solar abundances having a metallicity of approximately +0.5.
Solar System (1528) - Chemical abundances(224) - Jupiter(873) - Extrasolar gaseous giant planets(509) 0000-0002-4615-5885]Ananyo Bhattacharya
0000-0002-4880-788X]Cheng Li
0000-0002-4880-788X]Sushi K. Atreya
0000-0002-4880-788X]Paul G. Steffes
0000-0002-0703-070X]Steven M. Levin
0000-0002-0703-0888]Scott J. Bolton
0000-0002-0703-0888]Tristan Guillot
0000-0002-0703-088X]Pranika Gupta
0000-0002-0703-088X]Andrew P. Ingersoll
0000-0002-0703-088X]Jonathan I. Lunine
0000-0002-0703-088X]Glenn S. Orton
0000-0002-0703-088X]Fabiano A. Oyafuso
0000-0002-0703-088X]J. Hunter Waite
0000-0002-488-088X]Amadeo Belloti
0000-0002-488-088X]Michael H. Wong
## 1 Introduction
The alkali metals sodium and potassium have been previously detected in the atmospheres of hot Jupiters and a super-Neptune together with lithium [Chen et al. (2018)] in the latter. The detections show a large range of abundances from highly substellar to super-stellar values [Welbanks et al. (2019), Demory et al. (2011)]. Alkali metal abundances are important in understanding the formation of hot Jupiters and represent a bridge between the refractory and volatile elements, which in molecular form seed the growth of planets. Obtaining the abundance of alkali metals in Jupiter can potentially serve as a first constraint on the ratio of rocky to icy material in the interior of the solar system's largest planet when combined with the elemental and molecular abundances provided by the Galileo Probe Mass Spectrometer (GPMS) [Atreya et al. (1999), Wong et al. (2004), Atreya et al. (2019)] and Juno constraints on water [Li et al. (2020)]. Here we derive observationally based abundances of alkali metals in
Jupiter's atmosphere to determine whether they are enriched relative to solar like the other heavy elements or depleted.
To obtain these abundances requires knowing the deep structure of Jupiter's atmosphere. The shallower part of Jupiter's atmosphere has been previously investigated at microwave frequencies by the Very Large Array (VLA) telescope [de Pater and Dunn (2003), de Pater et al. (2019)]. VLA probes Jupiter at frequencies in the range of 74 MHz to 50 GHz [de Pater et al. (2019)]. However, confusion from Jupiter's powerful synchrotron radiation does not allow VLA to observe Jupiter's atmosphere below 5 GHz [de Pater and Dunn (2003)], limiting its reach to less than 5 bars, leaving the deep atmosphere of Jupiter inaccessible from microwave and radio frequency observatories from Earth. The orbit of Juno and the spin of the spacecraft allow the spacecraft to make observations at low frequencies, i.e. 0.6 GHz and 1.2 GHz, by avoiding the energetic electron belts around Jupiter from its field of view. Access to greater depths allows for the investigation of bulk elemental abundances of N and O in Jupiter [Janssen et al. (2017), Bolton et al. (2017), Steffes et al. (2017)].
The Microwave Radiometer (MWR) instrument onboard the Juno orbiter is a passive radiometer that is designed to measure the internal heat emitted by Jupiter's atmosphere at six different frequencies ranging from 0.6 GHz to 22 GHz [Janssen et al. (2017)]. The brightness temperature measured by MWR at these frequencies sounds different levels of Jupiter's atmosphere corresponding to pressures from 0.3 bar to 250 bar [Janssen et al. (2017)]. In addition, the highly inclined polar orbit and rotation of the Juno spacecraft aided in the high spatial resolution necessary for probing Jupiter's atmosphere at various latitudes [Bolton et al. (2017)].
Previous analysis of the MWR data at the 0.6 GHz found an unanticipated limb-darkening signal, which cannot be explained by nominal absorbers such as ammonia and water [Li et al. (2020)]. Based on investigation of thermodynamic models of Jupiter's deep atmosphere between 50 bar and 1 kbar [Fegley Jr and Lodders (1994), Weidenschilling and Lewis (1973)] we conjecture that the free electrons from thermally ionzied alkali metals may provide the missing opacity. Alkali metals are expected to undergo condensation to form clouds in the deep atmosphere [Visscher et al. (2006), Morley et al. (2012)]. Na\({}_{2}\)S and KCl are the first chemical species to condense in the above pressure range and thereby act as a sink for atomic sodium and potassium [Fegley Jr and Lodders (1994)]. Furthermore, high-temperature environments cause alkali metals to undergo ionization due to their low ionization energies [Bagenal et al. (2007)]. Density and temperature play a role in governing the electron densities according to the Saha ionization equation (Eq. 2). Electrons generated from alkali metal ionization act as a source of absorption at microwave frequencies that could affect the brightness temperatures at the 0.6 GHz frequency channel. Therefore, the objective of this study is to determine the alkali metal abundance in the deep atmosphere of Jupiter.
To facilitate comparison of our results on alkali metals with those of the extrasolar planets we express the abundances of non- hydrogen and helium elements using astronomical terminology, e.g., metallicity. The metallicity (_(M/H)_) of an element is the logarithm of the ratio of elemental abundance in a system to the stellar (or solar, for the solar system) elemental abundance. Generally, the metallicity of a star is defined in terms of the ratio of the number of Fe atoms to the number of hydrogen atoms. Here we define the metallicity in terms of alkali metal abundance in Jupiter to that of Sun e.g. for potassium, _[K/H]_ = log\({}_{10}\)(_N\({}_{K}\)/N\({}_{H}\))\({}_{Jupiter}\) - log\(10\)(_N\({}_{K}\)/N\({}_{H}\))\({}_{Sun}\). For the giant planets, iron and silicon is not measurable, emphasizing the importance of proxy indicators such as the alkali metals along with other elements measured by Galileo probe.
## 2 Methods
Brightness temperatures from 9 perijoves i.e. PJ 1,3-9, 12 have been taken into consideration for this article. Variations in brightness temperatures have been observed across the planetocentric latitudes from pole-to-pole at 0.6 and 1.2 GHz channels. These variations can be attributed to various sources of origin from the atmosphere and space environment. The most important sources of the observed variability are (i) changes in atmospheric structure and composition, (ii) Jupiter's synchrotron radiation in the microwave band, and (iii) variation in acceleration due to gravity due to the non-spherical shape of Jupiter. The latter sources, i.e. synchrotron and gravity need to be taken into account for proper interpretation of MWR observations. It will aid in investigating the true variability in
Jupiter's deep atmosphere.
The contribution of Jupiter's gravity can be corrected by taking into account the non-spherical shape of Jupiter. Brightness temperatures are corrected using a gravity correction factor defined as the ratio of theoretical _T\({}_{b}\)_ at a given latitude to that at the equator of Jupiter taking into consideration the acceleration due to gravity at the latitude. Thereby, it transforms the Juno observations at each latitude for equatorial gravity, which effectively removes variation in _T\({}_{b}\)_ due to changes in Jupiter's gravity from the equator to the poles.
Energetic electrons in Jupiter's space environment contribute to the synchrotron radiation [de Pater & Dunn (2003), Levin et al. (2001), Santos-Costa et al. (2017)]. The signature of the emission is observed in MWR data across all the perijoves which leads to anomalous changes in _T\({}_{b}\)_. Data at extremely high latitudes are polluted by synchrotron emission and thus, remain of no use for investigating Jupiter's deep atmosphere. Therefore, we only consider the MWR data between -60 to 60 deg. latitude. The correction for synchrotron and other sources of anomalous _T\({}_{b}\)_ is done by filtering the data at 0.6 and 1.2 GHz for each perijove. The process is carried out by sorting the deviations of _T\({}_{b}\)_ from the least value of T\({}_{b}\) in a group and removing the values greater than a filter cutoff temperature of the order of 2 K.
## 3 Results
### Sources of Microwave Opacity
The weighting function of Jupiter's atmospheric absorption and emission at a given microwave frequency determines the contribution of each region in the atmosphere to the observed brightness temperature at the given frequency. The peak structure of the weighting function gives the range of pressure levels corresponding to the measurements. The weighting function can be expressed as a function of microwave opacity of the atmosphere (1). Here, _T\({}_{b}\)_ is the brightness temperature, _W(p)_ is the weighting function as a function of pressure, and _T(p)_ is the physical temperature profile of the atmosphere.
\[T_{b}=\int_{-\infty}^{\infty}W(p)T(p)dlnP \tag{1}\]
Fig. 1 shows the relative weighting functions, i.e. weighting function divided by the maximum value of the function, at 0.6 GHz and 1.2 GHz with and without alkali metals. In the absence of alkali metals, the relative weighting functions peak at 100 bar and 30 bar, respectively[Janssen et al. (2017)]. At 0.6 GHz, the relative weighting function extends to the deeper atmosphere below the 100 bar level, and therefore, the _T\({}_{b}\)_ derived using this channel is sensitive to the sources of microwave opacity present in the deep atmosphere at \(p\) greater than 100 bar. The relative weighting function at 0.6 GHz channel shows a broad shape with a second maxima at kbar pressure levels which is attributed to the increase in mass absorption coefficients of water vapor with pressure. The mass absorption coefficient of ammonia decreases after a maximum near 1000 bar and eventually water vapor dominates the opacity in the deep atmosphere. Moreover, the inclusion of free electrons as sources of opacity due to alkali metal ionization causes a decrease in the value of the relative weighting function at 0.6 GHz around 100 bar, and a global maximum in the relative weighting function emerges at \(\sim\) 1 kbar pressure (magenta line). The shift of the global maximum can be attributed to the increase in opacity from free electrons with pressure as the ionization fraction of alkali metals increases with temperature under thermal equilibrium conditions [Saha (1920)] (described later in this section). Inclusion of lower amounts of alkali metals ([M/H] = -5) will lead to a peak at deeper levels (Fig. 1). However as the metallicity is increased to solar, the maximum drifts towards lower pressures around 1 kbar level. This could be attributed to the fact that higher abundance of alkali metals can produce higher amount of electrons at relatively lower pressures (magenta line), whereas low abundance of alkali metals in Jupiter would need to reach higher pressure (\(>\) 1 kbar) to produce equivalent opacity (blue line). Thereby the abundance of alkali metals directly affects the shape of weighting function.
The main sources of microwave opacity at 0.6 GHz and 1.2 GHz are ammonia, water vapor, free electrons, and collision-induced absorption by hydrogen and helium. Hydrogen-hydrogen and hydrogen-helium collisions are the dominant sources of collision-induced absorption processes in Jupiter. Their magnitude is well constrained due to the invariance of hydrogen and helium abundances in Jupiter's deep atmosphere. The microwave absorption behavior of
water and ammonia vapor has been investigated by laboratory experiments that show the pressure and temperature dependence of mass absorption coefficients (Devaraj et al. (2014), Karpowicz & Steffes (2011), Bellotti et al. (2016)). In addition, hydrogen, methane, and water vapor contribute to line broadening in the ammonia vapor absorption. The models based on laboratory experiments show significant divergent behavior when extrapolated to pressures greater than 50 bar and 550 K (Bellotti et al. (2016)). In order to obtain a robust estimate of the range of absorption coefficients at higher temperatures, we test a grid model describing a power scaling relationship with temperature based on the Hanley et al. (2009) model of ammonia absorption. For water vapor absorption at microwave frequencies, the laboratory models show divergence by orders of magnitude. However, recent laboratory measurements (Steffes et al. (2023)) at high pressure show that water vapor absorption can be explained by the Bellotti et al. (2016) model. Therefore, Bellotti et al. (2016) model is chosen to compute the water vapor opacity which incorporates water opacity measurements at high temperatures above 500 K.
Free electrons in the atmosphere can act as a source of opacity at microwave wavelengths through the process of free-free absorption in which electrons absorb photons during collisions with other ions and electrons. Electrons can be generated by the ionization of various elemental and molecular species in the atmosphere. Due to their low ionization energies, alkali metals i.e. Na, K are expected to be the major sources of free electrons in the atmosphere (Heays et al. (2017)). In Jupiter's atmosphere, the pressure and temperatures corresponding to the transition between the alkali metals and their compounds are calculated using an equilibrium cloud condensation model (ECCM) (Atreya
Figure 1: Relative weighting functions at 0.6 GHz (black) and 1.2 GHz (gray) for a Jupiter adiabat considering the Hanley model (Hanley et al. (2009)) for NH\({}_{3}\) absorption. The functions peak at 100 bar and 30 bar at 0.6 GHz and 1.2 GHz respectively without the inclusion of alkali metals. The inclusion of alkali metals (orange, magenta and blue) decreases the relative weighting function at \(\sim\) 100 bar and produces a second peak that is observed at \(\sim\) 1 kbar pressure due to the opacity contributed by free electrons from alkali metal ionization. As the metallicity of alkali metals increase, the global maximum of weighting function shifts towards lower pressure.
et al. (1999), Weidenschilling & Lewis (1973)] for Jupiter's adiabat with saturation vapor pressures of Na\({}_{2}\)S and KCl [Visscher et al. (2006), Morley et al. (2012)]. The condensation of alkali metals at solar abundance [Figure 2] takes place at 352 bar for KCl and 796 bar for Na\({}_{2}\)S, with corresponding temperatures of 967 K and 1234 K, respectively, assuming thermodynamic equilibrium. The condensation of Na\({}_{2}\)S at deeper levels, and a higher solar abundance of Na compared to K [Asplund et al. (2009)] will cause Na\({}_{2}\)S clouds to be significantly more massive than KCl clouds. Thermochemical equilibrium models indicate formation of metal hydrides and hydroxides in gas phase, however they are much lower in abundance [Fegley Jr & Lodders (1994)] as compared to the condensates, thereby they will not act as the primary sink of alkali metals in Jupiter. Condensation of the alkali metal compounds occurs when the partial pressure of a compound exceeds its saturation vapor pressure. If condensation occurs, it causes depletion in the alkali metal abundances at altitudes above the condensation level.
At high pressures 100 bar and beyond, alkali metals would undergo ionization to form cold plasma, and the electrons generated in the process would act as an additional source of opacity at microwave frequencies. The number density of free electrons due to the ionization of alkali metal atoms in the gas phase and is calculated using the Saha ionization [Saha (1920)] (Eq. 2) equation assuming Jupiter's atmosphere to be in a state of thermal equilibrium. The ionization equation itself assumes a single component gas phase system. Thereby, we add the electron densities from ionization of sodium and potassium to determine total number density of free electrons. Here, _N\({}_{e}\)_ is the electron density, \(N\) is number density, \(\epsilon\) is ionization energy, \(\lambda\) is De Broglie wavelength, _g\({}_{0}\)_ and _g\({}_{1}\)_ are statistical weights, _k\({}_{B}\)_ is Boltzmann
Figure 2: Condensation curves of NH\({}_{3}\), H\({}_{2}\)O, H\({}_{2}\)S and alkali metals Na\({}_{2}\)S and KCl at 1X solar abundance. Our calculations are based on the equilibrium cloud condensation model [Atreya et al. (1999)], and saturation vapor pressure corresponding to Na\({}_{2}\)S and KCl [Visscher et al. (2006), Morley et al. (2012)]. The cloud bases are at the levels where the condensation curves cross the adiabat considering T\({}_{1bar}\) = 166.1 K [Seiff et al. (1998)].
constant, _m\({}_{e}\)_ is mass of the electron and \(h\) is Planck's constant.
\[\frac{N_{e}^{2}}{N-N_{e}}=\frac{2}{\lambda^{3}}\frac{g_{1}}{g_{0}}e^{-\epsilon/k _{B}T} \tag{2}\]
\[\lambda=\sqrt{\frac{h^{2}}{2\pi m_{e}k_{B}T}} \tag{3}\]
The brightness temperatures correspond to electromagnetic radiation traveling from the interior of Jupiter radially outwards through the atmospheric layers. Thus, the transmission through the deep atmosphere is similar to the transmission through a cold plasma medium. The refractive index of microwaves propagating through a cold plasma media can be described by the Appleton- Hartree equation [Helliwell (2014)]. The formulation is applicable to low-temperature plasma medium both in the presence or absence of magnetic fields. At 100-1000 bar pressure levels, the contribution of the magnetic field is insignificant in the Appleton-Hartree formulation [Helliwell (2014)]. Therefore, a simplified version of the Appleton-Hartree equation (Eq. 4) is used to calculate the complex refractive index of the deep atmosphere using the electron number density calculated from the Saha ionization equation. For an unmagnetized cold plasma medium i.e. Jupiter's deep atmosphere, the Appleton- Hartree equation is simplified to:
\[n^{2}=1-\frac{X}{1-iZ} \tag{4}\]
\[\alpha=\frac{2\pi}{\lambda_{ch}Q} \tag{5}\]
Here, \(X=\frac{\omega_{0}^{2}}{\omega^{2}}\), \(Z=\frac{\nu}{\omega}\), \(\omega_{0}\) is electron plasma frequency, \(\omega\) is the angular frequency of microwave radiation, \(\omega_{h}\) is electron gyro frequency, \(\nu\) is electron- neutral collision frequency, \(\lambda_{ch}\) is the frequency of a given MWR channel, \(n\) is the refractive index, \(\alpha\) is the extinction coefficient and \(Q\) is the quality factor i.e. the ratio of squares of real and imaginary parts of the refractive index.
### Radiative Transfer Modeling
In order to draw a comparison between the MWR observations and theoretical knowledge of Jupiter's atmosphere, a benchmark model for the ideal Jupiter atmosphere is constructed using a moist hydrostatic adiabat following the ideal gas law [Li et al. (2018), Li et al. (2018)]. The specific heat of hydrogen is estimated from the mixing ratio of ortho and para hydrogen assuming thermal equilibrium between the ortho and para states. Moreover, the temperature profile of Jupiter's atmosphere is constructed for two cases of reference temperatures: (i) \(T\) = 166.1 K at the 1-bar pressure level from the Galileo probe [Seiff et al. (1998)] and (ii) \(T\) = 168.8 K at the 1-bar pressure level based on the reanalysis of the Voyager radio occultation experiment at Jupiter [Gupta et al. (2022)]. Ammonia and water vapor are considered vapors for the moist adiabat and their partial pressure is controlled by the cloud condensation process by forcing the partial pressures to be equal to their saturation vapor pressures. In the deep atmosphere of Jupiter, water and ammonia are not expected to form clouds; however, alkali metals are expected to undergo condensation. Therefore, a similar approach is applied to alkali metals to estimate the concentration of alkali metals present in the gas phase available for the ionization process.
Spectral radiance is proportional to the physical temperature of the atmosphere in the Rayleigh-Jeans limit. For microwave frequencies, we compute the brightness temperature (_T\({}_{b}\)_) from the physical temperature using Eq. (1). The opacity of Jupiter's atmosphere is the sum of opacities from individual sources discussed in the previous section i.e. ammonia, water, free electrons, and collision-induced absorption. The abundances of ammonia and water vapor have been assumed to be 2.7 and 5 times the solar abundance [Li et al. (2020), Li et al. (2017)]. Because there is no a priori information on the alkali metal abundance in Jupiter, Therefore, we compare two cases, one without alkali metals (baseline) and another with alkali metals (treatment) in order to provide a comparison between our current
knowledge of Jupiter and MWR data.
The spatial resolution of MWR data also provides the limb darkening coefficient at six microwave frequencies. Limb darkening (_L\({}_{d}\)_) is defined as the percent change in _T\({}_{b}\)_ at a given viewing angle relative to _T\({}_{b}\)_ at a position looking vertically down to the planet center i.e. nadir. For our simulations, we compute the limb darkening at a 45- degree angle from the nadir. The MWR channels at 0.6 GHz and 1.2 GHz are chosen to provide a comparison between theory and observations at higher pressures using _T\({}_{b}\)_ and _L\({}_{d}\)_ as the observables for comparison. The benchmark case of the ideal Jupiter atmosphere is compared with MWR observations as a function of latitude between -40 and 40 degrees planetocentric latitude. Data from higher latitudes are neglected due to the presence of signatures from synchrotron radiation that is inseparable from the atmospheric contribution.
A latitudinal variation in brightness temperatures is observed at both 0.6 and 1.2 GHz (Figure 3, panels (a) and (c)). The small- scale variations in _T\({}_{b}\)_ and _L\({}_{d}\)_ in all the panels can be attributed to variations in the atmospheric temperature structure and composition. It is important to note that the baseline case (without alkali metals) corresponds to two different temperature profiles of Jupiter's atmosphere for two different _T\({}_{1bar}\)_. There is an agreement between the baseline case and observations at 1.2 GHz in the equatorial region (panel (c)). On the other hand,
Figure 3: Limb darkening and brightness temperature MWR observations compared with simulation results at 0.6 GHz and 1.2 GHz corresponding to Jovian adiabats at (i) _T\({}_{1bar}\)_ = 166.1 K and (ii) _T\({}_{1bar}\)_ = 168.8 K, (a) _T\({}_{b}\)_ vs. latitude at 0.6 GHz, (b) _L\({}_{d}\)_ vs. latitude at 0.6 GHz, (c) _T\({}_{b}\)_ vs. latitude at 1.2 GHz, (d) _L\({}_{d}\)_ vs. latitude at 1.2 GHz.
brightness temperatures at 0.6 GHz are lower than the baseline case by 40-60 K at all latitudes (panel (a)) indicating the possibility of an additional source of opacity. Such a source is also supported by a depressed _L\({}_{d}\)_ observed by MWR; it is 4 percent less than the _L\({}_{d}\)_ magnitude of the ideal Jupiter atmosphere across all latitudes (panel (b)). The mismatch between the baseline and observations at 0.6 GHz is much greater than the uncertainty in measurements and variations in _T\({}_{b}\)_ and _L\({}_{d}\)_. Since the brightness temperatures correspond to different pressure regions in the atmosphere, the anomalous observations at 0.6 GHz must be attributed to the presence of an additional opacity source in the deep atmosphere or to a different opacity source that absorbs more effectively at 0.6 GHz than at 1.2 GHz. We test four confounding factors: (1) the distribution of ammonia, (2) the ammonia opacity at temperatures exceeding the range of laboratory measurements, (3) the opacity of water at high temperatures and (4) the contribution of alkali metals. The theoretical brightness temperature and limb darkening at 0.6 GHz and 1.2 GHz is shown in Fig. 3.
The latitudinal distribution of brightness temperatures and limb darkening from the forward model indicates the decrease in limb darkening from the equator to the pole at 0.6 GHz. It is opposite to the variation of limb darkening at 1.2 GHz across the latitudes. This effect could be attributed to the free electrons in the deep atmosphere which could be inferred from the shift in the contribution functions toward higher pressures in presence of alkali metals (Fig. 1). Alkali metals greatly affect the absorption behavior at 0.6 GHz which dominates the effect of gravitation on limb darkening.
### Ammonia, Water and Alkali Metals
Brightness temperature variations with latitude and the spectral inversion of brightness temperatures show a non-uniform distribution of ammonia vapor in Jupiter's atmosphere in the deep atmosphere region [Li et al. (2017), Ingersoll et al. (2017)]. Therefore, the non-uniform distribution of ammonia could contribute to variations in microwave opacity of the deep atmosphere. In order to estimate the effect of ammonia concentration variations, we perturb the ammonia profile in the model and use a scaling factor to vary the magnitude of ammonia vapor concentration in the model as described in Eq. (6).
\[q_{NH_{3}}(P)=q_{NH_{3},0}(P)-(q_{NH_{3},0}(P)-q_{NH_{3},MWR}(P))s \tag{6}\]
Here, _q\({}_{NH_{3}}\)_ is the ammonia mass mixing ratio at a given pressure \(P\), _q\({}_{NH_{3},0}(P)\)_ is the homogeneous ammonia mixing ratio which is set to 2.7 times solar abundance for NH\({}_{3}\)\(\sim\) 360 ppm [Li et al. (2017)] from the deep atmosphere till the NH\({}_{3}\) vapor saturation point. Above the saturation point, the mixing ratio follows the NH\({}_{3}\) saturation vapor pressure curve. _q\({}_{NH_{3},MWR}\)(P)_is the mixing ratio retrieved from MWR inversion. We use a scaling factor to vary the ammonia mixing ratio between the homogeneous case to MWR derived profiles. The scaling factor, s ranges from 0 to 1.5 where 0 is the case for homogeneous mixing ratio. Increasing s to 1 will change the ammonia profile to MWR inversion case for equator and mid-latitude regions. We also extend the scaling factor to 1.5, in order to take into account the low ammonia mixing ratio observed at the North Equatorial Belt (NEB) of Jupiter [Li et al. (2017)].
NH\({}_{3}\) opacity measurements are currently not available for high temperatures ( 550 K-300 K) corresponding to Jupiter's deep atmosphere and there is a decrease in the magnitude of absorption of NH\({}_{3}\) at high pressures. Thereby, we invoke a scaling factor to the NH\({}_{3}\) absorption coefficient to provide an estimation of the opacity at high temperatures. The mass absorption coefficient of ammonia is estimated by multiplying the temperature-scaling law to the absorption coefficient based on Hanley et al. (2009) (Eq. 7). In this equation, \(\alpha\) is the absorption coefficient of NH\({}_{3}\), h is the opacity factor, \(T\) is temperature and _T\({}_{c}\)_ is reference temperature equal to 750 K. The NH\({}_{3}\) opacity models show that the absorption coefficient peaks at 750 K and decreases at temperatures beyond 750 K. In the simulations, the scaling factor is multiplied to the NH\({}_{3}\) opacity at temperatures higher than _T\({}_{c}\)_. The power law index (h) is varied from 1 to 5 keeping the ammonia concentration constant, i.e., 2.7 times solar abundance. We also keep the water vapor constant at 5 times solar abundance as the laboratory measurements demonstrate that water vapor absorption does not show a significant increase with pressure and can be said to be relatively transparent when compared to the previous model of microwave absorption [Steffes et al. (2023)].
\[\alpha(NH_{3})\sim\left(\frac{T_{c}}{T}\right)^{h} \tag{7}\]
Changing the ammonia profile and introducing the additional temperature-dependent scaling factor produce brightness temperature and limb darkening divergent from MWR data at 0.6 GHz as shown in Figure 4a. The difference between _T\({}_{b}\)_ from the model and observations is in the range of 50-200 K at 0.6 GHz. Reducing the ammonia concentration causes a monotonic increase in _T\({}_{b}\)_ and a decrease in _L\({}_{d}\)_. Further, reducing the ammonia opacity shows a similar trend in _T\({}_{b}\)_, while a saturation in _L\({}_{d}\)_ is expected at a power law factor of 5. Changing the ammonia profile and ammonia opacity has a similar effect on _T\({}_{b}\)_ and _L\({}_{d}\)_ at 1.2 GHz. However, overall, the variation in the MWR observations at 1.2 GHz can be explained by these two factors and does not require the inclusion of alkali metals. The 1.2 GHz observations correspond to \(\sim\) 20 bar (Fig. 1), much above the cloud base of alkali metals and at relatively lower pressure levels. Therefore, the contribution of free electrons to opacity is expected to be less due to lower temperatures, and the opacity contribution of ammonia vapor dominates at 1.2 GHz. However, a comparison of MWR observations at both frequencies clearly implies that the variation in ammonia vapor opacity cannot solely explain the anomalous observations at the 0.6 GHz channel.//
Fig. 4c, 4d examines the overall effect of alkali metals and ammonia vapor on the _T\({}_{b}\)_ and _L\({}_{d}\)_ at 0.6 GHz and 1.2 GHz. We vary the alkali metal metallicities in a range from 0 to -7 (solar abundance of Na and K according to Asplund et al. (2009)) for each condition of NH\({}_{3}\) profile scaling and NH\({}_{3}\) opacity scaling. The volume mixing ratio of Na and K corresponding to abundance in solar photosphere [Asplund et al. (2009)] are 3.46 x 10\({}^{-6}\) (_[Na/H]_ = -5.76) and 2.14 x 10\({}^{-7}\) (_[K/H]_ = -6.97), respectively. Therefore, we simulate a wide range of ammonia opacity conditions for a given alkali metal abundance (colored dots). Both NH\({}_{3}\) profile and opacity scaling cause a change in _T\({}_{b}\)_ and _L\({}_{d}\)_, which is shown by the annotation in the figure. The variation in _T\({}_{b}\)_ and _L\({}_{d}\)_ is similar to the pattern in Fig. 4a. NH\({}_{3}\) profile scaling causes a decrease in _L\({}_{d}\)_, while the scaling in NH\({}_{3}\) vapor opacity causes _L\({}_{d}\)_ to increase at 0.6 GHz. For each case of metallicity, we then perform a scaling in ammonia vapor and ammonia opacity as described previously in this section. This provides us with a matrix of _T\({}_{b}\)_ and _L\({}_{d}\)_ to take into account all possible sources of opacity, i.e., collision-induced absorption, ammonia, water vapor, and free electrons from alkali metals. The free electron opacity is calculated from the Hartree-Appleton equation explained in the previous section.
When we compare the new model result with MWR observations (Fig. 4b), we observe that model matches with observations at 0.6 GHz for free electrons corresponding to alkali metal metallicities in the range of -2 to -5 (chocolate colored patches), i.e. 10\({}^{-2}\) to 10\({}^{-5}\) times the solar abundance. There is an agreement between the model and observations at 1.2 GHz for the same range of metallicities. The addition of free electrons from alkali metals dominates the effect of gravity (Fig. 5) and we expect the limb darkening to decrease from equator to the poles assuming uniform mixing ratio of water and ammonia vapor. It serves as a baseline to understand the sole effect of free electrons on latitudinal variation of microwave radiation from Jupiter's deep atmosphere.
## 4 Discussions
We infer metallicity of the alkali metals in Jupiter to be much lower than the solar value. A possible indication of low metallicity of the alkali metals in a hot Jupiter exoplanet was first proposed by Demory et al. (2011) as one plausible explanation for the high albedo of Kepler-7b. They derived an alkali metal abundance 10-100 times lower than the solar value. Since then the abundance of alkali metals has been derived for several other giant exoplanets, with abundances ranging from \(\sim\) 100 times below solar to \(\sim\) 100 times above solar, although the uncertainties are large. Recent observations of two hot Jupiters or Saturns with clear or mostly clear atmospheres were made. The alkali metal abundance for one such hot Jupiter (HAT-P-1b) is found to be sub-solar [Chen et al. (2022)], while it was found to be solar to greatly super-solar for the other (WASP-96b)[Nikolov et al. (2022)]. Considering the relatively small sample size of hot Jupiters with clear atmospheres, it is premature to make a meaningful comparison between their alkali metal metallicity and the metallicity in Jupiter presented in this paper. On the other hand, it is instructive to compare the abundance of alkali metals in Jupiter from this work with the abundance of the other heavy elements. While the opacity contribution from alkali metals suggest that Na and K are strongly depleted relative to solar at the level probed by MWR at 0.6 GHz, all other heavy elements are enriched by a factor of approximately three to five; while nitrogen is highly variable but enriched, and the water abundance remains uncertain [Atreya et al. (2019), Li
et al. (2020), Li et al. (2017), Mahaffy et al. (2000)]. The comparison to other heavy metal measurements from the Galileo probe corresponds to much lower pressures i.e. \(<\) 22 bars. The estimation of alkali metal metallicity from MWR implies lower metallicity at much higher pressures. The results (Fig. 4b) provide an important constraint on alkali metal abundance at pressures sensitive to 0.6 GHz channel. A [M/H] = -1 for alkali metals provides too much opacity while too little abundance or absence of alkali metals does not provide sufficient opacity to match the MWR
Figure 4: Comparison is drawn between the Juno MWR observations and the results of the radiative transfer model for _T\({}_{b}\)_ and _L\({}_{d}\)_ at 0.6 GHz and 1.2 GHz, keeping the water abundance constant \(\sim\) 5 times solar abundance. (a, b) Jupiter’s atmosphere in the absence of alkali metals with only variations in the NH\({}_{3}\) vapor profile and the NH\({}_{3}\) opacity, (c, d) Jupiter’s atmosphere in the presence of alkali metals with variations in the NH\({}_{3}\) vapor profile and the NH\({}_{3}\) opacity. The NH\({}_{3}\) profile of Jupiter’s atmosphere is varied using a scale from 0 to 1.5 to take into account the contribution of non-uniform distribution of NH\({}_{3}\) vapor observed by MWR [Li et al. (2017)]. NH\({}_{3}\) opacity at temperatures above 750 K undergoes power law scaling as a function of atmospheric temperature (Eq. 7). In the absence of alkali metals, the changes in NH\({}_{3}\) vapor profile and the scaling in NH\({}_{3}\) vapor opacity deviate significantly from Juno MWR observations at 0.6 GHz. However, in the presence of alkali metals of low metallicity, i.e., in the range of -2 to -5, there is an agreement between model results and MWR observations. Observations at 1.2 GHz can be explained by variations in the NH\({}_{3}\) vapor profile and the NH\({}_{3}\) opacity independent of opacity contributions from alkali metals.
observations at 0.6 GHz.
The low abundance of alkali metals indicated by MWR observations could be attributed to any of the following scenarios. (i) Initially enriched alkali metals, consistent with the other heavy elements in the atmosphere, are depleted by chemical reactions with other constituents deep in the atmosphere, resulting in a low abundance of Na and K at \(\sim\) 1 kilobar level sufficient to provide the free electrons to explain the MWR data at 0.6 GHz. Fegley and Lodders [Fegley Jr & Lodders (1994)] predict, for example, the formation of gas-phase species of Na and K in the atmosphere i.e. NaCl, NaOH, and KOH. Should there be chemical mechanisms that could selectively deplete K in the atmosphere, leaving Na to be the most significant contributor to free electrons in the deep atmosphere, the metallicity of Na would be expected to be in the range of 0 to -2 i.e. solar to highly sub- solar abundance (Appendix B). (ii) Unconventional planet formation processes, whereby Jupiter did not accrete a solar complement of alkali metals, or that the alkali metals are not well mixed at greater depths. If the depletion of alkali metals at \(\sim\) 1 kbar inferred in this paper is
Figure 5: Latitudinal variation of brightness temperature and limb darkening of Jupiter’s atmosphere at 0.6 GHz and 1.2 GHz at [M/H] = -3
representative of their bulk abundance, it could be indicative of the depletion of all rock- forming elements, with significant implications for the formation and evolution of Jupiter. Our conclusion of depletion is based on the data of the 0.6 GHz channel, whose weighting function peaks at 1 kilobar level with the inclusion of alkali metals. Thus, we are confident about the result only at this level. Alkali metals could well be more abundant deeper in the atmosphere and they could have been depleted by some as yet unknown mechanism before reaching the 1 kilobar level though the degree of depletion would have to be huge. Barshay & Lewis (1978) considered one such possibility, where silicates were found to be a way of sequestration of gas phase alkali metals. However, a later study by Fegley Jr & Lodders (1994) found it to be an ineffective mechanism. Further modeling and laboratory studies are needed to cover the full parameter space of combined thermochemistry of alkali metal and rock cloud forming species corresponding to the very high temperature and high pressure conditions of the deep atmosphere of Jupiter, together with any dynamical effects, before drawing any firm conclusions about depletion of alkali metals in bulk Jupiter below the level to which the MWR data of this paper are sensitive.
The new constraints on the abundance of alkalis are linked to their low ionisation potential, and the fact that the electrons that they provide directly affect opacities at 0.6 and 1.2 GHz (see Eq. 4). But when present, they are strong absorbers at visible wavelengths (e.g., Burrows et al. (2000)) and therefore directly affect the planetary radiative flux. The low abundances that we derive imply that a radiative zone may be present in Jupiter [Guillot et al. (1994), Guillot et al. (2004)]. Interestingly, this could explain at the same time the relatively low abundance of CO observed in Jupiter's atmosphere compared to expectations for a fully convective deep atmosphere [Cavalie et al. (2023)].
## 5 Software and Third Party Data Repository Citations
The software for the radiative transfer package will be available at zenodo archive ([https://doi.org/10.5281/zenodo.7893914](https://doi.org/10.5281/zenodo.7893914)) and the MWR data used in this work, and associated files for data visualization are available at archive ([https://doi.org/10.5281/zenodo.7893817](https://doi.org/10.5281/zenodo.7893817)). They can be made available upon request
High-performance Atmospheric Radiation Package (HARP) [Li et al. (2018b), Bhattacharya et al. (2023)]
## Appendix A Appendix A: Electron Density and Conductivity
The electron density of Jupiter's atmosphere is governed by two fundamental processes: (i) condensation of alkali metal condensates i.e. Na\({}_{2}\)S and KCl, and (ii) ionization of alkali metals in thermal equilibrium. Fig. 2 shows the pressure levels corresponding to the cloud base of Na\({}_{2}\)S and KCl based on their saturation vapor pressures. Cloud condensation reduces the amount of alkali metals available in gas phase that act as a source of free electrons, and restricts the abundance of Na and K corresponding to their respective saturation vapor pressure. In the cloud region, electron density is controlled by saturation vapor pressure of alkali metals whereas below the cloud base, electron densities are governed by metallicity of alkali metals. Thereby, it is evident that condensation controls the electron density and thereby, conductivity at low pressure levels. Condensation limited ionization is observed at low pressure (below 1 kbar) irrespective of the alkali metal abundance as the electron density lines converge (Fig. A.1 (a)). Fig. A.1 (a) and (b) show the presence of a kink in electron density and their respective conductivity at the cloud base corresponding to different alkali metal abundances. However, condensation does not play a significant role in governing the electron densities at \(\sim 1\) kbar pressure level corresponding to the global maxima in the weighting function at 0.6 GHz (Figure 1).
The electron density of the deep atmosphere is much lower than in the case of alkali metals at solar abundance. It is the true representation of the electron density of the deep atmosphere. At greater pressures, hydrogen behaves as a semiconductor and becomes the major contributor to the electron density [Liu et al. (2008)]. The electrical conductivity of the atmosphere is calculated using Drude's equation. It provides an estimate of the conductivity due to the free electrons provided by alkali metal ionization.
## Appendix B Appendix B: Selective depletion of alkali metals
Even though Na\({}_{2}\)S has a deeper condensation level compared to KCl, the cloud condensation is governed by atmospheric temperature, and does not reflect the chemical reactivity of alkali metals. K is more electropositive than Na
Figure A.1: (a) Electron density of Jupiter’s deep atmosphere at the solar abundance and _[M/H]_ = -3 and -4, (b) electrical conductivity of Jupiter’s deep atmosphere at the solar abundance and [M/H] = -3 and -4.
and thereby, is expected to be more reactive as compared to Na. Therefore, it is possible that there can be a chemical mechanism that could selectively deplete K into other compounds, leaving Na as the only source of free electrons in Jupiter. Under such conditions, we find that Na metallicity should be in the range of 0 to -3 to match the MWR observations. The increase in alkali metal metallicities can be attributed to two factors: (i) low ionization energy of K, and (ii) Na\({}_{2}\)S condenses much below KCl (Figure 2). Thereby, a larger amount of Na is required to produce enough free electrons to match the MWR brightness temperatures and limb darkening.
The elimination of K from the atmosphere highlights the role of the elemental abundance of Na required to match the MWR observations. The results of the forward model in Fig. B.1 indicate the possible solutions of Na metallicity under different conditions of ammonia vapor concentration profiles and microwave opacities. It is observed that the range of Na metallicity is expected to be from 0 to -3 i.e. solar abundance to highly sub-solar abundance. Thus, metallicity of Na required is expected to be higher than those considering both Na and K to be sources of free electrons.
## Appendix C Appendix C: Jovian adiabats and comparison of MWR with high temperature adiabat
Fig. 2 shows that brightness temperatures at 600 MHz from two adiabats differ by approximately 15 K. The relative weighting function for the adiabats is that of the ideal Jupiter's atmosphere without the inclusion of opacity due to free electrons from alkali metals. It shows a peak at \(\sim\) 100 bar. From the difference in physical temperature of the atmosphere of the two adiabats, it is seen that the difference reaches \(\sim\) 10-15 K at 100 bar level (Fig. C.1). The weighting function at 600 MHz also extends below 100 bar which could explain the difference in brightness temperatures. An interesting observation is that the difference in adiabat temperatures increases with increase in atmospheric pressure. This increase can be attributed to the temperature dependent specific heat of the atmospheric constituents.
The interior models of Jupiter generally use a high temperature in the range of 170-180 K at the outer boundary (1 bar pressure level) [Gupta et al. (2022), Miguel et al. (2022)]. These temperatures are about 10-15 K higher than the measurements from the Galileo probe (166.1 K)[Seiff et al. (1998)] and Voyager radio occultation reanalysis (168.8
Figure B.1: Limb darkening and brightness temperature comparison of MWR observations and forward model results at 600 MHz and 1.2 GHz for metallicities ranging from 0 to -7 at different ammonia vapor concentration profiles and opacities. It showcases the sole effect of free electrons due to the ionization of Na, without considering any contribution from K.
K)[Gupta et al. (2022)]. A simulation of brightness temperatures and limb darkening at 0.6 GHz and 1.2 GHz is carried out for all cases of alkali metal metallicities, ammonia concentration and opacity variation assuming _T\({}_{1bar}\)_ = 175 K. It can be clearly seen in Fig. C.2 that high temperature at 1 bar doesn't match with entire range of MWR observations for both the frequencies. Some alternate possibilities could be the presence of a non-adiabatic gradient or a radiative layer in Jupiter's deep atmosphere that can possibly account for a higher temperature at 1 bar level. However, the mismatch with MWR at 1.2 GHz poses a serious question on the assumption. The current measurements of temperature at 1 bar level are from limited radio occultation experiments. There is a need for radio science experiments from equator to the poles, in order to estimate the true variability in temperatures at 1 bar.
| 水分とアンモニアの蒸気は、ミクロ波長放射計（MWR）が観測した圧力レベルにおけるスペクトル吸収の主要な供給源です。ただし、MWRの最も長い波長チャンネル（50 cm、600 MHz）で観測された輝度温度と limb darkeningeは、ジェイプーの深層気候に新たな不透明性の存在を示唆しています（100 bar 以上の圧力）。アンモニアと水蒸気の吸収特性と、ジェイプーの気候におけるそれらの相対的濃度では、深層気候での透明度を説明するのに十分なものではありません。ここで、アルカリ金属のイオン化によって生じる自由電子が、太陽系に近い金属よりも低い金属licity（log based 10、相対的な濃度）のナトリウムとカリウムで、深層気候の不透明度を補完 |
2308.04850 | Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions | This paper investigates links between the eigenvalues and eigenfunctions of
the Laplace-Beltrami operator, and the higher Cheeger constants of smooth
Riemannian manifolds, possibly weighted and/or with boundary. The higher
Cheeger constants give a loose description of the major geometric features of a
manifold. We give a constructive upper bound on the higher Cheeger constants,
in terms of the eigenvalue of any eigenfunction with the corresponding number
of nodal domains. Specifically, we show that for each such eigenfunction, a
positive-measure collection of its superlevel sets have their Cheeger ratios
bounded above in terms of the corresponding eigenvalue.
Some manifolds have their major features entwined across several
eigenfunctions, and no single eigenfunction contains all the major features. In
this case, there may exist carefully chosen linear combinations of the
eigenfunctions, each with large values on a single feature, and small values
elsewhere. We can then apply a soft-thresholding operator to these linear
combinations to obtain new functions, each supported on a single feature. We
show that the Cheeger ratios of the level sets of these functions also give an
upper bound on the Laplace-Beltrami eigenvalues. We extend these level set
results to nonautonomous dynamical systems, and show that the dynamic Laplacian
eigenfunctions reveal sets with small dynamic Cheeger ratios. | Gary Froyland, Christopher P. Rock | 2023-08-09T10:26:23 | http://arxiv.org/abs/2308.04850v1 | # Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions
###### Abstract
This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.
Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.
## 1 Introduction
The classical static _Cheeger problem_ is an optimisation problem in Riemannian geometry, which has been studied extensively in relation to the eigenvalues of the Laplace-Beltrami operator [17, 43, 54, 9]. Given an \(n\)-dimensional Riemannian manifold \((M,g)\) with volume measure \(V\) and induced \(n-1\)-dimensional Hausdorff measure \(V_{n-1}\), the _Neumann Cheeger ratio_ of a set \(A\subset M\) with suitably smooth boundary is the ratio \(\mathcal{J}_{N}(A):=\frac{V_{n-1}(\partial A\cap\operatorname{int}M)}{V(A)}\). The Neumann Cheeger problem consists of finding a set that minimises \(\mathcal{J}_{N}(A)\) over sets \(A\subset M\) satisfying \(V(A)\leq\frac{V(M)}{2}\). The resulting minimal ratio is known as the _Neumann Cheeger constant for \(M\)_. For compact \(n\)-dimensional submanifolds \(M\subset\mathbb{R}^{n}\), a Neumann Cheeger ratio minimiser is a set \(A\subset M\) which is separated from \(M\backslash\overline{A}\) by an optimal 'bottleneck'. We give an example in Figure 1(a). The _Dirichlet Cheeger ratio_ of a set \(A\subset M\) with suitably smooth boundary is the ratio \(\mathcal{J}_{D}(A):=\frac{V_{n-1}(\partial A)}{V(A)}\), and the Dirichlet Cheeger problem consists of finding a set that minimises \(\mathcal{J}_{D}(A)\) over subsets \(A\subset M\). The resulting minimal ratio is known as the _Dirichlet Cheeger constant for \(M\)_. A Dirichlet Cheeger ratio minimiser is a region with an optimal balance between large volume and small boundary. For \(n\)-dimensional \(M\subset\mathbb{R}^{n}\) endowed with the Euclidean metric and \(A\subset M\), \(\mathcal{J}_{D}(A)\) decreases by a factor of \(s\) when we dilate \(A\) by a factor of \(s\) in each dimension, so minimisers for \(\mathcal{J}_{D}(A)\) always contact \(\partial M\) ([57, Theorem 3.5]). We give an example in Figure 1(b).
The Cheeger problem can be extended to seek _collections_ of subsets, each of which have small Cheeger ratios. Given a collection of \(k\) disjoint sets \(A_{1},\dots,A_{k}\subset M\), the Neumann and Dirichlet Cheeger ratios of \(\{A_{1},\dots,A_{k}\}\) are given by \(\mathcal{J}_{N}(\{A_{1},\dots,A_{k}\}):=\max_{1\leq i\leq k}\mathcal{J}_{N}(A_ {i})\) and \(\mathcal{J}_{D}(\{A_{1},\dots,A_{k}\}):=\max_{1\leq i\leq k}\mathcal{J}_{D}(A_ {i})\), respectively, i.e. the Cheeger ratio of a collection of disjoint subsets of \(M\) is the maximum Cheeger ratio among the subsets. For each \(k\geq 1\), the \(k\)_th Neumann_ or _Dirichlet Cheeger problem_ consists of finding a
collection of \(k\) disjoint sets \(\{A_{1},\ldots,A_{k}\}\) which minimises \(\mathcal{J}_{N}(\{A_{1},\ldots,A_{k}\})\) or \(\mathcal{J}_{D}(\{A_{1},\ldots,A_{k}\})\). The first Dirichlet Cheeger problem is exactly the classical Dirichlet Cheeger problem, while the second Neumann Cheeger problem corresponds to the classical Neumann Cheeger problem. The \(k\)th Cheeger problems for larger \(k\) are called the _higher Cheeger problems_, and the infima are called the _higher Cheeger constants_.
Exact minimisers for the Cheeger problem have only been computed for a few sets or classes of sets (see e.g. [5, 13, 46, 47]). In particular, [47, Theorem 1.4] obtains an expression for the Cheeger-minimising set of any subset of \(\mathbb{R}^{2}\) without a 'neck'. We are instead interested in using the Cheeger problem to identify necks, and the approach of [47] does not extend to sets with necks (see e.g. [47, Figs 1-2]). There are some algorithms for solving Cheeger problems numerically (see e.g. [10, 11, 12, 14, 42]), but these algorithms apply only to the classical Cheeger problems, not the versions with \(k\geq 2\) (in the Dirichlet case) or \(k\geq 3\) (in the Neumann case). These algorithms have not been studied on Riemannian manifolds other than full-dimensional subsets of \(\mathbb{R}^{n}\). Understanding the connectivity of more general Riemannian manifolds is important in settings such as manifold learning (e.g. [18, 36]), where one studies the geometry of a low-dimensional submanifold embedded in some high-dimensional Euclidean space. The second Dirichlet Cheeger problem is studied in [4], where the authors solve this problem for one specific subset of \(\mathbb{R}^{2}\) (an annulus).
Approximate minima and minimisers for the higher Cheeger problem, and upper bounds on the higher Cheeger constants, can be found using the eigenfunctions and eigenvalues of the (possibly _weighted_) _Laplace-Beltrami operator_. Miclo [53] and others have given upper bounds on the \(k\)th Cheeger constant on boundaryless manifolds, up to a non-explicit factor depending cubically on \(k\). Miclo improves this dependence on \(k\) to sub-logarithmic, by using (for example) the \(2k\)th eigenvalue to bound the \(k\)th Cheeger constant. We prove an alternative upper bound on the \(k\)th Cheeger constant (Theorem 3.7), extending a result from the graph setting [22, Theorem 5], in terms of the eigenvalue of any eigenfunction with \(k\) or more nodal domains, up to a small constant factor independent of \(k\). Thus, we can obtain a much tighter upper bound on the \(k\)th Cheeger constant whenever the appropriate eigenfunction has sufficiently many nodal domains. Our bound also applies to manifolds with nonempty boundary, under Neumann or Dirichlet boundary conditions. Moreover, our bound is constructive - we show that any (possibly weighted) Laplace-Beltrami eigenfunction has superlevel sets within each nodal domain whose Cheeger ratios are also bounded above. A similar approach is used in the graph setting in e.g. [40, sec 1.1], to obtain a \(2\)-partition of a graph with a low conductance from the first nontrivial graph Laplacian eigenvalue. Our approach is primarily useful in situations where Laplacian eigenfunctions on a manifold are calculated or approximated explicitly.
An important question in the study of nonautonomous dynamical systems is how to divide the phase space into regions which interact minimally with each other. In purely deterministic dynamics, any two disjoint regions have no interaction with each other, so we instead consider regions whose boundaries remain small, relative to their size, as they evolve with the deterministic dynamics. The ratio of a region's time-averaged boundary size to its overall size is called its _dynamic Cheeger ratio_. Sets with small dynamic Cheeger ratio are called _coherent sets_, and the infimal dynamic Cheeger ratio is called the _dynamic Cheeger constant_[25, 27]. We can obtain an upper bound on the dynamic Cheeger constants using the eigenvalues
Figure 1: Neumann and Dirichlet Cheeger minimisers for \(M\subset\mathbb{R}^{2}\) equipped with the Euclidean metric.
of an operator, which acts on the domain of the dynamical system, called the _dynamic Laplacian_. We show that \(k\) disjoint coherent sets with quality guarantees - upper bounds on their dynamic Cheeger ratios - can be obtained from any eigenfunction with \(k\) nodal domains (Theorem 3.19).
The remainder of this article is structured as follows. In section 2, we provide some basic definitions and define the higher Cheeger constants. In subsections 3.1-3.2, we summarise prior upper bounds on the Cheeger constants in terms of Laplace-Beltrami eigenvalues. We also state our own constructive upper bounds, which depend on properties of the eigenfunctions (Theorem 3.7 and Proposition 3.8). In subsection 3.4, we generalise these results to the dynamic setting. Lastly, in section 4, we give some examples comparing our bounds to bounds from the literature.
## 2 Preliminaries
### Higher Cheeger constants
Let \((M,g)\) be a smooth Riemannian manifold, possibly with nonempty boundary, i.e. a second-countable Hausdorff space where each point of \(M\) has a neighbourhood diffeomorphic to a relatively open subset of \(\{x\in\mathbb{R}^{n}:x_{n}\geq 0\}\). Except where otherwise noted, we assume all Riemannian manifolds are \(n\)-dimensional (\(n\geq 2\)), \(C^{\infty}\), compact and connected, and have smooth boundary if they have a nonempty boundary. Let \(V\) and \(\mathrm{d}V\) denote the volume measure and volume form on \(M\) induced by \(g\). Let \((M,g,\mu)\) be a _weighted manifold_, i.e. a Riemannian manifold \((M,g)\) equipped with a measure \(\mu\) satisfying \(\mathrm{d}\mu=e^{\phi}\,\mathrm{d}V\) for some \(\phi\in C^{\infty}(M)\). Note that we can treat any Riemannian manifold as a weighted manifold by taking \(\mu=V\), so all our results for weighted manifolds extend directly to unweighted manifolds (i.e. manifolds where \(\phi=0\) everywhere). On each \(n-1\)-dimensional submanifold \(\Sigma\subset M\), let \(V_{n-1}\) and \(\mathrm{d}V_{n-1}\) denote the \(n-1\)-dimensional Riemannian volume measure and volume form on \(\Sigma\), and let \(\mu_{n-1}\) be the measure satisfying \(\mathrm{d}\mu_{n-1}:=e^{\phi}\,\mathrm{d}V_{n-1}\).
For a set \(A\subset M\), we let \(\partial^{M}A\) denote the relative topological boundary of \(A\) in \(M\), i.e. the set of points \(p\in M\) such that every neighbourhood of \(p\) contains both points in \(A\) and points in \(M\backslash A\). For example, if \(M:=\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\leq 1\}\) and \(A:=\{(x,y)\in M:y>0\}\), then \(\partial^{M}A\) consists of the interval \(\{(x,0):-1\leq x\leq 1\}\) but not the semicircle \(\{(x,y)\in M:x^{2}+y^{2}=1\}\). We define the Neumann and Dirichlet Cheeger constants as follows.
**Definition 2.1**.: Let \(\mathscr{P}_{N}(M)\) denote the collection of nonempty, relatively open subsets \(A\subset M\) such that \(\partial^{M}A\) is a codimension-\(1\), \(C^{\infty}\) submanifold of \(M\) with boundary \(\partial(\partial^{M}A)=\partial^{M}A\cap\partial M\). Let \(\mathscr{P}_{D}(M)\) denote the collection of nonempty, relatively open subsets \(A\subset M\) such that \(\overline{A}\cap\partial M=\emptyset\), and \(\partial A\) is a codimension-\(1\), \(C^{\infty}\) submanifold of \(M\). Then for \(k\geq 1\), a _Neumann_, resp. _Dirichlet \(k\)-packing_ is a set \(\mathcal{A}_{k}:=\{A_{1},\ldots,A_{k}\}\) such that each \(A_{i}\in\mathscr{P}_{N}(M)\), resp. \(A_{i}\in\mathscr{P}_{D}(M)\), and the \(A_{i}\) are pairwise disjoint. Let \(\mathscr{P}_{k,N}(M)\), resp. \(\mathscr{P}_{k,D}(M)\) denote the set of Neumann, resp. Dirichlet \(k\)-packings for \(M\).
**Definition 2.2** (Higher Cheeger constants).: For \(k\geq 1\), the _Neumann Cheeger ratio_ of a Neumann \(k\)-packing \(\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k,N}(M)\) is
\[\mathcal{J}_{N}(\{A_{1},\ldots,A_{k}\}):=\max_{1\leq i\leq k}\frac{\mu_{n-1}( \partial^{M}A_{i})}{\mu(A_{i})}. \tag{1}\]
The _Dirichlet Cheeger ratio_ of a Dirichlet \(k\)-packing \(\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k,D}(M)\) is
\[\mathcal{J}_{D}(\{A_{1},\ldots,A_{k}\}):=\max_{1\leq i\leq k}\frac{\mu_{n-1}( \partial A_{i})}{\mu(A_{i})}. \tag{2}\]
The _\(k\)th Neumann_ and _Dirichlet Cheeger constants_ of \(M\) are
\[h_{k,N} :=\inf_{\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k,N}(M)}\mathcal{J} _{N}(\{A_{1},\ldots,A_{k}\}) \tag{3}\] \[h_{k,D} :=\inf_{\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k,D}(M)}\mathcal{J} _{D}(\{A_{1},\ldots,A_{k}\}). \tag{4}\]
We will sometimes write \(\mathcal{J}_{N}(A)\) and \(\mathcal{J}_{D}(A)\) instead of \(\mathcal{J}_{N}(\{A\})\) and \(\mathcal{J}_{D}(\{A\})\) for convenience. By this definition, we always have \(h_{1,N}=0\), aligning with our notation where \(\lambda_{1,N}=0\). In the special case \(\partial M=\emptyset\), we write \(\mathcal{J}_{\emptyset}\) and \(h_{k,\emptyset}\), respectively for \(\mathcal{J}_{N}\) and \(h_{k,N}\), respectively, and refer to \(\mathcal{J}_{\emptyset}\) and \(h_{k,\emptyset}\) as the _boundaryless Cheeger ratio_ and _constant_.
Our Dirichlet Cheeger constants generalises Cheeger's original constant for manifolds with boundary [17], while our Neumann Cheeger constants generalise the boundaryless Cheeger constant of [17], the Neumann Cheeger constant of [8], and the \(k\)th boundaryless Cheeger constants [53] for \(k\geq 1\). Our \(h_{k,\emptyset}\) is exactly that defined in [53, p.326]: Miclo requires that the \(A_{i}\) are connected and that each connected component of \(M\backslash A_{i}\) contains some \(A_{j}\) for \(j\neq i\), but Miclo notes that this does not change the value of \(h_{k,\emptyset}\).
Cheeger [17] and Buser [8] (see also [60, p.499]) consider \(h_{k,\emptyset}\) and \(h_{k,N}\) for \(k=2\) only, and they require that \(\{A_{1},A_{2}\}\) are a \(2\)-partition for \(M\) (up to sets of measure zero) with \(\partial^{M}A_{1}=\partial^{M}A_{2}\), instead of allowing \(2\)-packings of \(M\). This does not affect the value of \(h_{2,N}\). To see this, choose any \(\{A_{1},A_{2}\}\in\mathscr{P}_{2,N}(M)\) with \(\mu_{n-1}(\partial^{M}A_{1})\leq\mu_{n-1}(\partial^{M}A_{2})\), and define the \(2\)-packing \(\{\bar{A}_{1},\bar{A}_{2}\}\) by \(\bar{A}_{1}:=\overline{A_{1}}\backslash\partial^{M}\overline{A_{1}}\), \(\bar{A}_{2}:=M\backslash\overline{A_{1}}\). Then \(\partial^{M}\bar{A}_{1}=\partial^{M}\bar{A}_{2}\), and \(\{\bar{A}_{1},\bar{A}_{2}\}\) is a \(2\)-partition for \(M\). The fact \(\partial^{M}\bar{A}_{1}\subset\partial^{M}A_{1}\) implies \(\{\bar{A}_{1},\bar{A}_{2}\}\in\mathscr{P}_{2,N}(M)\), and since \(\mu_{n-1}(\partial^{M}\bar{A}_{2})\leq\mu_{n-1}(\partial^{M}A_{1})\leq\mu_{n- 1}(\partial^{M}A_{2})\) and \(\mu(\tilde{A}_{2})\geq\mu(A_{2})\), we have \(\mathcal{J}_{N}(\{\tilde{A}_{1},\tilde{A}_{2}\})\leq\mathcal{J}_{N}(\{A_{1},A_ {2}\})\).
Our Cheeger constants are defined slightly differently from those in [4, 24], who take the infimum over arbitrary packings of \(M\) and use _perimeter_ instead of Hausdorff measure. Bobkov and Parini's Cheeger constant is equal to \(h_{k,D}\) for unweighted full-dimensional submanifolds of \(\mathbb{R}^{n}\) ([4, Proposition 3.6] and e.g. [1, Proposition 3.62]), while \(h_{2,N}\) gives an upper bound on de Ponti and Mondino's Cheeger constant on unweighted Riemannian manifolds by [59, Proposition 2.37] ([24] defines perimeter differently to [24], but they are equal on unweighted manifolds by e.g. [59, remark on Definition 2.33 and Theorems 2.38-2.39]). Yau [60, p.499] also defines a variant of \(h_{2,N}\) which does not require each \(\partial A_{i}\) to be smooth.
### Eigenvalues of the weighted Laplace-Beltrami operator
Let \(W^{1,2}(M;\mu)\) denote the Sobolev space of \(L^{2}(M;\mu)\) functions \(f\) with \(L^{2}(M;\mu)\)-integrable weak derivatives \(\nabla f\), and let \(W^{1,2}_{0}(M;\mu)\) denote the completion in the Sobolev norm \(\|\cdot\|_{W^{1,2}(M;\mu)}^{2}:=\|\cdot\|_{L^{2}(M;\mu)}^{2}+\|\nabla\cdot\|_ {L^{2}(M;\mu)}^{2}\) of the set of \(C^{\infty}(M)\) functions with compact support in \(\operatorname{int}M\) (see e.g. [15, pp.14-15]).
For any \(C^{1}\) vector field \(X\) on \(M\), let \(\operatorname{div}X\) denote the _divergence_ of \(X\) with respect to \(\operatorname{d}V\) (defined in e.g. [33, p.96] or [16, Prop. III.7.1 and proof]). Writing the Radon-Nikodym derivative of \(\mu\) as \(\operatorname{d}\mu=e^{\phi}\operatorname{d}V\), let \(\operatorname{div}_{\mu}X\) denote the _weighted divergence_\(\operatorname{div}_{\mu}X:=e^{-\phi}\operatorname{div}(e^{\phi}X)\) (see e.g. [33, p.96]). Then the _weighted Laplace-Beltrami operator_\(\Delta_{\mu}\) is defined for \(f\in C^{2}(M)\) by
\[\Delta_{\mu}f:=\operatorname{div}_{\mu}\nabla f=e^{-\phi}\operatorname{div}(e^{ \phi}\nabla f). \tag{5}\]
We consider the _Neumann_ and _Dirichlet eigenproblems_ for \(\Delta_{\mu}\). The _Neumann eigenproblem_ is as follows: find \(u\in C^{\infty}(M)\) and \(\lambda\in\mathbb{R}\), such that
\[\Delta_{\mu}u=\lambda u, \tag{6}\]
subject to the _Neumann boundary condition_ (if \(\partial M\neq\emptyset\))
\[\frac{\partial u}{\partial\mathbf{n}}=0\quad\text{on }\partial M, \tag{7}\]
where \(\mathbf{n}\) denotes the outward unit normal to \(\partial M\). Solutions \(u\) and \(\lambda\) are called _eigenfunctions_ and _eigenvalues_ of \(\Delta_{\mu}\). There is an orthogonal Schauder basis for \(L^{2}(M;\mu)\) consisting of eigenfunctions of (6) satisfying (7) (see e.g. [41, Theorem 4.3.1] or [3, ch. III, Theorem 18]). The corresponding eigenvalues form a non-positive decreasing sequence accumulating only at \(-\infty\) (see e.g. [41, Theorem 4.3.1] or [37, Theorems 11.5.1-11.5.2]). We denote the eigenvalues as \(0=\lambda_{1,N}>\lambda_{2,N}\geq\lambda_{3,N}\geq\ldots\), or as \(0=\lambda_{1,\emptyset}>\lambda_{2,\emptyset}\geq\lambda_{3,\emptyset}\geq\ldots\) in the special case \(\partial M=\emptyset\). The eigenvalue ordering induces an ordering on the corresponding eigenfunctions, so we will occasionally write the basis of eigenfunctions as \(u_{1},u_{2},\ldots\).
The _Dirichlet eigenproblem_ consists of finding \(u\in C^{\infty}(M)\) and \(\lambda\in\mathbb{R}\) which solves (6), subject to the _Dirichlet boundary condition_,
\[u=0\quad\text{on }\partial M. \tag{8}\]
We assume \(\partial M\neq\emptyset\) when we consider Dirichlet boundary conditions. There is also an orthogonal Schauder basis for \(L^{2}(M;\mu)\) of eigenfunctions of (6) satisfying (8). In this case, the eigenvalues form a strictly negative decreasing sequence accumulating only at \(-\infty\), and we denote them \(0>\lambda_{1,D}>\lambda_{2,D}\geq\lambda_{3,D}\geq\ldots\).
The eigenvalues of \(\Delta_{\mu}\) have the following variational characterisation (the proof of [15, p.16] extends directly to the weighted case).
**Theorem 2.3**.: _Let \((M,g,\mu)\) be a weighted manifold, and let \(u_{1},u_{2},\ldots\) denote a complete orthogonal basis of Neumann (resp. Dirichlet) eigenfunctions of \(\Delta^{d}\) corresponding to \(\lambda_{1,N},\lambda_{2,N},\ldots\) (resp. \(\lambda_{1,D},\lambda_{2,D},\ldots\)). Then for each \(k\geq 1\), we have_
\[\lambda_{k,N}=-\inf_{\begin{subarray}{c}f\in W^{1,2}(M)\\ \int_{M}u_{i}f\,d\mu=0,\forall i\in\{1,\ldots,k-1\}\end{subarray}}\frac{\|| \nabla f||_{L^{2}(M;\mu)}^{2}}{\|f\|_{L^{2}(M;\mu)}^{2}}, \tag{9}\]
_resp._
\[\lambda_{k,D}=-\inf_{\begin{subarray}{c}f\in W^{1,2}_{0}(M)\\ \int_{M}u_{i}f\,d\mu=0,\forall i\in\{1,\ldots,k-1\}\end{subarray}}\frac{\|| \nabla f||_{L^{2}(M;\mu)}^{2}}{\|f\|_{L^{2}(M;\mu)}^{2}}, \tag{10}\]
_with equality if and only if \(f\) is a Neumann (resp. Dirichlet) eigenfunction of \(\Delta_{\mu}\) with eigenvalue \(\lambda_{k,N}\) (resp. \(\lambda_{k,D}\))._
A _nodal domain_ of a function \(f\in C^{0}(M)\) is a maximal connected component of \(M\) where \(f\) is positive or negative. The number of nodal domains in the \(k\)th eigenfunction of \(\Delta_{\mu}\) under Dirichlet or Neumann boundary conditions is bounded above by \(k\). Courant [20, p.452] proves this bound assuming each nodal domain has piecewise smooth boundary. Chavel [15, pp.19-23] gives a proof in the boundaryless and Dirichlet cases which avoids the piecewise smooth boundary requirement via an approximation argument. Using a more general version of Green's formula [35, Proposition 5.8 and remark after Proposition 5.10], we prove Courant's nodal domain theorem in the Neumann case without the piecewise-smooth boundary assumption, since we could not readily find this in the literature.
**Theorem 2.4** (Courant's nodal domain theorem).: _Let \((M,g,\mu)\) be a weighted manifold. Then the \(k\)th Neumann or Dirichlet eigenfunction \(u_{k}\) of \(\Delta_{\mu}\) on \(M\) has at most \(k\) nodal domains._
Proof.: We prove only the Neumann case; the proof in [15, pp.19-23] for the Dirichlet case extends immediately to weighted manifolds. Let \(G_{1},\ldots,G_{k},G_{k+1},\ldots\) denote the nodal domains of \(u_{k}\). For each \(j=1,\ldots,k\), define \(\psi_{j}\in W^{1,2}(M;\mu)\) by
\[\psi_{j}:=\begin{cases}u_{k}|_{G_{j}},&\text{on }G_{j},\\ 0,&\text{elsewhere}.\end{cases}\]
Using Chavel's approximation argument [15, pp.21-22] and the version of Green's formula in [35, Proposition 5.8 and remark after Proposition 5.10], as in (23)-(25) below, for each \(j\) we have \(\frac{\||\nabla\psi_{j}||_{L^{2}(G_{j};\mu)}^{2}}{\|\psi_{j}\|_{L^{2}(G_{j};\mu )}^{2}}=-\lambda_{k,N}\). One can select constants \(\alpha_{1},\ldots,\alpha_{k}\in\mathbb{R}\), not all zero, such that
\[f:=\sum_{j=1}^{k}\alpha_{j}\psi_{j}\]
satisfies
\[\int_{M}u_{i}f\,\mathrm{d}\mu=0,\]
for each \(i=1,\ldots,k-1\) (see e.g. [15, p.17]). Noting that the \(\psi_{j}\) are disjointly supported, we have
\[\frac{\||\nabla f|\|_{L^{2}(M;\mu)}^{2}}{\|f\|_{L^{2}(M;\mu)}^{2}}=\frac{\sum_{ j=1}^{k}\alpha_{j}^{2}\||\nabla\psi_{j}|\|_{L^{2}(M;\mu)}^{2}}{\sum_{j=1}^{k} \alpha_{j}^{2}\|\psi_{j}\|_{L^{2}(M;\mu)}^{2}}=\frac{\lambda_{k,N}\sum_{j=1}^ {k}\alpha_{j}^{2}\|\psi_{j}\|_{L^{2}(M;\mu)}^{2}}{\sum_{j=1}^{k}\alpha_{j}^{2} \|\psi_{j}\|_{L^{2}(M;\mu)}^{2}}=\lambda_{k,N}.\]
Thus, Theorem 2.3 implies \(f\) is an eigenfunction of \(\Delta_{\mu}\) with eigenvalue \(\lambda_{k,N}\) vanishing identically on \(G_{k+1}\). But then Aronszajn's unique continuation principle [2] implies that \(f\) vanishes identically on \(M\), which is a contradiction.
## 3 Classical and higher Cheeger inequalities
### Cheeger inequalities for the first nonzero eigenvalue
The classical Cheeger inequalities provide an explicit bound away from \(0\) for \(\lambda_{1,D}\) or \(\lambda_{2,N}\), in terms of \(h_{1,D}\) or \(h_{2,N}\). Cheeger [17] proves the boundaryless and Dirichlet cases, while Maz'ya [51] (summarised in English in e.g. [32, Sec. 6]) independently proves a slightly stronger result some years prior. Yau [60, Sec. 5, Corollary 1], and later Buser [8, Theorem 1.6], prove the Neumann case. The Cheeger inequality can also be extended to metric measure spaces (including weighted manifolds). De Ponti and Mondino [24, Theorem 3.6] and Funano [29, Lemma 7.1] give variants of the Cheeger inequality for metric spaces (including weighted manifolds), with a Rayleigh quotient in place of an eigenvalue.
Several other upper bounds on eigenvalues of \(\Delta\) exist, which do not depend on the Cheeger constant (see for example [34] and references therein). Fewer bounds exist on the Cheeger constant: Ledoux [43, Theorem 5.3] and Milman [54, Theorem 1.5] have obtained bounds on the Cheeger constant in terms of concentration inequalities, while Dai et al [21, Theorem 1.4] have obtained an upper bound on the Cheeger constant on convex manifolds in terms of the manifold's dimension, Ricci curvature and diameter.
**Theorem 3.1** (Cheeger's inequality).:
* _[_17_]__: Let_ \((M,g)\) _be an unweighted, boundaryless, compact smooth Riemannian manifold. Then_ \[\lambda_{2,\emptyset}\leq-\frac{1}{4}h_{2,\emptyset}^{2}.\] (11)
* _[_17, 51_]_ _(see also_ _[_32_, Sec. 6]__): Let_ \((M,g)\) _be an unweighted, connected, compact smooth Riemannian manifold with nonempty, smooth boundary. Then_ \[\lambda_{1,D}\leq-\frac{1}{4}h_{1,D}^{2}.\] (12)
* _[_60_, Sec. 5, Corollary 1],_ _[_8_, Theorem 1.6]__: Let_ \((M,g)\) _be an unweighted, compact smooth Riemannian manifold with nonempty, smooth boundary. Then_ \[\lambda_{2,N}\leq-\frac{1}{4}h_{2,N}^{2}.\] (13)
These results extend directly to weighted manifolds, and even to more general metric measure spaces (see e.g. [24, Theorem 3.6]).
We prove that some of the superlevel sets within any nodal domain of any eigenfunction of \(\Delta\) have an upper bound on their Cheeger ratio, in terms of the corresponding eigenvalue (Theorem 3.2). This yields
a constructive version of Theorem 3.1 (Corollary 3.3), and also allows us to prove a constructive higher Cheeger inequality (Theorem 3.7).
For any nodal domain \(G\) of a function \(f\in C^{0}(M)\), we let \(\operatorname{range}(f^{2}|_{G}):=\{s^{2}:s\in f(G)\}\), and for any \(s\in\operatorname{range}(f^{2}|_{G})\), we define the \(s\)_-superlevel set_ of \(f^{2}\) on \(G\) as
\[G_{s}:=\{p\in G:f(p)^{2}>s\}. \tag{14}\]
**Theorem 3.2**.: _Let \((M,g,\mu)\) be an \(n\)-dimensional weighted manifold. Let \(u\) be some nonconstant Neumann, resp. Dirichlet, eigenfunction of \(\Delta_{\mu}\), with eigenvalue \(\lambda\). Let \(G\subset M\) be any nodal domain of \(u\). Then the set_
\[S_{G}:=\bigg{\{}s\in\operatorname{range}(u^{2}|_{G}):G_{s}\in \mathscr{P}_{N}(M),\lambda\leq-\frac{1}{4}\mathcal{J}_{N}(G_{s})^{2}\bigg{\}}, \tag{15}\]
_resp._
\[S_{G}:=\bigg{\{}s\in\operatorname{range}(u^{2}|_{G}):G_{s}\in \mathscr{P}_{D}(M),\lambda\leq-\frac{1}{4}\mathcal{J}_{D}(G_{s})^{2}\bigg{\}}, \tag{16}\]
_has positive Lebesgue measure satisfying the lower bound (27)._
Proof.: We prove only the Neumann case; the Dirichlet case follows similarly. Firstly, we use the coarea formula to find an expression (20) for the weighted average (19) of \(\mathcal{J}_{N}(G_{s})\). Secondly, we use a Rayleigh quotient argument to bound \(\lambda_{2,N}\) in terms of this weighted average (equation (26)). Lastly, we obtain our lower bound on the measure of \(S_{G}\).
The coarea formula (see e.g. [7, 13.4.2]) implies
\[\int_{G}|\nabla(u^{2})|\,\mathrm{d}\mu=\int_{\operatorname{range}(u^{2}|_{G}) }\mu_{n-1}(\{p\in G:u^{2}(p)=s\})\,\mathrm{d}s. \tag{17}\]
It follows immediately from Sard's theorem (e.g. [44, Theorem 6.10]), [55, Theorem 6.2.8] and the reasoning for [55, Lemma 6.2.7] that \(G_{s}\in\mathscr{P}_{N}(M)\) and \(\partial^{M}G_{s}=\{p\in G:u(p)^{2}=s\}\) for almost every \(s\in\operatorname{range}(u^{2}|_{G})\). For such \(s\), we have \(\mu_{n-1}(\{p\in G:u(p)^{2}=s\})=\mu_{n-1}(\partial^{M}G_{s})=\mathcal{J}_{N}(G _{s})\mu(G_{s})\), by the definition (1). Hence, we have
\[\int_{G}|\nabla(u^{2})|\,\mathrm{d}\mu=\int_{\operatorname{range}(u^{2}|_{G}) }\mathcal{J}_{N}(G_{s})\mu(G_{s})\,\mathrm{d}s. \tag{18}\]
Define
\[\bar{h}:=\frac{1}{\|u\|_{L^{2}(G;\mu)}^{2}}\int_{\operatorname{range}(u^{2}|_{G })}\mathcal{J}_{N}(G_{s})\mu(G_{s})\,\mathrm{d}s, \tag{19}\]
then \(\bar{h}\) is the weighted average of \(\mathcal{J}_{N}(G_{s})\) over \(\operatorname{range}(u^{2}|_{G})\), according to the probability measure \(\mathbb{P}\) on \(\operatorname{range}(u^{2}|_{G})\) given by \(\mathbb{P}(L):=\int_{L}\frac{\mu(G_{s})}{\|u\|_{L^{2}(G;\mu)}^{2}}\,\mathrm{d}s\). Then (18) and (19) yield
\[\frac{\int_{G}|\nabla(u^{2})|\,\mathrm{d}\mu}{\|u\|_{L^{2}(G;\mu)}^{2}}=\bar{h}. \tag{20}\]
Now, the Cauchy-Schwarz inequality implies
\[2\||\nabla u|\|_{L^{2}(G;\mu)}\|u\|_{L^{2}(G;\mu)}\geq 2\int_{G}u|\nabla u|\, \mathrm{d}\mu=\int_{G}|\nabla(u^{2})|\,\mathrm{d}\mu. \tag{21}\]
Using (21) and (20), we obtain
\[\frac{\||\nabla u|\|_{L^{2}(G;\mu)}^{2}}{\|u\|_{L^{2}(G;\mu)}^{2}}\geq\frac{ \big{(}\int_{G}|\nabla(u^{2})|\,\mathrm{d}\mu\big{)}^{2}}{4\|u\|_{L^{2}(G;\mu) }^{4}}=\frac{1}{4}\bar{h}^{2}. \tag{22}\]
We can write \(\||\nabla u||_{L^{2}(G;\mu)}^{2}\) as
\[\||\nabla u||_{L^{2}(G;\mu)}^{2}=\int_{G}(\nabla u\cdot\nabla u)\, \mathrm{d}\mu=\int_{G}\nabla u\cdot(e^{\phi}\nabla u)\,\mathrm{d}V. \tag{23}\]
Applying Green's formula (e.g. [35, Proposition 5.8 and remark after Proposition 5.10]) to \(e^{\phi}u\nabla u\) on \(G\) via a short approximation argument1, recalling (5) and noting that \(u=0\) on \(\partial^{M}G\) and \(\frac{\partial u}{\partial\mathbf{n}}=0\) on \(\partial G\cap\partial M\) (where \(\mathbf{n}\) denotes the outward normal of \(M\)), we obtain
Footnote 1: We apply Green’s formula via an approximation argument, similarly to e.g. [15, pp21–22]. We showed above that \(G_{s}\in\mathscr{P}_{N}(M)\) for almost every \(s\in\operatorname{range}(u^{2}|_{G})\), but it does not follow that \(G\in\mathscr{P}_{N}(M)\), or that \(G\) has locally finite perimeter. Instead, choose some sequence \(s_{1},s_{2},\ldots\in\operatorname{range}(u^{2}|_{G})\) converging to \(0\), such that \(G_{s_{j}}\in\mathscr{P}_{N}(M)\) for each \(j\). Then taking \(u_{j}:=u-s_{j}\) and applying Green’s formula to \(u_{j}e^{\phi}\nabla u_{j}\) on \(G_{s_{j}}\), and recalling (5), yields \(\int_{G_{s_{j}}}\nabla u_{j}\cdot(e^{\phi}\nabla u_{j})\,\mathrm{d}V=-\int_{G _{s_{j}}}u_{j}\cdot\Delta_{\mu}u_{j}\,\mathrm{d}\mu+\int_{(\partial M\,G_{s_{ j}})\cup(\partial M\cap G_{s_{j}})}u_{j}\,\frac{\partial u_{j}}{\partial \mathbf{n}}\,\mathrm{d}\mu_{n-1}\), where \(\mathbf{n}\) is an outward unit normal to \(\partial M\) or \(\partial^{M}G_{s_{j}}\). But \(u_{j}=0\) on \(\partial^{M}G_{s_{j}}\) and \(\frac{\partial u_{j}}{\partial\mathbf{n}}=0\) on \(\partial M\cap G_{s_{j}}\), so the second integral disappears, and taking \(j\to\infty\), we obtain (24).
\[\int_{G}\nabla u\cdot(e^{\phi}\nabla u)\,\mathrm{d}V=-\int_{G}u \cdot\Delta_{\mu}u\,\mathrm{d}\mu+0. \tag{24}\]
Since \(u\cdot\Delta_{\mu}u=\lambda u^{2}\), we have
\[-\int_{G}u\cdot\Delta_{\mu}u\,\mathrm{d}\mu=-\lambda\|u\|_{L^{ 2}(G;\mu)}^{2}. \tag{25}\]
Hence (23)-(25) and (22) imply
\[\lambda=-\frac{\||\nabla u\|_{L^{2}(G;\mu)}^{2}}{\|u\|_{L^{2}(G; \mu)}^{2}}\leq-\frac{1}{4}\bar{h}^{2}. \tag{26}\]
But \(\bar{h}\) is a weighted average over \(s\in\operatorname{range}(u^{2}|_{G})\) of \(\mathcal{J}_{N}(G_{s})\), so the set \(S_{G}^{\prime}:=\{s\in\operatorname{range}(u^{2}|_{G}):\mathcal{J}_{N}(G_{s}) \leq\bar{h}\}\) has positive measure. By (26) and the definition (15), we have \(S_{G}^{\prime}\subseteq S_{G}\), so \(S_{G}\) must also have positive measure.
We can put a lower bound on the measure of \(S_{G}\), as follows. Let \(\mathrm{h}(s):=\mathcal{J}_{N}(G_{s})\). Then we have
\[\int_{S_{G}^{\prime}}(\bar{h}-\mathrm{h}(s))\frac{\mu(G_{s})}{ \|u\|_{L^{2}(G;\mu)}^{2}}\,\mathrm{d}s=\int_{S_{G}^{\prime}}(\bar{h}-\mathrm{h }(s))\,\mathrm{d}\mathbb{P}(s)=\frac{\|\bar{h}-\mathrm{h}\|_{L^{1}( \operatorname{range}(u^{2}|_{G});\mathbb{P})}}{2},\]
and
\[\int_{S_{G}^{\prime}}(\bar{h}-\mathrm{h}(s))\frac{\mu(G_{s})}{ \|u\|_{L^{2}(G;\mu)}^{2}}\,\mathrm{d}s \leq\int_{S_{G}^{\prime}}\frac{\mu(G_{s})}{\|u\|_{L^{2}(G;\mu)}^{ 2}}\,\mathrm{d}s\left(\bar{h}-\inf_{s\in\operatorname{range}(u^{2}|_{G})} \mathrm{h}(s)\right)\] \[\leq\operatorname{Leb}(S_{G}^{\prime})\frac{\mu(G)}{\|u\|_{L^{2}( G;\mu)}^{2}}\left(\bar{h}-\inf_{s\in\operatorname{range}(u^{2}|_{G})}\mathrm{h}(s)\right)\] \[\leq\operatorname{Leb}(S_{G})\frac{\mu(G)}{\|u\|_{L^{2}(G;\mu)}^{ 2}}\left(\bar{h}-\inf_{s\in\operatorname{range}(u^{2}|_{G})}\mathrm{h}(s) \right).\]
Thus, we have
\[\operatorname{Leb}(S_{G})\geq\frac{\|\bar{h}-\mathrm{h}\|_{L^{1}( \operatorname{range}(u^{2}|_{G});\mathbb{P})}\|u\|_{L^{2}(G;\mu)}^{2}}{2(\bar{h}- \inf_{s\in\operatorname{range}(u^{2}|_{G})}\mathrm{h}(s))\mu(G)}. \tag{27}\]
A similar result holds in the Dirichlet case, replacing \(\mathcal{J}_{N}\) with \(\mathcal{J}_{D}\) in the definition of \(\bar{h},\mathrm{h},\mathbb{P}\), and noting that \(\overline{G_{s}}\cap\partial M=0\) for all \(s\neq 0\).
**Corollary 3.3**.: _Let \((M,g,\mu)\) be a weighted manifold. For each Neumann eigenfunction \(u\) corresponding to \(\lambda_{2,N}\), there is a nodal domain \(G\) of \(u\) such that the set \(S_{G}\) defined in (15) has positive measure, and for each \(s\in S_{G}\), defining \(G_{s}\) as in (14), the 2-packing \(\{G_{s},M\backslash\overline{G_{s}}\}\) satisfies_
\[\lambda_{2,N}\leq-\frac{1}{4}\mathcal{J}_{N}(\{G_{s},M\backslash\overline{G_ {s}}\})^{2}. \tag{28}\]
_If \(\partial M\neq\emptyset\), there is a unique Dirichlet eigenfunction \(u\) corresponding to \(\lambda_{1,D}\) (up to scaling), and this \(u\) has only a single nodal domain \(G=M\backslash\partial M\). The set \(S_{G}\) defined in (16) has positive measure, and for each \(s\in S_{G}\), the set \(G_{s}\) defined in (14) satisfies_
\[\lambda_{1,D}\leq-\frac{1}{4}\mathcal{J}_{D}(G_{s})^{2}. \tag{29}\]
Proof.: By Theorem 2.4 and e.g. [41, Propositions 4.5.8-4.5.9], the eigenfunction corresponding to \(\lambda_{1,D}\) has one nodal domain \(G=M\backslash\partial M\), while each eigenfunction corresponding to \(\lambda_{2,N}\) has two nodal domains, and Theorem 3.2 immediately yields (29). In the Neumann case, let \(G\) denote whichever nodal domain of \(u\) satisfies \(\mu(G)\leq\mu(M\backslash\overline{G})\). Then for each \(s\in S_{G}\), the 2-packing \(\{G_{s},M\backslash\overline{G_{s}}\}\) satisfies \(\mathcal{J}_{N}(\{G_{s},M\backslash\overline{G_{s}}\})=\mathcal{J}_{N}(G_{s})\), and Theorem 3.2 yields (28).
### Higher Cheeger inequalities
On boundaryless manifolds, Miclo [53] and Funano [29] have proven Cheeger inequalities for \(h_{k,\emptyset}\) for all \(k\geq 3\). Both papers make use of higher Cheeger inequalities for the graph Laplacian on finite graphs [45], following a procedure outlined by Miclo in [52, Conjecture 13]. Miclo states these results for unweighted manifolds, but notes that they also apply to weighted manifolds with \(C^{\infty}\) measures [53, p.327].
**Theorem 3.4** ([53, Theorem 7]).: _There is a universal constant \(\hat{\eta}>0\) such that, for any boundaryless weighted manifold \((M,g,\mu)\) and for all \(k\geq 1\),_
\[\lambda_{k,\emptyset}\leq-\frac{\hat{\eta}}{k^{6}}h_{k,\emptyset}^{2}. \tag{30}\]
**Theorem 3.5** ([53, Theorem 13]).: _There is a universal constant \(\eta\) such that, for any boundaryless weighted manifold \((M,g,\mu)\) and for all \(k\geq 1\),_
\[\lambda_{2k,\emptyset}\leq-\frac{\eta}{\log(k+1)}h_{k,\emptyset}^{2}. \tag{31}\]
The factor of 2 in the \(\lambda_{2k,\emptyset}\) in the previous theorem is arbitrary. Indeed, one can obtain the following from Miclo's proof of the previous theorem: there is a universal constant \(\tilde{\eta}\) such that, for any boundaryless weighted manifold \((M,g,\mu)\) and for all \(k\geq 1\) and \(0<\delta<1\),
\[\lambda_{k,\emptyset}\leq-\frac{\tilde{\eta}\delta^{6}}{\log(k+1)}h_{\lceil(1 -\delta)k\rceil,\emptyset}^{2}. \tag{32}\]
In particular, taking \(\delta=\frac{1}{2}\), we have
\[\lambda_{2k-1,\emptyset}\leq-\frac{\tilde{\eta}}{64\log(k+1)}h_{k,\emptyset} ^{2}. \tag{33}\]
We have not aware of a closed-form expression for the constants in (30)-(32).
Parini [56, Theorem 5.4] notes that the classical proof of the \(k=2\) Neumann Cheeger inequality (13) extends to the \(k=2\) Dirichlet case. Parini states his inequality for eigenfunctions of the \(p\)-Laplacian for \(1<p<\infty\) on subsets of \(\mathbb{R}^{n}\) with Lipschitz boundary, but the same argument applies on weighted manifolds.
**Theorem 3.6**.: _Let \((M,g,\mu)\) be a weighted manifold. Then_
\[\lambda_{2,D}\leq-\frac{1}{4}h_{2,D}^{2}. \tag{34}\]
Parini's approach does not generalise directly to higher \(k\), since the eigenfunctions corresponding to \(\lambda_{k,N}\) or \(\lambda_{k,D}\) can sometimes have very few nodal domains. Indeed, for any boundaryless \(n\geq 3\)-dimensional manifold and any \(k\geq 1\), there is a metric \(g\) on \(M\) such that the second eigenspace is \(k\)-dimensional [19, p.254], and hence \(\lambda_{k+1,\emptyset}=\lambda_{2,\emptyset}\).
Madafiglio's (unpublished) Honours thesis [50] provides a generalisation of Theorem 3.6. Madafiglio observes that if some eigenfunction with eigenvalue \(\lambda_{k,D}\) has \(r_{k}\geq 2\) nodal domains, then \(\lambda_{k,D}\) gives an upper bound on \(h_{r_{k},D}\). The Neumann case follows by similar reasoning. Using Theorem 3.2, we can obtain a constructive version of Madafiglio's result.
**Theorem 3.7** (Higher Cheeger inequality).: _Let \((M,g,\mu)\) be a weighted manifold. For each \(k\geq 1\), let \(r_{k}\) denote the number of nodal domains in any Neumann (resp. Dirichlet) eigenfunction \(u\) of \(\Delta_{\mu}\) with eigenvalue \(\lambda\geq\lambda_{k,N}\) (resp. \(\lambda\geq\lambda_{k,D}\))._
1. _We have (_ (_36_)_ _due to_ _[_50_]__)_ \[\lambda_{k,N} \leq-\frac{1}{4}h_{r_{k},N}^{2},\] (35) \[\lambda_{k,D} \leq-\frac{1}{4}h_{r_{k},D}^{2}.\] (36)
2. _Let_ \(u\) _be any Neumann (resp. Dirichlet) eigenfunction of_ \(\Delta_{\mu}\) _with_ \(r_{k}\) _nodal domains, and let_ \(G^{1},\ldots,G^{r_{k}}\subset M\) _denote the nodal domains of_ \(u\)_. For each_ \(i\) _and each_ \(s\in\operatorname{range}(u^{2}|_{G^{i}})\)_, let_ \(G^{i}_{s}\) _denote the_ \(s\)_-superlevel set of_ \(u^{2}\) _on_ \(G^{i}\)_, and define_ \(S_{G^{i}}\) _as in (_15_) or (_16_). Then_ \(S_{G^{1}}\times\ldots\times S_{G^{r_{k}}}\) _has positive Lebesgue measure, and for each_ \(\{s_{1},\ldots,s_{r_{k}}\}\in S_{G^{1}}\times\ldots\times S_{G^{r_{k}}}\)_, the collection_ \(\mathcal{A}_{r_{k}}:=\{G^{1}_{s_{1}},\ldots,G^{r_{k}}_{s_{r_{k}}}\}\) _is a Neumann (resp. Dirichlet)_ \(r_{k}\)_-packing of_ \(M\) _satisfying_ \(\lambda_{k,N}\leq-\frac{1}{4}\mathcal{J}_{N}(\mathcal{A}_{r_{k}})^{2}\) _(resp._ \(\lambda_{k,D}\leq-\frac{1}{4}\mathcal{J}_{D}(\mathcal{A}_{r_{k}})^{2}\)_)._
Proof.: The sets \(G^{1}_{s_{1}},\ldots,G^{r_{k}}_{s_{r_{k}}}\) for each \(\{s_{1},\ldots,s_{r_{k}}\}\in S_{G^{1}},\ldots,S_{G^{r_{k}}}\) are pairwise disjoint, since \(G^{1},\ldots,G^{r_{k}}\) are pairwise disjoint, and each \(G^{i}_{s_{i}}\in\mathscr{P}_{N}(M)\) (resp. \(G^{i}_{s_{i}}\in\mathscr{P}_{D}(M)\)) by the definitions (15)-(16). Hence \(\mathcal{A}_{r_{k}}:=\{G^{1}_{s_{1}},\ldots,G^{r_{k}}_{s_{r_{k}}}\}\) is a Neumann \(r_{k}\)-packing for \(M\) satisfying \(\lambda=-\frac{1}{4}\mathcal{J}_{N}(\mathcal{A}_{r_{k}})^{2}\) (resp. a Dirichlet \(r_{k}\)-packing for \(M\) satisfying \(\lambda\leq-\frac{1}{4}\mathcal{J}_{D}(\mathcal{A}_{r_{k}})^{2}\)), and (35) (resp. (36)) follows immediately.
We can rewrite part 1 of Theorem 3.7 as follows: for \(k\geq 1\), let \(\tilde{r}_{k}\) be the index of a Neumann (resp. Dirichlet) eigenfunction of \(\Delta_{\mu}\) with \(\geq k\) nodal domains, when the eigenfunctions are ordered by decreasing eigenvalue. Then
\[\lambda_{\tilde{r}_{k},N}\leq-\frac{1}{4}h_{k,N}^{2} \tag{37}\]
and
\[\lambda_{\tilde{r}_{k},D}\leq-\frac{1}{4}h_{k,D}^{2}, \tag{38}\]
respectively. We can rewrite equations (75)-(76) similarly.
Theorem 3.7 is intended for situations where an eigenfunction of \(\Delta_{\mu}\) has been calculated explicitly, so that the number of nodal domains can be identified. In these cases, Theorem 3.7 has the twin advantages that it applies to manifolds with boundary, and that the constant in (35) is explicit and small. This allows relatively tight bounds on \(h_{k,N}\) or \(h_{k,D}\) to be computed be computed even for large \(k\), particularly when \(\tilde{r}_{k}\) is close to \(k\).
### Creating more nodal domains using linear combinations of eigenfunctions
Theorem 3.7 only allows us to obtain one feature from each of the nodal domains of a single eigenfunction of \(\Delta_{\mu}\). Sometimes, there are \(l\leq k\) features of interest which appear spread among the first \(k\) eigenfunctions, but no single eigenfunction has all \(l\) features appearing in separate nodal domains. One may be able to extract these \(l\) features and obtain a corresponding bound on \(h_{l,N}\) or \(h_{l,D}\), by applying an operator known as _soft thresholding_ to certain linear combinations of the first \(k\) eigenfunctions. Soft thresholding with parameter \(a>0\) is the map \(\tau_{a}:C^{0}(M)\to C^{0}(M)\), \(\tau_{a}(f)(p):=\operatorname{sign}(f(p))\max\{|f(p)|-a,0\}\). Soft thresholding does not increase \(W^{1,2}\)-norm, and is support-decreasing, in the sense that if \(f^{-1}(0)\not\in\{\emptyset,M\}\), then \(\operatorname{supp}(\tau_{a}(f))\subsetneq\operatorname{supp}(f)\). For some manifolds, there are parameters \(\alpha:=\{\alpha_{ij}:1\leq i\leq l,1\leq j\leq k\}\) for which the \(l\) linear combinations \(f_{i;\alpha}:=\sum_{j=1}^{k}\alpha_{ij}u_{j}\), \(i=1,\ldots,l\) of the first \(k\) (Neumann or Dirichlet) eigenfunctions of \(\Delta_{\mu}\) are \(L^{2}\)-close to a collection of \(l\) functions with pairwise disjoint supports [23, Theorem 19]. When the eigenfunctions can be computed or approximated explicitly, the parameters \(\alpha\) can be chosen using an algorithm such as _sparse eigenbasis approximation_[28], as discussed after the proof of Proposition 3.8. Each \(f_{i;\alpha}\) has support covering all of \(M\), as a consequence of the unique continuation theorem [2]2, but the thresholded functions \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\) may have pairwise disjoint supports. Increasing \(a\) decreases the supports of \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\), so one chooses \(a\) as small as required to achieve pairwise disjoint supports for \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\). In Proposition 3.8 below, we give upper bounds on \(h_{l,N}\) or \(h_{l,D}\), and prove that some of the level sets of \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\) yield Cheeger \(l\)-packings whose Cheeger ratios are bounded above, in terms of \(\lambda_{k,N}\) or \(\lambda_{k,D}\) and the proportion of mass lost in the thresholding step. We illustrate Proposition 3.8 in example 1.
Footnote 2: The function \(f_{i,\alpha}\) satisfies \(|\Delta_{\mu}f_{i,\alpha}|\leq|\lambda_{k,N}||f_{i,\alpha}|\) or \(|\Delta_{\mu}f_{i,\alpha}|\leq|\lambda_{k,D}||f_{i,\alpha}|\), so the main theorem of [2] implies that if \(f_{i,\alpha}\) cannot be zero in an open neighbourhood unless it is zero everywhere.
**Proposition 3.8**.: _For any weighted manifold \((M,g,\mu)\), let \(u_{1},\ldots,u_{k}\) denote the first \(k\) Neumann, resp. Dirichlet, eigenfunctions of \(\Delta_{\mu}\) on \(M\) for \(k\geq 1\). For any \(1\leq l\leq k\) and any \(\alpha\in\mathbb{R}^{l\times k}\), define \(f_{1,\alpha},\ldots,f_{l,\alpha}\) by \(f_{i,\alpha}:=\sum_{j=1}^{k}\alpha_{ij}u_{j}\). Suppose that for some \(a>0\), the functions \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\) are nonzero and have pairwise disjoint supports. Then each \(\tau_{a}(f_{i,\alpha})\) has a nodal domain \(\tilde{G}^{i}\) such that letting \(\tilde{G}^{i}_{s}\) for \(s\in\operatorname{range}(\tau_{a}(f_{i,\alpha})^{2}|_{\tilde{G}^{i}})\) denote the \(s\)-superlevel set of \(\tau_{a}(f_{i,\alpha})^{2}\) on \(\tilde{G}_{i}\), the set_
\[\tilde{S}_{\tilde{G}^{i}}:=\Big{\{}s\in\operatorname{range}(\tau_{a}(f_{i, \alpha})^{2}|_{\tilde{G}^{i}}):\tilde{G}^{i}_{s}\in\mathscr{P}_{N}(M),\frac{ \||\nabla\tau_{a}(f_{i,\alpha})|\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}{\|\tau_{a}( f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}\geq\frac{1}{4}\mathcal{J}_{N}( \tilde{G}^{i}_{s})^{2}\Big{\}}, \tag{39}\]
_resp._
\[\tilde{S}_{\tilde{G}^{i}}:=\Big{\{}s\in\operatorname{range}(\tau_{a}(f_{i, \alpha})^{2}|_{\tilde{G}^{i}}):\tilde{G}^{i}_{s}\in\mathscr{P}_{D}(M),\frac{ \||\nabla\tau_{a}(f_{i,\alpha})|\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}{\|\tau_{a}( f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}\geq\frac{1}{4}\mathcal{J}_{D}( \tilde{G}^{i}_{s})^{2}\Big{\}}, \tag{40}\]
_has positive measure and satisfies (45). Moreover, for each \(\{s_{1},\ldots,s_{l}\}\in\tilde{S}_{\tilde{G}^{1}}\times\ldots\times\tilde{S}_{ \tilde{G}^{l}}\), the collection \(\mathcal{A}_{l}:=\{\tilde{G}^{1}_{s_{1}},\ldots,\tilde{G}^{l}_{s_{l}}\}\) is a Neumann \(l\)-packing for \(M\) satisfying_
\[\lambda_{k,N}\leq-\frac{1}{4}\mathcal{J}_{N}(\mathcal{A}_{l})^{2}\max_{1\leq j \leq l}\frac{\|\tau_{a}(f_{j,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|f_{j,\alpha}\|^{2} _{L^{2}(M;\mu)}}\leq-\frac{1}{4}h^{2}_{l,N}\max_{1\leq j\leq l}\frac{\|\tau_{a }(f_{j,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|f_{j,\alpha}\|^{2}_{L^{2}(M;\mu)}}, \tag{41}\]
_resp. a Dirichlet \(l\)-packing for \(M\) satisfying_
\[\lambda_{k,D}\leq-\frac{1}{4}\mathcal{J}_{D}(\mathcal{A}_{l})^{2}\max_{1\leq j \leq l}\frac{\|\tau_{a}(f_{j,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|f_{j,\alpha}\|^{2} _{L^{2}(M;\mu)}}\leq-\frac{1}{4}h^{2}_{l,D}\max_{1\leq j\leq l}\frac{\|\tau_{a }(f_{j,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|f_{j,\alpha}\|^{2}_{L^{2}(M;\mu)}}. \tag{42}\]
**Example 1**.: Let \((M,g,\mu)\) denote the interval \([0,\pi]\) equipped with Euclidean distance and Lebesgue measure Leb, and let \(u_{1},u_{2},u_{3}\) denote the first three Dirichlet eigenfunctions of \(\Delta\) on \([0,\pi]\) (shown in Figure 2(a)). Using sparse eigenbasis approximation [28, Algorithm 3.1], we take \(\alpha:=\left(\begin{smallmatrix}0.77&0&-0.64\\ 0.45&-0.71&0.54\\ 0.45&0.71&0.54\\ \end{smallmatrix}\right)\). Then the linear combinations \(f_{1,\alpha}:=\sum_{j=1}^{3}\alpha_{ij}u_{j}\), \(j=1,2,3\) of \(u_{1},u_{2},u_{3}\) (shown in Figure 2(b)) are \(L^{2}\)-close to disjointly supported functions. Applying soft thresholding \(\tau_{a}\) with \(a:=0.84\) yields pairwise disjointly supported functions \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{3,\alpha})\) (shown in Figure 2(c)).
Each \(\tau_{a}(f_{i,\alpha})\) has a single nodal domain \(\tilde{G}^{i}\), and the corresponding positive-measure intervals \(\tilde{S}_{\tilde{G}^{i}}\) are given by \(\tilde{S}_{\tilde{G}^{1}}\approx(0,0.51]\), \(\tilde{S}_{\tilde{G}^{2}}=\tilde{S}_{\tilde{G}^{3}}\approx(0,0.55]\) (to two decimal places). We show some of the sets \(\tilde{G}^{i}_{s_{i}}\) for \(s_{i}\in\tilde{S}_{\tilde{Q}^{i}}\), \(i=1,2,3\), in Figure 2(d). Proposition 3.8 guarantees that each \(\tilde{S}_{\tilde{G}^{i}}\) has positive measure, and that each \(\mathcal{A}_{3}:=\{\tilde{G}^{1}_{s_{1}},\tilde{G}^{2}_{s_{2}},\tilde{G}^{3} _{s_{3}}\}\) for \(\{s_{1},s_{2},s_{3}\}\in\tilde{S}_{\tilde{G}^{1}}\times\tilde{S}_{\tilde{G}^ {2}}\times\tilde{S}_{\tilde{G}^{3}}\) satisfies \(\mathcal{J}_{D}(\mathcal{A}_{3})\leq 2\sqrt{-\lambda_{3,D}\frac{\|f_{3,\alpha}\|_{L^{2}( 0,\pi;L_{\mathrm{eb}})}}{\|\tau_{a}(f_{3,\alpha})\|_{L^{2}(0,\pi;L_{\mathrm{eb }})}}}=7.3\). Some choices for \(\{s_{1},s_{2},s_{3}\}\) give rise to packings \(\mathcal{A}_{3}\) with Cheeger ratios significantly smaller than this upper bound. Note that for this example, \(u_{3}\) already has \(3\) nodal domains, so we could use Theorem 3.7 to obtain a \(3\)-packing instead.
Proof.: We consider only the Neumann case; the proof of the Dirichlet case is similar. For each \(1\leq i\leq l\), let \(\tilde{G}^{i}:=\arg\min_{\tilde{G}}\frac{\||\nabla\tau_{a}(f_{i,\alpha})||^{2} _{L^{2}(\tilde{G},\mu)}}{\|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G},\mu)}}\), where the minimum is taken over nodal domains \(\tilde{G}\) of \(\tau_{a}(f_{i,\alpha})\). The level sets of \(\tau_{a}(f_{i,\alpha})\), other than \((\tau_{a}(f_{i,\alpha}))^{-1}(0)\), are level sets of \(f_{i,\alpha}\in C^{\infty}(M)\), so \(\tilde{G}^{i}_{s_{i}}\in\mathscr{P}_{N}(M)\) for almost every \(s\in\operatorname{range}(\tau_{a}(f_{i,\alpha})^{2}|_{\tilde{G}^{i}})\) by the reasoning after (17). By applying the reasoning from Theorem 3.2 ((17)-(22) and after (26)) to \(\tau_{a}(f_{i,\alpha})\) on \(\tilde{G}^{i}\), it follows immediately that \(\tilde{S}_{\tilde{G}^{i}}\) has positive measure satisfying (45) below,
and that \(\{\tilde{G}^{1}_{s_{1}},\ldots,\tilde{G}^{l}_{s_{l}}\}\in\mathscr{P}_{l,N}(M)\) for each \(\{s_{1},\ldots,s_{l}\}\in\tilde{S}_{\tilde{G}^{1}}\times\ldots\times\tilde{S}_{ \tilde{G}^{l}}\).
We now proceed to prove (41). Choose any \(i\in\{1,\ldots,l\}\). Note that for each nodal domain \(\tilde{G}\) of \(\tau_{a}(f_{i,\alpha})\), we have \(\||\nabla\tau_{a}(f_{i,\alpha})||^{2}_{L^{2}(\tilde{G};\mu)}\geq\|\tau_{a}(f_ {i,\alpha})\|^{2}_{L^{2}(\tilde{G};\mu)}\frac{\||\nabla\tau_{a}(f_{i,\alpha}) \|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}{\|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}( \tilde{G}^{i};\mu)}}\). Hence
\[\frac{\||\nabla\tau_{a}(f_{i,\alpha})||^{2}_{L^{2}(M;\mu)}}{\|\tau_{a}(f_{i, \alpha})\|^{2}_{L^{2}(M;\mu)}}=\frac{\sum_{\tilde{G}}\||\nabla\tau_{a}(f_{i, \alpha})\|^{2}_{L^{2}(\tilde{G};\mu)}}{\sum_{\tilde{G}}\|\tau_{a}(f_{i,\alpha })\|^{2}_{L^{2}(\tilde{G};\mu)}}\geq\frac{\||\nabla\tau_{a}(f_{i,\alpha})||^{2} _{L^{2}(\tilde{G}^{i};\mu)}}{\|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{ i};\mu)}}. \tag{43}\]
Recalling that \(f_{i,\alpha}=\sum_{j=1}^{k}\alpha_{ij}u_{j}\), we have that \(\lambda_{k,N}\leq-\frac{\||\nabla f_{i,\alpha}\|^{2}_{L^{2}(M;\mu)}}{\|f_{i, \alpha}\|^{2}_{L^{2}(M;\mu)}}\) (by e.g. [41, first equation of Proposition 4.5.4], which extends directly to the weighted case). Hence, since \(\||\nabla\tau_{a}(f_{i,\alpha})|\|_{L^{2}(M)}\leq\||\nabla f_{i,\alpha}|\|_{L ^{2}(M)}\), (43) implies
\[\lambda_{k,N}\leq-\frac{\||\nabla f_{i,\alpha}|\|^{2}_{L^{2}(M; \mu)}}{\|f_{i,\alpha}\|^{2}_{L^{2}(M;\mu)}}\leq -\frac{\||\nabla\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|\tau_{a}(f _{i,\alpha})\|^{2}_{L^{2}(M;\mu)}}\frac{\|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}( M;\mu)}}{\|f_{i,\alpha}\|^{2}_{L^{2}(M;\mu)}}\] \[\leq -\frac{\||\nabla\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i} ;\mu)}}{\|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}\frac{\|\tau_ {a}(f_{i,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|f_{i,\alpha}\|^{2}_{L^{2}(M;\mu)}}. \tag{44}\]
Hence, by the definition of \(\tilde{S}_{\tilde{G}^{i}}\) in the proposition statement, for each \(s_{i}\in\tilde{S}_{\tilde{G}^{i}}\), we have
\[\lambda_{k,N}\leq-\frac{1}{4}\mathcal{J}_{N}(\tilde{G}^{i}_{s_{i}})^{2}\frac{ \|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(M;\mu)}}{\|f_{i,\alpha}\|^{2}_{L^{2}(M; \mu)}}.\]
Applying this reasoning for each \(i\in\{1,\ldots,k\}\) and recalling that \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\) have pairwise disjoint supports yields (41).
Lastly, we state our lower bound on the measure of each \(\tilde{S}_{\tilde{G}^{i}}\). Similarly to Theorem 3.2, we define \(\overline{\tilde{h}_{i}}:=\frac{1}{\|\tau_{a}(f_{i,\alpha})\|_{L^{2}(\tilde{G} ^{i})^{2}}}\int_{\operatorname{range}\left(\tau_{a}(f_{i,\alpha})^{2}|_{\tilde {G}^{i}}\right)}\mathcal{J}_{N}(\tilde{G}^{i}_{s})\mu(\tilde{G}^{i}_{s})\, \mathrm{d}s\) and \(\tilde{\mathrm{h}}_{i}(s):=\mathcal{J}_{N}(\tilde{G}^{i}_{s})\), and we define the probability measure \(\tilde{\mathbb{P}}_{i}\) on \(\operatorname{range}(\tau_{a}(f_{i,\alpha})^{2}|_{\tilde{G}^{i}})\subset \mathbb{R}\) by \(\tilde{\mathbb{P}}_{i}(L):=\int_{L}\frac{\mu(\tilde{G}^{i}_{s})}{\|\tau_{a}(f _{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}\,\mathrm{d}s\). Then the reasoning for (27) implies
\[\operatorname{Leb}(\tilde{S}_{\tilde{G}^{i}})\geq\frac{\left\|\overline{\tilde{h }_{i}}-\tilde{\mathrm{h}}_{i}\right\|_{L^{1}\left(\operatorname{range}\left( \tau_{a}(f_{i,\alpha})^{2}|_{\tilde{G}^{i}}\right);\tilde{\mathbb{P}}_{i}\right)} \|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu)}}{2\Big{(}\overline{ \tilde{h}_{i}}-\inf_{s\in\operatorname{range}\left(\tau_{a}(f_{i,\alpha})^{2}|_ {\tilde{G}^{i}}\right)}\tilde{\mathrm{h}}_{i}(s)\Big{)}\mu(\tilde{G}^{i})}. \tag{45}\]
A similar result applies in the Dirichlet case, replacing \(\mathcal{J}_{N}\) with \(\mathcal{J}_{D}\) in the definitions of \(\overline{\tilde{h}_{i}},\tilde{\mathrm{h}}_{i},\tilde{\mathbb{P}}_{i}\).
In numerical calculations, the \(\alpha_{ij}\) in Proposition 3.8 can be readily computed using the sparse eigenbasis approximation algorithm of [28, Algorithm 3.1]. The orthogonal matrix \(R\) produced by that algorithm can be used as the matrix \(\alpha\). The resulting \(f_{1,\alpha},\ldots,f_{k,\alpha}\) form an orthogonal basis for \(\operatorname{span}\{u_{1},\ldots,u_{k}\}\), such that for some fixed \(a>0\), each \(\tau_{a}(f_{i,\alpha})\) (defined before Proposition 3.8) is _sparse_, i.e. each \(\operatorname{supp}(\tau_{a}(f_{i,\alpha}))\) is small. Using a larger \(a^{\prime}\geq a\) will create further support reductions.
### The dynamic Cheeger inequalities
#### 3.4.1 Preliminaries on higher dynamic Cheeger constants and dynamic Laplacian eigenvalues
We can generalise Theorems 3.4-3.7 into the setting of non-autonomous, advective dynamical systems. Many fluidic and geophysical flows can be modeled using purely advective dynamics. Such flows can be represented as a collection of time-indexed diffeomorphisms acting on an initial-time manifold, where each
diffeomorphism sends a point in the initial-time manifold to its position at the corresponding future time. These diffeomorphisms are physically meaningful, because they describe the fluid motion and evolve subsets of the initial-time manifold according to this motion.
The global behaviour of many fluidic and geophysical flows can be understood by separating the phase space (the physical space containing the fluid) into _coherent sets_[25], i.e. regions that are "as dynamically disconnected as possible" [26]. One approach in purely advective, finite-time nonautonomous systems is to identify subsets of the phase space whose boundary measures remain small over time, relative to the measures of those subsets. These volume ratios are known as _dynamic Cheeger ratios_[25, 26, 27], and sets which locally minimise this ratio are known as _coherent sets_. The infima of these ratios are known as the _dynamic Cheeger constants_[25, 26, 27]. The dynamic Cheeger constants generalise the (static) Cheeger constants of Definition 2.2.
Calculating a dynamic Cheeger constant exactly is generally impractical. Instead, approximate coherent sets can be obtained from the eigenfunctions of a specific weighted Laplace-Beltrami operator called the _dynamic Laplacian_. There are existing upper bounds on the first non-zero dynamic Cheeger constant in terms of the first non-zero eigenvalue of the dynamic Laplacian [25, 26, 27].
In practice, the higher eigenfunctions of the dynamic Laplacian reveal additional coherent sets (see e.g. [28]). Below, we introduce higher dynamic Cheeger constants, analogous to the (static) higher Cheeger constants of Definition 2.2, to quantify these additional coherent sets. We show that the higher dynamic Cheeger constants are bounded above by the eigenvalues of \(\Delta^{d}\) (Theorems 3.17, 3.18 and 3.19), and in particular that the eigenfunctions of \(\Delta^{d}\) reveal coherent sets whose dynamic Cheeger ratios are bounded above (Theorems 3.19 and 3.20).
**Definition 3.9**.: A _dynamical system_\(\mathcal{T}:=(\mathrm{T},\{(M_{t},g_{t})\}_{t\in\mathrm{T}},\{\Phi^{(t)}\}_{t \in\mathrm{T}})\) or \(\mathcal{T}:=(\mathrm{T},\{(M_{t},g_{t},\mu_{t})\}_{t\in\mathrm{T}},\)\(\{\Phi^{(t)}\}_{t\in\mathrm{T}})\) consists of the following:
* A time index set \(\mathrm{T}:=\{0,1,\ldots,t_{\max}\}\).
* A time-indexed family of Riemannian manifolds \(\{(M_{t},g_{t})\}_{t\in\mathrm{T}}\) or weighted manifolds \(\{(M_{t},g_{t},\mu_{t})\}_{t\in\mathrm{T}}\), where in the unweighted case, for \(t\in\mathrm{T}\) we take \(\mu_{t}\) to denote Riemannian volume on \(M_{t}\).
* A time-indexed family of \(C^{\infty}\) diffeomorphisms \(\{\Phi^{(t)}\}_{t\in\mathrm{T}}\), which are _measure-preserving_ in the sense \(\mu_{t}=\mu_{0}\circ(\Phi^{(t)})^{-1}\) (we call such \(\Phi^{(t)}\)_volume-preserving_ if each \(\mu_{t}\) is Riemannian volume).
We use the following notation. Since \(\Phi^{(t)}\) for \(t\in\mathrm{T}\) is a measure-preserving diffeomorphism, the _push-forward_\(\Phi^{(t)}_{*}:C^{\infty}(M_{0})\to C^{\infty}(M_{t})\) is given by \(\Phi^{(t)}_{*}f:=f\circ(\Phi^{(t)})^{-1}\), and the _pullback_\((\Phi^{(t)})^{*}:C^{\infty}(M_{t})\to C^{\infty}(M_{0})\) is given by \((\Phi^{(t)})^{*}f:=f\circ\Phi^{(t)}\). We also define the pullback Riemannian metric \((\Phi^{(t)})^{*}g_{t}\) given by \((\Phi^{(t)})^{*}g_{t}:=g_{t}(\mathrm{d}\Phi^{(t)}\,\cdot\,\mathrm{d}\Phi^{(t) }\,\cdot\,)\), where \(\mathrm{d}\Phi^{(t)}\) is the differential of \(\Phi^{(t)}\) (see e.g. [44, p.55]). For \(t\in\mathrm{T}\), we let \((\mu_{t})_{n-1}\) denote the \(n-1\)-dimensional Hausdorff measure on \(M_{t}\) constructed from \(\mu_{t}\) and \(g_{t}\). For \(s,s+t\in\mathrm{T}\), we write \(\Phi^{(t)}_{s}:=\Phi^{(s+t)}\circ(\Phi^{(s)})^{-1}\).
We define the higher dynamic Cheeger constants as follows.
**Definition 3.10** (Higher dynamic Cheeger constants).: Consider a dynamical system \(\mathcal{T}\). For \(k\geq 1\), the _dynamic Neumann Cheeger ratio_ of a \(k\)-packing \(\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k,D}(M_{0})\) is
\[\mathcal{J}^{d}_{N}(\{A_{1},\ldots,A_{k}\}):=\max_{1\leq i\leq k}\frac{\sum_{t= 0}^{t_{\max}}(\mu_{t})_{n-1}(\Phi^{(t)}(\partial^{M_{0}}A_{i}))}{|\mathrm{T} |\mu_{0}(A_{i})}. \tag{46}\]
The _dynamic Dirichlet Cheeger ratio_ of a Dirichlet \(k\)-packing \(\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k,D}(M_{0})\) is
\[\mathcal{J}^{d}_{D}(\{A_{1},\ldots,A_{k}\}):=\max_{1\leq i\leq k}\frac{\sum_{t= 0}^{t_{\max}}(\mu_{t})_{n-1}(\Phi^{(t)}(\partial A_{i}))}{|\mathrm{T}|\mu_{0} (A_{i})}. \tag{47}\]
The \(k\)_th dynamic Neumann_ and _dynamic Dirichlet Cheeger constants_ for \(\mathcal{T}\) are
\[h^{d}_{k,N} :=\inf_{\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k}(M_{0})}\mathcal{J }^{d}_{N}(\{A_{1},\ldots,A_{k}\}) \tag{48}\] \[h^{d}_{k,D} :=\inf_{\{A_{1},\ldots,A_{k}\}\in\mathscr{P}_{k}(M_{0})}\mathcal{J }^{d}_{D}(\{A_{1},\ldots,A_{k}\}). \tag{49}\]
For \(A\in\mathscr{P}_{N}(M_{0})\), resp. \(A\in\mathscr{P}_{D}(M_{0})\), we will occasionally write \(\mathcal{J}_{N}^{d}(A)\) instead of \(\mathcal{J}_{N}^{d}(\{A\})\), resp. \(\mathcal{J}_{D}^{d}(A)\) instead of \(\mathcal{J}_{D}^{d}(\{A\})\), for convenience.
The Neumann dynamic Cheeger constant \(h_{2,N}^{d}\) was originally defined requiring \(A_{1}\) and \(A_{2}\) to partition \(M_{0}\)[25], whereas (48) only requires them to form a packing of \(M_{0}\). This does not change the value of \(h_{2,N}^{d}\), by the reasoning after definition 2.2. Note that since the \(\Phi^{(t)}\) are measure-preserving, we have \(|\mathrm{T}|\mu_{0}(A_{i})=\sum_{t=0}^{t_{\mathrm{max}}}\mu_{t}(\Phi^{(t)}(A_ {i}))\), i.e. the denominators in (46)-(47) are \(|\mathrm{T}|\) times the time averages of the measures of the \(A_{i}\).
When considering dynamical systems, we let \(\Delta_{g_{t},\mu_{t}}\) denote the weighted Laplace-Beltrami operator on \((M_{t},g_{t},\mu_{t})\). The dynamic Laplacian [25, 27] is
\[\Delta^{d}:=\frac{1}{|\mathrm{T}|}\sum_{t=0}^{t_{\mathrm{max}}}(\Phi^{(t)})^{ *}\Delta_{g_{t},\mu_{t}}\Phi_{*}^{(t)}. \tag{50}\]
We consider Dirichlet and dynamic Neumann eigenproblems for \(\Delta^{d}\). The dynamic Neumann eigenproblem is to find \(u\in C^{\infty}(M_{0})\) and \(\lambda\in\mathbb{R}\), such that
\[\Delta^{d}u=\lambda u, \tag{51}\]
subject to the _dynamic Neumann boundary condition_ (if \(\partial M_{0}\neq\emptyset\))
\[\frac{1}{|\mathrm{T}|}\sum_{t=0}^{t_{\mathrm{max}}}\frac{\partial}{\partial \mathbf{n}_{t}}\left((\Phi^{(t)})_{*}u\right)=0\quad\text{on }\partial M_{0}, \tag{52}\]
where \(\mathbf{n}_{t}\) denotes an outward unit normal vector to \(\partial M_{t}\)[25, Theorem 4.1][27, Theorem 4.4]. Dynamic Neumann boundary conditions are the natural boundary condition as discussed in [25, pp.9-10] and [27, p.16]. There is an orthogonal Schauder basis for \(L^{2}(M_{0},\mu_{0})\) consisting of eigenfunctions for (51) satisfying (52) [27, Theorem 4.4]. The corresponding eigenvalues form a non-positive decreasing sequence accumulating only at \(-\infty\), and we denote them \(0=\lambda_{1,N}^{d}>\lambda_{2,N}^{d}\geq\lambda_{3,N}^{d}\geq\ldots\).
The Dirichlet eigenproblem is to find \(u\in C^{\infty}(M_{0})\) and \(\lambda\in\mathbb{R}\) satisfying (51), subject to
\[u=0\quad\text{on }\partial M_{0}. \tag{53}\]
By standard variational arguments as in e.g. [41, Theorem 4.3.1] and elliptic regularity theorems as in [30, Theorem 8.14], there is an orthogonal Schauder basis for \(L^{2}(M_{0},\mu_{0})\) of \(C^{\infty}(M_{0})\) eigenfunctions for (51) satisfying (53). The corresponding eigenvalues form a negative decreasing sequence accumulating only at \(-\infty\), and we denote them \(0>\lambda_{1,D}^{d}>\lambda_{2,D}^{d}\geq\lambda_{3,D}^{d}\geq\ldots\).
We have the following variational formula for the eigenvalues, in the dynamic Neumann setting [27].
**Proposition 3.11**.: _Let \(\mathcal{T}\) be a dynamical system, and let \(u_{1}^{d},u_{2}^{d},\ldots\) denote a complete orthogonal basis of dynamic Neumann eigenfunctions of \(\Delta^{d}\) corresponding to \(\lambda_{1,N}^{d},\lambda_{2,N}^{d},\ldots\) (resp. \(\lambda_{1,D}^{d},\lambda_{2,D}^{d},\ldots\)). Then for each \(k\geq 1\), we have_
\[\lambda_{k,N}^{d}=-\inf_{\begin{subarray}{c}f\in W^{1,2}(M_{0})\\ \int_{M_{0}}u_{i}^{d}\int\mathrm{d}\mu_{0}=0,\forall i\in\{1,\ldots,k-1\} \end{subarray}}\frac{\sum_{t=0}^{t_{\mathrm{max}}}\||\nabla_{g_{t}}\Phi_{*}^{(t )}f||_{L^{2}(M_{i};\mu_{i})}^{2}}{|\mathrm{T}|\|f\|_{L^{2}(M_{0};\mu_{0})}^{2}}, \tag{54}\]
_and the infimum is attained when \(f\) is a dynamic Neumann eigenfunction of \(\Delta^{d}\) with eigenvalue \(\lambda_{k,N}^{d}\)._
Extending the reasoning in e.g. [15, pp.16-17] to the dynamic case yields that the infimum in (54) is attained if and only if \(f\) is a dynamic Neumann eigenfunction of \(\Delta^{d}\) with eigenvalue \(\lambda_{k,N}^{d}\). This proposition also extends directly to the Dirichlet case, by similar arguments. Let \(u_{1}^{d},u_{2}^{d},\ldots\) denote a complete orthogonal
basis of Dirichlet eigenfunctions of \(\Delta^{d}\) corresponding to \(\lambda_{1,N}^{d},\lambda_{2,N}^{d},\ldots\) (resp. \(\lambda_{1,D}^{d},\lambda_{2,D}^{d},\ldots\)). Then for each \(k\geq 1\), we have
\[\lambda_{k,D}^{d}=-\inf_{\begin{subarray}{c}f\in W_{0}^{1,2}(M_{0})\\ \int_{M_{0}}u_{i}^{d}f\,\mathrm{d}\mu_{0}=0,\forall i\in\{1,\ldots,k-1\} \end{subarray}}\frac{\sum_{t=0}^{t_{\max}}\||\nabla_{g_{t}}\Phi_{*}^{(t)}f||_{ L^{2}(M_{t};\mu_{t})}^{2}}{|\Gamma|\|f\|_{L^{2}(M_{0};\mu_{0})}^{2}}, \tag{55}\]
and the infimum is attained if and only if \(f\) is a Dirichlet eigenfunction of \(\Delta^{d}\) with eigenvalue \(\lambda_{k,D}^{d}\). Since \(\Delta^{d}\) is an elliptic operator, Courant's nodal domain theorem (Theorem 2.4) extends to the eigenfunctions of \(\Delta^{d}\).
**Corollary 3.12** (to Theorem 2.4).: _For any dynamical system \(\mathcal{T}\), the \(k\)th dynamic Neumann (resp. Dirichlet) eigenfunction \(u_{k}\) of \(\Delta^{d}\) has at most \(k\) nodal domains._
Proof.: The proof is the same as that for Theorem 2.4, replacing \(M\), \(\mu\) and \(\lambda_{k,N}\) with \(M_{0}\), \(\mu_{0}\) and \(\lambda_{k,N}^{d}\), replacing the Rayleigh quotients as in Theorem 2.3 with dynamic Rayleigh quotients as in Proposition 3.11, and replacing (23)-(25) with (66) and the reasoning used to obtain (68).
The operator \(\Delta^{d}\) can be expressed as the weighted Laplace-Beltrami operator \(\Delta_{\bar{g},\mu_{0}}\) on \((M_{0},\bar{g},\mu_{0})\), where \(\bar{g}\) (called the _geometry of mixing metric_[38]) is the 'harmonic mean'3 of the pullbacks \((\Phi^{(t)})^{*}g_{t}\) of the metrics \(g_{t}\) to the initial-time manifold \(M_{0}\)[38]. Note that even if each \(\mu_{t}\) is Riemannian volume on \((M_{t},g_{t})\), \(\mu_{0}\) is not necessarily Riemannian volume on \((M_{0},\bar{g})\)[38, section 4.1.3].
Footnote 3: \(\bar{g}\) is defined via the _inverse metric_. The inverse metric of a Riemannian metric \(g\) on \(M_{0}\) is given by \(g^{-1}:T^{*}M_{0}\times T^{*}M_{0}\to\mathbb{R}\), \(g^{-1}(\eta,\omega):=g(\eta^{\sharp},\omega^{\sharp})\), where \(\sharp\) denotes raising an index (see e.g. [44, p.342]). Then \(\bar{g}\) is the unique metric on \(M_{0}\) for which \(\bar{g}^{-1}(\eta,\omega):=\frac{1}{|\Gamma|}\sum_{t=0}^{t_{\max}}((\Phi^{(t) })^{*}g_{t})^{-1}(\eta,\omega)\).
**Proposition 3.13** ([38, pp.1864, 1875]).: _In any dynamical system, \(\Delta^{d}\) is the weighted Laplace-Beltrami operator for the Riemannian manifold \((M_{0},\bar{g},\mu_{0})\), i.e._
\[\Delta^{d}=\Delta_{\bar{g},\mu_{0}}. \tag{56}\]
For any dynamical system \(\mathcal{T}\), let \(\nabla_{g_{t}}\) and \(\nabla_{\bar{g}}\) denote the gradient operator for the time-\(t\) manifold \((M_{t},g_{t},\mu_{t})\) and the geometry of mixing manifold \((M_{0},\bar{g},\mu_{0})\), respectively. It follows immediately from the definition of \(\bar{g}\) that \(|\nabla_{\bar{g}}f|^{2}=\frac{1}{|\Gamma|}\sum_{t=0}^{t_{\max}}\bigl{|}\nabla_ {g_{t}}\Phi_{*}^{(t)}f\bigr{|}^{2}\) for \(f\in W^{1,2}(M_{0})\). The Neumann boundary condition for the geometry of mixing manifold is the same as the dynamic Neumann boundary condition [38, p.1864]. For \(A\in\mathscr{P}_{N}(M_{0})\) or \(A\in\mathscr{P}_{D}(M_{0})\), respectively, we denote the (Neumann or Dirichlet) Cheeger ratio of \(A\) on the geometry of mixing manifold by \(\mathcal{J}_{N}(A;\bar{g},\mu_{0})\) or \(\mathcal{J}_{D}(A;\bar{g},\mu_{0})\), respectively. Then \(\mathcal{J}_{N}(\cdot;\bar{g},\mu)\) and \(\mathcal{J}_{D}(\cdot;\bar{g},\mu)\) give upper bounds on the dynamic Cheeger ratios and dynamic Cheeger constants [39, Proposition 4.3]:
\[\mathcal{J}_{N}^{d}(A) \leq\mathcal{J}_{N}(A;\bar{g},\mu),\quad\forall A\in\mathscr{P}_{ N}(M_{0}) \tag{57}\] \[\mathcal{J}_{D}^{d}(A) \leq\mathcal{J}_{D}(A;\bar{g},\mu),\quad\forall A\in\mathscr{P}_{ D}(M_{0}). \tag{58}\]
The bounds in Theorem 3.1 have been extended to the dynamic setting.
**Theorem 3.14** (Dynamic Cheeger inequality [25, 26, 27]).:
* _[_25_, Theorem 3.2]__,_ _[_27_, Theorem 4.5]__: For any dynamical system, we have_ \[\lambda_{2,N}^{d}\leq-\frac{1}{4}(h_{2,N}^{d})^{2}.\] (59)
* _[_26_, Theorem 2]_ _For any dynamical system such that each_ \((M_{t},g_{t},\mu_{t})\) _is an_ \(n\)_-dimensional,_ \(C^{\infty}\) _submanifold of_ \(\mathbb{R}^{n}\) _equipped with the Euclidean metric and Lebesgue measure, we have_ \[\lambda_{1,D}^{d}\leq-\frac{1}{4}(h_{1,D}^{d})^{2}.\] (60)
Combining the approach from [27] and [26], equation (59) extends to dynamical systems on arbitrary weighted Riemannian manifolds as in Definition 3.9.
Similarly to the static case, we can give constructive versions of the dynamic Cheeger inequality (Theorem 3.15 and Corollary 3.16). Specifically, we show that within any nodal domain of an eigenfunction \(u\) of \(\Delta^{d}\), a positive-measure collection of superlevel sets of \(u\) have their dynamic Cheeger ratio bounded above by the corresponding eigenvalue (Theorem 3.15). This immediately yields a constructive version of Theorem 3.14 (Corollary 3.16).
**Theorem 3.15**.: _Let \(\mathcal{T}\) be a dynamical system, and let \(u\) be some Neumann, resp. Dirichlet, eigenfunction of \(\Delta^{d}\) with eigenvalue \(\lambda\). Let \(G\subset M_{0}\) be any nodal domain of \(u\). Then, defining_
\[G_{s}:=\{p\in G:u(p)^{2}>s\}, \tag{61}\]
_the set_
\[S_{G}:=\bigg{\{}s\in\operatorname{range}(u^{2}|_{G}):G_{s}\in \mathscr{P}_{N}(M_{0}),\lambda\leq-\frac{1}{4}\mathcal{J}_{N}^{d}(G_{s})^{2} \bigg{\}}, \tag{62}\]
_resp._
\[S_{G}:=\bigg{\{}s\in\operatorname{range}(u^{2}|_{G}):G_{s}\in \mathscr{P}_{D}(M_{0}),\lambda\leq-\frac{1}{4}\mathcal{J}_{D}^{d}(G_{s})^{2} \bigg{\}}, \tag{63}\]
_has positive Lebesgue measure satisfying the lower bound (70)._
Proof.: The proof proceeds as for Theorem 3.2. For each \(t\in\mathrm{T}\), define \(\phi_{t}\in C^{\infty}(M_{t})\) via \(\mathrm{d}\mu_{t}=e^{\phi_{t}}\,\mathrm{d}V\), and observe that \(\operatorname{range}((\Phi_{*}^{(t)}u)^{2}|_{\Phi^{(t)}(G)})=\operatorname{ range}(u^{2}|_{G})\) and that for each \(s\in\operatorname{range}(u^{2}|_{G})\), \(\Phi^{(t)}(G_{s})\) is the superlevel set of \(\Phi_{*}^{(t)}u\) on \(\Phi^{(t)}(G)\). Replacing \((M,g,\mu)\), \(\phi\), \(G\) and \(u\), respectively, with \((M_{t},g_{t},\mu_{t})\), \(\phi_{t}\), \(\Phi^{(t)}(G)\) and \(\Phi_{*}^{(t)}u\), respectively, in each of (19), (22) and (23)-(24) yields
\[\bar{h}:=\frac{\int_{\operatorname{range}(u^{2}|_{G})}\mathcal{J}_{N}(\Phi^{( t)}(G_{s}))\mu_{t}(\Phi^{(t)}(G_{s}))\,\mathrm{d}s}{\|\Phi_{*}^{(t)}u\|_{L^{2}( \Phi^{(t)}(G);\mu_{t})}^{2}}, \tag{64}\]
\[\frac{\||\nabla_{g_{t}}\Phi_{*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G);\mu _{t})}^{2}}{\|\Phi_{*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G);\mu_{t})}^{2}} \geq\frac{1}{4}\bar{h}^{2}, \tag{65}\] \[\||\nabla_{g_{t}}\Phi_{*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G);\mu_{t})}^ {2} =\int_{\Phi^{(t)}(G)}\nabla_{g_{t}}\Phi_{*}^{(t)}u\cdot(e^{\phi_{ t}}\nabla_{g_{t}}\Phi_{*}^{(t)}u)\,\mathrm{d}V\] \[=-\int_{\Phi^{(t)}(G)}u\cdot(\Delta_{g_{t},\mu_{t}}\circ\Phi_{*} ^{(t)})u\,\mathrm{d}\mu_{t}+0. \tag{66}\]
Multiplying (65) by \(\|\Phi_{*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G);\mu_{t})}^{2}\), replacing \(\||\nabla_{g_{t}}\Phi_{*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G);\mu_{t})}^{2}\) with the right-hand side of (66), and then replacing \(\bar{h}\) with its definition (64), yields
\[-\int_{\Phi^{(t)}(G)}\Phi_{*}^{(t)}u\cdot\big{(}\Delta_{g_{t},\mu_ {t}}\circ\Phi_{*}^{(t)}\big{)}u\,\mathrm{d}\mu_{t} \geq\frac{1}{4}\bar{h}^{2}\|\Phi_{*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G );\mu_{t})}^{2}\] \[=\frac{\left(\int_{\operatorname{range}(u^{2}|_{G})}\mathcal{J}_{N }(\Phi^{(t)}(G_{s}))\mu_{t}(\Phi^{(t)}(G_{s}))\,\mathrm{d}s\right)^{2}}{4\|\Phi_ {*}^{(t)}u\|_{L^{2}(\Phi^{(t)}(G);\mu_{t})}^{2}}.\]
Since \(\Phi^{(t)}\) is measure-preserving, this is equivalent to
\[-\int_{G}u\cdot\big{(}(\Phi^{(t)})^{*}\circ\Delta_{g_{t},\mu_{t}}\circ\Phi^{(t)}_{ *}\big{)}u\,\mathrm{d}\mu_{0}\geq\frac{\Big{(}\int_{\mathrm{range}(u^{2}|_{G})} \mathcal{J}_{N}(\Phi^{(t)}(G_{s}))\mu_{0}(G_{s})\,\mathrm{d}s\Big{)}^{2}}{4\|u \|_{L^{2}(G;\mu_{0})}^{2}}. \tag{67}\]
Now, definition (50) and our choice of \(u\) imply \(\frac{1}{|T|}\sum_{t=0}^{t_{\mathrm{max}}}\big{(}(\Phi^{(t)})^{*}\circ\Delta_{ g_{*},\mu_{t}}\circ\Phi^{(t)}_{*}\big{)}u=\Delta^{d}u=\lambda u\), so summing (67) over \(t\) and dividing by \(-|\mathrm{T}|\|u\|_{L^{2}(G;\mu_{0})}^{2}\) yields
\[\lambda\leq-\frac{1}{4|\mathrm{T}|\|u\|_{L^{2}(G;\mu_{0})}^{4}}\sum_{t=0}^{t_ {\mathrm{max}}}\biggl{(}\int_{\mathrm{range}(u^{2}|_{G})}\mathcal{J}_{N}( \Phi^{(t)}(G_{s}))\mu_{0}(G_{s})\,\mathrm{d}s\biggr{)}^{2}. \tag{68}\]
Using the relation \(-\sum_{t=0}^{t_{\mathrm{max}}}x_{t}^{2}\leq-\frac{1}{|\Gamma|}\Bigl{(}\sum_{t =0}^{t_{\mathrm{max}}}x_{t}\Bigr{)}^{2}\) for \(x\in\mathbb{R}^{|\mathrm{T}|}\), this bound becomes
\[\lambda\leq-\frac{1}{4|\mathrm{T}|^{2}\|u\|_{L^{2}(G;\mu_{0})}^{4}}\biggl{(} \sum_{t=0}^{t_{\mathrm{max}}}\int_{\mathrm{range}(u^{2}|_{G})}\mathcal{J}_{N} (\Phi^{(t)}(G_{s}))\mu_{0}(G_{s})\,\mathrm{d}s\biggr{)}^{2}=-\frac{1}{4}(\bar{ h}_{G}^{d})^{2}, \tag{69}\]
where \(\bar{h}_{G}^{d}:=\frac{1}{\|u\|_{L^{2}(G;\mu_{0})}^{2}}\int_{\mathrm{range}(u^ {2}|_{G})}\mathcal{J}_{N}^{d}(G_{s})\mu_{0}(G_{s})\,\mathrm{d}s\). Thus, by the reasoning after (26), the set \(S_{G}\) defined in (62) has positive measure.
We can bound its measure as follows. Define the probability measure \(\mathbb{P}\) on \(\mathrm{range}(u^{2}|_{G})\) by \(\mathbb{P}(L):=\int_{L}\frac{\mu_{0}(G_{s})}{\|u\|_{L^{2}(G;\mu_{0})}^{2}} \,\mathrm{d}s\), and let \(\mathrm{h}^{d}(s):=\mathcal{J}_{N}^{d}(G_{s})\). Then the reasoning for (27) implies
\[\mathrm{Leb}(S_{G})\geq\frac{\|\bar{h}_{G}^{d}-\mathrm{h}^{d}\|_{L^{1}(\mathrm{ range}(u^{2}|_{G});\mathbb{P}^{d})}\|u\|_{L^{2}(G;\mu_{0})}^{2}}{2(\bar{h}_{G}^{d} -\inf_{s\in\mathrm{range}(u^{2}|_{G})}\mathrm{h}^{d}(s))\mu_{0}(G)}. \tag{70}\]
**Corollary 3.16**.: _For any dynamical system \(\mathcal{T}\), and for any dynamic Neumann eigenfunction \(u\) of \(\Delta^{d}\) corresponding to \(\lambda^{d}_{2,N}\), there is a nodal domain \(G\) of \(u\) such that the set \(S_{G}\) defined in (62) has positive measure, and for \(s\in S_{G}\), defining \(G_{s}\) as in (61), the 2-packing \(\{G_{s},M\backslash\overline{G_{s}}\}\) satisfies_
\[\lambda^{d}_{2,N}\leq-\frac{1}{4}\mathcal{J}_{N}^{d}(\{G_{s},M\backslash \overline{G_{s}}\})^{2}. \tag{71}\]
_If \(\partial M_{0}\neq\emptyset\), the leading Dirichlet eigenfunction \(\lambda^{d}_{1,D}\) of \(\Delta^{d}\) is simple, and the corresponding eigenfunction \(u\) has only a single nodal domain \(G=M_{0}\backslash\partial M_{0}\). The set \(S_{G}\) defined in (63) has positive measure, and for \(s\in S_{G}\), the set \(G_{s}\) defined in (14) satisfies_
\[\lambda^{d}_{1,D}\leq-\frac{1}{4}\mathcal{J}_{D}^{d}(G_{s})^{2}. \tag{72}\]
Proof.: In the Dirichlet case, we mostly follow the proof of [41, Proposition 4.5.8]. Corollary 3.12 ensures that any Dirichlet eigenfunction \(u\) of \(\Delta^{d}\) corresponding to \(\lambda^{d}_{1,D}\) has only one nodal domain, so the maximum principle (e.g. applying [58, Chapter 2, Theorem 5] in local coordinates) implies that \(u\) is strictly positive or strictly negative on \(M\backslash\partial M\). Hence there cannot be two orthogonal Dirichlet eigenfunctions of \(\Delta^{d}\) corresponding to \(\lambda^{d}_{1,D}\), i.e. \(\lambda^{d}_{1,D}\) is a simple eigenvalue of \(\Delta^{d}\), and (72) follows from Theorem 3.15. In the Neumann case, Corollary 3.12 yields that any dynamic Neumann eigenfunction \(u\) of \(\Delta^{d}\) corresponding to \(\lambda^{d}_{2,N}\) has at most two nodal domains. Since the constant function \(\mathbf{1}\) is a dynamic Neumann eigenfunction of \(\Delta^{d}\) orthogonal to \(u\), \(u\) has exactly two nodal domains \(G_{1},G_{2}\). One choice of \(G\in\{G_{1},G_{2}\}\) satisfies \(\mu(G)\leq\mu(M\backslash\overline{G})\), and (71) follows from Theorem 3.15.
#### 3.4.2 Higher dynamic Cheeger inequalities
We can extend our dynamic Cheeger inequalities of Section 3.2 directly to the dynamic setting. Our proofs of Theorem 3.7 and Proposition 3.8 carry over directly to the dynamic setting (Theorem 3.19 and Proposition 3.20). To extend Theorems 3.4 and 3.5 to the dynamic setting, we can avoid some technicalities by applying those theorems on the geometry of mixing manifold \((M_{0},\bar{g},\mu_{0})\), and applying (57).
**Theorem 3.17**.: _There is a universal constant \(\hat{\eta}\) such that for any dynamical system where \(M_{0}\) is boundary-less, for all \(k\geq 1\) we have_
\[\lambda_{k,\emptyset}^{d}\leq-\frac{\hat{\eta}}{k^{6}}(h_{k,\emptyset}^{d})^{2}. \tag{73}\]
Proof.: By Proposition 3.13, \(\lambda_{k,\emptyset}^{d}\) is the \(k\)th eigenvalue of \(\Delta_{\bar{g},\mu_{0}}\). Applying Theorem 3.4 to bound the \(k\)th Cheeger constant \(h_{k,\emptyset}\) on the geometry of mixing manifold yields \(\lambda_{k,\emptyset}^{d}\leq-\frac{\hat{\eta}}{k^{6}}h_{k,\emptyset}^{2}\). Then (57) and the definitions (3) and (48) imply \(-h_{k,\emptyset}\leq-h_{k,\emptyset}^{d}\), and (73) follows.
**Theorem 3.18**.: _There is a universal constant \(\eta\) such that for any dynamical system \(\mathcal{T}\) where \(M_{0}\) is boundaryless, for all \(k\geq 1\) we have_
\[\lambda_{2k,\emptyset}^{d}\leq-\frac{\eta}{\log(k+1)}(h_{k,\emptyset}^{d})^{2}. \tag{74}\]
Proof.: By Proposition 3.13, \(\lambda_{2k,\emptyset}^{d}\) is the \(2k\)th eigenvalue of \(\Delta_{\bar{g},\mu_{0}}\). Applying Theorem 3.5 to bound the \(k\)th Cheeger constant \(h_{k,\emptyset}\) on the geometry of mixing manifold yields \(\lambda_{2k,\emptyset}^{d}\leq-\frac{\eta}{\log(k+1)}h_{k,\emptyset}^{2}\). Then (57) and the definitions (3) and (48) imply \(-h_{k,\emptyset}\leq-h_{k,\emptyset}^{d}\), and (73) follows.
Our constructive, nodal domain-based higher Cheeger inequality, Theorem 3.7, generalises directly to the dynamic case.
**Theorem 3.19** (Higher dynamic Cheeger inequality).: _Let \(\mathcal{T}\) be a dynamical system. For each \(k\geq 1\), let \(r_{k}\) be the maximal number of nodal domains in any dynamic Neumann (resp. Dirichlet) eigenfunction \(u\) of \(\Delta^{d}\) with eigenvalue \(\lambda\geq\lambda_{k,N}^{d}\) (resp. \(\lambda\geq\lambda_{k,D}^{d}\))._
1. _We have_ \[\lambda_{k,N}^{d} \leq-\frac{1}{4}(h_{r_{k},N}^{d})^{2},\] (75) \[\lambda_{k,D}^{d} \leq-\frac{1}{4}(h_{r_{k},D}^{d})^{2}.\] (76)
2. _Let_ \(u\) _be an eigenfunction with eigenvalue_ \(\lambda\geq\lambda_{k,N}^{d}\) _(resp._ \(\lambda\geq\lambda_{k,D}^{d}\)_) and with_ \(r_{k}\) _nodal domains. Let_ \(G^{1},\ldots,G^{r_{k}}\subset M\) _denote the nodal domains of_ \(u\)_, and for each_ \(i\) _and each_ \(s\in\operatorname{range}(u^{2}|_{G^{i}})\)_, let_ \(G_{s}^{i}\) _denote the_ \(s\)_-superlevel set of_ \(u^{2}\) _on_ \(G^{i}\)_. For each_ \(i\)_, define_ \(S_{G^{i}}\) _as in (_62_) or (_63_). Then each_ \(S_{G^{i}}\) _has positive Lebesgue measure satisfying (_70_), and for each_ \(\{s_{1},\ldots,s_{r_{k}}\}\in S_{G^{1}}\times\ldots\times S_{G^{r_{k}}}\)_, the collection_ \(\mathcal{A}_{r_{k}}:=\{G_{s_{1}}^{1},\ldots,G_{s_{r_{k}}}^{r_{k}}\}\) _is a Neumann (resp. Dirichlet)_ \(r_{k}\)_-packing of_ \(M_{0}\) _satisfying_ \(\lambda_{k,N}^{d}\leq-\frac{1}{4}\mathcal{J}_{N}^{d}(\mathcal{A}_{r_{k}})^{2}\) _(resp._ \(\lambda_{k,D}^{d}\leq-\frac{1}{4}\mathcal{J}_{D}^{d}(\mathcal{A}_{r_{k}})^{2}\)_)._
Proof.: This theorem follows from Lemma 3.15, by the reasoning in the proof of Theorem 3.7.
We can also extend Proposition 3.8 to the dynamic setting, to obtain bounds on \(h_{l,N}^{d}\) or \(h_{l,D}^{d}\) for \(r_{k}\leq l\leq k\) in terms of thresholded functions obtained from linear combinations of the first \(k\) eigenfunctions of \(\Delta^{d}\).
**Proposition 3.20**.: _For any dynamical system \(\mathcal{T}\), let \(u_{1},\ldots,u_{k}\) denote the first \(k\) dynamic Neumann, resp. Dirichlet, eigenfunctions of \(\Delta^{d}\) for \(k\geq 1\). For any \(1\leq l\leq k\) and any \(\alpha\in\mathbb{R}^{l\times k}\), define \(f_{1,\alpha},\ldots,f_{l,\alpha}\) by \(f_{i,\alpha}:=\sum_{j=1}^{k}\alpha_{ij}u_{j}\). Suppose that for some \(a>0\), the functions \(\tau_{a}(f_{1,\alpha}),\ldots,\tau_{a}(f_{l,\alpha})\) are nonzero and have pairwise disjoint supports. Then each \(\tau_{a}(f_{i,\alpha})\) has a nodal domain \(\bar{G}^{i}\) such that letting \(\tilde{G}^{i}_{s}\) for \(s\in\operatorname{range}(\tau_{a}(f_{i,\alpha})^{2}|_{\bar{G}^{i}})\) denote the \(s\)-superlevel set of \(\tau_{a}(f_{i,\alpha})^{2}\) on \(\bar{G}^{i}\), the set_
\[\tilde{S}_{\bar{G}^{i}}:=\Big{\{}s\in\operatorname{range}(\tau_ {a}(f_{i,\alpha})^{2}|_{\bar{G}^{i}}):\tilde{G}^{i}_{s}\in\mathscr{P}_{N}(M_{ 0}),\] \[\frac{\sum_{t=0}^{t_{\max}}\||\nabla_{g_{t}}\Phi^{(t)}_{s}\tau_{a }(f_{i,\alpha})|\|^{2}_{L^{2}(\Phi^{(t)}(\bar{G}^{i});\mu_{t})}}{|\Gamma||\tau _{a}(f_{i,\alpha})\|^{2}_{L^{2}(\bar{G}^{i};\mu_{0})}}\geq\frac{1}{4}\mathcal{J }^{d}_{N}(\tilde{G}^{i}_{s})^{2}\Big{\}}, \tag{77}\]
_resp._
\[\tilde{S}_{\bar{G}^{i}}:=\Big{\{}s\in\operatorname{range}(\tau_ {a}(f_{i,\alpha})^{2}|_{\bar{G}^{i}}):\tilde{G}^{i}_{s}\in\mathscr{P}_{D}(M_{ 0}),\] \[\frac{\sum_{t=0}^{t_{\max}}\||\nabla_{g_{t}}\Phi^{(t)}_{s}\tau_{a }(f_{i,\alpha})|\|^{2}_{L^{2}(\Phi^{(t)}(\bar{G}^{i});\mu_{t})}}{|\Gamma||\tau _{a}(f_{i,\alpha})\|^{2}_{L^{2}(\bar{G}^{i};\mu_{0})}}\geq\frac{1}{4}\mathcal{ J}^{d}_{D}(\tilde{G}^{i}_{s})^{2}\Big{\}}, \tag{78}\]
_has positive measure and satisfies (81). Moreover, for each \(\{s_{1},\ldots,s_{l}\}\in\tilde{S}_{\bar{G}^{1}}\times\ldots\times\tilde{S}_{ \bar{G}^{l}}\), the collection \(\mathcal{A}_{l}:=\{\tilde{G}^{1}_{s_{1}},\ldots,\tilde{G}^{l}_{s_{l}}\}\) is a Neumann \(l\)-packing for \(M_{0}\) satisfying_
\[\lambda^{d}_{k,N}\leq-\frac{1}{4}\mathcal{J}^{d}_{N}(\mathcal{A}_{l})^{2}\max_ {1\leq j\leq l}\frac{\|\tau_{a}(f_{j,\alpha})\|^{2}_{L^{2}(M_{0};\mu_{0})}}{ \|f_{j,\alpha}\|^{2}_{L^{2}(M_{0};\mu_{0})}}\leq-\frac{1}{4}(h^{d}_{l,N})^{2} \max_{1\leq j\leq l}\frac{\|\tau_{a}(f_{j,\alpha})\|^{2}_{L^{2}(M_{0};\mu_{0}) }}{\|f_{j,\alpha}\|^{2}_{L^{2}(M_{0};\mu_{0})}}, \tag{79}\]
_resp. a Dirichlet \(l\)-packing for \(M_{0}\) satisfying_
\[\lambda^{d}_{k,D}\leq-\frac{1}{4}\mathcal{J}^{d}_{D}(\mathcal{A}_{l})^{2}\max_ {1\leq j\leq l}\frac{\|\tau_{a}(f_{j,\alpha})\|^{2}_{L^{2}(M_{0};\mu_{0})}}{ \|f_{j,\alpha}\|^{2}_{L^{2}(M_{0};\mu_{0})}}\leq-\frac{1}{4}(h^{d}_{l,D})^{2} \max_{1\leq j\leq l}\frac{\|\tau_{a}(f_{j,\alpha})\|^{2}_{L^{2}(M_{0};\mu_{0} )}}{\|f_{j,\alpha}\|^{2}_{L^{2}(M_{0};\mu_{0})}}. \tag{80}\]
Proof.: This result follows by the reasoning for Proposition 3.8 and Lemma 3.15. As in those proofs, we consider only the Neumann case. For each \(1\leq i\leq l\), we select \(\tilde{G}^{i}\) by \(\tilde{G}^{i}:=\operatorname*{arg\,min}_{\tilde{G}}\frac{\sum_{t=0}^{t_{\max} }\||\nabla_{g_{t}}\Phi^{(t)}_{s}\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\Phi^{(t)}( \tilde{G});\mu_{t})}}{|\Gamma||\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G} ^{i};\mu_{0})}},\) where the infimum is taken over nodal domains \(\tilde{G}\) of \(\tau_{a}(f_{i,\alpha})\). Then the reasoning for Theorem 3.2, modified as in the proofs of Proposition 3.8 and Theorem 3.15, imply that \(\tilde{S}_{\tilde{G}^{i}}\) has positive measure. The reasoning for (44) extends directly to the dynamic setting, and (79) follows as in the proof of Proposition 3.8.
Now, define \(\overline{\tilde{h}^{d}_{i}}:=\frac{1}{\|\tau_{a}(f_{i,\alpha})\|^{2}_{L^{2}(M_ {0};\mu_{0})}}\int_{\operatorname{range}\left(\tau_{a}(f_{i,\alpha})^{2}|_{ \tilde{G}^{i}}\right)}\mathcal{J}^{d}_{N}(\tilde{G}^{i}_{s})\mu_{0}(\tilde{G}^{i }_{s})\,\mathrm{d}s\) and \(\tilde{h}^{d}_{i}(s):=\mathcal{J}^{d}_{N}(\tilde{G}^{i}_{s})\), and define the probability measure \(\tilde{\mathbb{P}}_{i}\) on \(\operatorname{range}(\tau_{a}(f_{i,\alpha})^{2}|_{\tilde{G}^{i}})\) by \(\tilde{\mathbb{P}}_{i}(L):=\int_{L}\frac{\mu_{0}(\tilde{G}^{i}_{s})}{\|\tau_{a}( f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu_{0})}}\,\mathrm{d}s\). Then the reasoning for (70) implies
\[\operatorname{Leb}(\tilde{S}_{\tilde{G}^{i}})\geq\frac{\|\overline{\tilde{h}^{d }_{i}}-\tilde{\tilde{h}^{d}_{i}}\|_{L^{1}\left(\operatorname{range}\left(\tau_{a}(f _{i,\alpha})^{2}|_{\tilde{G}^{i}}\right);\tilde{\mathbb{P}}_{i}\right)}\|\tau_{a}( f_{i,\alpha})\|^{2}_{L^{2}(\tilde{G}^{i};\mu_{0})}}{2\Big{(}\overline{\tilde{h}^{d}_{i} }-\inf_{s\in\operatorname{range}\left(\tau_{a}(f_{i,\alpha})^{2}|_{\tilde{G}^{i} }\right)}\tilde{h}^{d}_{i}(s)\Big{)}\mu_{0}(\tilde{G}^{i})}. \tag{81}\]
A similar bound holds in the Dirichlet case, replacing \(\mathcal{J}^{d}_{N}\) with \(\mathcal{J}^{d}_{D}\) in the definitions of \(\overline{\tilde{h}^{d}_{i}},\tilde{\mathbb{P}}_{i}^{d},\tilde{\mathbb{P}}_{i}\).
## 4 Examples
We apply our higher Cheeger inequality (Theorem 3.7) to compare the Laplace-Beltrami eigenvalues to the higher Cheeger constants, on three manifolds: a torus (example 4.1), a cylinder using Neumann boundary
conditions (example 4.2) and a 3-ball using Dirichlet boundary conditions (example 4.3). Our Theorem 3.7 applies to manifolds with or without boundary, whenever we know the number of nodal domains in some eigenfunctions on those manifolds, i.e. to each of examples 4.1-4.3. Miclo's existing higher Cheeger inequalities (Theorems 3.4 and 3.5) apply only to manifold without boundary, i.e. to example 4.1. For that example, we obtain an asymptotically stronger bound on \(h_{k,\emptyset}\) using our Theorem 3.7 than using Miclo's Theorems 3.4 and 3.5. Using our higher dynamic Cheeger inequality (Theorem 3.19), we also compare the dynamic Laplacian eigenvalues to the dynamic Cheeger constants for one dynamical system, a cylinder with linear shear (example 4.4).
### Cheeger constants on a torus
Our first example is a flat torus \(\mathbb{T}^{2}:=2\pi\mathbb{S}^{1}\times 2\pi\mathbb{S}^{1}\), endowed with two-dimensional Lebesgue measure. Then \(\Delta\) has an orthogonal Hilbert basis of eigenfunctions on \(L^{2}(\mathbb{T}^{2},\mathrm{Leb})\), consisting of all functions of the form
\[u_{k_{1},k_{2},\zeta_{1},\zeta_{2}}(x,y):=\cos(k_{1}(x+\zeta_{1}))\cos(k_{2}(y +\zeta_{2})), \tag{82}\]
for \(k_{1},k_{2}=0,1,2,\ldots\) and \(\zeta_{1},\zeta_{2}\in\{0,\frac{\pi}{2}\}\), where we require \(\zeta_{1}=0\) if \(k_{1}=0\) and \(\zeta_{2}=0\) if \(k_{2}=0\) to ensure an orthogonal basis. Each eigenfunction \(u_{k_{1},k_{2},\zeta_{1},\zeta_{2}}\) has corresponding eigenvalue \(\lambda_{k_{1},k_{2},\zeta_{1},\zeta_{2}}=-k_{1}^{2}-k_{2}^{2}\), and we can globally order these eigenfunctions in order of decreasing eigenvalue (resolving ties arbitrarily).
To apply Theorem 3.7, we need to estimate the maximal number \(r_{k}\) of nodal domains of an eigenfunction with eigenvalue greater than or equal to the \(k\)th eigenvalue \(\lambda_{k,\emptyset}\). Each eigenfunction \(u_{k_{1},k_{2},\zeta_{1},\zeta_{2}}\) has \(\max\{4k_{1}k_{2},2k_{1},2k_{2},1\}\) nodal domains, by (82). It can be shown that for each \(k_{1}\geq 1\) and \(\zeta_{1},\zeta_{2}\in\{0,\frac{\pi}{2}\}\), any eigenfunction whose eigenvalue is greater than or equal to \(\lambda_{k_{1},k_{1},\zeta_{1},\zeta_{2}}\) has at most \(4k_{1}^{2}\) nodal domains. In this sense, the eigenfunctions \(u_{k_{1},k_{1},\zeta_{1},\zeta_{2}}\) maximise the number of nodal domains of an eigenfunction under an eigenvalue constraint. Thus, noting that \(\lambda_{6,\emptyset}=\lambda_{1,1,\zeta_{1},\zeta_{2}}\), we can obtain a lower bound on \(r_{k}\) for any \(k\geq 6\) by finding the largest \(k_{1}\) such that \(\lambda_{k,\emptyset}\leq\lambda_{k_{1},k_{1},\zeta_{1},\zeta_{2}}\) for some \(\zeta_{1},\zeta_{2}\in\{0,\frac{\pi}{2}\}\), and noting that \(r_{k}\) is bounded below by the number of eigenvalues in \(u_{k_{1},k_{2},\zeta_{1},\zeta_{2}}\). To estimate this \(k_{1}\) in terms of \(k\), we note that \(\lambda_{k,\emptyset}\geq\lambda_{k_{1}+1,k_{1}+1,\zeta_{1},\zeta_{2}}=-2(k_{1 }+1)^{2}\). Now, each integer pair in \(\mathcal{I}:=\{(i_{1},i_{2})\in\mathbb{Z}^{2}:i_{1},i_{2}\geq 1,-i_{1}^{2}-i_{2}^{2} \geq-2(k_{1}+1)^{2}\}\) corresponds to a unit-area square contained entirely in the nonnegative quadrant \(Q\) of the disk \(\{-x^{2}-y^{2}\geq-2(k_{1}+1)^{2}\}\). The quadrant \(Q\) has area \(\frac{\pi}{2}(k_{1}+1)^{2}\), so we have \(|\mathcal{I}|\leq\frac{\pi}{2}(k_{1}+1)^{2}\). Each integer pair in \(\mathcal{I}\) corresponds to 4 linearly independent eigenfunctions of the form (82) with different choices of \(\zeta_{1},\zeta_{2}\in\{0,\frac{\pi}{2}\}\), leading to at most \(2\pi(k_{1}+1)^{2}\) eigenvalues, counted with multiplicity, greater than or equal to \(\lambda_{k_{1}+1,k_{1}+1,\zeta_{1},\zeta_{2}}\).
There are also \(2\lfloor\sqrt{2}(k_{1}+1)\rfloor+1\) integer pairs in \(\mathcal{I}^{\prime}:=\{(i_{1},i_{2})\in\mathbb{Z}^{2}:i_{1},i_{2}\geq 0,i_{1}i_{2}=0,-i_{1 }^{2}-i_{2}^{2}\geq-2(k_{1}+1)^{2}\}\). Each such integer pair with \(i_{1}\geq 1\) or \(i_{2}\geq 1\) corresponds to 2 linearly independent eigenfunctions of the form (82) with different choices of \(\zeta_{1}\in\{0,\frac{\pi}{2}\}\) or \(\zeta_{2}\in\{0,\frac{\pi}{2}\}\) respectively, while the pair \((0,0)\) corresponds to only 1 eigenfunction. This leads to an additionally \(4\lfloor\sqrt{2}(k_{1}+1)\rfloor+1\) additional eigenvalues greater than or equal to \(\lambda_{k_{1}+1,k_{1}+1,\zeta_{1},\zeta_{2}}\). In total, we have at most \(2\pi(k_{1}+1)^{2}+4\sqrt{2}(k_{1}+1)+1\) eigenvalues greater than or equal to \(\lambda_{k_{1}+1,k_{1}+1,\zeta_{1},\zeta_{2}}\). The ordering of the eigenvalues \(\lambda_{i,\emptyset}\) implies there are at least \(k\) eigenvalues greater than or equal to \(\lambda_{k_{1}+1,k_{1}+1,\zeta_{1},\zeta_{2}}\), so \(k\leq 2\pi(k_{1}+1)^{2}+4\sqrt{2}(k_{1}+1)+1\). Applying the quadratic formula and noting \(\sqrt{\pi k+4-\pi}\geq\sqrt{\pi k}\) yields the bound \(k_{1}\geq\sqrt{\frac{k}{2\pi}}-1-\frac{\sqrt{2}}{\pi}\). Now, \(u_{k_{1},k_{1},\zeta_{1},\zeta_{2}}\) has \(4k_{1}^{2}\) nodal domains, so this bound on \(k_{1}\) and the fact \(k_{1}\geq 1\) imply \(r_{k}\geq 4k_{1}^{2}\geq\max\Bigl{\{}\frac{2k}{\pi}-4.7\sqrt{k}+8.4,4\Bigr{\}}\). Thus, Theorem 3.7 implies
\[\lambda_{k,\emptyset}\leq-\frac{1}{4}h_{r_{k},\emptyset}^{2}\leq-\frac{1}{4}h_{ \max\{\lceil\frac{2k}{\pi}-4.7\sqrt{k}+8.4\rceil,4\},\emptyset}^{2}. \tag{83}\]
To compare (83) to the bounds from Miclo's Theorems 3.4 and (31), we rewrite the outer inequality of (83) as a bound on \(h_{l,\emptyset}\) for \(l\geq 1\), and use Weyl's law. Let \(k^{*}(l):=\lceil\frac{\pi l}{2}+9.3\sqrt{l+0.3}+14.2\rceil\), then we can rearrange (83) to obtain
\[h_{l,\emptyset}\leq 2\sqrt{-\lambda_{k^{*}(l),\emptyset}}. \tag{84}\]
Now, from Weyl's law (see e.g. [41, p.118]), it follows that
\[\lambda_{k,\emptyset}=-\frac{k}{\pi}+O(\sqrt{k}). \tag{85}\]
This allows us to compare our bound (84) with the bounds obtained from Miclo's Theorems 3.4 and 3.5.
* Substituting (85) and the definition of \(k^{*}(l)\) into our bound (84), we obtain that as \(l\to\infty\), \[h_{l,\emptyset}\leq 2\sqrt{\frac{l}{2}+O(\sqrt{l})}=2\sqrt{\frac{l}{2}}+O(1).\] (86)
* Substituting (85) into Miclo's Theorem 3.4 [53, Theorem 7], the reasoning from (86) implies that as \(l\to\infty\), \[h_{l,\emptyset}\leq l^{3}\sqrt{-\frac{\lambda_{l,\emptyset}}{\hat{\eta}}}=l^{ 3}\sqrt{\frac{l}{\pi\hat{\eta}}}+O(l^{3}).\] (87) This is clearly asymptotically weaker than (86).
* Substituting (85) into Miclo's Theorem 3.5 [53, Theorem 13], the reasoning from 86 implies that as \(l\to\infty\), \[h_{l,\emptyset}\leq\sqrt{-\frac{\log(2l+1)\lambda_{2l,\emptyset}}{\eta}}=\sqrt {\frac{2l\log(2l+1)}{\pi\eta}}+O(\sqrt{\log(2l+1)}).\] (88) This is also asymptotically weaker than (86).
### Cheeger constants of a cylinder
Next, we consider a cylinder \(\mathcal{C}:=2\pi\mathbb{S}^{1}\times[0,\pi]\), endowed with two-dimensional Lebesgue measure. Then \(\mathcal{C}\) is a semiconvex subset of the torus \(\mathbb{T}^{2}\) from example 4.1, but \(\mathcal{C}\) is not a convex subset of any manifold since some pairs of points in \(\mathcal{C}\) are connected by two minimal geodesics contained in \(\mathcal{C}\). Under Neumann boundary conditions, \(\Delta\) has an orthogonal Hilbert basis of eigenfunctions on \(L^{2}(\mathcal{C},\mathrm{Leb})\), consisting of all functions of the form
\[u_{k_{1},k_{2},\zeta}(x,y):=\cos(k_{1}(x+\zeta))\cos(k_{2}y), \tag{89}\]
for \(k_{1},k_{2}=0,1,2,\ldots\) and \(\zeta\in\{0,\frac{\pi}{2}\}\), where we require \(\zeta=0\) whenever \(k_{1}=0\) to ensure an orthogonal basis. Each eigenfunction \(u_{k_{1},k_{2},\zeta}\) has corresponding eigenvalue \(\lambda_{k_{1},k_{2},\zeta}=-k_{1}^{2}-k_{2}^{2}\). To apply Theorem 3.7, we again need a lower bound for \(r_{k}\). First, we show that for each \(k_{1}\geq 1\), eigenfunctions of the form \(u_{k_{1},k_{1},\zeta}\) have the maximal number of nodal domains, among eigenfunctions of the form (89) for which \(\lambda_{i_{1},i_{2},\zeta}\geq-2k_{1}^{2}\). Each \(u_{i_{1},i_{2},\zeta}\) has \((i_{2}+1)\max\{2i_{1},1\}\) nodal domains by (89), so maximising the number of nodal domains in \(u_{i_{1},i_{2},\zeta}\) subject to \(\lambda_{i_{1},i_{2},\zeta}\,(=-i_{1}^{2}-i_{2}^{2})\geq-2k_{1}^{2}\) is equivalent to solving \(\max\{2i_{1}(i_{2}+1):(i_{1},i_{2})\in\mathbb{Z}_{\geq 0}^{2},i_{1}^{2}+i_{2}^{2} \leq 2k_{1}^{2}\}\). This can be solved via the relaxation \(\max\{2x(y+1):(x,y)\in([0,k_{1}]\cup[k_{1}+1,\infty))\times\mathbb{R}_{\geq 0},x^{2}+y^{2}\leq 2k_{1}^{2}\}\). Rearranging the constraint \(x^{2}+y^{2}\leq 2k_{1}^{2}\) and maximising \(y\) gives us \(y=\sqrt{2k_{1}^{2}-x^{2}}\). Substituting this into \(2x(y+1)\) gives us \(2x(\sqrt{2k_{1}^{2}-x^{2}}+1)\), which is strictly increasing for \(0\leq x\leq k_{1}\) and strictly decreasing for \(k_{1}+1\leq x\leq\sqrt{2}k_{1}\). Thus, since the objective is larger at \((x,y)=(k_{1},k_{1})\) than at \((x,y)=(k_{1}+1,\sqrt{k_{1}^{2}-2k_{1}-1})\), the maximum is uniquely attained at \((x,y)=(k_{1},k_{1})\). Hence the eigenfunctions \(u_{k_{1},k_{1},\zeta}\) for \(\zeta\in\{0,\frac{\pi}{2}\}\) maximise the number of nodal domains, among eigenfunctions \(u_{i_{1},i_{2},\zeta}\) of the form (89) satisfying \(\lambda_{i_{1},i_{2},\zeta}\leq-2k_{1}^{2}\).
Now, we bound \(r_{k}\) for each \(k\geq 5\) by finding the largest \(k_{1}\) such that \(\lambda_{k,N}\leq\lambda_{k_{1},k_{1},\zeta}\) for \(\zeta\in\{0,\frac{\pi}{2}\}\), noting that \(\lambda_{5,N}=\lambda_{1,1,\zeta}\). For this \(k_{1}\), we have \(\lambda_{k,N}\geq\lambda_{k_{1}+1,k_{1}+1,\zeta}=-2(k_{1}+1)^{2}\). Each integer pair in the set
from the previous example corresponds to two linearly independent eigenfunctions of the form (89), leading to at most \(\lfloor\pi(k_{1}+1)^{2}\rfloor\) eigenvalues \(\geq\lambda_{k_{1}+1,k_{1}+1,\zeta}\). There are also \(\lfloor\sqrt{2}(k_{1}+1)\rfloor\) nonnegative integer pairs in \(\mathcal{I}^{\prime}\) from the previous example with \(i_{1}>0\), each corresponding to \(2\) linearly independent eigenfunctions, and \(\lfloor\sqrt{2}(k_{1}+1)\rfloor+1\) such pairs with \(i_{1}=0\), each corresponding to only \(1\) linearly independent eigenfunction. These lead to at most an additional \(3\sqrt{2}(k_{1}+1)+1\) eigenvalues \(\geq\lambda_{k_{1}+1,k_{1}+1,\zeta}\). Thus, there are at most \(\pi(k_{1}+1)^{2}+3\sqrt{2}(k_{1}+1)+1\) eigenvalues \(\geq\lambda_{k_{1}+1,k_{1}+1,\zeta}\). Again, the ordering of the \(\lambda_{i,\emptyset}\) implies there are at least \(k\) eigenvalues \(ge\lambda_{k_{1}+1,k_{1}+1,\zeta}\), so \(k\leq\pi(k_{1}+1)^{2}+3\lfloor\sqrt{2}(k_{1}+1)\rfloor+1\). Then the quadratic formula and the fact \(\sqrt{4\pi k+18-4\pi}\geq\sqrt{4\pi k}\) yield \(k_{1}\geq\sqrt{\frac{k}{\pi}}-1-\frac{3}{\sqrt{2\pi}}\). Now, \(u_{k_{1},k_{1},\zeta}\) has \(2k_{1}(k_{1}+1)\) nodal domains, so this bound on \(k_{1}\) and the fact \(k_{1}\geq 1\) imply \(r_{k}\geq 2k_{1}(k_{1}+1)\geq\max\Bigl{\{}\frac{2k}{\pi}-2.7\sqrt{k}+2.2,4 \Bigr{\}}\). Thus, Theorem 3.7 implies that for \(k\geq 5\),
\[\lambda_{k,N}\leq-\frac{1}{4}h_{r_{k},N}^{2}\leq-\frac{1}{4}h_{\max\bigl{\{} \bigl{[}\frac{2k}{\pi}-2.7\sqrt{k}+2.2\bigr{]},4\bigr{\}},N}^{2}. \tag{90}\]
Note that we cannot apply Miclo's Theorems 3.4 or 3.5 to \(\mathcal{C}\), because \(\mathcal{C}\) has nonempty boundary.
### Cheeger constants on a 3-ball
Next, we consider the 3-ball \(\mathbb{B}:=\{\mathbf{x}\in\mathbb{R}^{3}:|\mathbf{x}|\leq 1\}\), equipped with 3-dimensional Lebesgue measure. We work in spherical coordinates \((r,\theta,\phi)\), where \(\theta\) is the polar angle and \(\phi\) is the azimuthal angle. Then \(\Delta\), under Dirichlet boundary conditions, has an orthogonal Hilbert basis of eigenfunctions on \(L^{2}(\mathbb{B},\mathrm{Leb})\), consisting of all functions of the form
\[u_{k_{1},k_{2},k_{3},\zeta}:=S_{k_{2}}(\alpha_{k_{1},k_{2}}r)P_{k_{2}}^{k_{3}} (\cos\theta)\cos(k_{3}(\phi+\zeta)) \tag{91}\]
for \(k_{1}=1,2,\ldots\); \(k_{2}=0,1,\ldots\); \(k_{3}=0,\ldots,k_{2}\); \(\zeta\in\{0,\frac{\pi}{2}\}\), where we require \(\zeta=0\) when \(k_{3}=0\) to ensure an orthonormal basis. The function \(S_{k_{2}}:\mathbb{R}_{+}\to\mathbb{R}\) is the \(k_{2}\)th _spherical Bessel function of the first kind_, \(\alpha_{k_{1},k_{2}}\) is the \(k_{1}\)th positive zero of \(S_{k_{2}}\), and \(P_{k_{2}}^{k_{3}}\) is the \(k_{2}\)th _associated Legendre polynomial of \(k_{3}\)th order_ (see e.g. [31, sec. 3.3] and [20, secs V.8 and VII.5]). The eigenfunction \(u_{k_{1},k_{2},k_{3},\zeta}\) has eigenvalue \(\lambda_{k_{1},k_{2},k_{3},\zeta}=-\alpha_{k_{1},k_{2}}^{2}\). The values \(\alpha_{k_{1},k_{2}}\) satisfy the bounds (simplified from [6, equations (1), (2), (5)])
\[\pi k_{1}+k_{2}-3.75<\alpha_{k_{1},k_{2}}<\pi k_{1}+\frac{\pi}{2}k_{2}+0.03- \frac{(k_{2}+\frac{1}{2})^{2}}{2\bigl{(}\pi k_{1}+\frac{\pi}{2}k_{2}+0.03\bigr{)}}. \tag{92}\]
To apply our Theorem 3.7, we first obtain a lower bound on \(r_{k}\). The function \(P_{k_{2}}^{k_{3}}(\cos\theta)\cos(k_{3}(\phi+\zeta))\) has \((k_{2}-k_{3}+1)\max\{2k_{3},1\}\) nodal domains (see e.g. [49, p.302]), while the function \(S_{k_{2}}(\alpha_{k_{1},k_{2}}r)\) has \(k_{1}\) nodal domains since \(\alpha_{k_{1},k_{2}}\) is the \(k_{1}\)th positive zero of \(S_{k_{2}}\). Thus, the eigenfunction \(u_{k_{1},k_{2},k_{3},\zeta}\) has \(k_{1}(k_{2}-k_{3}+1)\max\{2k_{3},1\}\) nodal domains. In particular, \(u_{k_{1},4k_{1}-1,2k_{1},\zeta}\) for \(k_{1}=1,2,\ldots\), \(\zeta\in\{0,\frac{\pi}{2}\}\), has \(8k_{1}^{3}\) nodal domains, i.e. it is a simple eigenfunction with a relatively high number of nodal domains for its eigenvalue. It can be shown using the second inequality in (92) that with \(c:=3\pi-\frac{8}{3\pi}\),
\[\lambda_{k_{1},4k_{1}-1,2k_{1},\zeta}=-\alpha_{k_{1},4k_{1}-1}^{2}\geq-(ck_{1}- 1.46)^{2}. \tag{93}\]
Thus, for each \(k\geq 18\), we can obtain a lower bound on \(r_{k}\) by finding the largest \(k_{1}\) such that
\[-(ck_{1}-1.46)^{2}\geq\lambda_{k,D}, \tag{94}\]
since we can confirm numerically that \(\lambda_{17,D}\geq-(c-1.46)^{2}\geq\lambda_{18,D}\). For this \(k_{1}\), we have \(\lambda_{k,D}\geq-(c(k_{1}+1)-1.46)^{2}\). By the first inequality in equation (92), we have \(\lambda_{i_{1},i_{2},i_{3},\zeta}\geq-(c(k_{1}+1)-1.46)^{2}\) only for \((i_{1},i_{2},i_{3},\zeta)\in\mathcal{I}:=\{(i_{1},i_{2},i_{3},\zeta):\pi i_{1}+i_{2 }-3.75\leq c(k_{1}+1)-1.46\}\). There are \(2i_{2}+1\) tuples \((i_{1},i_{2},i_{3},\zeta)\in\mathcal{I}\) for each pair \(i_{1},i_{2}\) such that \(\pi i_{1}+i_{2}\leq c(k_{1}+1)+2.29\). Using the formula for sums of squares, and writing
\(a:=c(k_{1}+1)+2.29\) for clarity, the cardinality of \(\mathcal{I}\) is bounded by
\[|\mathcal{I}| =\sum_{i_{1}=1}^{\left\lfloor\frac{a}{\pi}\right\rfloor}\sum_{i_{2} =0}^{\left\lfloor a-\pi i_{1}\right\rfloor}(2i_{2}+1)=\sum_{i_{1}=1}^{\left\lfloor \frac{a}{\pi}\right\rfloor}(\lfloor a-\pi i_{1}\rfloor+1)^{2}=\sum_{i_{1}=1}^{ \left\lfloor\frac{a}{\pi}\right\rfloor}\Bigl{(}\left\lfloor a-\pi\Bigl{(} \left\lfloor\frac{a}{\pi}\right\rfloor+1-i_{1}\Bigr{)}\right\rfloor+1\Bigr{)}^ {2}\] \[\leq\sum_{i_{1}=1}^{\left\lfloor\frac{a}{\pi}\right\rfloor}(\lfloor \pi i_{1}\rfloor+1)^{2}\leq\frac{a^{3}}{3\pi}+\biggl{(}\frac{1}{2}+\frac{1}{ \pi}\biggr{)}a^{2}+\biggl{(}1+\frac{\pi}{6}+\frac{1}{\pi}\biggr{)}a\leq\biggl{(} \frac{c}{\sqrt[3]{3\pi}}k_{1}+6.4\biggr{)}^{3}. \tag{95}\]
Every tuple in \(\mathcal{I}\) corresponds to at most one eigenvalue \(\lambda_{i_{1},i_{2},i_{3},\zeta}\) satisfying \(\lambda_{i_{1},i_{2},i_{3},\zeta}\geq-(c(k_{1}+1)-1.46)^{2}\), so there are at most \(\Bigl{(}\frac{c}{\sqrt[3]{3\pi}}k_{1}+6.4\Bigr{)}^{3}\) such eigenvalues. Hence \(k\leq\Bigl{(}\frac{c}{\sqrt[3]{3\pi}}k_{1}+6.4\Bigr{)}^{3}\), so
\[k_{1}\geq\max\Biggl{\{}\frac{\sqrt[3]{3\pi}}{c}(\sqrt[3]{k}-6.4),1\Biggr{\}}. \tag{96}\]
Now, equations (93) and (94) imply \(\lambda_{k,D}\leq\lambda_{k_{1},4k_{1}-1,2k_{1},\zeta}\), for \(\zeta\in\{0,\frac{\pi}{2}\}\). Thus, since \(u_{k_{1},4k_{1}-1,2k_{1}}\) has \(8k_{1}^{3}\) nodal domains, (96) and the fact \(k_{1}\geq 1\) imply \(r_{k}\geq 8k_{1}^{3}\geq\max\Bigl{\{}\frac{24\pi}{c^{3}}(\sqrt[3]{k}-6.4)^{3},8 \Bigr{\}}\geq\max\{0.119(\sqrt[3]{k}-6.4)^{3},8\}\). Hence Theorem 3.7 implies
\[\lambda_{k,D}\geq\frac{1}{4}h_{r_{k},D}^{2}\geq\frac{1}{4}h_{\max\bigl{\{} \left\lceil 0.119(\sqrt[3]{k}-6.4)^{3}\right\rceil},8\bigr{\}}^{2}. \tag{97}\]
As in the previous example, one cannot apply Miclo's Theorem 3.4 or 3.5 in this case, because \(\mathbb{B}\) has non-empty boundary.
### Dynamic Cheeger constant on a cylinder with linear shear
Finally, we consider a linear shear on the cylinder \(\mathcal{C}:=2\pi\mathbb{S}^{1}\times[0,\pi]\), similarly to [25, example 6.1]. We consider a dynamical system \(\mathcal{T}\) as in definition 3.9. We let \(\mathrm{T}:=\{0,1,\ldots,t_{\max}\}\) for some even \(t_{\max}\geq 2\), and for each \(t\), we let \(M_{t}:=\mathcal{C}\), and we define \(g_{t}\) as the Euclidean metric and \(V_{t}\) as two-dimensional Lebesgue measure. For some \(b>0\), we define each \(\Phi^{(t)}:\mathcal{C}\rightarrow\mathcal{C}\) by
\[\Phi^{(t)}(x,y):=\biggl{(}x+b\frac{t}{t_{\max}}y\;\;(\mathrm{mod}\;2\pi),y \biggr{)}. \tag{98}\]
The dynamics \(\Phi^{(t)}\) represents linear shear in the \(x\)-coordinate on the cylinder. The functions
\[u_{k_{1},k_{2},\zeta}^{d}(x,y):=\cos\biggl{(}k_{1}\biggl{(}x+\zeta-\frac{b}{2} y\biggr{)}\biggr{)}\cos(k_{2}y), \tag{99}\]
for \(k_{1},k_{2}=0,1,2,\ldots\), and \(\zeta\in\{0,\frac{\pi}{2}\}\), taking \(\zeta=0\) whenever \(k_{1}=0\), are a complete basis of eigenfunctions for \(\Delta^{d}\) under dynamic Neumann boundary conditions. This follows since for each \(t\in\mathrm{T}\), writing \(\tilde{x}_{t}:=x+\zeta+b\Bigl{(}\frac{t}{t_{\max}}-\frac{1}{2}\Bigr{)}y\) for brevity, we have \(\Phi_{*}^{(t)}u_{k_{1},k_{2},\zeta}^{d}(x,y)=\cos(k_{1}\tilde{x}_{t})\cos(k_{2 }y)\), so
\[\Delta\Phi_{*}^{(t)}u_{k_{1},k_{2},\zeta}^{d}(x,y)\] \[=-\frac{\partial}{\partial x}[k_{1}\sin(k_{1}\tilde{x}_{t})\cos( k_{2}y)]-\frac{\partial}{\partial y}\Bigl{[}k_{1}b\Bigl{(}\frac{t}{t_{\max}}- \frac{1}{2}\Bigr{)}\sin(k_{1}\tilde{x}_{t})\cos(k_{2}y)+k_{2}\cos(k_{1} \tilde{x}_{t})\sin(k_{2}y)\Bigr{]}\] \[=-\biggl{(}k_{1}^{2}\biggl{(}1+b^{2}\Bigl{(}\frac{t}{t_{\max}}- \frac{1}{2}\Bigr{)}^{2}\biggr{)}+k_{2}^{2}\biggr{)}\Phi_{*}^{(t)}u_{k_{1},k_{2}, \zeta}^{d}(x,y)+2k_{1}k_{2}b\Bigl{(}\frac{t}{t_{\max}}-\frac{1}{2}\Bigr{)} \sin(k_{1}\tilde{x}_{t})\sin(k_{2}y).\]
Then, since \(\sum_{t=0}^{t_{\max}}\Bigl{(}\frac{t}{t_{\max}}-\frac{1}{2}\Bigr{)}=0\) and \(\sum_{t=0}^{t_{\max}}\Bigl{(}\frac{t}{t_{\max}}-\frac{1}{2}\Bigr{)}^{2}=\frac{( t_{\max}+1)(t_{\max}+2)}{12t_{\max}}=\frac{|\Gamma|(|\Gamma|+1)}{12(|\Gamma|-1)}\), we have
\[\Delta^{d}u_{k_{1},k_{2},\zeta}(x,y)\] \[=-\Big{(}k_{1}^{2}\Bigl{(}1+\frac{b^{2}(|\Gamma|+1)}{12(|\Gamma|- 1)}\Bigr{)}+k_{2}^{2}\Big{)}u_{k_{1},k_{2},\zeta}^{d},\]
i.e. each \(u_{k_{1},k_{2},\zeta}^{d}\) is an eigenfunction with eigenvalue \(\lambda_{k_{1},k_{2},\zeta}^{d}:=-k_{1}^{2}\bigl{(}1+\frac{b^{2}(|\Gamma|+1)}{ 12(|\Gamma|-1)}\bigr{)}-k_{2}^{2}\). These eigenfunctions form a complete orthogonal Hilbert basis for \(L^{2}(\mathcal{C},\text{Leb})\), since for \(t^{*}=\frac{t_{\max}}{2}\), the \(L^{2}\)-isometry \(\Phi_{*}^{(t^{*})}:L^{2}(\mathcal{C})\to L^{2}(\mathcal{C})\) sends the functions (99) to the complete orthogonal Hilbert basis (89).
To apply Theorem 3.19, we need a lower bound for \(r_{k}\) for each sufficiently large \(k\). We consider \(k\geq\pi pq+\sqrt{2}(p+2q)+1\), where \(p\geq q\geq 1\) are integers for which \(1+\frac{b^{2}(|\Gamma|+1)}{12(|\mathcal{T}-1)}=\frac{p^{2}}{q^{2}}\). Then each eigenvalue \(\lambda_{k_{1},k_{2},\zeta}^{d}\) can be written
\[\lambda_{k_{1},k_{2},\zeta}^{d}=-\frac{p^{2}}{q^{2}}k_{1}^{2}-k_{2}^{2}. \tag{100}\]
We obtain our bound on \(r_{k}\) in the following steps. First, we show that for \(k_{1}\in\{q,2q,\ldots\}\), the eigenfunctions \(u_{k_{1},\frac{p}{q}k_{1},0}^{d}\) and \(u_{k_{1},\frac{p}{q}k_{1},\frac{p}{q}}^{d}\) have the maximum number of nodal domains, among eigenfunctions of the form (99) with eigenvalue \(\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\). Second, for each \(k_{1}\in\{q,2q,\ldots\}\), we obtain an upper bound for
\[\mathcal{E}(k_{1}):=\#\left\{\lambda_{i_{1},i_{2},\zeta}^{d}: \lambda_{i_{1},i_{2},\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\right\}, \tag{101}\]
the number of eigenvalues \(\lambda_{i_{1},i_{2},\zeta}^{d}\) (with multiplicity) satisfying \(\lambda_{i_{1},i_{2},\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\), and hence put an upper bound on the position of \(\lambda_{k_{1},\frac{p}{q}k_{1},0}^{d}\) in the eigenvalue ordering. Third, we use this bound to show that for each \(k\geq\pi pq+\sqrt{2}(p+2q)+1=(\pi q^{2}+\sqrt{2}q)\sqrt{1+\frac{b^{2}(|\Gamma |+1)}{12(|\Gamma|-1)}}+2\sqrt{2}q+1\), there is some \(k_{1}\in\{q,2q,\ldots\}\) such that \(\lambda_{k_{1},\frac{p}{q}k_{1},0}^{d}\geq\lambda_{k,N}^{d}\), and also to bound the largest such \(k_{1}\) from below. Finally, for this \(k\) and \(k_{1}\), we use the number of nodal domains in \(u_{k_{1},\frac{p}{q}k_{1},0}^{d}\) to give a lower bound on \(r_{k}\), and hence we use Theorem 3.19 to bound \(\lambda_{k,N}^{d}\) in terms of \(h_{r_{k},N}^{d}\).
_Step 1:_ We begin by proving that \(u_{k_{1},\frac{p}{q}k_{1},0}^{d}\) and \(u_{k_{1},\frac{p}{q}k_{1},\frac{p}{q}}^{d}\) have the maximal number of nodal domains among eigenfunctions \(u_{i_{1},i_{2},\zeta}^{d}\) of the form (99) for which \(\lambda_{i_{1},i_{2},\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\). Each eigenfunction \(u_{i_{1},i_{2},\zeta}^{d}\) has \(\max\{2i_{1},1\}(i_{2}+1)\) nodal domains by (99) (since \(\cos\bigl{(}i_{1}\bigl{(}x+\zeta-\frac{b}{2}y\bigr{)}\bigr{)}\) has \(\max\{2i_{1},1\}\) nodal domains and \(\cos(i_{2}y)\) has \(i_{2}+1\) nodal domains). Thus, by (100), maximising the number of nodal domains in \(u_{i_{1},i_{2},\zeta}^{d}\) subject to \(\lambda_{i_{1},i_{2},\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\) is equivalent to solving \(\max\{2i_{1}(i_{2}+1):(i_{1},i_{2})\in\mathbb{Z}_{>0},-\frac{p^{2}}{q^{2}}i_{1} ^{2}-i_{2}^{2}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\}\). By a similar relaxation argument to section 4.2, this is uniquely maximised by \((i_{1},i_{2})=(k_{1},\frac{p}{q}k_{1})\). Hence eigenfunctions \(u_{k_{1},\frac{p}{q}k_{1},\zeta}^{d}\) for \(\zeta\in\{0,\frac{\pi}{2}\}\) maximise the number of nodal domains, among eigenfunctions \(u_{i_{1},i_{2},\zeta}^{d}\) of the form (99) satisfying \(\lambda_{i_{1},i_{2},\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\).
_Step 2:_ Choose any \(k_{1}=q,2q,\ldots\). We can bound \(\mathcal{E}(k_{1})\) (defined in (101)) by considering three cases: eigenvalues \(\lambda_{i_{1},i_{2},\zeta}^{d}\) with \(i_{1},i_{2}\geq 1\), eigenvalues \(\lambda_{i_{1},0,\zeta}^{d}\) with \(i_{1}\geq 1\), and eigenvalues \(\lambda_{0,i_{2},0}^{d}\) for \(i_{2}\geq 0\).
The set \(\{\lambda_{i_{1},i_{2},\zeta}:\lambda_{i_{1},i_{2},\zeta}\geq-2\frac{p^{2}}{q^{2 }}k_{1}^{2},i_{1},i_{2}\geq 1\}\) is in bijection with the set \(\{(i_{1},i_{2},\zeta):\zeta\in\{0,\frac{\pi}{2}\},(i_{1},i_{2})\in\mathbb{Z}_{>0 },-\frac{p^{2}}{q^{2}}i_{1}^{2}-i_{2}^{2}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\}\), by (100). These tuples \((i_{1},i_{2})\) are in bijection with the grid points \((i_{1},i_{2})\) in the positive quadrant \(Q_{pq}\) of the ellipse \(\frac{x^{2}}{2k_{1}^{2}}+\frac{q^{2}y^{2}}{2p^{2}k_{1}^{2}}\leq 1\). The quadrant \(Q_{pq}\) has area \(\frac{\pi p}{2q}k_{1}^{2}\), and each grid
point \((i_{1},i_{2})\in Q_{pq}\) with \(i_{1},i_{2}\geq 1\) is associated with a unit area in \(Q_{pq}\). Therefore, there are at most \(\frac{\pi p}{2q}k_{1}^{2}\) grid points \((i_{1},i_{2})\), so there are at most \(\frac{\pi p}{q}k_{1}^{2}\) tuples \((i_{1},i_{2},\zeta)\), and hence at most \(\frac{\pi p}{q}k_{1}^{2}\) eigenvalues \(\lambda_{i_{1},i_{2},\zeta}^{d}\leq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\) with \(i_{1},i_{2}\geq 1\).
By (100), the eigenvalues \(\lambda_{i_{1},0,\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\) with \(i_{1}\geq 1\) are in bijection with the tuples \((i_{1},\zeta)\) with \(i_{1}\in\mathbb{Z}\cap[1,\sqrt{2}k_{1}]\) and \(\zeta\in\{0,\frac{\pi}{2}\}\), so there are \(2\lfloor\sqrt{2}k_{1}\rfloor\) such eigenvalues. Similarly, the eigenvalues \(\lambda_{0,i_{2},0}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\) are in bijection with the integers \(i_{2}\in\mathbb{Z}\cap[0,\sqrt{2}\frac{p}{q}k_{1}]\), so there are \(\lfloor\sqrt{2}\frac{p}{q}k_{1}\rfloor+1\) such eigenvalues. Combining these three cases, the number \(\mathcal{E}(k_{1})\) of eigenvalues \(\lambda_{i_{1},i_{2},\zeta}^{d}\geq-2\frac{p^{2}}{q^{2}}k_{1}^{2}\), counted with multiplicity, is bounded above by
\[\mathcal{E}(k_{1})\leq\frac{\pi p}{q}k_{1}^{2}+2\lfloor\sqrt{2}k_{1}\rfloor+ \left\lfloor\frac{\sqrt{2}p}{q}k_{1}\right\rfloor+1. \tag{102}\]
_Step 3:_ Equations (100)-(101) imply there are no more than \(\mathcal{E}(1)\) eigenvalues \(\geq\lambda_{q,p,0}^{d}\), so (102) implies there are no more than \(\pi pq_{+}\sqrt{2}(p+2q)+1\) such eigenvalues. Hence for each \(k\geq\pi pq+\sqrt{2}(p+2q)+1\), we have \(\lambda_{q,p,0}^{d}\geq\lambda_{k,N}^{d}\), i.e. for \(k_{1}=q\) we have \(\lambda_{k_{1},\frac{\pi}{q}\tilde{k}_{1},0}^{d}\geq\lambda_{k,N}^{d}\). Define
\[k_{1}:=\max\Bigl{\{}\tilde{k}_{1}\in\{q,2q,\ldots\}:\lambda_{\tilde{k}_{1}, \frac{\pi}{q}\tilde{k}_{1},0}^{d}\geq\lambda_{k,N}^{d}\Bigr{\}}, \tag{103}\]
then each multiple \(\tilde{k}_{1}\) of \(q\) greater than \(k_{1}\) satisfies \(\lambda_{k,N}^{d}\geq\lambda_{\tilde{k}_{1},\frac{\pi}{q}\tilde{k}_{1},0}^{d}\). In particular, by (100), we have \(\lambda_{k,N}^{d}\geq\lambda_{k_{1}+q,\frac{p}{q}(k_{1}+q),0}^{d}=-2\frac{p^{ 2}}{q^{2}}(k_{1}+q)^{2}\). Therefore, since \(\lambda_{k,N}^{d}\) is the \(k\)th-smallest eigenvalue in absolute value, (101) implies \(k\leq\mathcal{E}(k_{1}+q)\). Then (102) yields \(k\leq\frac{\pi p}{q}(k_{1}+q)^{2}+2\sqrt{2}(k_{1}+q)+\frac{\sqrt{2}p}{q}(k_{1} +q)+1\). Applying the quadratic formula yields \(k_{1}\geq\sqrt{\frac{qk}{\pi p}-\frac{q}{\pi p}+\frac{1}{2\pi^{2}}(\frac{2q}{ p}+1)^{2}}-(\frac{1}{\sqrt{2}\pi}+\frac{\sqrt{2}q}{\pi p}+q)\). Noting that \(\frac{1}{2\pi^{2}}(\frac{2q}{p}+1)^{2}-\frac{q}{\pi p}>\frac{1}{2\pi^{2}}( \frac{2q}{p}-1)^{2}>0+\), we obtain
\[k_{1}\geq\sqrt{\frac{qk}{\pi p}}-\Biggl{(}\frac{1}{\sqrt{2}\pi}+\frac{\sqrt{2} q}{\pi p}+q\Biggr{)}. \tag{104}\]
_Step 4:_ Choose \(k\) and \(k_{1}\) as in step 3, so that \(\lambda_{k_{1},\frac{p}{q}k_{1},0}^{d}\geq\lambda_{k,N}^{d}\) by (103). Then the number of nodal domains in \(u_{k_{1},\frac{p}{q}k_{1},0}^{d}\) gives a lower bound on \(r_{k}\). This eigenfunction has \(2k_{1}(\frac{p}{q}k_{1}+1)\) nodal domains by the reasoning in step 1, so \(r_{k}\geq 2k_{1}(\frac{p}{q}k_{1}+1)\). Substituting (104) into this expression gives \(r_{k}\geq 2\Bigl{(}\sqrt{\frac{qk}{\pi p}}-\Bigl{(}\frac{1}{\sqrt{2}\pi}+ \frac{\sqrt{2}q}{\pi p}+q\Bigr{)}\Bigr{)}\Bigl{(}\sqrt{\frac{pk}{\pi q}}-\frac {p}{q}\Bigl{(}\frac{1}{\sqrt{2}\pi}+\frac{\sqrt{2}q}{\pi p}+q\Bigr{)}+1 \Bigr{)}\). Expanding and noting that \(p\geq q\geq 1\) so \(2\sqrt{\frac{qk}{\pi p}}\Bigl{(}1-\frac{2\sqrt{2}}{\pi}\Bigr{)}>0\), \(\frac{2\sqrt{2}p}{\pi}+\frac{4\sqrt{2}q}{\pi}>2q+\frac{\sqrt{2}}{\pi}\) and \(\frac{p}{\pi^{2}q}+\frac{4q}{\pi^{2}p}+\frac{4}{\pi^{2}}>\frac{2\sqrt{2}q}{\pi p}\), we obtain \(r_{k}\geq\frac{2k}{\pi}-2.8\sqrt{pqk}+2pq\). Then the definition of \(p\) and \(q\) before (100) implies \(r_{k}\geq\frac{2k}{\pi}-2.8q\sqrt[4]{1+\frac{b^{2}(\lceil\Gamma\rceil+1)}{12( \lceil\Gamma\rceil-1)}}\sqrt{k}+2q^{2}\sqrt{1+\frac{b^{2}(\lceil\Gamma\rceil+1)}{ 12(\lceil\Gamma\rceil-1)}}\). Substituting \(k_{1}\geq q\) into \(r_{k}\geq 2k_{1}(\frac{p}{q}k_{1}+1)\) instead, we additionally obtain \(r_{k}\geq 2q(p+1)=2q^{2}\sqrt{1+\frac{b^{2}(\lceil\Gamma\rceil+1)}{12( \lceil\Gamma\rceil-1)}}+2q\). Hence, rewriting the definition of \(k\) from step 3 using the definition of \(p\) and \(q\) before (100), Theorem 3.19 implies that for each \(k\geq(\pi q^{2}+\sqrt{2}q)\sqrt{1+\frac{b^{2}(\lceil\Gamma\rceil+1)}{12( \lceil\Gamma\rceil-1)}}+2\sqrt{2}q+1\), we have
\[\lambda_{k,N}^{d}\leq-\frac{1}{4}(h_{r_{k}}^{d})^{2}\leq-\frac{1}{4}\Biggl{(}h_{ \max}^{d}\Bigl{\{}\Bigl{\lceil}\frac{2k}{\pi}-2.8q\sqrt[4]{1+\frac{b^{2}( \lceil\Gamma\rceil+1)}{12(\lceil\Gamma\rceil-1)}}\sqrt{k}+2q^{2}\sqrt{1+\frac{ b^{2}(\lceil\Gamma\rceil+1)}{12(\lceil\Gamma\rceil-1)}}\Bigr{\rceil}.2q^{2}\sqrt{1+\frac{b^{2}(\lceil\Gamma\rceil+1)}{12(\lceil\Gamma\rceil-1)}}+2q \Bigr{\}},N\Biggr{)}^{2}. \tag{105}\]
Asymptotically for large \(k\), this bound becomes \(\lambda_{k,N}^{d}\leq-\frac{1}{4}(h_{\frac{2k}{k}-O(\sqrt{k}),N}^{d})^{2}\), irrespective of the shear strength \(b\) and number of time steps \(|\mathrm{T}|\). Pre-asymptotically for intermediate-sized \(k\), this bound links \(\lambda_{k,N}^{d}\) to \(h_{j,N}^{d}\) for
progressively smaller \(j\) as the shear strength increases. This is because the domain behaves like a cylindrical domain with progressively more mismatched sides, so that gridlike packings of \(\mathcal{C}\) with the optimal aspect ratio for each packing element are rarer. We cannot apply Theorems 3.17 or 3.18, our dynamic versions of Theorems 3.4 and 3.5, because \(\mathcal{C}\) has non-empty boundary.
## 5 Summary
The sequence of the \(k\)th (Neumann or Dirichlet) Cheeger constants for a weighted Riemannian manifold (Definition 2.2) and the corresponding \(k\)-packings with small Cheeger ratio (Definition 2.1) together give a global geometric description of weighted Riemannian manifolds. There are no existing algorithms for computing \(k\)-packings for \(k\geq 2\) with small Cheeger ratio on arbitrary Riemannian manifolds. We proposed some methods for obtaining upper bounds on the Cheeger constants, and for finding packings with quality guarantees, i.e. upper bounds on their Cheeger ratios (Theorem 3.7 and Proposition 3.8). We showed that for any Neumann or Dirichlet eigenfunction, its eigenvalue gives an upper bound on the Cheeger constant corresponding to the number of nodal domains in the eigenfunction (Theorem 3.7). Moreover, we showed that positive-measure collections of the superlevel sets within each nodal domain give rise to packings whose Cheeger ratios are bounded above in terms of the eigenvalue (Proposition 3.8). This bound is straightforward to compute, but it only produces \(k\)-packings from eigenfunctions with \(k\) nodal domains. Sometimes, it is possible to combine geometric information from several eigenfunctions to obtain more features than the number of nodal domains in any single eigenfunction. One obtains disjointly supported functions, each supported on a single feature, by taking linear combinations of eigenfunctions and applying soft thresholding. The sparse eigenbasis approximation (SEBA) algorithm [28] can be used to find suitable linear combinations. We showed that if the separation into disjointly supported sparse functions is successful, then positive-measure collections of the resulting superlevel sets yield packings with an upper bound on their Cheeger ratios (Proposition 3.8). This bound depends only on the largest eigenvalue (in absolute value) and the effectiveness of the separation (i.e. the fraction of the \(L^{2}\) mass of the linear combinations that is preserved by the thresholding operation).
Coherent sets in nonautonomous dynamical systems are sets with small dynamic Cheeger ratio (Definition 3.10). We showed that positive-measure collections of the superlevel sets within each nodal domain of a dynamic Laplacian eigenfunction yield packings consisting of coherent sets, i.e. packings whose dynamic Cheeger ratios are bounded above (Theorem 3.19). Also, as in the static case, it is sometimes possible to obtain more coherent sets than the number of nodal domains in any single eigenfunction, by taking linear combinations of the first \(k\) eigenfunctions and applying soft thresholding. We showed (Proposition 3.20) that positive-measure collections of the resulting superlevel sets have their dynamic Cheeger ratios bounded above in terms of the largest eigenvalue (in absolute value), and the effectiveness of the separation (fraction of \(L^{2}\) mass preserved by soft thresholding).
| この論文は、ラプラシアン-ベルトラミ作用素の固有値と固有関数、そして滑らかな Riemannian マノフの、高次のチェーガー定数に焦点を当てています。特に、重み付きまたは境界を持つ場合もあります。高次のチェーガー定数は、マノフの主要な幾何学的特徴をゆるい形で記述しています。私たちは、その高次のチェーガー定数の構築可能な上限を、対応するノダ domains の数に依存する固有関数の固有値を用いて示しました。具体的には、各固有関数のすべての超レベルセットのチェーガー比は、対応する固有値の上限に制限されます。あるマノフは、複数の固有関数で主要な特徴が絡み合っているため、単一の固有関数にすべての主要な特徴が含まれることはありません。この場合、各固有関数には、単一の特徴に大きな |
2308.15785 | Collaborative, Code-Proximal Dynamic Software Visualization within Code
Editors | Software visualizations are usually realized as standalone and isolated tools
that use embedded code viewers within the visualization. In the context of
program comprehension, only few approaches integrate visualizations into code
editors, such as integrated development environments. This is surprising since
professional developers consider reading source code as one of the most
important ways to understand software, therefore spend a lot of time with code
editors. In this paper, we introduce the design and proof-of-concept
implementation for a software visualization approach that can be embedded into
code editors. Our contribution differs from related work in that we use dynamic
analysis of a software system's runtime behavior. Additionally, we incorporate
distributed tracing. This enables developers to understand how, for example,
the currently handled source code behaves as a fully deployed, distributed
software system. Our visualization approach enhances common remote pair
programming tools and is collaboratively usable by employing shared code
cities. As a result, user interactions are synchronized between code editor and
visualization, as well as broadcasted to collaborators. To the best of our
knowledge, this is the first approach that combines code editors with
collaboratively usable code cities. Therefore, we conducted a user study to
collect first-time feedback regarding the perceived usefulness and perceived
usability of our approach. We additionally collected logging information to
provide more data regarding time spent in code cities that are embedded in code
editors. Seven teams with two students each participated in that study. The
results show that the majority of participants find our approach useful and
would employ it for their own use. We provide each participant's video
recording, raw results, and all steps to reproduce our experiment as
supplementary package. | Alexander Krause-Glau, Wilhelm Hasselbring | 2023-08-30T06:35:40 | http://arxiv.org/abs/2308.15785v1 | # Collaborative, Code-Proximal
###### Abstract
Software visualizations are usually realized as standalone and isolated tools that use embedded code viewers within the visualization. In the context of program comprehension, only few approaches integrate visualizations into code editors, such as integrated development environments. This is surprising since professional developers consider reading source code as one of the most important ways to understand software, therefore spend a lot of time with code editors.
In this paper, we introduce the design and proof-of-concept implementation for a software visualization approach that can be embedded into code editors. Our contribution differs from related work in that we use dynamic analysis of a software system's runtime behavior. Additionally, we incorporate distributed tracing. This enables developers to understand how, for example, the currently handled source code behaves as a fully deployed, distributed software system. Our visualization approach enhances common remote pair programming tools and is collaboratively usable by employing shared code cities. As a result, user interactions are synchronized between code editor and visualization, as well as broadcasted to collaborators.
To the best of our knowledge, this is the first approach that combines code editors with collaboratively usable code cities. Therefore, we conducted a user study to collect first-time feedback regarding the perceived usefulness and perceived usability of our approach. We additionally collected logging information to provide more data regarding time spent in code cities that are embedded in code editors. Seven teams with two students each participated in that study. The results show that the majority of participants find our approach useful and would employ it for their own use. We provide each participant's video recording, raw results, and all steps to reproduce our experiment as supplementary package. Furthermore, a live demo of our tool is available online.1 We invite other researchers to extend our open-source software.2 Video URL: [https://youtu.be/3qZVSehnEug](https://youtu.be/3qZVSehnEug)
Footnote 1: [https://code.explorviz.dev](https://code.explorviz.dev)
software visualization, dynamic analysis, program comprehension, pair programming, integrated development
## I Introduction
Source code comprehension is still the primary method to come to an understanding of a software system's behavior [1]. This is not unexpected, because developers are trained to recognize recurring patterns and resulting behavior in source code. They even might spend most of their development time in integrated development environments (IDE) [2]. However, navigation in IDEs leads to a redundant but unavoidable overhead [3] and in terms of software visualization (SV) developers are concerned about the context switch caused by standalone SV tools [4]. As a result, code proximity is a necessary property for SV [5, 6] to succeed in its intended area, i.e., professional software development. Code proximity means the ability of the visualization tool to provide easy and fast access to the original, underlying source code [7]. In this context, research approaches have been shown in the past that embed SV into code editors and IDEs (both from now on referred to as _code editor_) to link source code with its visualization [8, 9, 10, 11].
In this paper, we introduce our collaboratively usable SV approach that can be embedded in code editors. In comparison to related approaches, we use dynamic analysis as source for rendering three-dimensional code cities [12, 13]. The SV is linked directly to the source code that is under development within the code editor and vice versa. Therefore, we directly connect runtime behavior with the related program elements, for example, Java methods. User interactions are synchronized between code editor and visualization, as well as broadcasted to collaborators. As proof of concept, we implemented a Visual Studio Code3 (VS Code) extension that realizes our design. We conducted a first-time user study to collect feedback regarding the perceived usefulness and perceived usability of our approach. Furthermore, we collected logging information to provide more data regarding usage statistics of SV that are embedded into code editors. In this study, seven teams with two students each collaboratively used our approach in an onboarding-related scenario. Overall, the results show a highly rated usefulness.
Footnote 3: [https://code.visualstudio.com](https://code.visualstudio.com)
The remainder of this paper is structured as follows. Section II presents the architectural overview and proof of concept implementation for our approach. We proceed by
introducing the envisioned usage scenarios for our approach in Section III. Afterwards, Section IV explains our experimental setup. Section V presents and discusses the results of our study. Then, Section VI introduces related work. Finally, we conclude this paper and present future work in Section VII.
## II Approach
In this section, we present the architectural design and proof of concept implementation for this research work. For that, we build upon our previously published approach named _Software Visualization as a Service_ (SVaaS), i.e., providing an online-accessible and on-demand service for collaborative program comprehension using SV. Due to space constraints, we refer readers to [14] for a description of the basic concepts of our approach.
### _Architectural Design_
Figure 1 shows (a simplified overview of) our approach's architectural design. It is technology independent with the exception of a browser-based SV component. As shown, it is divided into four stages (blue-striped areas). Figure 1-A and Figure 1-B depict the monitoring and analysis stages, respectively. These are the foundation of our SVaaS concept. The analysis pipeline for example can be horizontally scaled out to handle varying load of concurrent users, therefore positively influence the effectiveness of the overall tool [15]. Although data acquisition, analysis, and cloud technologies are important aspects of our concept, a detailed explanation is beyond the scope of this paper. Therefore, we refer readers to [14] for details and focus on the remaining two stages.
The Webserver (Figure 1-C) serves the static files that comprise the web-based SV, i.e., CSS, JavaScript, and HTML. Furthermore, it acts as reverse proxy for clients to connect to the backend services, e.g., to obtain data to be visualized. Users now have two options to link the SV with their code editor:
* For the first option, they can use the standalone SV that runs inside of their web browser and connects to an extension in their code editor (Figure 1-D). The latter acts as gateway between code editor and SV. This is similar to the 'classic' approach for code viewers embedded into SVs and relates to many other works (see Section VI). Interactions that should be linked between code editor and SV, e.g., 'open a class file in the code editor when the related visualization entity was clicked', are synchronized by the Code Editor Service (Figure 1-E).
* For the second option, users can install an extension in their code editor that already includes the Frontend (Figure 1-F). In this case, we do not need an external service to synchronize interaction events, but use a built-in communication mechanism between code editor and its extension. Therefore, we reduce the context switch overhead that occurs when switching from SV to code editor and vice versa [4]. Another advantage of the second option is that it can also be installed in cloud-based code editors that run in browsers. This can be beneficial in some use cases, e.g., onboarding of new developers, as shown in Section III.
Fig. 1: (Simplified) Architectural design of our approach.
Regardless of the selected option, users can collaboratively use the SV. To achieve this, the Collaboration Service (Figure 1-G) broadcasts events, e.g., 'user X opened package Y', to all clients of the same session except the one that triggered an event [16]. The clients then apply the received events to their SV, therefore synchronize their states.
### _Proof of Concept Implementation_
We have prototyped our approach within our SV tool ExplorViz.4 Our tool's development commenced in 2012 [17] and focused on several aspects throughout time, such as development concerns [18, 19] and extended reality [16, 20]. More recently, we use ExplorViz to research collaborative software visualization in the context of program comprehension [16, 21]. ExplorViz currently uses dynamic analysis as source for the visualization. Our depicted SV is configured to visualize the aggregated runtime behavior of ten seconds [14].
Footnote 4: [https://explorviz.dev](https://explorviz.dev)
Figure 2 shows a screenshot of our prototype implementation. We developed a VS Code extension that realizes the previously mentioned design. It can be used as gateway to link the external SV to the code editor or provide the embedded SV instead. Due to space constraints, we focus on the latter and refer readers to our supplementary video. The extension uses an HTML iFrame to embed the web-based Frontend, therefore SV, in VS Code (see Figure 1-F on the previous page). The embedded SV can be switched on or off via the ExplorViz logo button (Figure 2-A). It is automatically placed in a new editor group next to the source code (Figure 2-B). Users can select one of their (currently or previously) analyzed software systems (as shown in the supplementary video) and open the related SV. The latter is provided as three-dimensional code cities using Three.js5 for rendering (Figure 2-C). The embedded Frontend uses cross-origin communication based on the JavaScript Window object to interact with VS Code. Therefore, we do not need an external service that synchronizes the interaction events as it is the case when using the external Frontend or as shown in related works (see Section VI). Every tenth second the Frontend triggers a SV update. For that, it obtains the latest runtime data for the selected software system from the analysis pipeline and updates the visualization if required. Furthermore, the Frontend sends new data to VS Code, which then highlights Java classes and methods that have been used in the aggregated runtime behavior. This is shown by the gutter icons and code lenses in Figure 2-D. Users can click on a code lens to focus the related entity in the SV, e.g., a high-rise building visualizing a Java class. Vice versa, pressing for example on a communication line will cause the file to open and focus on the related method in VS Code. In terms of working together, users can join or host a collaborative session from within the embedded Frontend and use the collaborative features of the SV, e.g., pinging or shared popups (Figure 2-E), to interact with each other (please see [16] for more details). Furthermore, a collaborative session also enables remote pair programming. For VS Code in general, developers can for example use Microsoft's LiveShare extension for VS Code. LiveShare has great features and usability, but uses Microsoft servers that might be not available in the future or cannot be used due to compliance concerns. For the sake of our evaluation's reproducibility, we therefore decided against using an available product such as Microsoft's LiveShare, but developed our own solution (for the user study). This can be seen in Figure 2-F where the live text selection of another user is depicted (as yellow background of OwnerRepository). These text selection events are synchronized by an implementation of the external Code Editor Service (Figure 1-E) using WebSockets for almost real-time communication.
Footnote 5: [https://threejs.org](https://threejs.org)
## III Envisioned Usage Scenarios
Besides using advanced (web) technologies, our approach can be differentiated from related work by the use of dynamic analysis and collaborative SV features. Therefore, we now introduce envisioned usage scenarios that may follow from our approach and related future works.
Fig. 2: Proof of concept implementation – The editor of VS Code displays a Java class. The ExplorViz extension visualizes the associated runtime behavior and adds visual functions to the editor to directly link source code and visualization.
### Scenario 1 (SC1): Facilitate the Onboarding Process
In professional software development, companies utilize different techniques for the onboarding process of new developers. Peer support, product overview, and simple tasks are perceived as useful in that context [22], while finding documentation and technical issues, e.g., setting up a development environment, impede the onboarding process, especially for remote work [23]. We envision a scenario where cloud-based code editors with embedded SVs are prepared to guide new developers step-by-step through a software system's behavior. Users click on a use case of the analyzed (distributed) target system and understand its unfolding via SV. Furthermore, increasingly large portions of the source code (e.g., depending on experience) are directly linked to SV entities. This allows developers to understand which portion of the source code acts in which use cases. The approach can then be used for task-oriented onboarding, where developers also face small tasks to comprehend the software [22, 24]. At any time, users can invite other developers for collaborative comprehension or their mentor and ask for help. Next to voice communication, participants use collaborative features such as synchronized text selection and shared information popups to interact and exchange [16].
### Scenario 2 (SC2): Highlight changes during code reviews
Feature requests and resulting change-based code reviews are commonly used in professional software development [25]. However, reviewers tend to give vacuous feedback and generally report on review tools' limitations when used in complex scenarios [26]. In this context, we see another potential usage scenario for our approach that we outline in the following. A team member is supposed to review source code changes of a colleague. To do this, he or she can click on a link inside of the pull request that opens a prepared, cloud-based code editor with an embedded SV of the new program behavior (due to the source code change). Source code changes are color-coded in the IDE. For understanding the program behavior, it is possible to switch between old and new program behavior in the SV by pressing a button. The colleague who issued the pull request can be invited to the session such that the changes can also be discussed together.
### Scenario 3 (SC3): Integrate Runtime Information into Development Activities
Staging environments are used to test software systems in a production-like environment. We envision code editors informing selected developers about performance problems of a software system installed (e.g., in the staging area). A developer can click on this notification to open the embedded SV. The visualization depicts the runtime behavior which includes the performance problem. It also highlights the entity that introduces the problem, e.g., a method call that took too long to finish. Based on this, developers get runtime information displayed in their code editor and can analyze affected code lines.
## IV Experiment Design and Demographics
Effectiveness is one of the most common properties used to evaluate SV approaches. In that context, Merino et al. [27] present a systematic literature review of SV evaluation. Their work analyzes the literature body of full papers that were published in the SOFTVIS/VISSOFT conferences, resulting in the examination of 181 papers. The authors focus on evaluations that validate the effectiveness of their presented approach. It is mentioned that multiple evaluations omit other variables that can contribute to or generally influence the effectiveness [28], such as recollection and emotions. We share this opinion and argue that we must first evaluate properties such as perceived usefulness, perceived usability, or feature requests to potentially refine a new, exploratory approach. Only afterwards, we should evaluate effectiveness and efficiency with a sufficiently large number of participants in controlled experiments [29]. As a result, we decided to conduct an exploratory user-study first. We designed an experiment in which participants use and evaluate our approach in a task-oriented onboarding process, i.e., in a scenario similar to SC1 (see Section III). In the future, we will also evaluate our approach in other scenarios by using a similar experiment. In this paper however, we developed the experiment with a focus on SC1 due to the approach's prototype implementation, the exploratory nature of the study, and the duration of a single experiment run. As a result, our research questions (RQ) are not concerned about effectiveness or efficiency. Instead, we focus on several aspects to gather qualitative feedback and quantitative results, such as time spent in the embedded SV, to gain first insights into the use of our approach:
* **RQ1**: How do subjects use the embedded SV and code editor during task solving?
* **RQ2**: Is the code editor perceived as more useful than the embedded SV?
* **RQ3**: Do subjects recognize the usefulness of collaborative SV features for specific tasks?
* **RQ4**: What is the general perception of the usefulness and usability of the approach?
* **RQ5**: Is the approach perceived as useful in the envisioned usage scenarios?
We again emphasize that the findings of this contribution should be seen as first insights and indicators for refinements rather than statistically grounded results. However, by answering the research question, we can derive the following main **contributions** of our evaluation:
* Further insights regarding the perceived usefulness of software cities to comprehend runtime behavior.
* First quantitative and qualitative results regarding the perceived usefulness, perceived usability, and usage time for collaborative, code-proximal software cities.
* A supplementary package containing the evaluation's raw results, screen recordings of all participants, and detailed instructions as well as software packages for reproduction [30].
In the following, we now present the participants' demography and our experiment's procedure.
### _Participants_
We invited students of Kiel University that attend the Bachelor's or Master's program in computer science to participate in our user study [31]. The participation was voluntary. All participants could sign up for random group assignment or participate with a fellow student. Each group had the chance to win two out of ten 100 E gift cards for an e-commerce shop [32].
DistributionThe conducted user study included seven groups with two students each. The number of participants is therefore slightly larger than the median participant count (thirteen) in the related literature body [27], but too small to be effectively used in a controlled experiment [33, 29]. With the exception of one group, all other participants within their group knew each other. Five students attend the Master's program, the remaining students are undergraduates in computer science. All participants reported that they intend to become professional software developers.
ExperiencesFigure 3 shows participants' reported experiences with software development based on work experiences. The two students who indicated they had no experience are in the undergraduate program, and one of them also indicated (as only person) that the decision to become a software engineer is not final. The remaining twelve participants have either gained experiences while working as student employee or in private software development. Three participants are additionally involved in open source development. Figure 4 shows the results of various experiment-related aspects that were asked. All participants stated that they have knowledgeable or even better experiences in VS Code. Three persons rate their web development and software architecture experiences at beginner level. One of the participants with no software engineering work experience reported to have no experience in software architecture. Overall, the distribution of experiences match the courses of study, since SVs are often treated as seminar papers in the master's program, for example. However, we probably also see overestimation such as the persons that stated to be at expert level for VS Code and web development, as well as half of the participants stating to have at least knowledgeable experiences in SV. In this context, half of the participants have used ExplorViz at least once in the past. The participants of three groups each have different experiences with ExplorViz.
### _Target System and Task_
ExplorViz' SV visualizes a software system's runtime behavior. However, it is not limited to application tracing of monolithic software systems, but also supports distributed tracing,6 e.g., network requests between applications that use distributed architectures. Since distributed software systems are pretty common nowadays, we incorporated this fact in our experiment. To achieve that, we used the distributed version of the Spring PetClinic7 as target system for the experiment. As done in the past [16] we recorded traces during the execution of use cases within the PetClinic. For the experiment, these were then provided as so called snapshots, i.e., aggregated runtime behavior, to the Frontend, resulting in a structural,'static' SV of dynamic runtime behavior. We decided against using multiple snapshots so as not to overwhelm new users with the amount of features. However, this can be seen in the supplementary video of this work. The participants explored the target system by means of its source code as well as embedded SV and were asked to solve two tasks.
Footnote 6: [https://openteleometry.io](https://openteleometry.io)
Footnote 7: [https://github.com/spring-petclinic/spring-petclinic-microservices](https://github.com/spring-petclinic/spring-petclinic-microservices)
Table I depicts the program comprehension tasks that all participants had to solve during the experiment. We did not use metric analysis tasks such as 'find the class with the
Fig. 4: Participants’ reported experiences for different aspects.
Fig. 3: Participants’ reported experiences with software development based on work experiences (multi-choice).
highest instance count'. Instead, the chosen tasks instructed the participants to structurally comprehend the software and find analogies based on the depicted runtime behavior. Therefore, the tasks of the experiment refer to a scenario as presented in SC1, i.e., a guided, task-oriented introduction for onboarding. With the focus on SC1, we intend to investigate both the non-collaborative and collaborative onboarding process. Therefore, T1 had to be solved alone and served as an introduction to both the target system and the use of our approach. T2 introduced the collaborative features, e.g., shared SV and synchronized text selection events, and asked the participants to work together.
### _Procedure_
In the following, we present the experiment's procedure. For additional information, we refer readers to the second prepared video8 that demonstrates an exemplary experiment run. Overall, the experiment is divided into pre-questionnaire, mid-questionnaires, i.e., questions that had to be solved after each task completion, and post-questionnaire.
Footnote 8: [https://youtu.be/wdkcDDPXeQQ](https://youtu.be/wdkcDDPXeQQ)
The user study took place at Kiel University and included one instructor who also co-authored this paper. The instructor designed and implemented the approach as well as conducted the user study. Although our approach can be used remotely, we decided to have the study take place in one locality, so that the instructor could intervene if necessary. In each experimental run, the participants were first informed about the data that would be recorded and used for publication. After signing a consent form, the instructor gave a brief introduction to VS Code and the embedded SV. It was mentioned that all introduced features were additionally described on a cheat sheet, which was placed on the table in front of the subjects. Afterwards, the participants were told to openly ask questions if they had a problem. Furthermore, they were told that they could pause or abort the experiment at any time. They draw their login token for the survey tool LimeSurvey9 and started with the pre-questionnaire. Then T1 was introduced and all participants were redirected to browser-based VS Code instances by clicking a button inside of the LimeSurvey form. Each VS Code instance was specifically prepared for a given task and ready to use. It did not require any setup, so that the participants could completely focus on the task itself. They began by reading a markdown file that introduced the target system, controls, and the task itself. After answering T1 in LimeSurvey, all participants gave their feedback to the just used approach. T2 was introduced in the same way as T1. However, here participants were instructed to test the collaborative features first and then work together on solving T2. Again, the subjects gave their feedback and concluded with the post-questionnaire. During each experiment run, the instructor made a note of noticeable mentions stated by the participants.
Footnote 9: [https://www.limesurvey.org](https://www.limesurvey.org)
## V Results & Discussion
Our mid-questionnaires and post-questionnaire contained statements for which participants had to indicate their level of (dis)agreement on a 5-point Likert scale. The questionnaires also included free reply fields to leave a comment on any experiment-related matter. Additionally, the instructor made a note of observations such as rational usages of specific features as well as noticeable emotions [27] and mentions of the participants. In the following, we present and use the results of our conducted user study to revisit our posed research questions. Furthermore, we discuss the threats to validity of our evaluation. Although we use the term SV in this paper, we do not want it to be understood as a generalization of our results. We again emphasize that the results and their interpretation are restricted to our particular prototype using collaborative code cities and our experiment. Therefore, the findings should be seen as first insights and indicators for refinements rather than statistically grounded results.
Fig. 5: Total time spent & perceived difficulty per task.
\begin{table}
\begin{tabular}{c l l} \hline \hline ID & Category & Question \\ \hline T1 & Structural Understanding & What do you think is the reason that the ‘Owner’ class is instantiated multiple times, but the other classes in the relevant program flow are instantiated only once? \\ T2 & Software Insight & Name all Java classes that are involved in a program flow to show the visit screen with the new ‘select veterinarian’ feature. \\ \hline \hline \end{tabular}
\end{table} TABLE I: Program comprehension tasks that participants had to solve..
### Task evaluation
We measured an overall task correctness of 90 %. The related time spent solving the tasks is depicted in Figure 5. The average time spent on T1 is for both the mean and median 19 minutes. The fasted participant correctly solved T1 in seven minutes. This person was already familiar with ExplorViz. For T2, we see 29 minutes for the mean and 24 minutes for the median. Both tasks were without time limit, hence the outlier group for T2. Figure 5 also depicts the participants' perceived task difficulty. T1 and T2 were found to be difficult by four participants, with T1 also found to be very difficult by one person. Due to the overall distribution, we conclude that the tasks were neither too easy nor too difficult.
_RQ1: How do subjects use the embedded SV and code editor during task solving?_
To the best of our knowledge, this work presents a novel approach that combines code editors with remote pair programming techniques and embedded, collaborative code cities. Therefore, we first intend to understand how the participants in our study use the approach with free choice of the tool, i.e., embedded SV and code editor, as well as with tasks referenced to SC1. In that context, Figure 6 depicts the time spent using each tool per task. For measurement, a VS code event was used to capture the time at which participants clicked on the code editor or the ExplorViz extension, therefore switched their focused context. We would like to mention that it was technically (due to VS Code's limitations for extensions) only possible to measure the time spent between context switches. Thus, if a participant did not change the context but, for example, only used the SV, then our measurements indicate a time spent of one minute for the SV. This is the case for the fastest participant for T1 mentioned above, who actively interacted only with the SV during this task (as confirmed by the video recording). The average time spent using the SV for T1 is seven minutes and nine minutes for VS Code (both mean and median). During this task, participants comprehended the source code for the first time and probably spent more time reading it. It is therefore surprising that the time difference for the first task is already quite small. The reason for this is that code cities can facilitate the understanding of structures and are therefore suitable for the task of obtaining an overview [34, 35]. This was also explicitly mentioned by three participants in the free text fields. For T2, the average time spent using the SV is fifteen minutes and eight minutes for VS Code. The (almost) double amount of time spent using the SV results from the two outliers. For this task, however, the median for time spent using the SV is thirteen minutes and eight minutes for VS code. We suppose that this comes from the shared software cities and the ability to highlight objects in question. The instructor's notes mention the frequent use of shared popups within two groups. The video recordings confirm that these groups often use the popups as a basis for discussion. Also, participants often use the ping feature of our tool to highlight certain details for their collaborator. Therefore, they spent more time using the SV. However, collaboration is not the only reason for that. T2 explicitly requires to understand and extend a program flow. The SV provides a visual overview of the software system's structure and in our case also of a runtime behavior snapshot (see Section IV-B). As a result, it is far easier and obvious to use this available visualization and for example trace imaginary method calls with the mouse cursor (especially, when combined with collaborative features).
Figure 6 also presents the number of context switches for each task. We observe that for T1 the number of switches between SV and code editor is much more distributed among the participants than for T2. Again, the reason for that is presumably the collaboration in T2. Most of the time, the participants work together and therefore change their tool when initiated by the other collaborator. For both T1 and T2, the median of context switches is around forty, indicating that the amount of context switches is independent on our tasks and collaboration.
Since our approach incorporates the runtime behavior of the target system, we also intended to know how participants perceived the usefulness of the two tools to comprehend the posed program flow of T1. In this context, Figure 7 shows that the SV was perceived as more useful than the code editor. One participant mentioned that the communication lines are one of the most beneficial properties of the SV. In ExplorViz, the communication lines incorporate runtime information such as the method call's frequency in the visualized snapshot. These information are important to comprehend runtime behavior. Additionally, the SV already maps the runtime information that the users would otherwise have to find and understand on their own.
We conclude that the participants used the SV as supplement to the code editor for specific comprehension tasks.
Fig. 6: Time spent per tool & number of context switches performed per task.
_RQ2_: _Is the code editor perceived as more useful than the embedded SV?_
Traditionally, understanding a software system's behavior is primarily achieved by comprehending the source code [1]. For this experiment, the results related to RQ1 show that our approach was, for example, used by the participants to gain an overview of the target system. This is a common and suitable use case for SV, as shown in the past [34]. However, professional developers question the need for SV [15, 36]. In our opinion, one of the reasons for that is the lack of properties such as code proximity [5, 6] and the SV tool's setup [4]. In that context, we now examine how participants rate the usefulness of our approach.
Figure 7 depicts the results of the mid-questionnaires regarding the perceived usefulness of the tools for a task. For T1, overall 71 % agree with the posed statement 'SV helped with the task'. The usefulness of the code editor was slightly (one person difference) more agreed to. However, for the SV the number of participants who neither agree nor disagree is higher and those who disagree is lower. Regarding T2, we see that overall 86 % agree with the posed statement 'SV helped with the task'. In comparison, the code editor's usefulness was slightly (one person difference) less agreed to.
We conclude that the participants perceive code editor and SV as approximately equally useful (in the context of the task solving).
_RQ3_: _Do subjects recognize the usefulness of collaborative SV features for specific tasks?_
With RQ3, we expand the results of our previous work [16] regarding the perceived usefulness of collaborative code cities. In this context, we asked all participants to state their level of agreement with two statements posed.
Figure 8 presents the related results. We see that 43 % of the participants agree or strongly agree with the statement 'Collaborative SV features helped with the task', respectively. The one person that disagrees with the statement mentioned that the collaborative SV features did not help in his case, since there was barely any input from the other participant. However, he agrees that the communication would be a big help in pair programming supported development in the real world. Presumably due to the low contribution of his collaborator, the same person also disagrees with the second statement that is concerned about voice communication. Due to low input from his collaborator, the same person also disagrees with the second statement, which refers to the perceived usefulness of voice communication. Nevertheless, all of the remaining thirteen participants strongly agree that voice communication was helpful in the task. This is consistent with our previous findings indicating that voice communication is one of the most useful collaborative tools in SV [16].
We conclude that the majority of participants perceive the collaborative SV features as useful in the given task.
_RQ4_: _What is the general perception of the usefulness and usability of the approach?_
The post-questionnaire was designed to capture participants' overall perceptions of the approach's usefulness and usability. By answering RQ1, we have seen that the participants indeed use the SV as supplement during the comprehension task. For RQ2, we concluded that participants perceived code editor and SV to be about equally useful in the context of a real-world task. Finally, Figure 9 shows:
Fig. 8: Mid-questionnaire - T2 - Collaboration
Fig. 7: Mid-questionnaires - Perceived usefulness for tasks
All participants agree or strongly agree that the SV's code proximity is generally useful.
Collaboration is obviously dependent on many factors, e.g., mutual perception of collaborators or motivation. In our context, we have seen this for RQ3 or in previously published results [16]. The participants rate the collaborative SV features slightly different when to be evaluated independently of a task. Figure 9 shows a shift in the distribution of approval ratings. The one person who previously disagreed with the usefulness of the collaborative features now neither agrees nor disagrees. That fits his previous mentions. Compared to perceived usefulness for T2, overall perceived usefulness of collaborative SV features shows less strong agreement. As a matter of fact, we could not find a reason why two participants downgraded their level of agreement to 'agree'. However, the overall approval rate remains the same.
We conclude that the majority of subjects perceive the collaborative SV features as useful.
Although this evaluation is overall more concerned about the perceived usefulness of embedded SV, identified usability problems can help to identify desirable refinements. In this context, Figure 9 also presents the participant's perceived usability of our approach. The results show that 86 % of the participants find the used combination of embedded SV and code editor usable. There are some desirable improvements that are mentioned via text response, e.g., better performance. However, the biggest usability problem was the unintended minimization of the embedded SV. The reason for that is that VS code opens files that have been clicked in the package explorer in the currently focused editor group. This behavior can be disabled by locking an editor group. However, at the current time of writing, the lock mechanism cannot be triggered from within a VS Code extension. Figure 9 also shows that another 86 % would use this approach for private purposes such as collaborative program comprehension with fellow students.
We conclude that the majority of participants find our approach usable.
_Rq5: Is the approach perceived as useful in the envisioned usage scenarios?_
Our pilot study found that a single experiment run would take about an hour to complete. In order not to discourage potential participants due to the time to be spent, we decided to ignore the other usage scenarios and only use tasks in the experiment based on SC1. Nevertheless, the post-questionnaire was also used to capture participants' perceived usefulness in applying the approach in the remaining, envisioned scenarios. In this case, they were described in text and subjects were asked to state their agreement on a 5-point Likert scale. Figure 10 depicts the related results. The complete scenario descriptions are available in the supplementary package of this paper [30], but essentially summarize the envisioned usage scenarios in Section III. The participants rated SC1 with the highest overall agreement and strong agreement, respectively. The experiment's tasks and their introduction originate from SC1. SC2 has the highest amount of neutrality and disagreement. One person that answered with neither agreement nor disagreement mentioned that code changes are usually reviewed before deploying them. Since our approach only shows runtime behavior, he is not sure how changes will be visualized for the code review. This detail was in fact omitted in the textual description of SC2. We believe that this uncertainty is the reason for the highest amount of neutrality and disagreement for SC2. However, the majority consensus was positive for all scenarios.
We conclude that the majority of subjects find the application of our approach useful in the posed scenarios.
Fig. 10: Post-questionnaire – Perceived usefulness of the approach when applied in a described scenario. (see Section III)
Fig. 9: Post-questionnaire - Perceived usefulness and usability
### _Threats to Validity_
Remote pair programming solutionAs mentioned in Section II-B, we decided to implement our own remote pair programming approach, so that the reproducibility of our evaluation is not dependent on the availability of external services. However, this custom implementation lacks useful features compared to full-fledged solutions for remote pair programming. For example, one participant mentioned that he was unable to draw the attention of the collaborator to a specific code part. Although our study did not aim to evaluate whether one tool is better than the other, this custom implementation may have influenced the perceived usefulness or usability of the SV or code editor. In contrast, Figure 7 shows that the participants find the SV to be more suitable to understand dynamic program flows. With that being said, we conclude that more empirical research is required in this context.
Experiment durationThe average time spent on the user study was about one hour, both median and mean. It follows that the attention span of the participants and thus the results might have been influenced. To mitigate this, we told participants during the introduction that breaks could be taken at any time and participation could be aborted. Moreover, T2 was solved collaboratively and therefore presumably relieved the experimental situation.
Target systemThe prepared target system contains 26 application logic-related Java files that are distributed among four Maven subprojects. As a result, the small project size may have influenced the perceived usability of the SV, as also mentioned by one participant. We agree, but also emphasize that we did not intend to evaluate usability based on the scalability of the visualization, but on the overall concept. Overall, this evaluation is more concerned about the perceived usefulness of SV incorporating distributed tracing for the onboarding process. In addition, we argue that a real-world application of the onboarding scenario with SV should guide new developers through the software system's behavior with increasingly large portions of the code base.
ParticipantsThe use of students in experiments is a valid simplification that is often said to possibly compromise external validity [31]. In our case, the participants' experiences might have influenced their perception regarding the usefulness of the SV as well as their time spent using the SV. In this context, professional developers can benefit from their experience, e.g. with the Spring framework, and can understand the source code faster. As a result, we will repeat the experiment with professional developers.
## VI Related Work
Code proximity is often not clearly emphasized in SV publications, but follows from the mentions of code viewers in the text itself. Therefore, there are numerous research approaches that use embedded code viewers within SV such as code cities [34]. This also applies to more recent and often collaboratively usable virtual reality approaches [37, 38, 39, 40, 41]. Other publications present different use cases for embedded SV in code editors, such as dependency browsing [42] or sketching [43, 44]. Few approaches enable developers to modify source code via embedded code editors [45]. Due to space limitations, we cannot address and discuss all related approaches [10, 8, 11], but focus below on what we consider to be the most comparable work.
In 2015, Balogh et. al presented a refined version of their tool CodeMetropolis [9]. Their approach uses static analysis of a software systems' source code and visualizes the result as 3D code city. The related rendering is achieved using a modded version of the video game Minecraft. Thanks to the multiplayer mode of Minecraft, the code city can also be explored collaboratively. Overall, CodeMetropolis and ExplorViz share the aspects of collaboration and code editor integration. However, these features are implemented differently in each case. For example, in CodeMetropolis users navigate through the same instance of a given code city using the first-person perspective. They can see the avatars of collaborators and interact based on Minecraft's limitations. In ExplorViz, the collaboration is achieved using collaborative SV features, e.g., shared popups. Regarding the code editor integration, both CodeMetropolis and ExplorViz provide an extension than can be installed in Eclipse and VS Code, respectively. In this context, both extensions provide a comparable set of features, e.g., open Java class in the SV. However, our extension is also able to embed the SV in the actual code editor, whereas the Metropolis approach can only be used as external SV that links to the code editor (see Section II-A).
## VII Conclusions & Future Work
In this paper, we presented the architectural design of our approach for collaborative, code-proximal dynamic software cities within code editors. The main idea is to link collaborative SVs directly to the source code that is under development within a code editor and vice versa. We have prototyped this approach within our SV tool ExplorViz. The result is a VS Code extension that either embeds three-dimensional software cities in the code editor or acts as gateway between code editor and external SV. Therefore, we directly link runtime behavior with the related program elements, for example, Java methods. Users can collaboratively explore the SV from within their code editor using synchronized software cities and collaborative SV features, e.g., shared popups. In addition to the implementation, we sketched three envisioned usage scenarios.
We conducted an initial user study to collect first-time feedback regarding the perceived usefulness and perceived usability of our approach. The results show that the majority of participants generally perceive the approach as useful and usable. In this context, participants rated code editor and SV as equally useful in solving the given program comprehension tasks. The measured time spent in each tool, i.e., SV and code editor, indicates that the participants indeed use the SV as supplementary tool.
In the future, we will implement useful features and refinements. Additionally, we plan to repeat the experiment with professional developers.
## Acknowledgment
The authors would like to thank Malte Hansen and Lennart Ideler for their contributions with implementing and evaluating some of the features presented in this paper. | ソフトウェア可視化は、通常、独立したツールとして実現され、コードビューアーを埋め込み、コードを可視化するためのツールとして使用されます。プログラム理解のコンテキストでは、可視化がコードエディターに統合されるアプローチは限られており、例えば統合開発環境などです。これは、プロフェッショナルな開発者はソースコードを読み込むことがソフトウェアを理解する上で重要な方法であると考えており、多くの時間をコードエディターで過ごしていることに対する驚きの事実です。この論文では、コードエディターに埋め込むことができるソフトウェア可視化アプローチの設計と実証的な実装を導入します。本アプローチとの関連研究との違いは、ソフトウェアシステムのランタイム挙動の動的分析を用いることです。また、分散追跡も組み込まれています。これにより、開発者は、例えば、現在処理されているソースコードがデプロイされた、分散型のソフトウェアシステムとしてどのように動作しているか理解することができます。私たちの可視 |
2304.03232 | Computationally-efficient Motion Cueing Algorithm via Model Predictive
Control | Driving simulators have been used in the automotive industry for many years
because of their ability to perform tests in a safe, reproducible and
controlled immersive virtual environment. The improved performance of the
simulator and its ability to recreate in-vehicle experience for the user is
established through motion cueing algorithms (MCA). Such algorithms have
constantly been developed with model predictive control (MPC) acting as the
main control technique. Currently, available MPC-based methods either compute
the optimal controller online or derive an explicit control law offline. These
approaches limit the applicability of the MCA for real-time applications due to
online computational costs and/or offline memory storage issues. This research
presents a solution to deal with issues of offline and online solving through a
hybrid approach. For this, an explicit MPC is used to generate a look-up table
to provide an initial guess as a warm-start for the implicit MPC-based MCA.
From the simulations, it is observed that the presented hybrid approach is able
to reduce online computation load by shifting it offline using the explicit
controller. Further, the algorithm demonstrates a good tracking performance
with a significant reduction of computation time in a complex driving scenario
using an emulator environment of a driving simulator. | Akhil Chadha, Vishrut Jain, Andrea Michelle Rios Lazcano, Barys Shyrokau | 2023-04-06T17:10:07 | http://arxiv.org/abs/2304.03232v1 | # Computationally-efficient Motion Cueing Algorithm via Model Predictive Control
###### Abstract
Driving simulators have been used in the automotive industry for many years because of their ability to perform tests in a safe, reproducible and controlled immersive virtual environment. The improved performance of the simulator and its ability to recreate in-vehicle experience for the user is established through motion cueing algorithms (MCA). Such algorithms have constantly been developed with model predictive control (MPC) acting as the main control technique. Currently, available MPC-based methods either compute the optimal controller online or derive an explicit control law offline. These approaches limit the applicability of the MCA for real-time applications due to online computational costs and/or offline memory storage issues. This research presents a solution to deal with issues of offline and online solving through a hybrid approach. For this, an explicit MPC is used to generate a look-up table to provide an initial guess as a warm-start for the implicit MPC-based MCA. From the simulations, it is observed that the presented hybrid approach is able to reduce online computation load by shifting it offline using the explicit controller. Further, the algorithm demonstrates a good tracking performance with a significant reduction of computation time in a complex driving scenario using an emulator environment of a driving simulator.
Motion cueing algorithm, driving simulator, model predictive control
## I Introduction
Driving simulators are frequently used for development and testing in the automotive domain [1]. The virtual environment of the simulator helps to recreate the in-vehicle experience without any damage dealt to the real vehicle. To achieve this, an MCA is used acting as the control technique for the driving simulator's movements. It governs the process allowing the simulator to function properly so that a similar feeling of motion is experienced by the user and to maximise the workspace utilization [2].
In motion cueing, driver input is sent to the vehicle model which generates the reference signal to be tracked. The MCA computes the desired platform motion to follow these reference signals and commands it to the platform as specific forces and rotational accelerations. The notion of specific force is exploited for the recreation of in-vehicle experience. The sensed specific force \(f_{spec,s}\) comprises two components: platform translational acceleration, \(a_{tran,p}\) and the gravitational acceleration, which allows us to study the human body's movement in space during the cueing process. This is compared with the actual specific force value \(f_{spec,a}\) obtained from the real vehicle. The computed error is then fed back into the MCA to improve results for the next time step. Based on this, several kinds of MCAs have been developed, which differ in terms of control techniques used.
Conventional filter-based algorithms use the concept of high and low pass filters to reproduce the on-road experience within the virtual environment [3]. They operate using three main channels. The first is the translational channel which takes in translational accelerations as input. It uses a high pass filter to filter out sustained low-frequency accelerations, which can drive the simulator to its physical limits [3, 4, 5, 6]. These filtered low-frequency accelerations are then recreated using tilt-coordination in the tilt channel [7]. Lastly, a rotational channel is present, which is similar to the translational channel.
Based on the same principle, other kinds of conventional algorithms have been developed such as the optimal and adaptive washout algorithms. The main drawback of such algorithms is their inability to take explicit constraints into account, leading to poor workspace utilization. Furthermore, some of these approaches like the classical washout algorithm are feed-forward techniques which result in poor performance. To overcome these problems, MPC-based MCAs are commonly used.
MPC has been used in MCAs for over a decade considering two different approaches. The first approach is the implicit controller, which solves the optimization problem online at each time step. Initially, linear MPC-based MCAs were developed [7, 8]. They outperformed the conventional methods but provided sub-optimal results, as the non-linear dynamics were not taken into account. Further, they employed constraints in the driver reference frame, to keep the problem linear, resulting in difficulties in realizing the available workspace. To solve these issues, nonlinear MCAs have been proposed [9], constraining the actuator lengths and showing performance improvement compared with the linear MPC-based MCA. A nonlinear MPC-based MCA with actuator constraints was also developed in [6]. Perception thresholds were applied to reduce false cues, additionally, adaptive weights were introduced for washout effect, which improved tracking performance. A
similar algorithm has been developed involving perception thresholds, which uses a separate optimal control problem to predict future driver behaviour [10]. Such MPC-based MCAs provide better performance compared to conventional and linear MPC-based algorithms; however, they suffer from high online computation costs resulting in these algorithms not being real-time implementable.
To reduce computational costs, an alternative approach of MPC-based cueing has been proposed. Explicit MPC has been developed, which pre-computes the solution and then uses it in the form of a look-up table online. This method significantly reduces online computation time [11]. A 2 DoF MCA was developed which was later extended by incorporating a vestibular model in [12]. Although this technique reduces online computation time, it suffers from memory storage issues along with restrictions in using large prediction horizons \(N_{p}\) with fast sampling rates. This is due to the exponential increase in control region computation time with an increase in the complexity and scope of the problem.
To overcome issues faced by implicit and explicit MPCs, a hybrid approach has been developed by Zeilinger [13]. An explicit controller provides an initial guess for the online optimization problem. The guess acts as a warm-start resulting in faster computation of the optimal control input. Since its inception, this technique has been used in applications such as curve tilting [14] and lateral motion stabilisation [15].
he main contribution of the paper is a hybrid motion cueing approach using explicit and implicit MPCs. The proposed algorithm increases the computational efficiency without degradation of the tracking performance. The algorithm outperforms the state-of-the-art MPC-based MCA described in subsection III-A), in terms of computational performance.
The paper is structured as follows. In Section II, the controller design is explained including information about both MPCs in the hybrid scheme. The test setup and simulations performed are presented in Section III. Conclusions and recommendations are listed in Section IV.
## II Methodology
### _Hybrid Scheme_
The design of the hybrid MPC-based MCA comprises two main components: initialisation using explicit MPC and online computation using the implicit controller. A general scheme of the MCA is shown in Figure 1. As the first step, the initial states and reference values are sent to the explicit MPC. This block searches for the corresponding control region related to the states and reference values, the associated control inputs are then provided to the online nonlinear solver (implicit MPC), as the initial guess. With the information of the initial guess along with the current states and the reference signals, the implicit controller computes the optimised control inputs. These inputs are fed to the plant model and the states are updated for the next time step. Once the state update is complete, the entire process is repeated.
### _Explicit MPC_
The explicit MPC is used to compute a look-up table that provides the online solver's initial guess. This comprises states and reference values stored in the form of control regions. Each control region corresponds to a particular control input value generated as follows [16, 17]:
\[U(x)=F_{i}x+G_{i}\text{ if }x\in\mathcal{CR}_{i}. \tag{1}\]
where, \(\mathcal{CR}_{i}\) are the control regions, to which the vectors \(F_{i}\) and \(G_{i}\) correspond.
To generate the look-up table, an MCA is designed considering 4 DoFs of the driving simulator using the Multi Parametric Toolbox (MPT). The algorithm is a simplified version of the online implicit controller to provide an educated guess for the warm-start strategy. Eight states are used in the model considering the platform displacement \(s_{p}\), platform velocity \(v_{p}\), tilt angle \(\theta_{p}\) and tilt rate \(\omega_{p}\) for pitch-surge and sway-roll DoFs. The state space equations are shown in (2):
\[\dot{x}(k)=\left\{\begin{array}{l}\dot{\omega}_{p,long}=a_{p,long,rot}\\ \dot{\theta}_{p,long}=\omega_{p,long}\\ \dot{v}_{p,long}=a_{p,long,tran}\\ \dot{s}_{p,long}=v_{p,long}\\ \dot{\omega}_{p,lat}=a_{p,lat,rot}\\ \dot{\theta}_{p,lat}=\omega_{p}\\ \dot{v}_{p,lat}=a_{p,lat,tran}\\ \dot{s}_{p,lat}=v_{p,lat}\\ \end{array}\right. \tag{2}\]
Here, the subscripts \({}^{\prime}long^{\prime}\) and \({}^{\prime}lat^{\prime}\) refer to the pitch-surge (longitudinal) and sway-roll (lateral) DoFs respectively. Also, this problem contains four control inputs \(u(k)\), comprising translational and rotational accelerations acting in both longitudinal and lateral directions:
\[u(k)=[a_{p,long,rot},a_{p,long,tran},a_{p,lat,rot},a_{p,lat,tran}] \tag{3}\]
Thus, the combined system can be represented as follows:
\[\dot{x}(k)=f(x(k),u(k)) \tag{4}\]
Constraints are applied in the MCA to limit the movements of the motion platform, according to the driving simulator's capabilities. Firstly, the tilt rate is constrained according to the perception thresholds of pitch and roll movements, to ensure that the driver does not perceive the tilting action. Generally,
Fig. 1: Hybrid MPC scheme for the proposed motion cueing algorithm
a lower value in the range of \(2\)-\(4\) deg/s is used [6, 10, 18] and for the proposed MCA, \(3\) and \(2.6\) deg/s were chosen for pitch and roll tilt rates respectively.
Secondly, constraints are applied to the platform displacement to limit the platform within the workspace envelope. Since 4 DoFs are considered, the workspace envelope can be represented by \(\sqrt{s_{p,long}^{2}+s_{p,lat}^{2}}\leq s_{max}^{2}\). The explicit MPC is defined using the MPT toolbox where non-linear constraints can not be added. Thus, the constraint described above is only applied in implicit MPC. For the explicit MPC, the platform displacement limits are imposed separately with a value of \(\sqrt{s_{max}^{2}/2}\) in both the longitudinal and lateral directions. As the explicit controller only provides the initial guess for the actual solution, using a marginally different constraint for the displacement does not affect the final solution. The implicit controller produces the final solution. The constraints used in the problem are summarised below:
\[\begin{split}-3deg/s&\leq\omega_{p,long}\leq 3 deg/s\\ -2.6deg/s&\leq\omega_{p,lat}\leq 2.6deg/s\\ -30deg&\leq\theta_{p}\leq 30deg\\ -7.2m/s&\leq v_{p}\leq 7.2m/s\\ -0.35m&\leq s_{p}\leq 0.35m\\ -9.81m/s^{2}&\leq a_{p}\leq 9.81m/s^{2}\end{split} \tag{5}\]
Here, subscript \({}^{\prime}p^{\prime}\) alone (without \({}^{\prime}long^{\prime}\) or \({}^{\prime}lat^{\prime}\)) represents that both longitudinal and lateral counterparts have the same constraint limit. The goal of this MCA is to track the reference specific force defined by the two vector components: translational and gravitational tilt accelerations. Rotations with respect to the \(x\) and \(y\) axis are used in deriving the gravitational tilt components which are as follows:
\[g_{tilt}=\left\{\begin{array}{l}g_{long}=g\sin\theta_{p,long}\\ g_{lat}=-g\cos\theta_{p,long}\sin\theta_{p,lat}\end{array}\right. \tag{6}\]
Taking the translational accelerations into account, the specific force is given by:
\[y(k)=\left\{\begin{array}{l}f_{spec,long}=a_{p,long,tran}+g\sin\theta_{p, long}\\ f_{spec,lat}=a_{p,lat,tran}-g\cos\theta_{p,long}\sin\theta_{p,lat}\end{array}\right. \tag{7}\]
Furthermore, the objective function consists of weighted states, specific forces and control inputs. As the states are already constrained, a value of \(0\) is assigned to allow freedom of movement in the available workspace. Further, the highest weights are given to the specific forces to achieve their tracking. Thus for the objective function, a weight of 1 is selected for specific force (output), and the inputs namely translational and angular accelerations are penalised with a weight of \(1e-3\). The cost function can be defined as:
\[J_{ex}=\sum_{i=0}^{N_{c}}[y_{k}-y_{ref}]^{T}w_{f}[y_{k}-y_{ref}]+x_{p}^{T}\ w_{x}\ x_{p}+u^{T}\ w_{u}\ u \tag{8}\]
where \(y_{ref}\) is the reference specific force, \(w_{f}\) is the weight for specific force tracking, \(x_{p}\) are the states of the motion platform, \(w_{x}\) are weights on the states to obtain washout effect. Lastly, \(u\) are the control inputs and \(w_{u}\) corresponds to the weights on the inputs to restrict them.
### _Implicit MPC_
The second part of the hybrid approach is the online implicit MPC-based algorithm. This algorithm is able to take nonlinear constraints into account and is designed using ACADO optimisation toolbox in MATLAB. The formulation of the implicit MCA is as follows:
\[\begin{split}\min_{u\,N_{p}}& J\left(x_{0},u\right) \\ \text{s.t.},& x(k+1)=f(x(k),u(k))\\ & x\in\chi_{i}\\ & x(N)\in\mathbb{X}_{f}\end{split} \tag{9}\]
The cost function in Equation 9 is defined as:
\[J_{im}=\sum_{i=0}^{N_{c}}[y_{k}-y_{ref}]^{T}w_{f}[y_{k}-y_{ref}]+x_{p}^{T}\ w _{x}\ x_{p}+u^{T}\ w_{u}\ u \tag{10}\]
The cost function of the implicit MPC is similar to explicit MPC, apart from the addition of a few extra states corresponding to commanded inputs. The states \(x(k)\) of the cueing algorithm are also updated by adding the platform accelerations, previously used as the control inputs. Commanded acceleration values that include a first-order time delay are now employed as control inputs. The state space model is presented in Equation 11:
\[\dot{x}(k)=\left\{\begin{array}{l}\dot{\omega}_{p,long}=a_{p,long,rot}\\ \dot{\theta}_{p,long}=\omega_{p,long}\\ \dot{v}_{p,long}=a_{p,long,tran}\\ \dot{s}_{p,long}=v_{p,long}\\ \dot{\omega}_{p,lat}=a_{p,lat,rot}\\ \dot{\theta}_{p,lat}=\omega_{p}\\ \dot{v}_{p,lat}=a_{p,lat,tran}\\ \dot{s}_{p,lat}=v_{p,lat}\\ \dot{a}_{p,long,tran}=\frac{a_{cmd,long,tran}-a_{p,long,tran}}{T_{s}}\\ \dot{a}_{p,long,rot}=\frac{a_{cmd,long,rot}-a_{p,long,rot}}{T_{s}}\\ \dot{a}_{p,lat,tran}=\frac{a_{cmd,lat,tran}-a_{p,lat,tran}}{T_{s}}\\ \dot{a}_{p,lat,rot}=\frac{a_{cmd,lat,rot}-a_{p,lat,rot}}{T_{s}}\end{array}\right. \tag{11}\]
The implicit controller allows us to consider the constraints of the working envelope directly. Apart from this, additional braking constraints are incorporated [11]. As the workspace limits approach, braking constraints help in slowing down the platform velocity and tilt rate. Two sets of constraints are used:
one for platform displacement and the other for the tilt angle as follows:
\[s_{p,min}\leq s_{p}+c_{v}v_{p}T_{brk,p}+0.5c_{u}a_{p,ran}T_{brk,p}^{2}\leq s_{p,max} \tag{12}\]
\[\theta_{p,min}\leq\theta_{p}+c_{w}\omega_{p}T_{brk,\theta}+0.5c_{u}a_{p,rot}T_{ brk,\theta}^{2}\leq\theta_{p,max} \tag{13}\]
where, \(c_{v}=1,c_{w}=1,c_{u}=0.45,T_{brk,\theta}=0.5\), \(T_{brk,p}=2.5\) and \(s_{p},\theta_{p}\) thresholds are 0.5m and 30 deg respectively.
The constraints used in the model are presented in Table I
where
\[s_{br,long}= s_{p,long}+c_{v}v_{p,long}T_{brk,p}\] \[+0.5c_{u}a_{p,long,ran}T_{brk,p}^{2}\] \[s_{br,lat}= s_{p,lat}+c_{v}v_{p,lat}T_{brk,p}\] \[+0.5c_{u}a_{p,lat,ran}T_{brk,p}^{2}\] \[\theta_{br,long}= \theta_{p,long}+c_{w}\omega_{p,long}T_{brk,\theta}\] \[+0.5c_{u}a_{p,long,rot}T_{brk,\theta}^{2}\] \[\theta_{br,lat}= \theta_{p,lat}+c_{w}\omega_{p,lat}T_{brk,\theta}\] \[+0.5c_{u}a_{p,lat,rot}T_{brk,\theta}^{2}\]
Finally, washout effect is introduced. The usage of constant weight penalisation requires weight re-tuning for different scenarios to obtain desirable performance. On the other hand, the application of adaptive weights allows a unique configuration for various driving scenarios. The formulation of the adaptive weight for these two states can be seen in (14) and (15). Figure 2 shows how the weight changes based on the platform's position. A high weight is applied when the platform is close to its limit, and a low weight when it is near the neutral position, allowing a washout effect to take place.
\[W_{s_{p}}= w_{s,1}+w_{s,2}\left(\frac{abs(s_{p,i})}{w_{s,5}}\right)+w_{s,3} \left(\frac{abs(s_{p,i})}{w_{s,5}}\right)^{2} \tag{14}\] \[+w_{s,4}\left(\frac{abs(s_{p,i})}{w_{s,5}}\right)^{4}\] \[W_{\omega_{p}}= w_{\omega,1}+w_{\omega,2}\left(\frac{abs(\omega_{p,i})}{w_{ \omega,5}}\right)+w_{\omega,3}\left(\frac{abs(\omega_{p,i})}{w_{\omega,5}} \right)^{2}\] \[+w_{\omega,4}\left(\frac{abs(\omega_{p,i})}{w_{\omega,5}} \right)^{4} \tag{15}\]
where, the parameters are \(w_{s,1}=0.01,w_{s,2}=20,w_{s,3}=20,w_{s,4}=20,w_{s,5}=0.5,w_{\omega,1}=0.0001,w _{\omega,2}=0.7,w_{\omega,3}=0.7,w_{\omega,4}=0.7,\) and \(w_{\omega,5}=3\). The parameter selection is heuristic and based on the analysis of various driving scenarios.
The weighting for the objective function remains consistent with the explicit MPC, apart from the added adaptive washout weights. A more detailed description of the weight selection is in [19].
## III Simulation Results
### _Simulation Setup_
To analyse the effectiveness of the algorithm in reducing computational costs, a general set of test conditions is taken into account. This includes specific force signals to be tracked in the form of sine waves and step signals along with multiple event waves (step signal + sine wave), for a range of amplitude (\(0.5-2\) m/\(s^{2}\)) and frequency (\(0.1-0.8\) Hz) values. Only the latter scenario is shown in Figure 3 and Figure 4).
While computing the explicit solution, a \(N_{p}\) of \(2\) is selected with a sampling time of \(0.25\) s to ensure the look-ahead time of \(0.5\) s. A higher \(N_{p}\) with a faster sampling time cannot be achieved due to the exponential increase in the computation time of the explicit solution. The online version of the MCA (implicit MPC) is able to operate at a faster sampling time and higher prediction horizon \(N_{p}\). Thus, \(N_{p}\) of \(50\) with a \(T_{s}\) of \(0.01\) s is used to maintain the same look-ahead time as used in the explicit controller. It is to be noted that explicit MPC only gives the initial guess for the hybrid MPC setup. Thus, the numerical stability of the method is ensured by selecting a time step of 0.01s for the implicit MPC.
Different MCA algorithms were analysed to compare their performance and listed as follows:
* Implicit MPC without any initial guess.
* Implicit MPC with the first control input. The first control input from the trajectory prediction is applied for the entire horizon as the initial guess for the next optimisation step.
* Hybrid MCA with the first explicit MPC control input. The first control input from the explicit MPC is applied for the entire prediction horizon.
Fig. 2: Adaptive weights for platform displacement \(r_{p}\)
* Hybrid MCA with all explicit MPC control inputs. All control inputs obtained from the explicit MPC controller are used for the entire horizon. Since the sampling time is different in both controllers, the explicit MPC inputs are applied in equal intervals throughout the larger prediction horizon of the implicit MPC. For e.g. with a \(N_{p,eMPC}\) of 5, the five control inputs are applied ten times each (\(1^{st}\) from 1-10, \(2^{nd}\) from 11-20 and so on) for a \(N_{p,iMPC}\) of 50.
Simplified actuator dynamics of the motion platform were considered for the comparison of the algorithms shown in the next section. The extended description of the simulation parameters can be found in [19].
### _Motion Cueing Performance_
Using the defined reference signals, the comparison has been conducted focusing on specific force tracking and online computation time. In Figure 3 and Figure 4 the specific force tracking performance for a multiple event wave is shown. This comprises an initial step signal followed by a sine wave, both of amplitude \(0.5\) m/\(s^{2}\). From the results, it can be observed that the MCA is able to track the reference signal.
Furthermore, the cueing algorithms were evaluated from the point of computational costs across different reference signal scenarios mentioned earlier. All the hybrid models were compared with the implicit MPC-based cueing algorithm, which is the current state-of-the-art MCA. The obtained results are presented in Figure 5. The average tracking performance in both longitudinal and lateral directions for all scenarios is also shown in Figure 5. It can be observed that the developed hybrid models need less time to compute the optimized control input. The hybrid model with all explicit MPC control inputs performs best amongst all the models analysed. The highest improvement in mean iterations from the implicit algorithm is by \(30\%\) while keeping similar tracking performance in both longitudinal and lateral directions. Also, while performing the simulations the maximum iterations are set to \(200\). This ensures faster computation with marginal sub-optimal results (\(<0.3\%\)).
### _Emulator Track Performance_
To evaluate the performance and computational costs, the software emulator has been used developed by the motion platform supplier E2M Technologies B.V. The multibody modeling and the coordinate system are described in [20].
This emulator represents the actual dynamics of the Delft Advanced Vehicle Simulator (DAVSi). The DAVSi is a 6 DoF driving simulator and using its emulator interface, tests can be performed without imparting any damage to the real system.
Full-track simulation tests were performed using this virtual environment. First, IPG CarMaker (a high-fidelity virtual vehicle simulation environment) was used to simulate a vehicle driving around the Hockenheim race track, limited to a speed of \(120\) km/h. Then, acceleration values were extracted and passed through the perception model [18] before using them as reference signals. This was done to ensure that only the perceived acceleration values are sent to the MCA for performing the simulations with the emulator interface. Additionally, the perception model also scaled down the accelerations which makes the signals fit to be recreated in the driving simulator. Figure 6 and Figure 7 show that the MCA is capable of tracking the reference signal in a desirable manner. An RMSE of \(0.42\), \(0.21\) is observed in both directions respectively.
Fig. 4: Specific force tracking for lateral motion for multiple event wave
Fig. 5: Mean iterations along with respective standard deviation and tracking performance for all scenarios
Fig. 3: Specific force tracking for longitudinal motion for multiple event wave
Further, a similar trend in mean iterations is observed with the hybrid MCA improving online computation time performance. An improvement of \(9\%\) can be observed with the hybrid model using all control inputs, whereas the other hybrid and implicit models show an improvement of \(5.9\%\) and \(5.1\%\) respectively. Thus, the developed algorithm can be implemented and used with real track data in motion-based driving simulators.
## IV Conclusion
In this study, a hybrid MCA is proposed using a combination of explicit and implicit MPC techniques. The explicit MPC provides an initial guess used by the implicit MPC to warm-start the algorithm and computes the optimized control input. Amongst the considered state-of-the-art motion cueing algorithms, the best computation time performance is observed from the proposed algorithm taking all explicit MPC control inputs as the initial guess. Moreover, to improve motion cueing, braking constraints are used for workspace management of the simulator when it is about to reach its physical displacement limits. Adaptive washout weights are also implemented to reduce false cues by bringing the simulator to its neutral position. Overall, the proposed algorithm maintains similar tracking performance across the considered state-of-the-art motion cueing algorithms, but it helps to reduce online computation time by 30%. The performance of the proposed algorithm has been demonstrated in complex track driving. Future work focuses on human-in-the-loop experiments for subjective assessment of the proposed algorithm.
For better results/performance of the adaptive weights law, feasibility analysis of the adaptive weight should be conducted. This is considered as the scope for future work in this paper.
| ```
自動車産業において、運転シミュレーターは、安全で再現性があり、制御された没入型の仮想環境でテストを行うことができるため、長年にわたり使用されてきました。シミュレーターの性能向上と、ユーザーの車内体験を再現する能力は、モーションキューイングアルゴリズム（MCA）によって確立されています。このようなアルゴリズムは、常にモデル予測制御（MPC）をメイン制御技術として開発されています。現在、有効なMPCベースの方法では、オンラインで最適な制御量を計算したり、オフラインで明示的な制御法を導出したりするケースがあります。このアプローチは、リアルタイムの適用におけるMCAの適用性を制限するオンラインの計算コストと/またはオフラインのメモリ保存の問題につながります。この研究では、オフラインとオンラインの解法を解決するための解決策を提供します。そのために、明示的なMPCを使用し、インプット値を caliente として提供し、非 |
2306.15229 | Baryogenesis via flavoured leptogenesis in a minimal type-II seesaw
model | We study baryogenesis via leptogenesis in an extension of the Standard Model
by adding one right-handed neutrino and one triplet scalar. These heavy
particles contribute to the generation of tiny neutrino mass through seesaw
mechanism. The contribution of the heavy particles to the neutrino masses is
inversely proportional to their corresponding masses. Considering leptogenesis
is achieved by the decay of the right-handed neutrino, the new source of CP
asymmetry comes solely from the decay of the right-handed neutrino with
one-loop vertex diagram involving the triplet scalar. The predictiveness of the
model is enhanced by introducing Fritzsch-type textures for the neutrino mass
matrix and charged lepton mass matrix. We execute the parameter space study
following the latest neutrino oscillation data. We study baryogenesis via
leptogenesis in the two-flavoured regime, using the zero textures, and show
that there is an enhancement in baryon asymmetry as compared to the unflavoured
regime. For two-flavour leptogenesis we consider the suitable temperature
regime $T\subset\left[10^{10},10^{11}\right]$ GeV. We also study the common
correlation of CP violation between low and high-energy regimes using the
geometrical description of CP violation in terms of unitarity triangle. | Sreerupa Chongdar, Sasmita Mishra | 2023-06-27T06:07:51 | http://arxiv.org/abs/2306.15229v1 | # Baryogenesis via flavoured leptogenesis in a minimal type-II seesaw model
###### Abstract
We study baryogenesis via leptogenesis in an extension of the Standard Model by adding one right-handed neutrino and one triplet scalar. These heavy particles contribute to the generation of tiny neutrino mass through seesaw mechanism. The contribution of the heavy particles to the neutrino masses is inversely proportional to their corresponding masses. Considering leptogenesis is achieved by the decay of the right-handed neutrino, the new source of CP asymmetry comes solely from the decay of the right-handed neutrino with one-loop vertex diagram involving the triplet scalar. The predictiveness of the model is enhanced by introducing Fritzsch-type textures for the neutrino mass matrix and charged lepton mass matrix. We execute the parameter space study following the latest neutrino oscillation data. We study baryogenesis via leptogenesis in the two-flavoured regime, using the zero textures, and show that there is an enhancement in baryon asymmetry as compared to the unflavoured regime. For two-flavour leptogenesis we consider the suitable temperature regime \(T\subset\left[10^{10},10^{11}\right]\) GeV. We also study the common correlation of CP violation between low and high-energy regimes using the geometrical description of CP violation in terms of unitarity triangle.
pacs: 11.15.Ha, 12.30.-k, 12.30.-k
Introduction
The two enigmatic problems of non-zero neutrino mass and baryon asymmetry of the Universe find a common solution in the Standard Model (SM) augmented with a certain choice of additional heavy fields. Through the seesaw mechanism one can provide a possible theoretical explanation of nonzero neutrino mass, confirmed by the experimental observation of neutrino flavour oscillation [1]. The observed baryon asymmetry of the Universe (BAU) could also be explained through leptogenesis [2] via out-of-equilibrium decay of the same heavy fields taking part in the seesaw mechanism. The former is one low-energy observation (after electroweak symmetry breaking) whereas the latter, a high-energy phenomenon (before electroweak symmetry breaking). Thus seesaw mechanism provides a nontrivial link between the generation of light neutrino mass and baryogenesis through leptogenesis. In this work, we study the generation of BAU through flavoured leptogenesis in the minimally extended SM by one fermion singlet and one scalar triplet. The phenomena of CP violation is inevitable both in neutrino oscillation and leptogenesis. We also establish a link between low- and high-energy CP violations.
One of the famous frameworks of neutrino mass generation via seesaw mechanism and leptogenesis via out-of-equilibrium decay of a heavy beyond SM field is the one where the SM is extended with heavy right-handed neutrinos. One needs more than just one right handed neutrino to account for light neutrino mass generation, compatible with experimental data and leptogenesis. The addition of right-handed neutrinos is also consistent with the theories inspired by Grand Unification such as Left-right symmetry [3; 4; 5], Pati-Salam [6] and SO(10) [7; 8]. However, in such theories, heavy fields such as scalar triplets and fermion triplets arise naturally and can establish the connection between light neutrino mass generation via seesaw mechanism and leptogenesis.
Keeping minimal extension of the SM in mind, we study a minimal type-II seesaw model, where the SM is extended with one \(SU(2)_{L}\) triplet scalar, \(\Delta\) and one right-handed singlet fermion, \(N\). So in our study, there are two mass scales involved: the mass of the right-handed singlet, \(M\) and that of scalar triplet, \(M_{\Delta}\). While both the fields contribute to light neutrino mass via seesaw mechanism, considering hierarchical mass limits, the out-of-equilibrium decay of one field is responsible for creating lepton asymmetry and hence BAU via leptogenesis. The requirement of CP violation being essential for baryogenesis via
leptogenesis [9], it is possible to produce CP violation in two ways in such a model. The vertex diagram involving one right-handed neutrino and one triplet scalar can be of two types, depending on the relative hierarchy of their masses:
1. If we have the triplet scalar as the lightest seesaw state (\(M_{\Delta}\ll M\)), then the decay of the triplet scalar dominates the CP asymmetry in the presence of a virtual right-handed neutrino present in the vertex diagram.
2. If the right-handed neutrino is lighter than the triplet scalar (\(M\ll M_{\Delta}\)), the CP asymmetry is produced predominantly from the decay of the right-handed neutrino in the presence of a virtual triplet scalar in the vertex diagram.
In the SM, considering there are not so light scalars we choose to work in the hierarchy \(M\ll M_{\Delta}\). In a model with \(n\) number of right-handed neutrinos and one triplet scalar, the neutrino mass can be generated from two sources,
\[m_{\nu}=m_{\nu}^{(I)}+m_{\nu}^{(II)}, \tag{1}\]
where \(m_{\nu}^{(I)}\) is the right-handed neutrino contribution coming from the type-I seesaw mechanism, and \(m_{\nu}^{(II)}\) is the triplet scalar contribution coming from the type-II seesaw mechanism. If we consider the lightest right-handed neutrino \(N_{1}\) to be the lightest seesaw state, then the CP asymmetry can be obtained from the decay of \(N_{1}\) in the presence of the other right-handed neutrinos (\(N_{2},...N_{n}\)) or the triplet scalar \(\Delta\) in the vertex diagram. There is no one-to-one correspondence by each seesaw state, between the amount of contribution to neutrino mass and CP asymmetry (as the neutrino mass matrix is a \(3\times 3\) complex matrix, not a number). But the contribution of the seesaw states to the production of CP asymmetry is found to be proportional to their respective contribution to the neutrino mass generation [10]. In that case, in the limit, \(m_{\nu}^{(I)}\gg m_{\nu}^{(II)}\), the contribution of the triplet scalar in the production of CP asymmetry can be safely ignored. However, in a minimal type-II seesaw model, the CP asymmetry can be obtained only from the one-loop vertex diagram involving the triplet scalar as there is only one right-handed neutrino. So we will consider the contribution of triplet scalar in the CP asymmetry production.
In the study of baryogenesis through leptogenesis flavour effects are known to induce some novel feature as compared to unflavoured case especially due to the nature of wash-out effects along different directions in flavour space. At higher temperature, \(T\gtrsim 10^{12}\) GeV, the charged
lepton flavours \((e,\mu,\tau)\) are out of thermal equilibrium and thus indistinguishable. In this case, the leptogenesis can be successfully expressed in an unflavoured regime. However, at a temperature \(T\lesssim 10^{12}\) GeV, the processes induced by \(\tau\)-Yukawa are in thermal equilibrium. It breaks the coherence between \(\tau\)-lepton and the other two leptons \((e,\mu)\). The lepton asymmetries are expressed through \(Y_{\Delta_{a}}\) and \(Y_{\Delta_{\tau}}\) in this temperature range, where \(a=e+\mu\) is the superposition of the flavours \(e\) and \(\mu\). Further below \(T\sim 10^{9}\) GeV, the interactions induced by the \(\mu\)-Yukawa are in thermal equilibrium, completely breaking the flavour coherence. So, leptogenesis needs to be studied in terms of fully flavoured lepton asymmetries, \(Y_{\Delta_{e}}\), \(Y_{\Delta_{\mu}}\), and \(Y_{\Delta_{\tau}}\) at temperature \(T\lesssim 10^{9}\) GeV.
In seesaw models, generally, there are many free parameters as long as the coupling matrices are concerned. There are not enough experimental constraints to fix the parameters. The number of free parameters can be reduced by imposing texture zeros in the coupling matrices. Often they are motivated by imposing new symmetries on the particle content of the model. In this way, the model becomes predictive by fitting the parameters to low-energy neutrino data. Taking Fritzsch-type textures [11; 12; 13; 14; 15] into account we show that the BAU can be enhanced to comply with observational value, considering flavoured effects as compared to unflavoured case. This feature is helpful in bringing down the scale of leptogenesis. We study the two-flavoured leptogenesis in the temperature range \(T\sim[10^{10},10^{11}]\) GeV. We find that the lepton asymmetries obtained through leptogenesis can lead to baryon asymmetry of the order \(\sim 10^{-10}\), and the proper flavour consideration enhances the production of baryon asymmetry. It is believed that the CP violation in low-energy (e.g. neutrino oscillation) and high-energy regimes (e.g. leptogenesis) are in general not related. Using a geometrical interpretation of CP violation at low-energy, we also show a common link between low and high energy CP violation in this setup.
The paper is organized as follows. In section (II), a minimal type-II neutrino mass model is introduced. In section (II.1), a detailed parameter space study of the neutrino mass matrix elements is carried out, which covers the diagonalization of matrices and the interesting correlation among different mixing angles arising from the model. The allowed parameter space indicated in this section is further used to obtain CP asymmetry parameters. In section (III), the processes like baryogenesis and leptogenesis are discussed from a cosmological point of view. Section (III.1) contains the results and discussions. It gives a comparative study among these regimes of leptogenesis, generically judged based on different right-handed
neutrino masses \(M\). The common origin of CP violations in low- and high-energy sectors is analyzed in section (IV). Finally, in section (V), we give our conclusions of the study.
## II Neutrino mass model
With addition of a triplet scalar, \(\Delta\) and \(n\)-number of right-handed Majorana neutrinos, \(N_{i}\) (\(i=1,....,n\)), the extended SM Lagrangian can be written as
\[-\mathcal{L}\supset Y_{l\alpha\beta}\overline{L}_{\alpha}l_{R\beta}\phi+Y_{ \nu\alpha i}\overline{L}_{\alpha}N_{i}\phi+\frac{1}{2}M_{i}\bar{N}_{i}N_{i}+Y _{\Delta_{\alpha\beta}}\bar{L}_{\alpha}^{C}i\tau_{2}\Delta L_{\beta}-\mu\phi^{T }i\tau_{2}\Delta\phi+M_{\Delta}^{2}\text{Tr}\Delta^{\dagger}\Delta+\text{h.c.}, \tag{2}\]
where \(L_{\alpha}=\left(\nu_{\alpha},l_{\alpha}\right)^{T}\) and \(l_{R\beta}\), \(\left(\alpha,\beta=e,\mu,\tau\right)\) are the left- and right-handed SM leptons respectively. The SM Higgs doublet is represented as \(\phi=\left(\phi^{0},\phi^{-}\right)^{T}\). The triplet scalar, \(\Delta\) can be represented in \(SU(2)_{L}\) adjoint representation as,
\[\Delta=\begin{pmatrix}\frac{\delta^{+}}{\sqrt{2}}&\delta^{++}\\ \delta^{0}&-\frac{\delta^{+}}{\sqrt{2}}\end{pmatrix}.\]
The Dirac-type Yukawa coupling matrix of the right-handed neutrinos and charged leptons with the SM leptons and Higgs scalar are represented as \(Y_{\nu}\) and \(Y_{l}\) respectively. The coupling matrix, \(Y_{\Delta}\) is a \(3\times 3\) Majorana-type Yukawa coupling matrix of the triplet scalar with the SM lepton doublets. After electroweak symmetry breaking, due to the vacuum expectation value (vev) developed by the neutral component of the doublet Higgs, \(v=\langle\phi_{0}\rangle\), the neutral component of \(\Delta\) also acquires a vev, \(v_{\Delta}=\langle\delta_{0}\rangle\simeq\frac{\mu^{*}v^{2}}{M_{\Delta}^{2}}\). The triplet vev, \(v_{\Delta}\) is seesaw suppressed for heavy triplet scalar. Once the heavy degrees of freedom are integrated out, as a consequence, there are two sources of light neutrino masses,
\[m_{\nu}=m_{\nu}^{(I)}+m_{\nu}^{(II)}=-Y_{\nu}^{*}\frac{1}{M}Y_{\nu}^{\dagger}v ^{2}+2Y_{\Delta}v_{\Delta}. \tag{3}\]
The first and second terms are mass terms due to type-I and II seesaw induced masses respectively, with \(v=174\) GeV. The low-energy neutrino oscillation experiments provide data for six parameters: three mixing angles, two mass-squared differences, and one CP violation phase. In our model, the Dirac-type Yukawa coupling for right-handed neutrinos has 6 (3 moduli and 3 phases for one right-handed neutrino) and the Majorana-type Yukawa which is complex symmetric, has 12 (6 moduli and 6 phases) independent parameters. In order to make the model predictive we assume the Fritzsch-type textures [11; 12; 13; 14; 15] of charged
lepton mass matrix, \(m_{l}\) and triplet scalar induced neutrino mass matrix, \(m_{\nu}^{(II)}\). Following the same texture for both,
\[m_{l}=v\left(\begin{array}{ccc}0&C_{l}e^{i\alpha_{l}}&0\\ C_{l}e^{i\alpha_{l}}&0&B_{l}e^{i\beta_{l}}\\ 0&B_{l}e^{i\beta_{l}}&A_{l}e^{i\gamma_{l}}\end{array}\right), \tag{4}\]
and
\[m_{\nu}^{(II)}=v_{\Delta}\left(\begin{array}{ccc}0&C_{\nu}e^{i\alpha_{\nu}}& 0\\ C_{\nu}e^{i\alpha_{\nu}}&0&B_{\nu}e^{i\beta_{\nu}}\\ 0&B_{\nu}e^{i\beta_{\nu}}&A_{\nu}e^{i\gamma_{\nu}}\end{array}\right). \tag{5}\]
Such textures arise in left-right symmetric models, where right-handed neutrinos and triplet scalars arise naturally and were studied in Ref. [16]. Also, in the construction of models based on Froggatt-Nielsen mechanism [17], texture zeros arise in the neutrino Yukawa coupling due to the assignment of different charges under an additional symmetry to particles of different generations. The consequences of texture zeros in \((1,1),(1,2)\) and \((1,3)\) positions of Yukawa matrix have interesting consequences in flavoured leptogenesis as shown in Ref.[18]. For example in the case of texture zero in \((1,1)\) position, \(e\)-CP asymmetry is weakly washed out while the \(\mu\)- and \(\tau\)-CP asymmetries are strongly washed out. In the quark sector, the simultaneous presence of zeros in the \((1,1)\) elements of symmetric up and down quark mass matrices leads to the prediction of Cabibbo angle \(\theta_{C}\simeq\sqrt{m_{d}/m_{s}}\)[19].
In our case, we have one right-handed neutrino. So the corresponding Yukawa coupling matrix is a column matrix, which can be set as [20],
\[Y_{\nu}=iy_{0}\left(0,r,1\right)^{T}. \tag{6}\]
The purpose of introducing imaginary unit \(i\) is to cancel the minus sign of the type-I term for convenience. The appearance of zero in the \((1,1)\) position of the above Yukawa matrix ensures two-zero texture in the total light neutrino mass matrix as can be seen in the subsequent equation. Now using the equations (5) and (6) in Eq.(3), the neutrino mass matrix turns out to be
\[m_{\nu}=m_{0}\left(\begin{array}{ccc}0&\hat{C_{\nu}}e^{i\alpha_{\nu}}&0\\ \hat{C_{\nu}}e^{i\alpha_{\nu}}&r^{2}&r+\hat{B_{\nu}}e^{i\beta_{\nu}}\\ 0&r+\hat{B_{\nu}}e^{i\beta_{\nu}}&1+\hat{A_{\nu}}e^{i\gamma_{\nu}}\end{array} \right). \tag{7}\]
Here \(m_{0}\equiv v^{2}y_{0}^{2}/M\) and \(\hat{A_{\nu}}\equiv v_{\Delta}A_{\nu}/m_{0}\) and similarly for \(\hat{B_{\nu}}\) and \(\hat{C_{\nu}}\).
### Parameter space determination by confronting with neutrino data
To study the neutrino mass generation and to produce optimum CP asymmetry from the neutrino mass model, we need to study the parameter space offered by the model.
There are three important steps to be followed for diagonalizing the matrices and make a connection with experimental observations [21; 22].
1. In the first step the charged and neutral lepton mass matrices are decomposed in terms of diagonal phase matrix, \(P_{l,\nu}\) and real symmetric matrix, \(\bar{m}_{l,\nu}\) so that \[m_{l,\nu}=P_{l,\nu}^{T}\bar{m}_{l,\nu}P_{l,\nu},\] (8) where \[P_{l,\nu}=\begin{pmatrix}e^{i\theta_{l,\nu}}&0&0\\ 0&e^{i\phi_{l,\nu}}&0\\ 0&0&e^{i\psi_{l,\nu}}\end{pmatrix}.\] (9)
2. In the second step the real symmetric matrix, \(\bar{m}_{l,\nu}\) is diagonalized following unitary transformation: \[U_{l}^{T}\bar{m}_{l}U_{l}=\text{Diag}\left(m_{e},m_{\mu},m_{\tau}\right),\ U_{\nu}^{T}\bar{m}_{\nu}U_{\nu}=\text{ Diag}\left(m_{1},m_{2},m_{3}\right),\] (10) where \(m_{e},m_{\mu}\) and \(m_{\tau}\) are the masses of \(e,\mu\) and \(\tau\) leptons respectively. The mass eigenvalues of the light neutrino mass matrix are represented as \(m_{1},m_{2}\) and \(m_{3}\).
3. The lepton flavour mixing matrix, \(V\) then arises from the mismatch between the diagonalization of the charged and neutral mass matrices: \(V=U_{l}^{T}\left(P_{l}^{*}P_{\nu}\right)U_{\nu}^{*}\). The elements of the matrix can be written as \[V_{pq}=U_{l1p}{U_{\nu}}^{*}_{1q}e^{i\theta}+U_{l2p}{U_{\nu}}^{*}_{2q}e^{i\phi} +U_{l3p}{U_{\nu}}^{*}_{3q}e^{i\psi},\] (11) where \(p\equiv(e,\mu,\tau)\), \(q\equiv(1,2,3)\). The phases are defined as, \(\theta=\left(\theta_{\nu}-\theta_{l}\right)\), \(\phi=\left(\phi_{\nu}-\phi_{l}\right)\), \(\psi=\left(\psi_{\nu}-\psi_{l}\right)\). The elements of \(U_{\nu}\) and \(U_{l}\) are given in the appendices (A) and (B).
4. The elements of the mixing matrix \(V\) depends on only two combinations of three phases, \((\theta,\phi,\psi)\) as the overall phase of \(V\) has nothing to do with experimental observable. The elements of the matrices \(U_{l,\nu}\) also depend on the mass ratios of the charged
and neutral leptons, as given below,
\[x_{l}=\frac{m_{e}}{m_{\mu}},\quad y_{l}=\frac{m_{\mu}}{m_{\tau}}, \tag{12}\]
\[x_{\nu}=\frac{m_{1}}{m_{2}},\quad y_{\nu}=\frac{m_{2}}{m_{3}}. \tag{13}\]
The charged lepton ratios \(x_{l}\), \(y_{l}\) are now determined with better accuracy, as
\[x_{l}\simeq 0.00484,\quad y_{l}\simeq 0.0594. \tag{14}\]
So, there are only four free parameters (two phases and \(x_{\nu}\) and \(y_{\nu}\)) that can be constrained from neutrino oscillation data.
The three mixing angles \(\theta_{ij}\) of the neutrino oscillation parameters can be expressed in terms of the lepton flavour mixing matrix \(V\), as follows:
\[\sin^{2}2\theta_{12}=4\left|V_{e1}\right|^{2}\left|V_{e2}\right|^{2}, \tag{15}\]
\[\sin^{2}2\theta_{23}=4\left|V_{\mu 3}\right|^{2}\left(1-\left|V_{\mu 3}\right|^{2 }\right), \tag{16}\]
\[\sin^{2}2\theta_{13}=4\left|V_{e3}\right|^{2}\left(1-\left|V_{e3}\right|^{2} \right). \tag{17}\]
The parameter space of \([x_{\nu},y_{\nu}]\) can be restricted by the bounds on mixing angles. The experimental constraints are given by [23],
\[31.27^{\circ}<\theta_{12}<35.87^{\circ},\ 39.7^{\circ}<\theta_{23}<50.9^{\circ}, \tag{18}\]
\[8.25^{\circ}<\theta_{13}<8.98^{\circ},\ 144^{\circ}<\delta<350^{\circ}. \tag{19}\]
Even though the Dirac CP phase \(\delta\) is bounded as
\[144^{\circ}<\delta<350^{\circ}, \tag{20}\]
we have used the full range of \(\delta\sim[0:360^{\circ}]\) for the parameter space study. The latest bounds on the two mass-squared differences are given by
\[6.82\times 10^{-5}\ {\rm eV}^{2}<\Delta m_{21}^{2}<8.04\times 10^{-5}\ {\rm eV }^{2},\]
\[2.430\times 10^{-3}\ {\rm eV}^{2}<\Delta m_{31}^{2}<2.593\times 10^{-3}\ {\rm eV }^{2}. \tag{21}\]
Once the values of \(x_{\nu}\) and \(y_{\nu}\) are constrained the absolute values of three neutrino masses can be found by using the mass-squared differences,
\[\Delta m^{2}_{21}=m^{2}_{2}-m^{2}_{1}=m^{2}_{2}\left|1-x^{2}_{\nu}\right|,\quad \Delta m^{2}_{31}=m^{2}_{3}-m^{2}_{1}=m^{2}_{3}\left|1-y^{2}_{\nu}\right|. \tag{22}\]
In order to diagonalize the charged lepton and neutrino mass matrices given in equations (4) and (7) and determine the parameter space using experimental data we follow the steps laid above. In order to write the lepton mass matrices \(m_{l}\) and \(m_{\nu}\) in factorized form as given in Eq.(8), we make two assumptions here: \(r=\sqrt{m_{2}/m_{0}}\) and \(\arg\left(1+\hat{A}_{\nu}e^{i\gamma_{\nu}}\right)=2\arg\left(r+\hat{B_{\nu}} \right)e^{i\beta_{\nu}}\)[21] and then obtain
\[\hat{A}_{\nu}=\left[\frac{(m_{3}-m_{1})^{2}}{m^{2}_{0}}-\sin^{2}\gamma_{\nu} \right]^{\frac{1}{2}}-\cos\gamma_{\nu}, \tag{23}\]
\[\hat{B_{\nu}}=\left[\frac{m_{1}m_{3}(m_{3}-m_{1}-m_{2})}{m^{2}_{0}(m_{3}-m_{1 })}-r^{2}\sin^{2}\beta_{\nu}\right]^{\frac{1}{2}}-r\cos\beta_{\nu}, \tag{24}\]
\[\hat{C}_{\nu}=\left[\frac{m_{1}m_{2}m_{3}}{m^{2}_{0}(m_{3}-m_{1})}\right]^{ \frac{1}{2}}. \tag{25}\]
Similarly, the elements of the charged lepton mass matrix, shown in Eq.(4), can be expressed in terms of three charged lepton masses \(m_{e}\), \(m_{\mu}\) and \(m_{\tau}\).
\[A_{l}=\left(m_{\tau}-m_{\mu}+m_{e}\right), \tag{26}\]
\[B_{l}=\left[\frac{(m_{\mu}-m_{e})(m_{\tau}-m_{\mu})(m_{e}-m_{\tau})}{(m_{\tau} -m_{\mu}+m_{e})}\right]^{\frac{1}{2}}, \tag{27}\]
\[C_{l}=\left[\frac{m_{e}m_{\mu}m_{\tau}}{(m_{\tau}-m_{\mu}+m_{e})}\right]^{ \frac{1}{2}}. \tag{28}\]
In order to determine the parameter space of \(x_{\nu}\) and \(y_{\nu}\) we assume the general range of the lightest neutrino mass in normal hierarchy (NH), as \(m_{1}\sim\left[0.001:0.05\right]\) eV. Hence, the other neutrino masses can be calculated directly from Eq.(22). The plot in Fig.(1) shows how different bounds restrict the parameter space of \(x_{\nu}\) and \(y_{\nu}\). In Fig.(1), the scattered points represent the allowed values of \(x_{\nu}\) and \(y_{\nu}\) that satisfy total neutrino mass \(\Sigma=m_{1}+m_{2}+m_{3}<0.12\) eV [24], mass-squared differences and mixing angles as given in equations (18) - (21). One can see a commonly allowed parameter space as the allowed ranges of \(x_{\nu}\) and \(y_{\nu}\) are given by,
\[x_{\nu}\sim\left[0.80-0.90\right],\quad y_{\nu}\sim\left[0.30-0.35\right]. \tag{29}\]
We shall use these ranges to estimate the CP asymmetry parameter for the study of leptogenesis. The parameter space for the neutrino mass matrix elements \(\hat{A}_{\nu}\), \(\hat{B}_{\nu}\) and \(\hat{C}_{\nu}\) can be further constrained using the latest neutrino oscillation parameter data using equations (23 - 25), as the model contributes to the formation of baryon asymmetry through these matrix elements.
## III Baryogenesis and Leptogenesis
The baryon asymmetry of the Universe, \(\eta_{B}=\left(n_{B}-n_{\overline{B}}\right)/n_{\gamma}\) is constrained from the Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background Radiation (CMBR) data [25], and is given by,
\[4.7\times 10^{-10}\leq\eta_{B}\leq 6.5\times 10^{-10}, \tag{30}\]
where \(n_{B}\), \(n_{\overline{B}}\), and \(n_{\gamma}=\frac{2\zeta(3)}{\pi^{2}}T^{3}\) denote the number densities of baryons, antibaryons, and gamma photons respectively. Theoretically, baryogenesis through leptogenesis [26] provides a framework for generating the required BAU following three Sakharov's conditions [9]. In the seesaw model under consideration, they are satisfied as follows: (i) The lepton number violation comes into the scenario from the decay of right-handed neutrino considering neutrinos as Majorana particles. (ii) CP violation is ensured from the interference of the tree-level and one-loop diagrams of the decay of the right-handed neutrino. (iii) The out-of-equilibrium conditons is achieved by determining the particle asymmetries, while considering the decay and inverse decay processes in an expanding early Universe through a set of Boltzmann equations. The lepton asymmetry is converted to the baryon asymmetry through \(B+L\) violating Sphaleron processes [27].
If the temperature is as high as \(T\sim 10^{12}\) GeV or more, the individual flavours of the leptons do not appear to be of much importance, and in this case, solving a set of flavour-independent or unflavoured Boltzmann equations proves to be sufficient to study the leptogenesis. On the other hand, in the temperature range \(T\subset[10^{10},10^{12}]\) GeV, \(\tau\)-Yukawa interactions are faster than the rate of expansion of the Universe, making \(\tau\)-leptons decouple from the flavour coherent lepton state. Hence, it is essential to study leptogenesis in a two-flavoured regime. Below \(T\sim 10^{9}\) GeV, the \(\mu\)-leptons decouple and completely break down the flavour coherence of the leptons. Hence, the study of leptogenesis requires three-flavour consideration. So we have a set of flavour-independent and flavour-specific Boltzmann equations given below, and we solve it numerically to obtain the unflavoured, and flavoured lepton asymmetries. As mentioned before, the Boltzmann equations [28; 29; 30; 20] take care of the dynamics of particle abundancies, taking care of the production of CP asymmetries and washing out of the asymmetry due to the interplay between decay and inverse decay processes. These equations are expressed in terms of particle asymmetries of species
where \(n_{x}\) is the number density and \(s\) is the entropy density.
\[\frac{dY_{N}}{dz}=-\frac{\gamma_{D}}{sHz}\left(\frac{Y_{N}}{Y_{N}^{eq}}-1\right) \tag{31}\]
\[\frac{dY_{\Delta_{i}}}{dz}=-\frac{\gamma_{D}}{sHz}\left[\left(\frac{Y_{N}}{Y_{N }^{eq}}-1\right)\epsilon_{i}+K_{0}^{i}\sum_{j}\frac{1}{2}\left(C_{ij}^{l}+C_{j }^{H}\right)\frac{Y_{\Delta_{j}}}{Y_{l}^{eq}}\right] \tag{32}\]
where \(z=\frac{M}{T}\), the decay and inverse decay rate of the right handed neutrino (\(N\to l\phi\)), \(\gamma_{D}=DsHzY_{N}^{eq}\), \(D=Kz\frac{K_{1}(z)}{K_{2}(z)}\), \(s\) is the entropy density, \(K_{1}\) and \(K_{2}\) are modified Bessel functions, \(Y_{\Delta_{i}}\) being the i-flavoured lepton asymmetry, \(Y_{N}^{eq}=\frac{45}{2\pi^{4}g_{*}}z^{2}K_{2}(z)\), and \(Y_{l}^{eq}=\frac{15}{4\pi^{2}g_{*}}\). The decay or wash-out parameter,
\[K\equiv\frac{\sum_{\alpha}\Gamma(N\to L_{\alpha}\phi)}{H(M)}=\frac{\tilde{m} }{m_{*}}, \tag{33}\]
where \(\tilde{m}\) is the effective neutrino mass \(\tilde{m}\equiv\frac{(Y^{\dagger}Y)_{11}v^{2}}{M}\) and is proportional to the total decay rate of the right-handed neutrino. \(H(M)\) is the Hubble parameter evaluated at a temperature \(T=M\); \(m_{*}\sim 10^{-3}\) eV is equilibrium neutrino mass. We have considered the case of strong wash-out (\(\tilde{m}>m_{*}\)) for our study. \(K_{0}^{i}\) are the flavour projection operators where \(i=a,\tau\) in two-flavour configuration, and \(i=e,\mu,\tau\) in three-flavour configuration. It can be expressed as [18; 29; 31; 32; 33; 34],
\[K_{0}^{i}=\frac{\left(Y^{*}\right)_{i1}\left(Y\right)_{i1}}{\left(Y^{\dagger} Y\right)_{11}}. \tag{34}\]
The importance of flavour projection operators is discussed further in the appendix (C).
In the case of flavour-independent lepton asymmetry, Eq.(32) can be written as
\[\frac{dY_{\Delta}}{dz}=-\frac{\gamma_{D}}{sHz}\left[\left(\frac{Y_{N}}{Y_{N}^ {eq}}-1\right)\epsilon+\frac{1}{2}\frac{Y_{\Delta}}{Y_{l}^{eq}}\right], \tag{35}\]
where \(Y_{\Delta}\) represents the unflavoured lepton asymmetry.
In this study, the lepton number violating decays of the heavy states introduced in the seesaw mechanism act as new sources of CP violation. For our model, one right-handed neutrino \(N\) and one heavier triplet scalar \(\Delta\) are chosen as the two heavy states. The lepton number violating effects produced by the heavier state get washed out due to the same produced by the lighter state. So, here the CP asymmetry is getting generated from the decay of the right-handed Majorana neutrino, with the assumption \(M\ll M_{\Delta}\). The CP asymmetry can be calculated from the interaction of ordinary tree-level decay and the three diagrams shown in Fig.(2). The first two figures are self-energy and vertex diagrams
mediated by one extra heavy right-handed neutrino. Nevertheless, there is only one right-handed neutrino in the model under consideration. So the third figure representing the vertex diagram mediated by the Higgs triplet scalar \(\Delta\) contributes to the generation of CP asymmetry. In the temperature range \(T>10^{12}\) GeV, the lepton flavours states can be approximated as a single or unflavoured state. In that temperature range, we write the flavour-independent CP asymmetry parameter as [20]
\[\epsilon_{N}^{\Delta}\simeq\frac{3}{16\pi}\frac{M}{v^{2}}\frac{\sum_{\alpha \beta}\text{Im}\left[Y_{1\alpha}^{\dagger}Y_{1\beta}^{\dagger}(m_{\nu}^{(II) \ast})_{\alpha\beta}\right]}{\left(Y^{\dagger}Y\right)_{11}}. \tag{36}\]
Again, for temperature \(T\ll 10^{12}\) GeV, the lepton flavours become distinguishable, and the flavour consideration becomes important. Then the flavour-specific CP asymmetry parameter can be written as [35; 36]
\[\epsilon_{N,\alpha}^{\Delta}\simeq\frac{3}{16\pi}\frac{M}{v^{2}}\frac{\sum_{ \beta}\text{Im}\left[Y_{1\alpha}^{\dagger}Y_{1\beta}^{\dagger}(m_{\nu}^{(II) \ast})_{\alpha\beta}\right]}{\left(Y^{\dagger}Y\right)_{11}}. \tag{37}\]
For simplicity, we write the unflavoured CP asymmetry as \(\epsilon\) and the flavour specific CP asymmetry as \(\epsilon_{i}\), where \(i=e\), \(\mu\), \(\tau\) or \(a=e+\mu\). From Eq.(36), we obtain the expression of the unflavoured CP asymmetry parameter, using the neutrino mass matrix elements
\[\epsilon=\frac{3}{16\pi}\frac{M}{v^{2}\left(1+r^{2}\right)}m_{0}\left(\hat{A} _{\nu}\sin\gamma_{\nu}+2r\hat{B}_{\nu}\sin\beta_{\nu}\right). \tag{38}\]
From Eq.(37), we can further obtain the two-flavoured CP asymmetries \(\epsilon_{a}\), (\(a=e+\mu\)) and \(\epsilon_{\tau}\),
\[\epsilon_{a}=\frac{3}{16\pi}\frac{M}{v^{2}\left(1+r^{2}\right)}rm_{0}\hat{B} _{\nu}\sin\beta_{\nu},\]
Figure 2: The one-loop Feynman diagrams of \(N_{i}\) decay in a model with \(n\) number of right-handed neutrinos and one triplet, \(\Delta\). In our case, we have one right-handed neutrino, so CP violation comes from the interference of tree-level diagram with the third diagram.
\[\epsilon_{\tau}=\frac{3}{16\pi}\frac{M}{v^{2}\left(1+r^{2}\right)}m_{0}\left(r\hat{ B}_{\nu}\sin\beta_{\nu}+\hat{A}_{\nu}\sin\gamma_{\nu}\right), \tag{39}\]
which is relevant in the temperature range \(T>10^{9}\) GeV.
The CP asymmetry parameters are to be determined from the model to further investigate different lepton asymmetries through a set of Boltzmann equations. In the temperature range \(10^{11}\text{GeV}\lesssim\text{T}\lesssim 10^{12}\text{GeV}\), under two-flavoured leptogenesis regime,
\[C^{H}=\frac{1}{230}\left(41,56\right), \tag{40}\]
\[C^{l}=\frac{1}{460}\begin{pmatrix}196&-24\\ -9&156\end{pmatrix}. \tag{41}\]
Finally, we can estimate the baryon asymmetry after solving the suitable set of Boltzmann equations numerically and obtain the lepton asymmetry. In the case of unflavoured leptogenesis, we use the expression,
\[\eta_{B}=-7.04\times Y_{B},\quad Y_{B}=-1.38\times 10^{-3}\epsilon\eta, \tag{42}\]
to calculate baryon asymmetry [28], where
\[\eta=\frac{Y_{\Delta}\left(z\gg 1\right)/\epsilon}{Y_{N}^{eq}\left(0\right)}, \tag{43}\]
is known as the efficiency factor [37]. In the case of flavoured leptogenesis, the expression of baryon asymmetry given in Eq.(42) is replaced by [28]
\[\eta_{B}=-7.04\times Y_{B},\quad Y_{B}=-1.38\times 10^{-3}\sum_{i}Y_{\Delta_{i}} \left(z\gg 1\right). \tag{44}\]
In the next subsection we make a quantitative analysis of the BAU in the model.
### Baryon asymmetry determination: Result
In this section, we make quantitative analysis of baryon asymmetry by calculating the CP asymmetry and solving the set of Boltzmann equations, both for unflavoured and flavoured leptogenesis. The CP asymmetries are calculated by using equations (38)- (39) for unflavoured and flavoured leptogenesis respectively. Similarly the corresponding set of Boltzmann equations (31), (32) and (35) are taken into account to calculate the final BAU using
equations (42) and (44). It is known that the generation of BAU can be enhanced by taking flavour effects into account over unflavoured leptogenesis. Here, we show that Fritzsch-type textures can be used to verify the above characteristic of leptogenesis mechanism. So an integrated scenario of generation of baryon asymmetry and experimentally compatible values of neutrino mixing parameters can be achieved in this framework.
Although unflavoured leptogenesis is viable above temperature \(T\gtrsim 10^{12}\) GeV, we make comparison between unflavoured and flavoured leptogenesis by bringing down the leptogenesis scale to \(10^{10}-10^{11}\) GeV, suitable for studying two-flavoured leptogenesis. In order to study the cases of unflavoured and two-flavoured leptogenesis, we choose different values of \(M\) in the range \(4\times 10^{10}\) GeV \(\leq M\leq 5\times 10^{11}\) GeV, and four benchmark sets are formed corresponding to those \(M\) values. For each benchmark set, the values of neutrino mass eigenvalue \(m_{1}\), phases \(\gamma_{\nu}\), \(\beta_{\nu}\), and \(y_{0}\) are made to be fixed for calculating the CP asymmetry. The values are consistent with the oscillation data. Using the chosen values for these parameters the effective neutrino mass \(\tilde{m}=\frac{(Y^{\dagger}Y)_{11}v^{2}}{M}=\frac{y_{0}^{2}(1+r^{2})}{M}\), neutrino mass matrix elements \(\hat{A_{\nu}}\), \(\hat{B_{\nu}}\), \(\hat{C_{\nu}}\) (using expressions shown in equations (23), (24) and (25)) are calculated for each set and shown in table (1). For each set, the values of the parameters are chosen so that the neutrino mass eigenvalues follow normal hierarchy and the sum of neutrino masses, \(\Sigma\) is in agreement with its observational value.
For the hierarchical mass spectrum of light neutrinos, the renormalization group running between low energy and the high energy seesaw scale has a nominal impact on the neutrino parameters, except from an overall scaling of the light neutrino masses [38; 39; 40]. The effect of scaling can be taken care of by multiplying by a factor of 1.2 in the low energy values of
\begin{table}
\begin{tabular}{|l||l||l||l||l||l||l||l||l||l||} \hline Set no. & \(M\)(GeV) & \(y_{0}\) & \(\tilde{m}\)(eV) & \(\gamma_{\nu}\) & \(\beta_{\nu}\) & \(\hat{A_{\nu}}\) & \(\hat{B_{\nu}}\) & \(\hat{C_{\nu}}\) & \(\Sigma\) \\ & & & & & & (eV) & (eV) & (eV) \\ \hline I & \(4\times 10^{11}\) & 0.0201 & 0.043379 & 57.32\({}^{\circ}\) & 343.95\({}^{\circ}\) & \(-\)0.317 & \(-\)0.177 & 0.461 & 0.0614 \\ \hline II & \(2\times 10^{11}\) & 0.0201 & 0.074759 & 171.97\({}^{\circ}\) & 171.97\({}^{\circ}\) & 1.507 & 0.733 & 0.231 & 0.0684 \\ \hline III & \(8\times 10^{10}\) & 0.0049 & 0.022743 & 343.95\({}^{\circ}\) & 57.32\({}^{\circ}\) & 2.172 & 0.692 & 1.528 & 0.0644 \\ \hline IV & \(4\times 10^{10}\) & 0.0048 & 0.031625 & 57.32\({}^{\circ}\) & 114.65\({}^{\circ}\) & 0.827 & 0.669 & 0.796 & 0.0645 \\ \hline \end{tabular}
\end{table}
Table 1: Neutrino mass matrix elements for different values of right-handed neutrino mass \(M\). These values are used to estimate the CP asymmetry and wash-out parameters.
the parameters [41].
The purpose of setting up these benchmark points is to determine and compare the production baryon asymmetry via unflavoured and two-flavoured leptogenesis. Hence, washout parameter \(K=\frac{\hat{m}}{m_{\star}}=\frac{y_{0}^{2}(1+r^{2})v^{2}}{Mm_{\star}}\) and the CP asymmetries \(\epsilon\), \(\epsilon_{a}\), and \(\epsilon_{\tau}\) are calculated corresponding to different \(M\) values. For each set, baryon asymmetry is calculated via both unflavoured and two-flavoured leptogenesis. For the purpose of having a comparative study, we have kept the values of washout parameter \(K\) and the CP asymmetry parameter \(\epsilon(\epsilon_{i})\) consistent for the study of unflavoured and two-flavoured leptogenesis. The consistency is ensured by the relation \(\epsilon=\epsilon_{a}+\epsilon_{\tau}\). In case of two-flavoured leptogenesis, the flavour projection operators \(K_{0}^{i}\), expressed in Eq.(34), are calculated as
\[K_{0}^{a}=\frac{(Y^{*})_{e1}(Y)_{e1}}{(Y^{\dagger}Y)_{11}}+\frac{(Y^{*})_{\mu 1 }(Y)_{\mu 1}}{(Y^{\dagger}Y)_{11}}=\frac{r^{2}}{1+r^{2}}, \tag{45}\]
and
\[K_{0}^{\tau}=\frac{(Y^{*})_{\tau 1}(Y)_{\tau 1}}{(Y^{\dagger}Y)_{11}}=\frac{1}{ 1+r^{2}}, \tag{46}\]
for each benchmark set.
The flavour independent, as well as flavour-specific CP asymmetry parameters arising from the model, mentioned in equations (38 - 39), respectively, are functions of the neutrino mass eigenvalues \(m_{1}\), \(m_{2}\), \(m_{3}\) through the neutrino mass matrix elements \(\hat{A_{\nu}}\), \(\hat{B_{\nu}}\), \(\hat{C_{\nu}}\). The mass eigenvalues \(m_{2}\), \(m_{3}\) and thereby the elements \(\hat{A_{\nu}}\), \(\hat{B_{\nu}}\), \(\hat{C_{\nu}}\) are calculated using the equations (22- 25) for different values of \(m_{1}\). The allowed ranges of \(x_{\nu}\) and \(y_{\nu}\), given in Eq.(29), are used to estimate the CP asymmetry parameters.
The suitable set of Boltzmann equations are solved numerically and they are shown in the appendix(E). The final lepton asymmetries thus obtained are used to calculate the final baryon asymmetries. The final results corresponding to different sets are enlisted in table (2) and table (3), for unflavoured and two-flavoured leptogenesis, respectively. The corresponding plots are shown in Fig.(6) in the appendix (E). The baryon asymmetries \(|\eta_{B}|\) are compared in table (4) to realize the enhancement in the result after considering leptogenesis in an appropriate flavoured regime.
In addition to using the benchmark points, to observe the importance of the flavour effects in producing baryon asymmetry through unflavoured and two-flavoured leptogenesis, we numerically vary the right-handed neutrino mass \(M\) values in the range \([2-6]\times 10^{10}\) GeV, for different values of \(K\) and \(\epsilon\) (\(K\), \(K_{0}^{i}\) and \(\epsilon_{i}\) for flavoured leptogenesis) and show the
results in Fig.(3). In Fig.(3), baryon asymmetries are plotted against \(M\). After eliminating the results showing over-production of baryon asymmetry \(|\eta_{B}|\), it shows that within the experimental bound \((4.7-6.5)\times 10^{-10}\), the results coming from two-flavoured leptogenesis are enhanced over the results coming from unflavoured leptogenesis.
## IV Relating CP Violation in Low and High Energy Phenomena
Baryogenesis through leptogenesis can easily be implemented in the seesaw models. CP violation in lepton sector which are measurable in low-energy experiments like neutrino factories can have profound consequences in high-energy phenomena like leptogenesis. It is believed that the CP violation in the two sectors is in general not related to each other. The difficulty in setting up the relation is due to lack of low-energy data available to quantify the parameters of the seesaw model. There are interesting studies [42] which show that it may be possible in specific Grand Unification inspired models up to size and sign of the observed BAU to CP violation at low energies. The link between CP violations in leptogenesis and
\begin{table}
\begin{tabular}{|l||l||l||l||l||l|} \hline Set no. & \(M\)(GeV) & K & \(\epsilon\) & \(|\eta_{B}|/10^{-10}\) \\ \hline I & \(4\times 10^{11}\) & 43.379 & \(-3.45\times 10^{-6}\) & 1.78 \\ \hline II & \(2\times 10^{11}\) & 74.759 & \(6.12\times 10^{-6}\) & 1.67 \\ \hline III & \(8\times 10^{10}\) & 22.473 & \(4.73\times 10^{-7}\) & 0.54 \\ \hline IV & \(4\times 10^{10}\) & 31.625 & \(1.42\times 10^{-6}\) & 1.07 \\ \hline \end{tabular}
\end{table}
Table 2: Baryon asymmetries from unflavoured leptogenesis for different values of \(M\) and \(K\).
\begin{table}
\begin{tabular}{|l||l||l||l||l||l||l||l|} \hline Set no. & \(M\)(GeV) & K & \(\epsilon_{a}\) & \(\epsilon_{\tau}\) & \(K_{0}^{a}\) & \(K_{0}^{\tau}\) & \(|\eta_{B}|/10^{-10}\) \\ \hline I & \(4\times 10^{11}\) & 43.379 & \(5.43\times 10^{-7}\) & \(-3.99\times 10^{-6}\) & 0.294911 & 0.705088 & 2.02 \\ \hline II & \(2\times 10^{11}\) & 74.759 & \(9.61\times 10^{-7}\) & \(5.16\times 10^{-6}\) & 0.181698 & 0.818301 & 5.10 \\ \hline III & \(8\times 10^{10}\) & 22.473 & \(4.19\times 10^{-7}\) & \(5.38\times 10^{-8}\) & 0.588344 & 0.411655 & 1.56 \\ \hline IV & \(4\times 10^{10}\) & 31.625 & \(4.28\times 10^{-7}\) & \(9.92\times 10^{-7}\) & 0.429678 & 0.570321 & 3.26 \\ \hline \end{tabular}
\end{table}
Table 3: Baryon asymmetries from two-flavoured leptogenesis for different values of \(M\), \(K\), \(\epsilon_{i}\) and \(K_{0}^{i}\).
low-energy observable like neutrino less double beta decay, lepton flavour violation, Jarlskog invariant has been studied in [43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54]. In our work, we encounter a common origin of CP
\begin{table}
\begin{tabular}{|l||l||l|} \hline \(M\)(GeV) & \(|\eta_{B}|\) from unflavoured leptogenesis & \(|\eta_{B}|\) from two-flavoured leptogenesis \\ \hline \(4\times 10^{11}\) & \(1.78\times 10^{-10}\) [Set-I, 6a] & \(2.02\times 10^{-10}\) [Set-I, 6b] \\ \(2\times 10^{11}\) & \(1.67\times 10^{-10}\) [Set-II, 6c] & \(5.10\times 10^{-10}\) [Set-II, 6d] \\ \(8\times 10^{10}\) & \(5.38\times 10^{-11}\) [Set-III, 6e] & \(1.56\times 10^{-10}\) [Set-III, 6f] \\ \(4\times 10^{10}\) & \(1.07\times 10^{-10}\) [Set-IV, 6g] & \(3.26\times 10^{-10}\) [Set-IV, 6h] \\ \hline \end{tabular}
\end{table}
Table 4: Baryon asymmetries from unflavoured and two-flavoured leptogenesis for different values of \(M\), and \(\epsilon(\epsilon_{i})\). We have shown a comparative study on how the flavour effects enhance the baryon asymmetry. The different sets in the table correspond to the corresponding plots, where the set of Boltzmann equations are numerically solved and lepton asymmetries are shown.
Figure 3: Comparing the Baryon asymmetry results coming from unflavoured and two-flavoured leptogenesis regimes. The blue points correspond to baryon asymmetries via unflavoured leptogenesis. On the other hand, the magenta points correspond to baryon asymmetries via two-flavoured leptogenesis. The region between the two red lines signifies the experimentally obtained range of baryon asymmetry, i.e. \([4.7-6.5]\times 10^{-10}\).
violation in low-energy neutrino experiments in terms of \(J_{\rm CP}\) and in high-energy sector in terms of CP asymmetry parameter, required for leptogenesis. It can be explained in a geometrical interpretation of CP violation with Majorana neutrinos.
Analogous to the CKM matrix in the quark sector, in the lepton sector, six unitarity triangles can be formed known as leptonic unitarity triangles, from the orthogonality of the rows and columns of the \(3\times 3\) PMNS matrix. These triangles are analogous to the quark unitarity triangles used for studying various manifestations of CP violation. However, in the case of Majorana neutrinos, there is an important difference which will be discussed here. The unitarity condition of \(V\) is given by,
\[V^{\dagger}V=VV^{\dagger}=1. \tag{47}\]
Under the rephasing transformation of lepton fields \(L_{\alpha}\to e^{i\phi_{\alpha}}L_{\alpha}\), the matrix \(V\) transforms as \(V_{\alpha i}\to V^{\prime}_{\alpha i}=e^{i\phi_{\alpha}}V_{\alpha i}\). The vector \(V_{\alpha i}V^{*}_{\beta i}\to e^{i(\phi_{\alpha}-\phi_{\beta})}V_{\alpha i}V^ {*}_{\beta i}\) rotates in the complex plane, whereas the vector \(V_{\alpha i}V^{*}_{\alpha j}\) remains invariant. Based on this observation, the unitarity triangles are classified into Dirac triangles and Majorana triangles which are discussed below.
### Dirac Unitarity Triangles
From the orthogonality of rows of the mixng matrix \(V\),
\[\Sigma V_{\alpha i}V^{*}_{\beta i}=0\quad(\alpha\neq\beta), \tag{48}\]
we obtain the expressions of three Dirac triangles,
\[T_{e\mu}:V_{e1}V^{*}_{\mu 1}+V_{e2}V^{*}_{\mu 2}+V_{e3}V^{*}_{\mu 3}=0, \tag{49}\]
\[T_{e\tau}:V_{e1}V^{*}_{\tau 1}+V_{e2}V^{*}_{\tau 2}+V_{e3}V^{*}_{\tau 3}=0, \tag{50}\]
\[T_{\mu\tau}:V_{\mu 1}V^{*}_{\tau 1}+V_{\mu 2}V^{*}_{\tau 2}+V_{\mu 3}V^{*}_{ \tau 3}=0. \tag{51}\]
The orientation of the Dirac triangles has no physical significance since under the rephasing of the charged-lepton fields these triangles exhibit rotation in the complex plane. The Dirac triangles share a common area \(A=\frac{1}{2}J_{CP}\) and the vanishing area of the Dirac triangles indicates vanishing Jarlskog Invariant \(J_{CP}=0\) but it does not guarantee the conservation of CP symmetry. It only indicates that the Dirac CP phase is zero, but the two Majorana
phases can still violate CP. Thus Dirac triangles fail to completely describe CP violation [55].
Nevertheless, the quantity \(J_{\rm CP}\) can be determined through
\[J_{CP}=Im\left(V_{11}V_{22}V_{12}^{*}V_{21}^{*}\right), \tag{52}\]
by using the explicit form of \(V\). In this case, it is not possible to get a compact form of \(J_{\rm CP}\), we make an approximate analytical study in appendix(D), which shows that it depends on the phases \(\beta_{\nu}\) and \(\gamma_{\nu}\). Also, it can be observed from equations (38) and (39), the same phases appear in the CP asymmetry parameter. We numerically calculate the values of \(J_{\rm CP}\) and \(\epsilon\) as shown in Fig.(4).
In order to calculate \(\epsilon\) the value of \(m_{1}\) is chosen to be \(0.021\) eV, and \(m_{2}\) and \(m_{3}\) are further calculated following the relation given in Eq.(13), using the allowed parameter space for \([x_{\nu},y_{\nu}]\) given in Eq.(29). The allowed neutrino mass eigenvalues \(m_{1}\), \(m_{2}\), \(m_{3}\) are found by imposing the bound on total neutrino mass \(\Sigma=m_{1}+m_{2}+m_{3}<0.12\) eV. The parameter \(y_{0}\) is varied in the range \([0.0101:0.0201]\). The right-handed neutrino mass \(M\) is chosen to be \(8\times 10^{10}\) GeV. The phases \([\gamma_{\nu},\beta_{\nu}]\) are both varied in the range \([0:2\pi]\). The maximum order of the obtained CP asymmetry is found to be \(|\epsilon|\sim 10^{-5}\).
On the other hand, in the expression of \(J_{CP}\) in Eq.(52), the mixing matrix \(V\) is a function of \(U_{\nu}\), \(U_{l}\) and phases \((\theta,\phi,\psi)\) as it can be seen in Eq.(11). Here \(U_{\nu}\) and \(U_{l}\) are given in appendix (A) and (B), respectively. The elements of \(U_{l}(U_{\nu})\) are functions of charged (neutral) lepton mass ratios \([x_{l}(x_{\nu}),y_{l}(y_{\nu})]\). The charged lepton mass ratios are determined, as given in Eq.(14). Also keeping in mind the factorization of lepton mass matrices given in Eq.(8), can be written, when the the condition imposed on elements of the neutrino mass matrix given in Eq.(7) was \(\arg\left(1+\hat{A_{\nu}}e^{i\gamma_{\nu}}\right)=2\arg\left(r+\hat{B_{\nu}}e^{ i\beta_{\nu}}\right)\). Using this condition one can see \(\psi_{\nu}\) is related to the phases \([\gamma_{\nu},\beta_{\nu}]\) as,
\[\frac{\hat{A_{\nu}}\sin\gamma_{\nu}}{1+\hat{A_{\nu}}\cos\gamma_{\nu}}=\tan 2 \psi_{\nu},\ \ \ \frac{\hat{B_{\nu}}\sin\beta_{\nu}}{r+\hat{B_{\nu}}\cos\beta_{\nu}}=\tan \psi_{\nu}. \tag{53}\]
Therefore, it is understood that \(J_{CP}\) is a function of the phases \([\gamma_{\nu},\beta_{\nu}]\) through \(\psi_{\nu}\). Since, \(\hat{A_{\nu}}\), \(\hat{B_{\nu}}\), \(\hat{C_{\nu}}\) and \(r\) depend on \(M\), \(m_{1}\), \(m_{2}\), \(m_{3}\) and \(y_{0}\), we vary \(m_{1}\sim[0.001-0.05]\)eV and \(y_{0}\sim[0.0001-1]\) and obtained \(m_{2}\), \(m_{3}\) using Eq.(13) within the allowed parameter space for \([x_{\nu},y_{\nu}]\) mentioned in Eq.(29). We also varied \(\theta\), \(\phi\), \(\psi_{l}\) and \([\gamma_{\nu},\beta_{\nu}]\) in the range \([0:2\pi]\), and determined the allowed range for \(\psi_{\nu}\) and thereby for \([\gamma_{\nu},\beta_{\nu}]\), which satisfy the conditions given in Eq.(53) for obtaining a scattered plot of \(J_{CP}\) as a function of \([\gamma_{\nu},\beta_{\nu}]\). The maximum low energy CP violation through \(|J_{CP}|\) for \(M=8\times 10^{10}\) GeV is found to be \(\sim 0.032\).
### Majorana Unitarity Triangles
From the orthogonality of columns of the mixing matrix, \(V\),
\[\Sigma V_{\alpha i}V_{\alpha j}^{*}=0\ \ \ (i\neq j), \tag{54}\]
we obtain the expressions of three Majorana triangles,
\[T_{12}:V_{e1}V_{e2}^{*}+V_{\mu 1}V_{\mu 2}^{*}+V_{\tau 1}V_{\tau 2}^{*} =0, \tag{55}\] \[T_{13}:V_{e1}V_{e3}^{*}+V_{\mu 1}V_{\mu 3}^{*}+V_{\tau 1}V_{\tau 3}^{*} =0,\] (56) \[T_{23}:V_{e2}V_{e3}^{*}+V_{\mu 2}V_{\mu 3}^{*}+V_{\tau 2}V_{\tau 3}^{*} =0. \tag{57}\]
Since the Majorana triangles remain invariant under the rephasing, the orientation of the Majorana triangles has physical significance. These Majorana triangles provide the necessary and sufficient conditions for CP conservation in the lepton sector. The absence of CP violation is guaranteed by
1. Vanishing of the common area \(A=\frac{1}{2}J_{CP}\) of the Majorana triangles.
2. Orientation of all Majorana triangles along the direction of the real or imaginary axes.
The first condition implies that the three triangles collapse into lines in the complex plane and the vanishing of the Dirac phase. The second condition implies that the Majorana phases do not violate CP. Hence the three Majorana triangles are capable to provide a complete description of CP violation, unlike the Dirac triangles. If each one of the sides of the Majorana triangles is not parallel to one of the axes, then it is a signal for CP non-conservation, contrarily to the Dirac triangle case where only a nonzero area signifies CP violation [55].
The low energy CP violation can be obtained through the lepton unitarity triangles as discussed earlier. We consider the Majorana triangle, \(T_{13}\) as expressed in Eq.(56), and define a parameter \(Z\) as a side of the Majorana triangle-\(T13\) after resizing it so that the base of the triangle becomes of unit length. In analogy with the quark sector, the triangle corresponding to the unitary conditions in the first and third columns with proper rescaling is shown in a figure in NuFIT 5.0 [23]. The figure indicates the absence of CP violation would imply a flat triangle i.e., \({\rm Im}(Z)=0\), where
\[Z=-\frac{V_{e1}V_{e3}^{*}}{V_{\mu 1}V_{\mu 3}^{*}}={\rm Re}(Z)+i\ {\rm Im}(Z). \tag{58}\]
As long as the area of the triangle is non-zero and all Majorana triangles are oriented along neither real nor imaginary axes, the CP symmetry will be violated. In order to understand the dependence of the low energy CP violation through the \(Z\)-parameter, on the CP violating phases arising in the neutrino mass matrix, the real and imaginary counterpart of \(Z\)-parameter is plotted as functions of the two phases \((\gamma_{\nu},\beta_{\nu})\) as depicted in Fig.(5). Here, the mixing matrix \(V\) is found from the Eq.(11). From the Eq.(53), it can be seen that \(Z\)-parameter is a function of the phases \([\gamma_{\nu},\beta_{\nu}]\) through \(\psi_{\nu}\).
For right-handed neutrino mass \(M=8\times 10^{10}\) GeV, we varied \(m_{1}\sim[0.001-0.05]\) eV and \(y_{0}\sim[0.0001-1]\) and obtained \(m_{2}\), \(m_{3}\) through Eq.(13) within the allowed parameter space for \([x_{\nu},y_{\nu}]\) given in Eq.(29). We also varied \(\theta\), \(\phi\), \(\psi_{l}\) and \([\gamma_{\nu},\beta_{\nu}]\) in the range \([0:2\pi]\), and determined the allowed range for \(\psi_{\nu}\) and thereby for \([\gamma_{\nu},\beta_{\nu}]\) which satisfy the conditions given in Eq.(53) for obtaining a scattered plot of \({\rm Re}(Z)\) and \({\rm Im}(Z)\) as a function of \([\gamma_{\nu},\beta_{\nu}]\).
## V Conclusion
In the minimal extension of the SM with right-handed neutrinos and scalar triplet, baryogenesis can be achieved through leptogenesis from the CP violating decay of either the lightest right-handed neutrino or triplet scalar. We have studied a minimal type-II seesaw model where the SM is extended with one right-handed neutrino and one triplet scalar. There are two mass scales involved in this case; the mass of the right-handed neutrino, \(M\), and that of the triplet scalar, \(M_{\Delta}\). Considering there are no heavy scalars in the theory we choose to work in the hierarchical mass \(M\ll M_{\Delta}\). In this case, for leptogenesis, the sources of CP violation can be found, that are mediated by the decay of the heavy right-handed neutrino. In the absence of extra right-handed neutrinos, non-vanishing CP asymmetry is sourced from the interference of the tree-level and one-loop diagrams mediated by the Higgs triplet scalar, which is taken to be heavier than the right-handed neutrino. By lowering the mass scale of the heavy right-handed neutrino, its mass is taken to be in the range \(M\subset[10^{10},10^{11}]\) GeV for successful baryogenesis via leptogenesis. In this mass range, we have studied the leptogenesis in a two-flavoured regime. It is seen that for \(M\subset[10^{10},10^{11}]\) GeV, the obtained lepton asymmetries lead to baryon asymmetry within the desired range of \(\eta_{B}\sim(4.7-6.5)\times 10^{-10}\). We show that incorporating appropriate flavour consideration,
Figure 5: 3D scattered plots of \(\text{Re}(Z)(\gamma_{\nu},\beta_{\nu})\) and \(\text{Im}(Z)(\gamma_{\nu},\beta_{\nu})\): The plots show the dependence of the \(Z\)-parameter on the CP violating phases arising in the neutrino mass matrix, for the case of Majorana triangle-\(T_{13}\) (Eq.(56)). The figure in left(right) shows \(\text{Re}(Z)(\text{Im}(Z))\) as functions of the two phases \((\gamma_{\nu},\beta_{\nu})\) appearing in neutrino mass matrix.
the results show an enhancement in the baryon asymmetry as compared to unflavoured case. Thus the minimal type-II seesaw model with only one heavy right-handed neutrino and one heavier Higgs triplet scalar can provide a viable explanation for neutrino mass generation and baryon asymmetry of the Universe through leptogenesis. We show this feature by using the Fritzsch-type texture. Using geometrical interpretation of low-energy CP violation we also show there is a common link between CP violation of both low- and high-energy regimes. These features can further be implemented in more predictive models like left-right symmetric models.
## Appendix A Diagonalizing neutrino mass matrix \(m_{\nu}\) with two-zero texture
The elements of the matrix \(U_{\nu}\) are given in terms of the ratios \(x_{\nu}\), \(y_{\nu}\):
\[U_{\nu 11}=i\left[\frac{1}{\left(1+x_{\nu}\right)\left(1-x_{\nu}^{2}y_{\nu}^{2} \right)}\right]^{\frac{1}{2}},\]
\[U_{\nu 12}=+\left[\frac{x_{\nu}\left(1-y_{\nu}-x_{\nu}y_{\nu}\right)}{\left(1+x _{\nu}\right)\left(1-y_{\nu}\right)\left(1-x_{\nu}y_{\nu}\right)}\right]^{ \frac{1}{2}},\]
\[U_{\nu 13}=+\left[\frac{x_{\nu}^{2}y_{\nu}^{3}}{\left(1-y_{\nu}\right)\left(1- x_{\nu}^{2}y_{\nu}^{2}\right)}\right]^{\frac{1}{2}},\]
\[U_{\nu 21}=-i\left[\frac{x_{\nu}}{\left(1+x_{\nu}\right)\left(1+x_{\nu}y_{ \nu}\right)}\right]^{\frac{1}{2}},\]
\[U_{\nu 22}=+\left[\frac{\left(1-y_{\nu}-x_{\nu}y_{\nu}\right)}{\left(1+x_{\nu} \right)\left(1-y_{\nu}\right)}\right]^{\frac{1}{2}},\]
\[U_{\nu 31}=+i\left[\frac{x_{\nu}^{2}y_{\nu}\left(1-y_{\nu}-x_{\nu}y_{\nu} \right)}{\left(1+x_{\nu}\right)\left(1-x_{\nu}^{2}y_{\nu}^{2}\right)}\right]^{ \frac{1}{2}},\]
\[U_{\nu 32}=-\left[\frac{x_{\nu}y_{\nu}}{\left(1+x_{\nu}\right)\left(1-y_{\nu} \right)\left(1-x_{\nu}y_{\nu}\right)}\right]^{\frac{1}{2}},\]
\[U_{\nu 33}=+\left[\frac{1-y_{\nu}-x_{\nu}y_{\nu}}{\left(1-y_{\nu}\right)\left(1 -x_{\nu}^{2}y_{\nu}^{2}\right)}\right]^{\frac{1}{2}}.\]
## Appendix B Diagonalizing charged-lepton mass matrix \(m_{l}\) with three-zero texture
The elements of the matrix \(U_{l}\) are given in terms of the ratios \(x_{l}\), \(y_{l}\):
\[U_{l11}=+\left[\frac{1-y_{l}}{\left(1+x_{l}\right)\left(1-x_{l}y_{l}\right) \left(1-y_{l}+x_{l}y_{l}\right)}\right]^{\frac{1}{2}},\]
\[U_{l12}=-i\left[\frac{x_{l}\left(1+x_{l}y_{l}\right)}{\left(1+x_{l}\right) \left(1+y_{l}\right)\left(1-y_{l}+x_{l}y_{l}\right)}\right]^{\frac{1}{2}},\]
\[U_{l13}=+\left[\frac{x_{l}y_{l}^{3}\left(1-x_{l}\right)}{\left(1-x_{l}y_{l} \right)\left(1+y_{l}\right)\left(1-y_{l}+x_{l}y_{l}\right)}\right]^{\frac{1}{ 2}},\]
\[U_{l21}=+\left[\frac{x_{l}\left(1-y_{l}\right)}{\left(1+x_{l}\right)\left(1- x_{l}y_{l}\right)}\right]^{\frac{1}{2}},\]
\[U_{l22}=+i\left[\frac{1+x_{l}y_{l}}{\left(1+x_{l}\right)\left(1+y_{l}\right) }\right]^{\frac{1}{2}},\]
\[U_{l23}=+\left[\frac{y_{l}\left(1-x_{l}\right)}{\left(1-x_{l}y_{l}\right) \left(1+y_{l}\right)}\right]^{\frac{1}{2}},\]
\[U_{l31}=-\left[\frac{x_{l}y_{l}\left(1-x_{l}\right)\left(1+x_{l}y_{l}\right) }{\left(1+x_{l}\right)\left(1-x_{l}y_{l}\right)\left(1-y_{l}+x_{l}y_{l}\right) }\right]^{\frac{1}{2}},\]
\[U_{l32}=-i\left[\frac{y_{l}\left(1-x_{l}\right)\left(1-y_{l}\right)}{\left(1 +x_{l}\right)\left(1+y_{l}\right)\left(1-y_{l}+x_{l}y_{l}\right)}\right]^{ \frac{1}{2}},\]
\[U_{l33}=+\left[\frac{\left(1-y_{l}\right)\left(1+x_{l}y_{l}\right)}{\left(1- x_{l}y_{l}\right)\left(1+x_{l}\right)\left(1-y_{l}+x_{l}y_{l}\right)}\right]^{ \frac{1}{2}}.\]
## Appendix C The importance of flavour projectors in leptogenesis
Before deepening the elaborate execution of flavoured leptogenesis, we will see how lepton flavours play a role in constructing CP asymmetry. In general, in the case of leptogenesis, CP violation can be observed from \(N_{1}\) decay in two cases:
1. If the rate of production of leptons and anti-leptons differ, expressed as \[\Gamma\neq\overline{\Gamma},\] where \(\Gamma\) is the decay rate of the process \(N_{1}\longrightarrow l+\phi^{\dagger}\), and \(\overline{\Gamma}\) is the decay rate of the process \(N_{1}\longrightarrow\overline{l^{\prime}}+\phi\). The CP asymmetry can be expressed in terms of these decay rates, as \[\epsilon=\frac{\Gamma-\overline{\Gamma}}{\Gamma+\overline{\Gamma}}.\] (122)
2. If the lepton flavour effects are included, it can affect the CP asymmetry construction in two ways, 1. It suppresses the washout, as the interaction of the leptons with Higgs, during inverse decay, gets fragmented in terms of different flavour states. In this context, we define a parameter called flavour projector[28, 32, 56, 57], \[K_{i}=\left|<l|l_{i}>\right|^{2}=\frac{\Gamma_{i}}{\Gamma},\quad i=e,\mu,\tau\] and \[\overline{K}_{i}=\left|<\overline{l^{\prime}}\right|\overline{l}_{i}>|^{2}= \frac{\overline{\Gamma}_{i}}{\overline{\Gamma}}.\] Here, \(\Gamma_{i}\) is the partial decay rate of the process \(N_{1}\longrightarrow l_{i}+\phi^{\dagger}\), and \(\overline{\Gamma}_{i}\) is the partial decay rate of the process \(N_{1}\longrightarrow\overline{l}_{i}+\phi\). The concept of total decay rate \(\Gamma=\sum_{i}\Gamma_{i}\), and \(\overline{\Gamma}=\sum_{i}\overline{\Gamma}_{i}\) leads to the realisation that \(\sum_{i}K_{i}=\sum_{i}\overline{K}_{i}=1\). Due to these flavour effects, we need to consider individually flavoured CP asymmetries as \[\epsilon_{i}=\frac{\Gamma_{i}-\overline{\Gamma}_{i}}{\Gamma_{i}+\overline{ \Gamma}_{i}}.\] (C2) 2. If the state \(|\overline{l^{\prime}}>\) is not the CP conjugate state of the state \(|l>\), which arises from misalignment in the flavour space due to loop-effects, then it gives an additional source of CP violation. To understand this effect, we introduce projector difference, as \[\Delta K_{i}=K_{i}-\overline{K}_{i}.\] The CP asymmetry can be further modified, incorporating \(K_{i}=K_{0}^{i}+\frac{\Delta K_{i}}{2}\) and \(\overline{K}_{i}=K_{0}^{i}-\frac{\Delta K_{i}}{2}\), as \[\epsilon_{i}\sim\epsilon K_{0}^{i}+\frac{\Delta K_{i}}{2}.\] (C3) Here, \(K_{0}^{i}=\frac{K_{i}+\overline{K}_{i}}{2}\) is the tree level contributions to the projections.
In the Eq.(C3), the first term on the right-hand side, proportional to \(\epsilon\), comes from the type of contribution discussed in the first case. On the other hand, the second term proportional to \(\Delta K_{i}\) comes from the contribution mentioned in the second case, as the term vanishes when we have \(|\overline{l^{\prime}}\rangle\) as the CP conjugate state of \(|l\rangle\). From Eq.(C3), it can be easily shown that \(\epsilon=\sum_{i}\epsilon_{i}\). As we are particularly interested in a temperature range where lepton flavour interaction becomes important, we have incorporated flavour projectors, especially the tree
level contribution \(K_{0}^{i}\) in the flavoured Boltzmann equations, which we have described in the next section. Flavour projectors become important to segregate the flavour regimes of leptogenesis in terms of lepton flavour alignment or non-alignment. In the context of our chosen temperature range, the lepton flavour non-alignment problem will be relevant, which refers to the situation when no flavour state can be found to be perfectly aligned with the states \(\left|l\right\rangle\) and \(\left|l^{\prime}\right\rangle\)[28].
When only the \(\tau\)-Yukawa processes are in thermal equilibrium, then the concept of flavour projectors suggests,
\[\sum_{i=a,\tau}K_{i}=K_{a}+K_{\tau}=1,\]
and
\[\sum_{i=a,\tau}\overline{K_{i}}=\overline{K}_{a}+\overline{K}_{\tau}=1,\]
where, \(\left|l_{a}\right\rangle\) and \(\left|\overline{l^{\prime}}_{a}\right\rangle\) are two entangled states of flavours \(e\) and \(\mu\). Generally, this condition appears as a two-flavoured leptogenesis scenario. On the other hand, when \(\tau\) and \(\mu\)-Yukawa processes are in thermal equilibrium, no entangled flavour states can be formed. The thermal bath becomes populated with the CP-conjugate flavour states \(\left|l_{i}\right\rangle\) and \(\left|\overline{l}_{i}\right\rangle\) (with \(i=e\), \(\mu\), \(\tau\)). Here arises typically the case of three-flavoured or fully-flavoured leptogenesis. From the flavour projector condition, we obtain,
\[\sum_{i=e,\mu,\tau}K_{i}=K_{e}+K_{\mu}+K_{\tau}=1,\]
and
\[\sum_{i=e,\mu,\tau}\overline{K}_{i}=\overline{K}_{e}+\overline{K}_{\mu}+ \overline{K}_{\tau}=1.\]
## Appendix D Low scale CP violation and Jarlskog invariant \(J_{cp}\)
In simplified form,
\[J_{CP}\sim\text{Im}\left[V_{13}\right], \tag{12}\]
where \(V_{13}\) can be obtained from Eq.(11) as,
\[V_{13}=U_{l11}U_{\nu 13}^{\;\;*}e^{i\theta}+U_{l21}U_{\nu 23}^{\;\;*}e^{i \phi}+U_{l31}U_{\nu 33}^{\;\;*}e^{i\psi}. \tag{13}\]
From appendix(A), it is clear that \(U_{l11}\), \(U_{l21}\), \(U_{l31}\), \(U_{\nu 13}\), \(U_{\nu 23}\), \(U_{\nu 33}\) are all real, and \(U_{l11}\), \(U_{l21}\), \(U_{l31}\) are function of \(x_{l}=\frac{m_{e}}{m_{\nu}}\), \(y_{l}=\frac{m_{\mu}}{m_{\tau}}\). Since, the charged lepton masses are already
known, the ratios \(x_{l}\) and \(y_{l}\) have definite values, and so the values of \(U_{l11}\), \(U_{l21}\), \(U_{l31}\) are also known. Hence, after further simplification of equations (45) and (46), we obtain
\[J_{CP}\sim U_{l11}U_{\nu 13}\sin\theta+U_{l21}U_{\nu 23}\sin\phi+U_{l31}U_{\nu 3 3}\sin\psi. \tag{47}\]
Leading from the definition of \(x_{\nu}\) and \(y_{\nu}\) given in Eq.(13) and combining equations (23), (24), and (25), the parameters \(U_{\nu 13}\), \(U_{\nu 23}\), \(U_{\nu 33}\) can be written as functions of the two arbitrary CP violating phases \(\gamma_{\nu}\) and \(\beta_{\nu}\) (introduced in Eq.(7)) as \(U_{\nu 13}\to g_{1}\left(\gamma_{\nu},\beta_{\nu}\right)\), \(U_{\nu 23}\to g_{2}\left(\gamma_{\nu},\beta_{\nu}\right)\) and \(U_{\nu 33}\to g_{3}\left(\gamma_{\nu},\beta_{\nu}\right)\). Hence,
\[J_{CP}\sim U_{l11}g_{1}\left(\gamma_{\nu},\beta_{\nu}\right)\sin\theta+U_{l21 }g_{2}\left(\gamma_{\nu},\beta_{\nu}\right)\sin\phi+U_{l31}g_{3}\left(\gamma_ {\nu},\beta_{\nu}\right)\sin\psi. \tag{48}\]
The functions \(g_{1}\left(\gamma_{\nu},\beta_{\nu}\right)\), \(g_{2}\left(\gamma_{\nu},\beta_{\nu}\right)\), \(g_{3}\left(\gamma_{\nu},\beta_{\nu}\right)\) are given by,
\[g_{1}\left(\gamma_{\nu},\beta_{\nu}\right)=r\left(f_{1}-f_{2}\right)\left[ \frac{1}{2f_{1}f_{2}(f_{1}+f_{2}-2r^{2})}\right]^{\frac{1}{2}}, \tag{49}\]
\[g_{2}\left(\gamma_{\nu},\beta_{\nu}\right)=\left[\frac{f_{2}(f_{1}^{2}-f_{2}^ {2})}{2f_{1}f_{2}(f_{1}+f_{2}-2r^{2})}\right]^{\frac{1}{2}}, \tag{50}\]
\[g_{2}\left(\gamma_{\nu},\beta_{\nu}\right)=\left(f_{1}+f_{2}\right)\left[ \frac{(f_{2}-r^{2})}{2f_{1}f_{2}\left(f_{1}+f_{2}-2r^{2}\right)}\right]^{ \frac{1}{2}}, \tag{51}\]
where
\[f_{1}\to f_{1}\left(\gamma_{\nu},\beta_{\nu}\right)=\left[1+\hat{A}_{\nu}^{2} +4\left(r^{2}+\hat{B}_{\nu}^{2}+\hat{C}_{\nu}^{2}\right)+2\hat{A}_{\nu}\cos \gamma_{\nu}+8r\hat{B}_{\nu}\cos\beta_{\nu}\right]^{\frac{1}{2}}, \tag{52}\]
and
\[f_{2}\to f_{2}\left(\gamma_{\nu}\right)=\left[1+\hat{A}_{\nu}^{2}+2\hat{A}_{ \nu}\cos\gamma_{\nu}\right]^{\frac{1}{2}}. \tag{53}\]
## Appendix E Numerical solution of Boltzmann Equations
The solutions of the set of Boltzmann equations are depicted in the figures below (Fig.(6)).
Figure 6: Evolution of lepton asymmetries for different choices of right-handed neutrino mass \(M\), in unflavoured (left column) and two-flavoured (right column) leptogenesis regimes. | **Standard Model** extension by adding one right-handed neutrino and one triplet scalar. These heavy particles contribute to the generation of tiny neutrino mass through the seesaw mechanism. The contribution of the heavy particles to the neutrino masses is inversely proportional to their corresponding masses. Considering leptogenesis achieved by the decay of the right-handed neutrino, the new source of CP asymmetry comes solely from the decay of the right-handed neutrino with a one-loop vertex diagram involving the triplet scalar. The predictiveness of the model is enhanced by introducing Fritzsch-type textures for the neutrino mass matrix and charged lepton mass matrix. We execute the parameter space study following the latest neutrino oscillation data. We study baryogenesis via leptogenesis in the two-flavor regime, using the zero textures, and show that there is an enhancement in baryon asymmetry as compared to the unflavored regime. For two-flavor leptogenesis, we consider the suitable temperature regime $T\subset[10^{10}, |
2305.14467 | FLAIR #2: textural and temporal information for semantic segmentation
from multi-source optical imagery | The FLAIR #2 dataset hereby presented includes two very distinct types of
data, which are exploited for a semantic segmentation task aimed at mapping
land cover. The data fusion workflow proposes the exploitation of the fine
spatial and textural information of very high spatial resolution (VHR)
mono-temporal aerial imagery and the temporal and spectral richness of high
spatial resolution (HR) time series of Copernicus Sentinel-2 satellite images.
The French National Institute of Geographical and Forest Information (IGN), in
response to the growing availability of high-quality Earth Observation (EO)
data, is actively exploring innovative strategies to integrate these data with
heterogeneous characteristics. IGN is therefore offering this dataset to
promote innovation and improve our knowledge of our territories. | Anatol Garioud, Apolline De Wit, Marc Poupée, Marion Valette, Sébastien Giordano, Boris Wattrelos | 2023-05-23T18:47:19 | http://arxiv.org/abs/2305.14467v1 | # FLAIR: French Land cover from Aerospace ImageRy.
###### Abstract
According to a report by the Food and Agriculture Organization of the United Nations (FAO) in 2015 [1], a significant portion of the world's soil resources are in a condition that can be classified as fair, poor, or very poor. This degradation of soils, coupled with the loss of biodiversity, has far-reaching implications for the state of ecosystems and their long-term sustainability. Soils play a vital role in providing a range of ecosystem services. They serve as natural habitats for numerous plant and animal species, act as a crucial carbon sink by absorbing CO\({}_{2}\) (to the extent that they are the largest carbon sink, surpassing the atmosphere and all vegetation and animals on Earth's surface), filter rainwater, support food production, and function as the planet's largest water reservoir. The degradation of soils and biodiversity can be attributed in large part to the process of land artificialization, with urban sprawl being a significant contributing factor. This growing phenomenon has raised concerns among public authorities, who recognize the importance of monitoring the state of territories. Artificialization is defined as the long-term deterioration of the ecological functions of soil, including its biological, hydrological, climatic, and agronomic functions, resulting from its occupation or use [2].
The French National Institute of Geographical and Forest Information (IGN) [3], in response to the growing availability of high-quality Earth Observation (EO) data, is actively exploring innovative strategies to integrate these data with heterogeneous characteristics. As part of their initiatives, the institute employs artificial intelligence (AI) tools to monitor land cover across the territory of France and provides reliable and up-to-date geographical reference datasets.
The FLAIR #1 dataset, which focused on aerial imagery for semantic segmentation, was released to facilitate research in the field. Building upon this datset, the FLAIR #2 dataset extends the capabilities by incorporating a new input modality, namely Sentinel-2 satellite image time series, and introduces a new test dataset Both FLAIR #1 and #2 datasets are part of the currently explored or exploited resources by IGN to produce the French national land cover map reference _Occupation du sol a grande echelle_ (OCS-GE).
The growing importance of EO in the monitoring and understanding of Earth's physical processes, and the diversity of data now publicly available naturally favours multi-modal approaches that take advantage of the distinct strengths of this data pool. Remote sensing data have several main characteristics that are of crucial importance depending on the intended purpose. Spatial, temporal and spectral resolutions will influence the choice of data and their importance in a process. The complexity of integrating these different data tend to promotes the use of machine learning for their exploitation.
This FLAIR #2 challenge organized by IGN proposes the development of multi-resolution, multi-sensor and multi-temporal aerospace data fusion methods, exploiting deep learning computer vision techniques.
The FLAIR #2 dataset hereby presented includes two very distinct types of data, which are exploited for a semantic segmentation task aimed at mapping land cover. The data fusion workflow proposes the exploitation of the fine spatial and textural information of very high spatial resolution (VHR) mono-temporal aerial imagery and the temporal and spectral richness of high spatial resolution (HR) time series of Copernicus Sentinel-2 [4] satellite images, one of the most prominent EO mission. Although less spatially detailed, the information contained in satellite time series can be helpful in improving the inter-class distinction by analyzing their temporal profile and different responses in parts of the electromagnetic (EM) spectrum.
**Spatial and temporal domains definition**
**Spatial domains and divisions**: as for the FLAIR #1 dataset, a spatial domain is equivalent to a French 'departement' which is a french sub-region administrative division. While the spatial domains can be geographically close, heavy pre-processing of the radiometry of aerial images independently per 'departement' create important differences (see [5]). Each domain has a varying number of areas subdivided in patches of same size across the dataset.
While these areas were initially defined to contain sufficient spatial context by taking into account aerial imagery, the strong difference in spatial resolution with satellite data means that they consist of few Sentinel-2 pixels. Therefore, in order to also provide a minimum of context from the satellite data, a buffer was applied to create _super-areas_. This allows, for every patch of the dataset to be associated to a _super-patch_ of Sentinel-2 data with sufficient size through a large footprint. Figure 1 illustrates the different spatial units of the dataset.
**Temporal domains**: they are twofold, on the one hand the date of acquisition of the aerial imagery (which varies in terms of year, month, days) and on the other hand by the satellite acquisitions, varying in terms of months and days.
**Dataset extent**: The dataset includes 50 spatial domains (Figure 2) representing the different landscapes and climates of metropolitan France. The train dataset constitute 4/5 of the spatial domains (40) while the remaining 1/5 domains (10) are kept for testing. This test dataset introduces new domains compared to the FLAIR #1 test dataset. Some domain are in common but areas within those domains are distinct. The FLAIR #2 dataset covers approximately 817 km 2 of the French metropolitan territory.
Footnote 2: [https://www.face.com/face](https://www.face.com/face)
For details about aerial images (ORTHO HR(r)) and associated elevation data, as well as pre-processing, refer to the FLAIR #1 datapapaper [5].
Technical details about Sentinel-2 can be found in [4]. The images were downloaded from the Sinergise API [6] as Level-2A products (L2A) which are atmospherically corrected using the Sen2Cor algorithm [7]. L2A products provide Bottom-Of-the-Atmosphere (BOA) reflectances, corresponding to a percentage of the energy the surface reflects. L2A products also deliver pixel-based cloud (CLD) and snow (SNW) masks at 20 m spatial resolution. Sentinel-2 images are typically provided as 110\(\times\)110 km (with 10 km overlay) squared ortho-image in UTM/WGS84 projection. However, in order to limit the size of the data and due to the wide extent of the dataset, only the super-areas were downloaded. Concerning Sentinel-2 pre-processing, the 20 m spatial resolution bands are first resampled during data retrieval to 10 m by the nearest interpolation method. Same approach is adopted for the cloud and snow masks. Due to the relative orbits of Sentinel-2 some images contain nodata pixels (reflectances at 0). As all Sentinel-2 images during the aerial image acquisition year are gathered all dates containing such nodata were removed. It must be remarked that the length of time series and the acquisition dates thus varies for each super-area. Table II provides information about the number of dates included in the filtered Sentinel-2 time series for the train and test datasets. In average, each area is acquired on 55 dates over the course of a year by the satellite imagery.
Note that cloudy dates are not suppressed from the time series. Instead, the masks are provided and can be used to filter the cloudy dates if needed. The resulting Sentinel-2 time series are subsequently reprojected into the Lambert-93 projection (EPSG:2154) which is the one of the aerial imagery.
**Data description, naming conventions and usage**
The FLAIR #2 dataset is composed of 77,762 aerial imagery patches, each 512\(\times\)512 pixels, along with corresponding annotations, resulting in a total of over 20 billion pixels. The patches correspond to 916 areas distributed across 50 domains and cover approximately 817 km\({}^{2}\). The area sizes and the number of patches per area vary but are always a multiple of 512 pixels at a resolution of 0.20 meters. Additionally, the dataset includes 55,244 satellite super-areas acquisitions that have a buffer of 5 aerial patches (512 m) surrounding each aerial area. Description of the data is provided bellow:
* The **aerial input patches (IMG)** consist of 5 channels, similar to the FLAIR #1 dataset. These channels include blue, green, red, near-infrared, and elevation bands, all encoded as 8-bit unsigned integer datatype. The aerial patches are named as _IMG_ID_, with a unique identifier (ID) across the dataset assigned to each patch. A file named _flair_aerial_metadata_json_ contains metadata for each of the aerial patches. This JSON file provides detailed information such as the date and time of acquisition, the geographical location of the patch centroid (x, y), the mean altitude of the patch (z), and the type of camera used. For more in-depth descriptions of these metadata attributes, please refer to the documentation provided in [5].
* _data_, _masks_, _products_ and a _JSON_ file to match aerial and satellite imagery
- : * the super-area reflectance time series is stored in the _SEN2_xxxx_data.npy_ files. These files contain 4D NumPy arrays with a shape of \(T\times\)C\(\times\)H\(\times\)_W_, where \(T\) represents the acquisition dates (which can vary for each file), \(C\) represents the 10 spectral bands of Sentinel-2, and \(H\) and \(W\) denote the height and width dimensions of the data, respectively. The data is stored as uint16 datatype, which differs from the acquisition datatype mentioned in the Senflub reference provided [6]. It's important to note that the data in these files is provided without any additional processing or modifications. * the super-area cloud and snow masks are stored in the _SEN2_xxxx_masks.npy_ files. These files have a similar shape as the data files, with a 4D array format of \(T\times\)C\(\times\)_H\(\times\)W_. However, they consist of only two channels, representing the snow masks and cloud masks, respectively, in that order. The values in the masks range from 0 to 100 and indicate the probability of cloud or snow occurrence for each pixel. A value of 100 indicates a high probability. * the names of the Sentinel-2 time series products are listed in the _SEN2_xxxx_products.txt_ file. This file provides additional information for each acquisition, including the Sentinel-2 platform (S2A or S2B), the acquisition date (which corresponds to the first date mentioned in the product name), the acquisition time, the orbit number and tile name associated with the product. These details help identify and differentiate the specific products within the Sentinel-2 time series dataset.
\begin{table}
\begin{tabular}{l|c c c|c} & \multicolumn{4}{c}{ acquisitions per super-area} \\ \cline{2-5} Sentinel-2 time series (1 year) & min & max & mean & total \\ \hline train dataset & 20 & 100 & 55 & 757 \\ test dataset & 20 & 114 & 55 & 193 \\ \hline \end{tabular}
\end{table} TABLE II: Number of acquisitions (dates) in the Sentinel-2 times series of one year (corresponding to the year of aerial imagery acquisition).
Additionally, _flair-2_centroids_sp_to_patch_json_ file is provided alongside the data. This file plays a role in dynamically cropping the satellite super-areas into super-patches during the data loading process. The JSON file uses the aerial patch name (_e.g._, IMG_077413) as the key and provides a list of two indexes (_e.g._, [13, 25]) that represent the data-coordinates of the aerial patch centroids. Using these coordinates and a specified number of pixels (referred to as _sat_superpatch_size_), super-patches are extracted from the satellite data. For the experiments, the default _sat_superpatch_size_ is set to 40, resulting in super-patches with a spatial size of 40*40 pixels. This size corresponds approximately to two aerial patches on each side of the centroid.
The pattern \(\mathbf{xxxx}\) in the file names corresponds to the format domain_year-areanumber_arealandcoverletters (_e.g._, D077_2021-Z9_AF). The _arealandcoverletters_ represent the two broad types of land cover present in the area. For more detailed information about the specific land cover types, please refer to [5].
* The **annotation patches (MSK)** consist of a single channel with values ranging from 1 to 19, encoded as an 8-bit unsigned integer datatype. These files are named as _MSK_ID_, where ID corresponds to the same identifier used for the aerial imagery patches. It is important to note that annotations are limited to the boundaries of aerial imagery areas and do not extend to satellite super-areas. In addition, annotations derived from aerial imagery correspond to the specific date the images were captured. However, certain evolving classes may not accurately reflect the current state of the features as observed in Sentinel imagery. For instance, the banks of a watercourse, delineated based on aerial imagery, may undergo changes over time, spanning a year. These changes can result from various factors such as natural
Fig. 4: Example of input and supervision data: true color composition, near-infrared color composition, elevation band, Sentinel-2 true color composition super-patch and supervision masks. The data from the first three columns are retrieved from the IMG files, the super-patch from SEN numpy files while the last column corresponds to the MSK files.
processes or human activities, causing the banks to shift or erode. Consequently, the annotations based on older aerial imagery may not capture these temporal variations.
Figure 4 gives an example of aerial patches, corresponding extracted super-patch (with the aerial patch footprint in the outlines) and annotation patches. The interest of the extended spatial information provided by the Sentinel-2 super-patches is particularly visible in the last two rows of Figure 4. Indeed, the location on a beach or on a lake is difficult to determine from the aerial image alone, and could easily be confused with the sea for example in the last row.
The current test dataset has a different sampling than FLAIR #1. The use of satellite time series to inject temporal information is especially relevant for natural surfaces with _e.g._ a seasonal variation. Therefore, the classes of forests (coniferous and deciduous), agricultural land and herbaceous cover were favored, accounting for 72.98% of the test dataset.
## Benchmark architecture
**Network definition**: to capture both spatial and temporal information from very high resolution aerial images and high-resolution satellite images, we propose a two-branch architecture called **U-T&T**, for _Textural_ and _Temporal_ information. The model allows enables the fusion of learned time series-related information with the low-level representations of mono-date learned information. The U-T&T model combines two commonly used architectures:
* **U-Net (spatial/texture branch)**: to handle the aerial imagery patches, a U-Net architecture [8] is adopted. The encoder is using a ResNet34 backbone model [9] which has been pre-trained on the ImageNet dataset [10]. The U-Net branch has \(\approx\) 24.4 M parameters. Ith closely resembles to the model described in the FLAIR #1 datapapper [5], ensuring consistency and comparability with prior work.
* **U-TAE (spatio-temporal branch)**: a U-TAE [11] architecture focuses on extracting and incorporating both spatial and temporal information from the Sentinel-2 time series data. This architecture is based on U-Net but incorporates a Temporal self-Attention Encoder (TAE) component taking as input the lowest resolution features of the convolutional encoder to generate set of attention masks that capture the temporal dependencies within the time series data. These attention masks are then applied at all resolutions upon the decoding process, enabling the model to capture spatio-temporal patterns in the data.
Fig. 5: Class distribution of the train dataset (_top_) and test dataset (_bottom_).
\begin{table}
\begin{tabular}{l c c c}
**Class** & **MSK** & **Pixels** & **\%** \\ \hline building & 1 & 1,453,245,093 & 7.13 \\ pervious surface & 2 & 1,495,168,513 & 7.33 \\ impervious surface & 3 & 2,467,133,374 & 12.1 \\ bare soil & 4 & 629,187,886 & 3.09 \\ water & 5 & 922,004,548 & 4.52 \\ coniferous & 6 & 873,397,479 & 4.28 \\ deciduous & 7 & 3,531,567,944 & 17.32 \\ brushwood & 8 & 1,284,640,813 & 6.3 \\ vineyard & 9 & 612,965,642 & 3.01 \\ herbaceous vegetation & 10 & 3,717,682,095 & 18.24 \\ agricultural land & 11 & 2,541,274,397 & 12.47 \\ plowed land & 12 & 703,518,642 & 3.45 \\ other & **\textgreater{13}** & 153,055,302 & 0.75 \\ \hline \end{tabular}
\end{table} TABLE III: Semantic classes of the main nomenclature of the FLAIR #2 dataset and their corresponding MSK values, frequency in pixels and percentage among the entire dataset.
Figure 6 provides an overview of the proposed method, which combines the U-TAE and U-Net architectures. The main idea behind this approach is to incorporate features learned by the U-TAE branch, which considers the temporal dimension and a wider spatial context, into the U-Net branch, which focuses on aerial imagery. However, a key constraint is the significant difference in spatial resolution between the satellite and aerial data. With the satellite imagery having a spatial resolution 50 times lower than the aerial imagery (10 m versus 0.2 m), early and late fusion strategies (_i.e._, fusion at input or prediction levels) are not viable due to the large size disparity. To address this, a _Fusion Module_ is introduced, depicted in Figure 7, which enables mid-stage fusion of features from both branches:
* **Fusion Module**: the fusion module takes as input the U-TAE embedding (last feature maps of the U-TAE decoder, shown in blue in Figure 6) and is applied to each stage of the U-Net branch. Within the _Fusion Module_, two sub-modules have different purposes and focus on distinct aspects:
* _Cropped_: this sub-module aims at incorporating information from the U-TAE super-patch embedding into the spatial extent of the aerial parches. The U-TAE embedding is first cropped to match the extent of the aerial patch. This cropped embedding is then fed to a single convolution layer, which produces a new channel dimension size that aligns with the one of the
Fig. 6: _Texture and Time_ extraction network including two branches: i) a U-TAE network applied to the Sentinel-2 super-patch time series and ii) a U-Net network applied to the mono-date aerial imagery patch. The last decoder layer yielded features from the U-TAE branch are used as embeddings added to the features of the U-Net branch, integrating temporal information from the time series and spatial information from the extended super-patch. The light-blue fusion type modules are enabled or not and varying according to the fusion method.
Fig. 7: Fusion module taking as input the last U-TAE embeddings. This module is applied to each stages of the U-Net encoder feature maps. _out_ corresponds to the channel size of the U-Net encoder feature map and \(H\) and \(W\) to the corresponding spatial dimensions.
U-Net encoder feature maps channel size. The output of this convolutional layer is then passed through an interpolation layer that utilizes bilinear resampling. This interpolation ensures that the spatial dimensions matches those of the U-Net feature maps.
* _Collapsed_: this sub-module is designed to preserve spatial information from the extended super-patch, which will be integrated into the U-Net feature maps. Initially, the spatial dimension of the U-TAE is collapsed into a single value per channel, typically by taking the mean. The resulting vector is then fed into a shallow Multi-Layer Perceptron (MLP) consisting of three linear layers with dropout regularization and Rectified Linear Unit (ReLU) activation. The output size of the MLP is adjusted to match one of the U-Net encoder feature maps channel size.Subsequently, for each value in the obtained vector, the value is duplicated across the spatial dimension of the corresponding U-Net encoder feature maps.
Both the _cropped_ and _collapsed_ sub-modules produce a mask of size _out\(\times H\times W\)_, where _out_, \(H\), and \(W\) correspond to the targeted feature map dimensions of the U-Net model. These masks, generated separately, are initially added together to integrate spatio-temporal information from the Sentinel-2 satellite time series. The resulting combined mask is added to the feature maps of the U-Net model. This integration step allows the spatio-temporal information captured by the _cropped_ and _collapsed_ sub-modules from the Sentinel-2 satellite time series to be incorporated into the U-Net's feature representation.
**Network supervision**: a single \(\mathcal{L}_{\mathcal{TLT}}\) loss is used to monitor the training, which is the sum of two auxiliary losses \(\mathcal{L}_{sat}\) and \(\mathcal{L}_{aerial}\), obtained respectively from the U-TAE and U-Net branches. The two branches are using a categorical Cross Entropy (CE) cost-function, suitable for multi-class supervised classification task :
\[\mathcal{L}_{\mathcal{CE}}=-\sum_{i=1}^{n}t_{i}\log(p_{i})\quad,\]
\[\mathcal{L}_{TkT}=\mathcal{L}_{CE\ aerial}+\mathcal{L}_{CE\ sat}\]
where \(t_{i}\) is the MSK label and \(p_{i}\) the Softmax probability of the \(i^{th}\) class.
The MSK files in the FLAIR #2 dataset are provided at a spatial resolution of 0.2 m. The output of the U-TAE branch corresponds to a super-patch, which lacks annotations for most of its parts. To address this, the U-TAE outputs are initially cropped to match the extent of the corresponding aerial patch. Subsequently, they are interpolated to fit the spatial dimensions of the MSK files (512\(\times\)512 pixels). This interpolation ensures compatibility before calculating the \(\mathcal{L}_{sat}\) loss.
**Benchmark metric**
The evaluation methodology for the semantic segmentation task follows the approach used in the FLAIR #1 challenge [5]. Initially, confusion matrices are calculated per patch, and then aggregated across the test dataset to create a single confusion matrix. To assess the performance of each semantic class, the Intersection over Union (IoU) metric, also known as the Jaccard Index, is computed. The IoU is calculated using the formula:
\[IoU=\frac{|U\cap V|}{|U\cup V|}=\frac{TP}{TP+FP+FN}\]
where U denotes the intersection, V the union, TP the true positives, FP the false positives and FN the false negatives.
The mean Intersection over Union (**mIoU**) is then determined by taking the average of the per-class IoU values. However, since the _other_ class is not well-defined and is equivalent to void, it is excluded from the IoU calculations. Consequently, the mIoU is computed as the average of the IoUs from the remaining 12 classes.
**Benchmark framework and settings**
The baselines are calculated using the efficient _PyTorch Lightning_ framework [12]. For the implementation of the U-Net model, the _segmentation-models-pytorch_ library [13] is exploited, while the U-TAE network is obtained from [11]. The U-TAE parameters are kept at their default values (as provided in the GitHub implementation), except for the encoder and decoder widths.
For the training process, the train dataset consists of 40 domains, out of which 32 are used for training the model, while the remaining 8 domains are used for validation. The optimization technique employed is the stochastic gradient descent (SGD) with a learning rate of 0.001. A reduction strategy is implemented with a patience value of 10, allowing for adaptive adjustments to the learning rate during training. The maximum number of epochs is set to 100, but to prevent overfitting and save computational resources, an early stopping method is utilized with a patience of 30 epochs. A batch size of 10 is used for the baselines.
To ensure reproducibility and consistent results, all randomness is controlled by fixing the seed using the _seed_everything_ function from the PyTorch library, with the seed value set to 2022. Twelve NVIDIA Tesla V100 GPUs with 32 GB memory each, located on a High-Performance Computing (HPC) cluster, are used to speed up experiments. The distributed data parallel (ddp) strategy is employed to
leverage these computational resources efficiently, allowing for parallel training across multiple GPUs.
In the context of the U-TAE and U-Net models, both of which utilize CE loss, per class weighting is employed. When assigning weights to the classes, the _other_ class is explicitly set to 0, indicating that it does not contribute to the loss calculation. The remaining classes are assigned a weight of 1. However, in the case of the U-TAE model, the _plowed land_ class is also assigned a weight of 0 for the U-TAE CE loss. This decision is made because the _plowed land_ class is specifically designed for mono-temporal data. The inclusion of time series data introduces ambiguity with agricultural land, and therefore, setting the weight of the _plowed land_ class to 0 helps to mitigate this confusion.
In addition to these general hyperparameters, there are several other parameters and strategies that have been or could be explored further:
* the **size of super-patches** refers to the dimensions, in terms of pixels, of the patches that are cropped from the super-areas. Different sizes can be tested, allowing for experimentation with smaller or larger super-patch sizes. However, it is important to note that there is a limit of 110 pixels for edge patches. The choice of super-patch size has an impact on the spatial context provided to both the U-TAE and U-Net branches through the _collapsed_ fusion sub-module. _Baselines:_ the number 40 has been empirically determined and set as the baseline for this specific parameter.
* with the exception of the _other_ and _plowed land_ classes, no specific distinction or weighting has been applied during training between the classes and the network branches. However, it is possible to introduce **per-class weights** for both the \(\mathcal{L}sat\) and \(\mathcal{L}aerial\) losses. These weights can be determined based on expert knowledge to encourage specialization of one branch or the other on certain classes. Another approach is to apply weights during the summation of both losses to obtain \(\mathcal{L}_{T\&T}\). _Baselines:_ the _other_ class is assigned a weight of 0 for both branches, and the _plowed land_ class is assigned a weight of 0 for the U-TAE branch. The remaining classes are assigned a weight of 1. Additionally, no weights are applied during the summation of the \(\mathcal{L}sat\) and \(\mathcal{L}aerial\) losses.
* to prevent overfitting of the U-TAE branch and enhance the learned aerial features, we incorporate a **modality dropout mechanism**. This involves generating a random single value for each batch. If the generated value exceeds a specified threshold, provided as an input parameter, the U-TAE modality is dropped out, and only the U-Net branch is used for that particular batch. _Baselines:_ considering the coarse spatial resolution of Sentinel-2 data, we set the modality dropout threshold relatively high, at a value of 0.5. This ensures that a significant portion of the batches will exclusively utilize the U-Net branch, thereby emphasizing the importance of the aerial imagery.
* to address the potential impact of cloud or snow in the Sentinel-2 time series, two strategies are implemented using the provided masks files. The first strategy, called **filter clouds**, involves examining the probability of cloud occurrence in the masks. If the number of pixels above a certain probability threshold exceeds a specified percentage of all pixels in the image, that particular date is excluded from the training process. This helps to mitigate the influence of cloudy or snowy images on the training data. The second strategy, known as **monthly average**, is specifically implemented to alleviate potential challenges faced by the U-TAE branch due to a large number of dates in the time series. In this strategy, a monthly average is computed using cloudless dates. If no cloudless dates are available for a specific month, fewer than 12 images may be used as input to the U-TAE branch. _Baselines:_ a probability threshold of 0.5 is employed for filtering clouds or snow in the masks. Additionally, to be considered for exclusion, the clouds or snow must cover at least 60% of the super-patch.
* similar to the FLAIR #1 approach, **metadata associated with each aerial patch** are integrated into the model. These metadata are encoded using positional encoding or one-hot encoding techniques (see [5]). The encoded metadata are then passed through a MLP before being added to each U-Net encoder feature map. _Baselines:_ a positional encoding of size 32 is used specifically for encoding the geographical location information.
* **data augmentation techniques** usually prevent overfitting and help generalization capabilities of a network. Simple geometric transformations are applied during the training process. These transformations include vertical and horizontal flips as well as random rotations of 0, 90, 180, and 270 degrees. This approach aligns with the methodology used in the FLAIR #1 challenge. _Baselines:_ a data augmentation probability of 0.5 is used.
## Benchmark results
Firstly, an evaluation is conducted on a U-Net model that incorporates only aerial imagery, resembling the approach used in the FLAIR #1 challenge. The evaluation involves assessing the model's performance using the code provided in the GitHub repository (accessible at [14]). Following this, the results obtained from applying the two-branches U-T&T model are reported. Additionally, various parameters and strategies mentioned earlier are tested.
The models used in the evaluation were trained using a consistent train/validation/test split and the parameters previously specified. The training dataset consisted of 61,712 aerial imagery patches, and for the U-T&T approach, an additional 41,029 (unfiltered) Sentinel-2 acquisitions are included. During the inference phase, the models were applied to 16,050 patches of aerial imagery and 10,215 (unfiltered) satellite acquisitions from the test dataset. The reported results represent the average mIoU scores obtained from five separate runs of each model configuration. Additionally, the standard deviation of the mIoU scores across the five runs is provided, indicating the degree of variability in the performance of the models.
The results obtained from the different experiments are presented in Table IV. When using only aerial imagery and a U-Net model, the highest mIoU score of 0.5517 is achieved by integrating aerial metadata and employing data augmentation techniques. In the case of jointly utilizing aerial and satellite imagery with the U-T&T model, the baseline model yields a slightly better mIoU score compared to the aerial-only baseline (0.5490 versus 0.5467), but it also exhibits a higher standard deviation in the results.
Table IV also includes the results obtained when implementing additional strategies individually, as described in the Benchmark framework and settings section. It is observed that using modality dropout leads to a decrease in the mIoU score. Integrating aerial metadata into the U-Net branch only marginally improves the results. However, for the remaining three strategies, namely filtering the dates using cloud and snow masks, performing a monthly average of Sentinel-2 acquisitions, and applying data augmentation, the mIoU scores improve. By combining these three strategies, a mIoU score of 0.5623 is achieved, corresponding to a 2.85% increase compared to the U-Net baseline.
The per-class IoU scores for three models are provided in Table V. The three models considered are the U-Net baseline, the U-T&T baseline, and the U-T&T model with dates filtering of Sentinel-2, monthly average, and data augmentation. These models were selected based on achieving the highest mIoU scores among the five runs. Among the 12 classes, the U-Net baseline outperforms the other models by having a higher IoU score only for the _plowed land_ class, with a marginal improvement of 0.02 points compared to the U-T&T best model. On the other hand, the U-T&T baseline model performs better in predicting the _water_ and _brushwood_ classes, but the differences in IoU scores are quite close to the other models. For the remaining nine classes, the U-T&T best model surpasses the other models, exhibiting notable improvements in classes such as _buildings_, _impervious surfaces_, _bare soil_, _coniferous_, and _vineyards_. These improvements highlight the effectiveness of the U-T&T model with the integrated strategies of dates filtering, monthly average, and data augmentation.
Figure 8 illustrates the confusion matrix of the best U-T&T model. This confusion matrix is derived by combining all
\begin{table}
\begin{tabular}{l c c c c c c c c} & **INPUT** & **FILT.** & **AVG M.** & **MDR** & **MTD** & **AUG** & **PARA.** & **EP.** & **mIoU** \\ \hline
**U-Net** & aerial & - & - & - & ✗ & ✗ & 24.4 & 62 & **0.5467\(\pm\)0.0009** \\ \hline +_MTD_ & aerial & - & - & - & ✓ & ✗ & 24.4 & 59 & **0.5473\(\pm\)**0.0017** \\ \hline +_MTD +AUG_ & aerial & - & - & - & ✓ & ✗ & 24.4 & 52 & **0.5517\(\pm\)**0.0013** \\ \hline
**U-T& aerial+sat** & ✗ & ✗ & ✗ & ✗ & ✗ & 27.3 & 9 & **0.5490\(\pm\)**0.0072** \\ \hline +_FILT_ & aerial+sat & ✓ & ✗ & ✗ & ✗ & ✗ & 27.3 & 11 & **0.5517\(\pm\)**0.0135** \\ \hline +_AVG M_ & aerial+sat & ✗ & ✗ & ✗ & ✗ & ✗ & 27.3 & 10 & **0.5504\(\pm\)**0.0067** \\ \hline +_MD DR_ & aerial+sat & ✗ & ✗ & ✗ & ✗ & ✗ & 27.3 & 27 & **0.5354\(\pm\)**0.0104** \\ \hline +_MTD_ & aerial+sat & ✗ & ✗ & ✗ & ✗ & ✗ & 27.3 & 7 & **0.5494\(\pm\)**0.0064** \\ \hline +_AUG_ & aerial+sat & ✗ & ✗ & ✗ & ✗ & ✗ & 27.3 & 22 & **0.5554\(\pm\)**0.0146** \\ \hline +_FILT +AVG M +M DR +MTD +AUG_ & aerial+sat & ✓ & ✗ & ✗ & ✗ & 27.3 & 36 & **0.5523\(\pm\)**0.0016** \\ \hline \end{tabular}
\end{table} TABLE IV: Baseline results of ResNet34/U-Net architecture with aerial imagery only and U-T&T with aerial and satellite imagery on the FLAIR #2 test set. Results are averages of 5 runs of each configuration. **FILT**: filter Sentinel-2 acquisition with masks (clouds & snow); **AVG M**: monthly average of all Sentinel-2 acquisitions; **MDR**: modality dropout of the U-TAE branch; **MTD**: metadata for aerial imagery added; **AUG**: geometric data augmentation for aerial imagery; **PARA.**: number of parameters of the network; **EP**: best validation loss epoch.
individual confusion matrices per patch and is normalized by rows. The analysis of the confusion matrix shows that the best U-T&T model achieves accurate predictions with minimal confusion in the majority of classes. However, when it comes to natural areas such as _bare soil_ and _brushwood_, although there is improvement due to the use of Sentinel-2 time series data, a certain level of uncertainty remains. These classes exhibit some confusion with semantically similar classes, indicating the challenge of accurately distinguishing them.
Figure 9 showcases an example that illustrates the results of both the U-net baseline andU-T&T baseline models in relation to the aerial imagery and the corresponding annotations.
The experiments conducted in this study were performed using HPC/AI resources provided by GENCI-IDRIS (Grant 2022-A0131013803). This work was supported by the European Union through the project "Copernicus / FPCUP," as well as by the French Space Agency (CNES) and Connect by CNES. The authors would like to acknowledge the valuable support and resources provided by these organizations.
## Data access
The dataset and codes used in this study will be made available after the completion of the FLAIR #2 challenge at the following website: [https://ignf.github.io/](https://ignf.github.io/) FLAIR/.
| FLAIR #2 データセットは、現在提出されるもので、2つの非常に異なる種類のデータを含んでいます。これらのデータは、土地の覆いに関するセマンティック分割タスクに利用されます。データ融合ワークフローでは、非常に高解像度（VHR）単一時間的な空中写真における精細空間と質感情報の利用、そしてCopernicus Sentinel-2 satellit画像の時間とスペクトル的な豊富さを利用します。フランス国家地理情報と森林情報研究所（IGN）は、高品質な地球観測（EO）データの増加に伴い、これらのデータの統合に革新的な戦略を積極的に探索しています。IGNはこのデータセットを提供することで、イノベーションを促進し、私たちの領域に関する知識を向上させることを目的としています。 |
2301.06594 | Exploring the nature of neutrinos in a dissipative environment | We study the possibility of determining the nature of neutrinos in a
dissipative environment. In the presence of environmental decoherence, the
neutrino oscillation probabilities get modified and accommodate the Majorana
phase. In this context, we analyse the transition probabilities that are of
interest in current and upcoming long baseline neutrino oscillation
experiments. Additionally, we explore the measure of quantumness in a two
flavour neutrino oscillation framework via Bell-CHSH inequality, steering
inequality and non-local advantage of quantum coherence (NAQC). We further show
that non-local advantage of quantum coherence (NAQC) serves as stronger quantum
quantifier and it takes different values for Dirac and Majorana neutrinos in a
dissipative environment. | Chinmay Bera, K. N. Deepthi | 2023-01-16T20:24:00 | http://arxiv.org/abs/2301.06594v1 | # Exploring the nature of neutrinos in a dissipative environment.
###### Abstract
We study the possibility of determining the nature of neutrinos in a dissipative environment. In the presence of environmental decoherence, the neutrino oscillation probabilities get modified and accommodate the Majorana phase. In this context, we analyse the transition probabilities that are of interest in current and upcoming long baseline neutrino oscillation experiments. Additionally, we explore the measure of quantumness in a two flavour neutrino oscillation framework via Bell-CHSH inequality, steering inequality and non-local advantage of quantum coherence (NAQC). We further show that non-local advantage of quantum coherence (NAQC) serves as stronger quantum quantifier and it takes different values for Dirac and Majorana neutrinos in a dissipative environment.
Introduction
One of the unresolved puzzles in neutrino physics is whether neutrinos are Dirac or Majorana fermions. In the Dirac description, neutrinos are different from their antiparticles and the lepton number (L) is conserved. Whereas in the Majorana picture, neutrinos and anti-neutrinos are not physically distinguishable. If neutrinos are Majorana fermions, neutrinoless double beta decay (\(0\nu 2\beta\)) process (\(X_{A}^{Z}\to Y_{Z+2}^{A}+2e^{-}\)) could occur with the violation of lepton number (\(\Delta L=2\)) [1; 2]. To this day, there is no experimental evidence for \(0\nu 2\beta\) process. The most recent combined analysis of KamLAND-Zen 400 (2011-2015) and KamLAND-Zen 800 (started 2019) data predicts the half-life of \(T_{0\nu 2\beta}^{1/2}>2.3\times 10^{26}\) yr at 90% C.L. and this corresponds to an effective neutrino mass (\(m_{\beta\beta}\)) less than 36-156 meV [3]. The smallness of neutrino mass has been successfully explained in beyond the standard model (BSM) physics, where neutrinos are assumed to be Majorana particles. Therefore, to resolve this puzzle, holds a strong theoretical motivation for BSM physics.
Experimental data from nearly two decades has established neutrino oscillations as a leading mechanism for neutrino flavour transitions. However, sub-leading effects like non-standard interactions, quantum decoherence, neutrino decay, still hold a torch for new physics scenarios beyond the standard model. In this precision era of neutrino physics experiments, it is the need of the hour to investigate the implications of these sub-leading effects. In this context, we study the effect of quantum decoherence on the neutrino oscillation probabilities in an open quantum system framework. The open quantum system is modelled by considering the interaction of the neutrino subsystem with the environment [4; 5; 6; 7; 8; 9]. The interactions of this kind could originate from the effects of quantum gravity [10; 11; 12; 13], strings and branes [14; 15; 16] at the Plank scale. Consequently, they manifest as dissipation effects and modify the neutrino oscillation probabilities. Moreover, in ref. [17; 18; 19; 20; 9], it has been shown that one can probe the nature of neutrinos when the neutrino system interacts with the environment. The authors in ref [18] have presented that Leggett-Garg \(K_{3}\) quantity takes different values for Majorana and Dirac neutrinos in a dissipative environment.
In the present work, we analyse the effect of a dissipative environment on the neutrino oscillation probabilities in a two flavour picture including the matter effect. The probabilities depend on the Majorana phase and thus provide a window to probe the nature of neutrinos for different baselines. In this context, we study the transition probabilities at different baselines of T2K, NOvA and
the upcoming experiments ESS\(\nu\)SB, T2HKK, DUNE experiments and their dependency on the Majorana phase. In addition, we verify how the measures of quantum correlation like Bell-CHSH inequality, Steering inequality and Non-local advantage of quantum coherence (NAQC) vary with respect to the neutrino beam energy in the case of all the five experiments T2K, NOvA, ESS\(\nu\)SB, T2HKK, DUNE. We show, that NAQC serves as a stronger and legitimate quantifier among all three to study the quantumness of a system. We further define a new parameter \(\Delta\)NAQC and analyse how it varies with respect to the Majorana phase for different baselines.
This paper is structured as follows: In sec. II, we present a basic formalism to determine the neutrino oscillation probabilities in matter while assuming decoherence among neutrino mass eigen states. We discuss in sec. III, the short description of the neutrino oscillation experiments that have been used in this study. In sec. IV, we show the oscillation probabilities relevant to these experiments and discuss their implications by considering two cases of decoherence matrix. In sec. V, we present a brief account of several non-classical measures of quantum coherence and further study probe the nature of neutrinoa using a new parameter \(\Delta NAQC\). We finally summarise our findings in sec. VI.
## II Formalism of neutrino oscillations in matter assuming decoherence
In an open quantum system framework, the neutrino subsystem interacts weakly with the environment leading to a loss of coherence in the subsystem. The decoherence phenomenon in Markovian systems is described by the Lindblad-Kossakowski master equation [21; 22]
\[\mathcal{L}\rho(t)=\frac{\partial\rho}{\partial t}=-i[H_{eff},\rho(t)]+ \mathcal{D}[\rho(t)]. \tag{1}\]
Here, the infinitesimal generator \(\mathcal{L}\) and its action on the density matrix \(\rho(t)\) depend on effective Hamiltonian \(H_{eff}\) and the dissipative term \(\mathcal{D}\). The dissipative factor \(\mathcal{D}\) has the following form,
\[\mathcal{D}[\rho]=-\frac{1}{2}\sum_{j=0}^{N^{2}-1}(A_{j}^{\dagger}A_{j}\rho+ \rho A_{j}^{\dagger}A_{j})+\sum_{j=0}^{N^{2}-1}A_{j}\rho A_{j}^{\dagger}. \tag{2}\]
Here \(A_{j}\) are the operators that depend on the system dimensions. In an N-level system, \(A_{j}\) has \(N\times N\) dimension and they form \(N^{2}-1\) linearly independent basis without including identity. For
the two flavor mixing, \(A_{j}\) are the Pauli matrices \(\sigma_{i}\) and that for the three flavor are represented by Gell-Mann matrices \(\lambda_{i}\). In the former case the density matrix \(\rho\) and operator \(A_{j}\) in eq. (2.2), can be written as, \(\rho=\frac{1}{2}\rho_{\mu}\sigma_{\mu}\), \(A_{j}=a_{\mu}^{j}\sigma_{\mu}\) where, \(\mu=0,1,2,3\), \(\sigma_{0}\) is \(2\times 2\) identity matrix and \(\sigma_{i=1,2,3}\) are the Pauli matrices.
In this work, we consider ultrarelativistic electron neutrinos (\(\nu_{e}\)) and muon neutrinos (\(\nu_{\mu}\)) in two dimensional Hilbert space. In two generation model of neutrinos, the flavor states (\(\nu_{e}\), \(\nu_{\mu}\)) are related to the mass states (\(\nu_{1}\), \(\nu_{2}\)) by a \(2\times 2\) unitary mixing matrix U
\[\begin{pmatrix}\nu_{e}\\ \nu_{\mu}\end{pmatrix}=U\begin{pmatrix}\nu_{1}\\ \nu_{2}\end{pmatrix}, \tag{2.3}\]
where
\[U=\begin{pmatrix}\cos\theta&\sin\theta\ e^{-i\phi}\\ -\sin\theta\ e^{i\phi}&\cos\theta\end{pmatrix}\, \tag{2.4}\]
\(\theta\) is the mixing angle and the phase \(\phi\) represents the Majorana phase.
In eq. (2.1), one can note that the evolution of \(\rho\) depends on a time independent effective Hamiltonian \(H_{eff}\), which is combination of vacuum Hamiltonian (\(H_{vac}\)) and matter interaction Hamiltonian (\(H_{mat}\))
\[H_{eff}=H_{vac}+H_{mat}=h_{\mu}\sigma_{\mu}\, \tag{2.5}\]
here \(h_{\mu}\) are the components of \(H_{eff}\). If neutrinos propagate in vacuum, the free Hamiltonian will be in the following form,
\[H_{vac}=\begin{pmatrix}-\omega&0\\ 0&\omega\end{pmatrix} \tag{2.6}\]
where \(\omega=\frac{\Delta m^{2}}{4E}\) contains the square mass difference of two mass eigenstates (\(\Delta m^{2}\)) and \(E\) represents the neutrino energy. The interaction of the neutrinos with the electrons in the medium is given by the interaction Hamiltonian,
\[H_{mat}=U^{\dagger}\begin{pmatrix}A&0\\ 0&0\end{pmatrix}U \tag{2.7}\]
\[=A\begin{pmatrix}\cos^{2}\theta&\cos\theta\sin\theta\;e^{-i\phi}\\ \cos\theta\sin\theta\;e^{i\phi}&\sin^{2}\theta\end{pmatrix}. \tag{2.8}\]
where \(A=\sqrt{2}G_{F}n_{e}\). Here \(G_{F}\) is called the Fermi constant and the electron number density in the medium is denoted by \(n_{e}\).
The dissipative matrix \(\mathcal{D}_{mn}\) (based on positivity and trace-preserving conditions) depends on six real and independent parameters \(\alpha,\ \beta,\ \gamma,\ a,\ b,\ c.\ \mathcal{D}_{mn}\) is given by,
\[\mathcal{D}_{mn}=-2\begin{pmatrix}0&0&0&0\\ 0&\alpha&\beta&\gamma\\ 0&\beta&a&b\\ 0&\gamma&b&c\end{pmatrix}\, \tag{2.9}\]
where \(m,n=0,1,2,3\).
Now, the time evolution of the density matrix \(\rho\) in eq. (2.1) can be expressed through Schr\(\ddot{o}\)dinger like equation
\[\frac{d}{dt}\left|\rho(t)\right\rangle=-2\mathcal{H}\left|\rho(t)\right\rangle\, \tag{2.10}\]
with
\[\mathcal{H}=2\begin{pmatrix}0&0&0&0\\ 0&\alpha&\beta+\xi&\gamma-\lambda\sin\phi\\ 0&\beta-\xi&a&b+\lambda\cos\phi\\ 0&\gamma+\lambda\sin\phi&b-\lambda\cos\phi&c\end{pmatrix} \tag{2.11}\]
and
\[\xi=\frac{A}{2}\cos 2\theta-\omega,\ \ \ \lambda=\frac{A}{2}\sin 2\theta \tag{2.12}\]
After imposing trace preserving condition \(\dot{\rho_{0}}(t)=0\), eq. (2.10) will take the following form
\[\begin{pmatrix}\dot{\rho_{1}}(t)\\ \dot{\rho_{2}}(t)\\ \dot{\rho_{3}}(t)\end{pmatrix}=-2\mathcal{H}\begin{pmatrix}\rho_{1}(t)\\ \rho_{2}(t)\\ \rho_{3}(t)\end{pmatrix}. \tag{2.13}\]
Hence the evolved state at time t is written as
\[\begin{pmatrix}\rho_{1}(t)\\ \rho_{2}(t)\\ \rho_{3}(t)\end{pmatrix}=\mathcal{M}(t)\begin{pmatrix}\rho_{1}(0)\\ \rho_{2}(0)\\ \rho_{3}(0)\end{pmatrix} \tag{2.14}\]
where \(\mathcal{M}(t)=e^{-2\mathcal{H}t}=\mathcal{S}e^{-2\mathcal{H}^{\prime}t} \mathcal{S}^{-1}\) is a \(3\times 3\) matrix. Here, \(\mathcal{S}\) is the similarity transformation matrix and \(\mathcal{H}^{\prime}\) is the diagonal of the eigenvalues of \(\mathcal{H}\).
The density matrix at arbitrary time t is
\[\rho(t)=\frac{1}{2}\begin{pmatrix}\rho_{0}(t)+\rho_{3}(t)&\rho_{1}(t)-i\rho_{2 }(t)\\ \rho_{1}(t)+i\rho_{2}(t)&\rho_{0}(t)-\rho_{3}(t)\end{pmatrix}. \tag{2.15}\]
Using eq. (2.4), the density matrix at initial time t = 0 can be obtained as
\[\rho_{\nu_{e}}(0)=\begin{pmatrix}\cos^{2}\theta&\frac{1}{2}\sin 2\theta \ e^{i\phi}\\ \frac{1}{2}\sin 2\theta\ e^{-i\phi}&\sin^{2}\theta\end{pmatrix} \tag{2.16}\]
and
\[\rho_{\nu_{\mu}}(0)=\begin{pmatrix}\sin^{2}\theta&-\frac{1}{2}\sin 2\theta \ e^{i\phi}\\ -\frac{1}{2}\sin 2\theta\ e^{-i\phi}&\cos^{2}\theta\end{pmatrix}. \tag{2.17}\]
Further, the evolved state \(\rho_{\nu_{\mu}}(t)\) can be obtained using eq. (2.14)-(2.17). The probability of transition of an initial state \(\nu_{\alpha}\) to a final state \(\nu_{\beta}\) can be evaluated using
\[P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(t)=Tr[\rho_{\nu_{\alpha}}(t)\rho_{\nu _{\beta}}(0)]. \tag{2.18}\]
Upon substitution, the appearance \(P_{\nu_{\mu}\rightarrow\nu_{e}}(t)\) and disappearance \(P_{\nu_{\mu}\rightarrow\nu_{\mu}}(t)\) probabilities are obtained to be
\[\begin{split} P_{\nu_{\mu}\rightarrow\nu_{e}}(t)& =\frac{1}{2}\Big{[}1-M_{33}\cos^{2}(2\theta)+\sin(2\theta)\cos(2 \theta)\big{\{}(M_{23}+M_{32})\sin\phi\\ -(M_{13}+M_{31})\cos\phi\big{\}}&+\sin^{2}(2\theta) \big{\{}(M_{12}+M_{21})\sin\phi\cos\phi\\ -(M_{11}\cos^{2}\phi+M_{22}\sin^{2}\phi)\big{\}}\Big{]}\,\end{split} \tag{2.19}\]
\[\begin{split} P_{\nu_{\mu}\rightarrow\nu_{\mu}}(t)& =\frac{1}{2}\Big{[}1+M_{33}\cos^{2}(2\theta)-\sin(2\theta)\cos(2 \theta)\big{\{}(M_{23}+M_{32})\sin\phi\\ -(M_{13}+M_{31})\cos\phi\big{\}}&-\sin^{2}(2\theta) \big{\{}(M_{12}+M_{21})\sin\phi\cos\phi\\ -(M_{11}\cos^{2}\phi+M_{22}\sin^{2}\phi)\big{\}}\Big{]}& =1-P_{\nu_{\mu}\rightarrow\nu_{e}}(t)\,\end{split} \tag{2.20}\]
where \(M_{ij}\) are the matrix elements of \({\cal M}\) and \(i,j=1,2,3\).
The probability of antineutrinos under the same conditions can be obtained from the probability of neutrinos by replacing the \(A\rightarrow-A\) and \(U\to U^{*}\) in eq. (8). Then \(P_{\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}}(t)\) and \(P_{\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}}(t)\) are deduced as
\[\begin{split} P_{\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}}(t)=& \frac{1}{2}\Big{[}1-M_{33}\cos^{2}(2\theta)-\sin(2\theta)\cos(2 \theta)\big{\{}(M_{13}+M_{31})\cos\phi\\ +(M_{23}+M_{32})\sin\phi\big{\}}-\sin^{2}(2\theta)\big{\{}(M_{12} +M_{21})\sin\phi\cos\phi\\ +(M_{11}\cos^{2}\phi+M_{22}\sin^{2}\phi)\big{\}}\Big{]}\,\end{split} \tag{21}\]
and
\[P_{\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}}(t)=1-P_{\bar{\nu}_{\mu} \rightarrow\bar{\nu}_{e}}(t). \tag{22}\]
The eqns.(19 - 22) show that in the presence of decoherence the neutrino and anti-neutrino oscillation probabilities depend on the Majorana phase \(\phi\). This modified oscillation probabilities open a window to explore the nature of neutrinos in the current and the upcoming neutrino oscillation experiments. In this context, we analyse the probabilities of various long baseline neutrino oscillation experiments and their dependency on the Majorana phase \(\phi\) in the presence of decoherence. In the following section, we give a brief account of the five experiments considered in this study.
## III Experimental Details
T2K and T2HKK : Tokai to Kamiokande (T2K) experiment [23] is an off-axis (off-axis angle (OAA) of 2.5\({}^{\circ}\)) oscillation experiment with Japan Proton Accelerator Research Complex (J-PARC) based \(\nu_{\mu}\) beam facility. The far detector of volume 22.5 kt is placed at Kamiokande with a baseline of 295 km. The motivation of this experiment is to precisely measure oscillation parameters, \(\theta_{13}\), \(\theta_{23}\) and \(\Delta m^{2}_{32}\). The T2HKK experiment [24] is proposed to have an off-axis \(\nu_{\mu}\) beam (OAA ranging from 1-3\({}^{\circ}\)) from J-PARC facility, to travel a distance of 1100 km before it reaches the Water Cherenkov detector of 187 kt based in Korea.
NOvA : The NuMI Off-axis \(\nu_{e}\) Appearance Experiment [25] is an ongoing neutrino oscillation experiment with a baseline of 810 km. It has an off-axis (OAA 0.8\({}^{\circ}\)) muon neutrino beam with a peak energy of \(\sim\)2 GeV. NOvA has a near detector at the Fermilab site and NuMI beam focussed towards a far detector of volume 14 kt, placed at Minnesota. The main goal is to understand the atmospheric neutrino flavor transition and also measure atmospheric mass square difference to the higher precision level (\(10^{-4}\ eV^{2}\)).
ESS\(\nu\)SB : The future neutrino experiment European Spallation Source Neutrino Super Beam (ESS\(\nu\)SB) experiment [26] is a long baseline experiment (baseline 540 km). The peak energy of the neutrino beam is around 0.2 GeV which is the energy corresponding to the second oscillation maxima. A megaton Water Cherenkov far detector is placed underground in ESS site at Lund. This experiment is sensitive to observe leptonic CP-violation phase at 5\(\sigma\) confidence level.
DUNE : The Deep Underground Neutrino Experiment (DUNE) [27] is an on-axis accelerator based long baseline neutrino experiment. The DUNE mainly consists of a beamline, a near detector at Fermilab and a far detector at Sanford Underground Research Facility (SURF) which is 1300 km away from the near detector. It will measure the neutrino events having broad range of energy (1-8 GeV). The neutrino flux is peaked around 2.8 GeV corresponding to the energy at the first oscillation maxima. This experiment will help to determine charge-parity (CP) violation phase and the neutrino mass ordering with very high precision.
The baselines and the corresponding peak neutrino beam \(\nu_{\mu}\) energy values of the above ex
\begin{table}
\begin{tabular}{l r r} \hline
**Experiment** & **L(km)** & **E(GeV)** \\ \hline T2K & 295 & 0.6 \\ ESS & 540 & 0.2 \\ NOvA & 810 & 2.0 \\ T2HKK & 1100 & 0.7 \\ DUNE & 1300 & 2.8 \\ \hline \end{tabular}
\end{table}
Table 1: The baseline length and the peak neutrino beam energy for different experiments are considered in this work.
periments are tabulated in table 1. The details of the oscillation parameters and the decoherence parameters considered in this work are listed in table 2.
## IV Numerical analysis
We consider the two-flavour neutrino oscillation analysis of \(\nu_{\mu}\)-\(\nu_{e}\) oscillations in a dissipative medium presented in section-II. We provide the probability plots corresponding to the eqns. (2.19 - 2.22) as these are the major oscillation channels to be studied in the experiments T2K, NOvA, ESS\(\nu\)SB, T2HKK and DUNE. We further substitute \(t=L\) in natural units, where L is the distance travelled by the neutrino beam. We assume a constant matter density profile with \(\rho=2.8\ g/cm^{3}\) corresponding to the matter density potential \(A=1.01\times 10^{-13}\ eV\). In this work, we analyse two cases of decoherence matrix. Firstly, for simplicity, we assume all the off-diagonal elements of the decoherence matrix in eq. (2.9) to be zero and secondly we consider non-zero diagonal and two non-zero off-diagonal elements. We assume the neutrino mass ordering as normal ordering throughout the work unless otherwise mentioned.
### _Non-zero diagonal elements in \(D_{mn}\):_
Firstly, we assume that the decoherence matrix \(D_{mn}\) in eq. 2.9 has only non-zero diagonal elements and impose that the off-diagonal elements are zero. Under this assumption \(D_{mn}\) in
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline \(\Delta m^{2}_{32}\) & \(\pm\) 2.54 \(\times 10^{-3}\ eV^{2}\) \\ \(\theta_{23}\) & 49.2\({}^{\circ}\) \\ \(\alpha\) = \(a\) = \(c\) & 7.8 \(\times 10^{-24}\) GeV \\ \(\beta\) & 3.8 \(\times 10^{-24}\) GeV \\ \(b\) & 3.0 \(\times 10^{-24}\) GeV \\ \hline \hline \end{tabular}
\end{table}
Table 2: Standard neutrino oscillation parameters and decoherence parameters. Plus and minus sign in \(\Delta m^{2}_{32}\) refers to the normal and inverted ordering respectively.
eq. 2.9 takes a simple form
\[{\cal D}_{mn}=-2\begin{pmatrix}0&0&0&0\\ 0&\alpha&0&0\\ 0&0&a&0\\ 0&0&0&c\end{pmatrix}\, \tag{4.1}\]
where \(\alpha=a=c\). Now, using the density matrix formalism presented in sec.II, we numerically obtain the oscillation probabilities \(P_{\nu_{\alpha}\rightarrow\nu_{\beta}}\) and \(P_{\bar{\nu}_{\alpha}\rightarrow\bar{\nu}_{\beta}}\) where \((\alpha,\beta)=(e,\mu)\).
In fig 1, we plot the transition probabilities with respect to the neutrino beam energy E. We plot the appearance probability \(P_{\mu e}=P_{\nu_{\mu}\rightarrow\nu_{e}}\) and the disappearance probability \(P_{\mu\mu}=P_{\nu_{\mu}\rightarrow\nu_{\mu}}\) of the \(\nu_{\mu}\) beam with respect to E, in the top left and right panels respectively. Additionally, in the
Figure 1: Appearance and disappearance probabilities of neutrinos (upper panel) and anti-neutrinos (lower panel) with respect to energy E (assuming normal mass ordering). The orange, green, red, cyan and black curves correspond to the probability versus energy E for the baselines 295 km, 540 km, 810 km, 1100 km and 1300 km respectively.
bottom panel we show \(P_{\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}}\) and \(P_{\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}}\). The solid lines correspond to \(\phi\neq 0\) which represents the non-zero Majorana phase while dashed lines correspond to \(\phi=0\) which is the case with Dirac neutrinos. We consider maximal phase \(\phi=\pi/2\) as a representative value in these plots. The orange, green, red, cyan and black curves correspond to the probability versus energy E for the baselines 295 km, 540 km, 810 km, 1100 km and 1300 km respectively. As can be seen from the orange curves, in the case of T2K experiment where baseline length is 295 km the dashed curve and the solid curve are almost overlapping with each other, indicating minimal effect of decoherence at this L and E. As the baseline increases, we can see from the green (540 km), red (810 km), cyan (1100 km) and black (1300km) curves that the impact of decoherence increases and reaches maximum for DUNE experiment. However, the ESS\(\nu\)SB experiment (540 km) and T2HKK (1100 km) are designed to work at the second oscillation maxima i.e. 0.2 GeV and 0.7 GeV respectively. At these energy values one can see negligible difference between the Dirac (solid curve) and Majorana (dashed curve) cases from the green and cyan curves. While the solid and dashed black curves corresponding to DUNE baseline 1300 km show maximum sensitivity to discriminate between Dirac and Majorana phase.
We further study the effect of decoherence on the determination of neutrino nature at the five experiments considered, by defining two quantities \(\Delta P_{\alpha\beta}\) and \(\Delta P_{\bar{\alpha}\bar{\beta}}\). The two quantities \(\Delta P_{\alpha\beta}\) and \(\Delta P_{\bar{\alpha}\bar{\beta}}\) in the case of neutrinos and anti-neutrinos are defined as below
\[\Delta P_{\alpha\beta}=|P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(Dirac)-P_{\nu_ {\alpha}\rightarrow\nu_{\beta}}(Majorana)|, \tag{4.2}\]
\[\Delta P_{\bar{\alpha}\bar{\beta}}=|P_{\bar{\nu}_{\alpha}\to\bar{\nu}_{\beta}}(Dirac)-P_{ \bar{\nu}_{\alpha}\to\bar{\nu}_{\beta}}(Majorana)|. \tag{10}\]
The first term in eq. (11) and eq. (10) is obtained by assuming \(\phi=0\) while the second is obtained by taking \(\phi\subset[0,2\pi]\). When \(\alpha=\beta=\mu\) in eq. (11), one can derive that \(\Delta P_{\mu\mu}=\Delta P_{\mu e}\) since our analysis is done for two flavour neutrino oscillation picture. Similarly, from eq. (10), we obtain \(\Delta P_{\bar{\mu}\bar{\mu}}=\Delta P_{\bar{\mu}\bar{e}}\). Therefore, we numerically obtain two quantities \(\Delta P_{\mu e}\) and \(\Delta P_{\bar{\mu}\bar{e}}\) with respect to \(\phi\) from eq. (11) and eq. (10).
Fig. 2, shows \(\Delta P_{\mu e}\) vs \(\phi\) in the left panel and \(\Delta P_{\bar{\mu}\bar{e}}\) vs \(\phi\) in the right panel. To obtain this figure, we fix the peak neutrino beam energy values as per the experimental specifications in table-I and vary the Majorana phase \(\phi\subset[0,2\pi]\). For all non-zero values of \(\phi\), the green curve corresponding to the 540 km baseline (ESS\(\nu\)SB) and the black curve corresponding to the 1300 km baseline (DUNE) show minimum and maximum potential to determine the nature of neutrinos. The green (540 km) and cyan (1100 km) curves show lesser sensitivity to discriminate between Dirac and Majorana neutrinos for all values of \(\phi\neq 0\). This can be explained from fig 1, for \(\phi=\pi/2\) we have the green, cyan solid (Majorana case) and the dashed (Dirac) curves almost overlapping at the second oscillation maximum. Since, ESS\(nu\)SB and T2HKK are planned at second oscillation maximum, the effect of decoherence is lesser than that at the first oscillation maxima. For this reason NOvA (810 km) will be more sensitive to determining the nature of neutrinos, next to DUNE as can be seen from the red curve.
### _Non-zero off diagonal elements in \(D_{mn}\)_
In this case, we assume that the decoherence matrix \(D_{mn}\) in eq. 9 has two non-zero off-diagonal elements and the diagonal elements are equal i.e. \(\alpha=a=c\), \(\gamma=0\), \(\beta\) and \(b\) non-zero. Similar to the previous case, we further obtain the oscillation probabilities under this approximation.
In fig 3, we plot the transition probabilities with respect to the beam energy E. We show the appearance probability \(P_{\nu_{\mu}\to\nu_{e}}\) and the disappearance probability \(P_{\nu_{\mu}\to\nu_{\mu}}\) of the \(\nu_{\mu}\) beam with respect to E, in the top left and right panels respectively. In the bottom left(right) panel, we present \(P_{\bar{\nu}_{\mu}\to\bar{\nu}_{e}}\) (\(P_{\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}}\)) with respect to anti-neutrino \(\bar{\nu_{\mu}}\) beam energy. We assume the neutrino mass ordering as normal ordering. The solid lines correspond to \(\phi\neq 0\) which represents the case
of non-zero Majorana phase while the dashed lines correspond to \(\phi=0\), the Dirac neutrinos. We consider maximal phase \(\phi=\pi/2\) as a representative value in these plots. The orange, green, red, cyan and black curves correspond to the probability versus energy E for the baselines 295 km, 540 km, 810 km, 1100 km and 1300 km respectively. In comparison between fig 1 and fig 3, we can see that the additional non-zero off-diagonal elements do not change the probabilities vs E significantly. Accordingly, the conclusions drawn for fig 1, also apply in this case. That is there is maximum difference between the Dirac and the Majorana case for 1300 km baseline when one assumes decoherence in a two flavour neutrino framework. The green and cyan curves corresponding to ESS and T2HKK baselines, show that at second oscillation maxima 0.2 GeV and 0.7 GeV the difference between probabilities for Majorana (solid curve) and Dirac (dashed curve) case is almost negligible.
Figure 3: Appearance and disappearance probabilities of neutrinos (upper panel) and anti-neutrinos (lower panel) with respect to energy E (assuming normal mass ordering). The orange, green, red, cyan and black curves correspond to the probability versus energy E for the baselines 295 km, 540 km, 810 km, 1100 km and 1300 km respectively.
We further obtain \(\Delta P_{\mu e}\) and \(\Delta P_{\bar{\mu}\bar{e}}\) using eq. (20) and eq. (21) respectively, for this case of non-zero off-diagonal elements. In fig 4, we plot \(\Delta P_{\mu e}\) vs \(\phi\) in the left panel and \(\Delta P_{\bar{\mu}\bar{e}}\) vs \(\phi\) in the right panel. We fix the L and E values as listed in table-I. From the black curve, one can see that for all non-conserving values of \(\phi\subset[0,2\pi]\), DUNE shows maximum \(\Delta P_{\mu e}\) and \(\Delta P_{\bar{\mu}\bar{e}}\) values and is followed by NOvA experiment. T2K and T2HKK have same \(\Delta P_{\mu e}\) and \(\Delta P_{\bar{\mu}\bar{e}}\) irrespective of their baselines because T2HKK is planned at second oscillation maxima. From the left plot we note that in the case of neutrino beam, the sensitivity to the Majorana phase is seen for all non-zero values. However for the anti-neutrino beam, the \(\Delta P_{\bar{\mu}\bar{e}}\) values are non-negligible for smaller range of \(\phi\) values (\([0.6\pi\) to \(2.6\pi]\), \([4\pi\) to \(5.6\pi]\)).
## V Quantum quantifiers
To study a quantum system, it is essential to explore the quantumness of the system. It has been shown that various measures of quantum coherence such as, Bell holocality [28; 29; 30], Leggett-Garg inequality (LG-I) [31; 32], entanglement [33; 34], steering inequality [35; 36; 37], non-local advantage of quantum coherence (NAQC) [38] can be expressed in terms of neutrino oscillation probabilities [39; 40] and are useful to shed some light on the unknown neutrino oscillation parameters. Recently, some progress has been made where non-classical correlations are defined to analyze charge-parity (CP) violating phase and the neutrino mass hierarchy problem [41].
In this section, we briefly discuss the various well established measures of quantum correlations
and their relations with neutrino oscillation probability. The two flavour neutrino oscillation framework, considered in this work is ideal to study the quantum quantifiers like Bell-CHSH inequality, Steering inequality and non-local advantage of quantum coherence (NAQC), as they are defined for a two-qubit system. We show that these nonclassical entities take different forms for Majorana and Dirac neutrinos. Therefore, one can consider them as potential candidates to probe the nature of neutrinos.
Firstly, we give a brief account of the measures of quantum correlations Bell-CHSH inequality, Steering inequality and NAQC. Later, we numerically obtain these quantities in terms of neutrino oscillation parameters and plot them with respect to the neutrino beam energy E. For analysis purposes we show the results for three baselines, smallest - 295 km, intermediate - 540 km and largest - 1300 km baselines corresponding to T2K, ESS\(\nu\)SB, DUNE.
_Bell-CHSH Inequality :_ The 2022 Nobel Prize in Physics has been awarded to Alain Aspect, John F. Clauser and Anton Zeilinger for establishing the violation of Bell's inequalities in experiments with entangled photons. In 1963, it was shown by John Bell that, quantum mechanics is not compatible with local hidden variable theory [42]. The generalisation of Bell's theorem presented by Clauser, Horne, Shimony and Holt, to apply on realistic experiment is popularly known as Bell-CHSH inequality [28]. This inequality is used to test the quantum correlation between experimentally measurable quantities which are operated on two spatially separated systems. The CHSH inequality in terms of the Bell operator \(B_{CHSH}\) is given by
\[|\langle B_{CHSH}\rangle|\leq 2. \tag{5.1}\]
The violation of the inequality in eq. (5.1), can be rewritten in terms of a quantity \(M(\rho)=\max_{i\neq j}(u_{i}+u_{j})\) where \(u_{i=1,2,3}\) are the eigenvalues of a real matrix \(T^{\dagger}T\) (T - correlation matrix) [29], as
\[|\langle B_{CHSH}\rangle|=2\sqrt{M(\rho)}>2 \tag{5.2}\]
In the context of two flavour neutrino oscillations, A and B can be considered as neutrinos with two different flavours. The authors of [39] have derived \(M(\rho)\) in terms of the neutrino oscillation probability \(P_{\rm osc}\) and the survival probability \(P_{\rm sur}\) as
\[M(\rho)=1+4P_{\rm sur}P_{\rm osc}. \tag{5.3}\]
Therefore, from eq. (5.2)
\[|\langle B_{CHSH}\rangle|=2\sqrt{1+4P_{osc}P_{sur}}\ >2. \tag{5.4}\]
_Steering Inequality :_ The notion of quantum steering was first proposed by Schrodinger in
Figure 5: Bell-CHSH inequality versus energy (E) in first row, steering inequality versus energy (E) in second row and NAQC versus energy (E) in third row for T2K (left), ESS\(\nu\)SB (middle), DUNE (right). Solid line, dashed line and dot-dashed line represents decoherence including matter effect, only matter effect (without decoherence) and vacuum respectively.
1935 [43] to investigate the Einstein-Podolsky-Rosen (EPR) paradox [44]. In a system with two parties, quantum steering is a phenomenon where one party influences/steers the outcome of a measurement on another party. The steering criteria of a bipartite state \(\Gamma_{AB}\) shared between two parties (Alice and Bob) can be diagnosed by an inequality that was developed in a seminal paper by Cavalcanti-Jones-Wiseman-Reid [45]. If two parties, both are permitted to measure \(n\) observables, then the inequality is formulated as [36]
\[S_{n}(\Gamma_{AB},\varsigma)=\frac{1}{\sqrt{n}}\left|\sum_{i=1}^{n}\langle A_{ i}\otimes B_{i}\rangle\right|\leq 1 \tag{5.5}\]
with \(\langle A_{i}\otimes B_{i}\rangle\) = Tr\((\Gamma_{AB}A_{i}\otimes B_{i})\), where \(A_{i}=\hat{x}_{i}\cdot\hat{\sigma}\), \(B_{i}=\hat{y}_{i}\cdot\hat{\sigma}\), \(\hat{x}_{i}\in\mathbb{R}^{3}\) are unit vectors, \(\hat{y}_{i}\in\mathbb{R}^{3}\) are orthonormal vectors, \(\varsigma=\{\hat{x}_{i},...,\hat{x}_{n},\hat{y}_{i},...,\hat{y}_{n}\}\) represents set of measurement directions.
In the case of two flavor neutrino oscillation the steering inequality can be represented as [40]
\[S_{n}(\Gamma_{AB},\varsigma)=\sqrt{1+8P_{osc}P_{sur}}\ \leq 1. \tag{5.6}\]
_Non-local Advantage of Quantum Coherence (NAQC)_ : Recently, authors in ref. [38] have proposed a quantifier called non-local advantage of quantum coherence (NAQC). This NAQC parameter is based on \(l_{1}\)-norm of coherence measure and it represents the quantumness of a system in terms of entanglement and steerability. In [40] the authors have shown that NAQC acts as a stronger quantifier when compared to Bell-CHSH and steering inequality.
The NAQC-parameter \(N_{l_{1}}\) measures the quantumness of a system using steerability and quantum coherence criteria. For a two flavor neutrino system, this parameter can be derived in terms of oscillation (\(P_{osc}\)) and survival probability (\(P_{sur}\)) as [40]
\[N_{l_{1}}=2+2\sqrt{P_{osc}P_{sur}}. \tag{5.7}\]
Thus using the above definition of \(N_{l_{1}}\), the condition to achieve NAQC for two flavor neutrino system is \(N_{l_{1}}\geq 2.45\). In fig. 5 we present the variation of the Bell-CHSH (top panel), steering inequality (middle panel) and NAQC inequality (bottom panel) with respect to neutrino energy E using eqn. (5.4), eqn. (5.6) and eqn. (5.7) to investigate the quantumness of neutrino oscillations. The oscillation and the decoherence parameters considered are listed in table 2. In each row, we show the plots for three different baselines 295 km (left), 540 km (middle) and 1300 km (right). The solid lines in each plot are obtained by considering the neutrino propagation over a distance
L, assuming decoherence and matter interactions while the dashed and the dot-dashed lines correspond to the cases of neutrino propagation with matter effect (no decoherence) and propagation in vacuum respectively. From the first row of fig 5, it can be seen that the Bell-CHSH inequality is violated i.e. \(B_{CHSH}\geq 2\), for all E values in the case of all the three baselines. Second row of fig 5 shows that the flavor states violate steering inequality i.e. \(S_{3}\geq 1\) for all the energies in the case of T2K, ESS\(\nu\)SB and DUNE baselines. However, in the third row it is illustrated that NAQC i.e. \(N_{l_{1}}\geq 2.45\) is not achieved for some energy ranges by the neutrino flavor states in the case of all the three baselines. Hence, neutrino flavour states attaining NAQC can also achieve Bell nonlocality and quantum steering. NAQC can be considered as a stronger quantifier than steering and Bell-CHSH quantity to reveal the quantumness of the neutrino system. Similar conclusions have been reported by Fei Ming et al [40] examining the neutrino oscillations in vacuum. Basing on this conclusion, we consider the NAQC quantifier as a better candidate to throw some light on
the nature of neutrinos.
From eq. (19) and eq. (57) it is obvious that \(N_{l_{1}}\) is a function of the Majorana phase (\(\phi\)). When we consider \(\phi=0\) we can obtain \(P_{osc}\) and thus \(N_{l_{1}}\) for the case of Dirac neutrinos. Here we propose a candidate \(\Delta NAQC\), to determine the nature of neutrinos
\[\Delta NAQC=|N_{l_{1}}(Dirac)-N_{l_{1}}(Majorana)|. \tag{58}\]
We use the same values for the oscillation and decoherence parameters to calculate \(\Delta NAQC\), as given in table 2. In fig 6 we plot \(\Delta NAQC\) as a function of \(\phi\) imposing some extra conditions on decoherence parameters. We consider the following combinations for the sake of completeness: in fig 6a both \(\beta\) and \(b\) are zero (only non-zero diagonal elements); in fig 6b, \(\beta\) is non-zero and \(b\) is zero (one non-zero off-diagonal element); in fig 6c \(b\) is non-zero and \(\beta\) is zero (one non-zero off-diagonal element); and in fig 6d both \(\beta\) and \(b\) are non-zero (two non-zero off-diagonal elements). The orange, green, red, cyan and black lines are assigned for T2K, ESS\(\nu\)SB, NOvA, T2HKK and DUNE experiments respectively. In fig 6, the \(\Delta NAQC\) values in the presence of decoherence indicate that one can probe the nature of neutrinos using DUNE, NOvA and T2K but the possibility is low for the baselines T2HKK and ESS\(\nu\)SB. Additionally, it is visible from fig 6 that the \(\Delta NAQC\) for 295 km baseline (orange curve) shows more sensitivity to the nature of neutrinos when compared to \(\Delta P_{\mu e}\) plotted in fig 2 and fig 4. Thus reinforcing the conclusion that NAQC, quantum quantifiers in general, can be used as a measure to understand the nature of neutrinos.
## VI Conclusions
We analyse the two flavour neutrino oscillations in matter in a dissipative environment, by assuming two scenarios for the decoherence matrix. Firstly, for simplicity, we consider the case where we have non-zero diagonal elements and zero off-diagonal elements in the decoherent matrix. Whereas in the second case, we assume non-zero diagonal and non-zero off-diagonal elements. In these two scenarios, we see that the transition probabilities depend explicitly on the Majorana phase \(\phi\). We further study the oscillations of Dirac and Majorana neutrinos at L (baseline) and E (neutrino beam energy) values corresponding to five long-baseline neutrino oscillation experiments
T2K, NOvA, ESS\(\nu\)SB, T2HKK and DUNE. We find that, in principle, one can probe the nature of neutrinos at these experiments under the assumption that neutrino subsystem interacts with the environment. Moreover, we extend our analysis to estimate the quantum correlations in the system using quantifiers - Bell-CHSH, steering and NAQC for these L, E values. We note that the neutrino flavour states achieving NAQC attain Bell-CHSH, steering inequalities for all E. Thus, we consider NAQC as a stronger quantifier for a quantum system and analyse its sensitivity to the nature of neutrinos. The NAQC quantifier carries different values for Majorana and Dirac neutrinos. Hence, the non-vanishing difference \(\Delta NAQC\) can be used to discriminate between Dirac and Majorana neutrinos. In the presence of matter effect, a dissipative matrix presents non-vanishing \(\Delta NAQC\), which in principle allows us to differentiate between the Dirac and Majorana neutrinos at neutrino oscillation experiments.
**Acknowledgements:** K. N. Deepthi would like to thank DST-SERB SIRE program for the financial support to visit Indiana University, Bloomington, Indiana, USA.
| *Please use the most accurate and natural Japanese translation possible.*
## Translation:
**νの性質を決定できる可能性を研究する。環境の decoherence の存在下では、νの振る舞い確率が変化し、マジョラン位相を対応させる。この背景に、長距離ベース neutrino の振る舞い実験における興味のある遷移確率を分析する。さらに、Bell-CHSH 不等式、量子相関の非局所優勢（NAQC）を用いて、2つの性質を持つνの振る舞いにおける量子性の測定を探索する。量子相関の非局所優勢（NAQC）は、ディラックとマジョランナνにおいて異なる値を示す。**
---
I hope this helps! Let me know if you have any other sentences you'd like me to translate! |
2308.05596 | You Only Prompt Once: On the Capabilities of Prompt Learning on Large
Language Models to Tackle Toxic Content | The spread of toxic content online is an important problem that has adverse
effects on user experience online and in our society at large. Motivated by the
importance and impact of the problem, research focuses on developing solutions
to detect toxic content, usually leveraging machine learning (ML) models
trained on human-annotated datasets. While these efforts are important, these
models usually do not generalize well and they can not cope with new trends
(e.g., the emergence of new toxic terms). Currently, we are witnessing a shift
in the approach to tackling societal issues online, particularly leveraging
large language models (LLMs) like GPT-3 or T5 that are trained on vast corpora
and have strong generalizability. In this work, we investigate how we can use
LLMs and prompt learning to tackle the problem of toxic content, particularly
focusing on three tasks; 1) Toxicity Classification, 2) Toxic Span Detection,
and 3) Detoxification. We perform an extensive evaluation over five model
architectures and eight datasets demonstrating that LLMs with prompt learning
can achieve similar or even better performance compared to models trained on
these specific tasks. We find that prompt learning achieves around 10\%
improvement in the toxicity classification task compared to the baselines,
while for the toxic span detection task we find better performance to the best
baseline (0.643 vs. 0.640 in terms of $F_1$-score). Finally, for the
detoxification task, we find that prompt learning can successfully reduce the
average toxicity score (from 0.775 to 0.213) while preserving semantic meaning. | Xinlei He, Savvas Zannettou, Yun Shen, Yang Zhang | 2023-08-10T14:14:13 | http://arxiv.org/abs/2308.05596v1 | You Only Prompt Once: On the Capabilities of Prompt Learning on Large Language Models to Tackle Toxic Content
###### Abstract
The spread of toxic content online is an important problem that has adverse effects on user experience online and in our society at large. Motivated by the importance and impact of the problem, research focuses on developing solutions to detect toxic content, usually leveraging machine learning (ML) models trained on human-annotated datasets. While these efforts are important, these models usually do not generalize well and they can not cope with new trends (e.g., the emergence of new toxic terms). Currently, we are witnessing a shift in the approach to tackling societal issues online, particularly leveraging large language models (LLMs) like GPT-3 or T5 that are trained on vast corpora and have strong generalizability. In this work, we investigate how we can use LLMs and prompt learning to tackle the problem of toxic content, particularly focusing on three tasks; 1) Toxicity Classification, 2) Toxic Span Detection, and 3) Detoxification. We perform an extensive evaluation over five model architectures and eight datasets demonstrating that LLMs with prompt learning can achieve similar or even better performance compared to models trained on these specific tasks. We find that prompt learning achieves around 10% improvement in the toxicity classification task compared to the baselines, while for the toxic span detection task we find better performance to the best baseline (0.643 vs. 0.640 in terms of \(F_{1}\)-score). Finally, for the detoxification task, we find that prompt learning can successfully reduce the average toxicity score (from 0.775 to 0.213) while preserving semantic meaning.1
Footnote 1: Our code is available at [https://github.com/xinlieibe/toxic-prompt](https://github.com/xinlieibe/toxic-prompt).
**Disclaimer. This paper contains uncensored toxic content that might be offensive or disturbing to the readers.**
## 1 Introduction
In online platforms, toxic content can be defined as rude, disrespectful, or unreasonable content that may result in users leaving the conversation [6]. It has been a long-standing problem affecting our society [53, 10, 14, 5]. To tackle this problem, researchers and companies leverage large-scale labeled datasets to train powerful machine learning (ML) models for toxicity detection and mitigation [61, 63, 4, 10, 36, 66].
One major obstacle in the development of accurate and generalizable toxic content classifiers is the lack of a comprehensive labeled dataset that contains different types of toxic content. This is mainly because the data collection and labeling process for the creation of such datasets is costly, which hinders the development of effective methods for detecting toxic content. Also, previous work [5, 61] has shown that the toxicity detection model trained on one dataset is less effective when applied to other datasets. Moreover, due to the fast evolution of language (new phrases, words, style, etc.), it is crucial to develop a toxicity detection mechanism that can quickly adapt to different circumstances.
With the success of pre-trained language models (LMs), a dominant way to adapt the model to downstream tasks is fine-tuning, where the whole model or part of the model is optimized to better fit the downstream tasks. Recently, large language models (LLMs) like GPT-3 [7] and T5 [44] have shown promising performance in downstream tasks without updating at all the model's parameters by directly querying the model using natural language, an emerging paradigm called _prompt learning_. With the help of prompt learning, the LLM can generate an output that aims to solve a specific task, all with a natural language task instruction (e.g., using a prompt: "Translate it from English to French" for machine translation) and a few samples as the task input. Besides the handcrafted fixed prompts, recent work [28, 30] shows that prompt tuning is an efficient way to achieve more promising performance on various tasks with restricted computational resources, limited datasets, and bounded time. Concretely, instead of fine-tuning the LLM, prompt tuning freezes the LLM and only optimizes the prompt (e.g., the way that the prompt is written) in such a way that the LLM's performance is optimized for the specific task at hand. Given that prompt learning is a promising way to use LLM for various tasks, here we aim to use prompt learning to tackle the problem of toxic content and assess how prompt learning-based approaches compare to state-of-the-art methods of tackling toxic content.
**Our Work.** In this work, we conduct the first systematic analysis focusing on how prompt learning can help tackle the problem of toxic content. Concretely, we focus on three tasks, i.e., toxicity classification, toxic span detection, and detoxification (see Table 1 for examples of these tasks). Specifically, for the first task (toxicity classification), given a
sentence, we first map its label into the word "Yes" or "No" and fine-tune the prompt to better guide the LLM to conduct the task. For the second task (toxic span detection), with prompt tuning, given a sentence with toxic spans, we aim to first generate the sentence without the toxic spans, then subtract the original sentence with the generated sentence to obtain the spans. Finally, for the third task (detoxification), we tune the prompt to rephrase the toxic sentence into a non-toxic version while preserving the semantic meaning.
Extensive evaluation of eight datasets and five model architectures shows that prompt tuning has comparable or even better performance than the baselines. For instance, for the toxicity classification task, prompt tuning gains more than 10% \(F_{1}\)-score improvement on average (see Table 3). For the toxic span detection task, our method achieves 0.643 \(F_{1}\)-score, which is better than the best result provided by SPAN-BERT (0.640), but with much less training time. Regarding the detoxification task, we find that our method can successfully detoxify the text (e.g., the average toxicity score drops from 0.775 to 0.213 on ParaDetox) while preserving the semantic information to a large extent. In general, one major advantage of prompt tuning is that it can adapt to different tasks with fewer training samples/steps. For online services such as social media, these improvements and cost reductions are significant (given billions of posts per day). This also fits the purpose of green AI [3, 49] for making AI research more environmentally friendly and inclusive.
In summary, we make the following contributions:
* To the best of our knowledge, we perform the first systematic evaluation using prompt tuning to tackle the problem of toxic content.
* We leverage prompt tuning to solve the three most representative tasks in this domain, i.e., toxicity classification, toxic span detection, and detoxification.
* Extensive evaluations show that our prompt tuning methods can achieve comparable or even better performance than the SOTA methods. Also, we observe that prompt tuning has promising performance on fast adaptation to different tasks, i.e., with fewer training samples/epochs.
**Implications.** Our work has important implications for various stakeholders involved in understanding and mitigating online abuse, hate, and harassment. First, we make our code and annotated dataset available, enabling social media operators to implement solutions to detect and moderate toxic content. Our approach is superior to previous efforts when considering the annotated data requirements, the performance, the time cost, and the robustness/transferability of the proposed solution. Additionally, our work can be used to build explainable toxic detection/moderation tools, given our method's outstanding performance on the toxic span detection and detoxification tasks. Third, we argue that our work can assist and motivate the research community in leveraging the prompt tuning approach for solving other emerging socio-technical issues, such as the spread of misinformation online. Overall, our work is an important step towards understanding the power and generalizability of LLM in solving hard tasks (e.g., online toxicity), which is an important and timely issue, given the extensive popularity of LLM and chatbots powered by LLM (e.g., ChatGPT).
**Ethical Considerations.** We emphasize that in this work we work exclusively with publicly available datasets focusing on toxicity classification, toxic span detection, and detoxification tasks. Also, we use publicly available large language models to assess their performance on these tasks and how our work compares to previous efforts. We acknowledge that since we model all three tasks as generation tasks, the model may generate toxic content, however, we took the following steps to minimize harm: 1) we do not share the generated content with people or online users; and 2) all annotations required for our work were done by the authors of this study. Finally, in this work, we show that using prompt-tuning, large language models can detoxify content with acceptable performance. At the same time, however, adversaries might use large language models and prompt tuning to do the opposite task (i.e., toxifying content). We believe that this potential abuse is outside of the scope of this work. Yet, it highlights the need for the implementation and use of appropriate safeguards (e.g., similar to Stable Diffusion's Safety Filter2), to ensure that large language models and prompt tuning can not be used for malicious purposes (e.g., generation and dissemination of toxic content).
Footnote 2: [https://stability.ai/blog/stable-diffusion-public-release](https://stability.ai/blog/stable-diffusion-public-release).
## 2 Preliminary
**Prompt Learning.** With the advance of pre-trained LLM such as GPT-2/3, the previous "pre-train, fine-tune" procedure is replaced by the "pre-train, prompt, and predict" paradigm [31]. Concretely, given a downstream task, fine-tuning requires the training objective to be specified beforehand and the model needs to be updated. In contrast, prompt
\begin{table}
\begin{tabular}{l l} \hline \hline
**Toxicity Classification** & **Answer** \\ \hline your reading comprehension is more fucked up than a football bat. & Toxic \\ \hline
**Toxic Span Detection** & **Answer** \\ \hline keep hiring imbeciles like this jerk and you will end up with a no firearms for rent-a-cops bill next session. & keep hiring imbeciles like this jerk and you will end up with a no firearms for rent-a-cops bill next session.
learning [7] uses a _prompt_ that contains the task-specific description and text examples in a natural language way as the input to the model. In this way, the downstream task can be formulated as a [MASK]language modeling problem (i.e., predict masked text pieces based on the context) and does not need to update the parameters in the underlying model. Prompt learning is especially suitable for few-shot downstream tasks when limited training examples are available and fine-tuning the pre-trained model is costly. In general, prompt learning can be broadly grouped into two categories - manual prompt and learnable prompt (soft prompt).
**Manual Prompt.** The natural way to create prompts is to manually design intuitive textual templates based on human/domain knowledge [7]. For example, if the task is to classify the sentiment of a movie review "Absolutely terrible writing and dragged-out unnecessary dialogue", we can append a prompt "The review is" to the content and get "Absolutely terrible writing and dragged-out unnecessary dialogue. The review is [MASK]". We expect the language model to _generate_ "horrible" than "great" to replace [MASK]. Manual prompts have been proven to solve various tasks with decent accuracy [31]. However, handcrafted prompts need to be customized based on the downstream tasks, inevitably introducing artificial bias and leading to sub-optimal results.
**Learnable Prompt.** In contrast to the manual prompts, learnable prompt methods automatically learn to prompt from a larger searching space for the candidate prompts to better fit the downstream tasks. Prefix tuning [30] is one of the most promising techniques for prompt tuning. Concretely, it adds a prefix (i.e., a sequence of continuous task-specific vectors) before the input, which can be considered as a set of "virtual tokens". Given the downstream task, the prefix will be optimized while the parameters \(\theta\) of LM are frozen. This is extremely efficient compared to fine-tuning the whole model as for different downstream tasks, only different prefixes instead of different models will be updated. Formally, the prefix matrix \(M_{\phi}\) parameterized by \(\phi\) can be updated via the following log-likelihood objective:
\[\max_{\phi}logP(\mathbf{y}|\mathbf{x};\theta;\phi)=\max_{\phi}\sum_{y_{1}}logP(y_{i}|h _{<i};\theta;\phi) \tag{1}\]
where \(h_{<i}=[h_{<i}^{(1)},\cdots;h_{<i}^{(n)}]\) is a function of the trainable parameters at time step \(i\). It is directly copied from \(M_{\phi}\) if the time step is within the prefix (\(h_{i}\) is \(M_{\phi}[i]\)), otherwise it is computed with the LM. Similarly, Lester et al. [28] propose a more efficient method that adds several tunable tokens as the prefix and optimizes the embeddings of those tunable tokens directly. It has fewer tunable parameters as it does not involve additional tunable parameters in each network layer. Note that the learnable prompt (prefix matrix) is the embedding of a set of "virtual words" which can be optimized. The embeddings have mathematical meanings but cannot be mapped into real words.
## 3 Tasks
In this work, we consider three tasks that are related to toxicity: 1) toxicity classification (detect whether the text is toxic), 2) toxic span detection (detect which parts of the text are toxic), and 3) detoxification (eliminate toxicity in the text while preserving its semantics). The three tasks handle toxicity in different levels: toxicity classification only detects whether the whole text is toxic or not; toxic span detection aims to detect the exact character offset of the spans that make the text to be toxic, and detoxification's goal is to eliminate the toxic content from the text while preserving its semantic meaning.
### Task1: Toxicity Classification
**Goal.** We frame this task as a binary classification task, where the input is a piece of text and the output is _whether the given text is toxic or not_. An example of toxicity classification is shown in Table 1.
**Existing Methods.** Existing toxicity classification methods usually leverage a labeled dataset (a text is annotated as toxic or not) to train classifiers or fine-tune an LM. Early efforts widely use feature engineering (e.g., dictionaries, bag-of-words, etc.) to extract features from text and detect toxic language or phrases [12]. With the advance of deep neural networks (DNNs), recent efforts have been focusing on training toxicity classification models based on recurrent neural networks (RNNs) [38], convolutional neural networks (CNNs) [15], and transformers (e.g., BERT) [1]. The very latest trend of toxicity classification is using LLMs that are pre-trained on large unlabeled corpora and then fine-tuning them to tailor them for the toxicity classification task [64]. The drawback of these methods is that they require a large annotated corpus to train or fine-tune an LM and their detection effectiveness is limited by either the size of the labeled dataset or the time to fine-tune the pre-trained LMs.
**Our Method.** Given the language model parameterized by \(\theta\), a set of texts \(\{\mathbf{x}|\mathbf{x}\in X\}\) and the corresponding label \(\{\mathbf{y}\in Y\}\), we aim to learn the prefix matrix \(M_{\phi}\) so that the prompt consist with \(M_{\phi}\) (parameterized by \(\phi\)) and \(\mathbf{x}\) can successfully retrieve label \(\mathbf{y}\) from the language model \(\theta\). Our optimization goal is summarized in Equation 2.
\[\phi^{*} = \underset{\phi}{\text{arg min}}\quad\mathcal{L}(f(X,\phi,\theta),Y) \tag{2}\]
where \(\mathcal{L}\) is our loss function (e.g., binary cross-entropy loss) and \(f\) is our toxicity classification model. It is important to note that our model does not fine-tune the language model parameterized by \(\theta\).
### Task2: Toxic Span Detection
**Goal.** The toxic span detection aims to identify the specific spans (i.e., the character offsets) that make the text toxic. For instance, in the example shown in Table 1, the toxic span detection task should return two spans - one for "imbeciles" (starting at 13 and ending at 21) and one for "jerk" (starting at 33 and ending at 36). It is another important task as it can assist users in better understanding how the toxicity is reflected in the text (e.g., the highlighted toxic span can assist annotators to support their decisions). Formally, given an
input text \(t\), our goal is to determine the exact toxic spans \(\{S^{t}\}\) in the text.
**Existing Methods.** Toxic span detection can be seen as a case of attribution or rationale extraction [39]. Most of previous work [12, 18, 22] frame this task as a sequence labeling task. Concretely, given the labeled toxic span corpus, an LM can be trained to label each word as toxic or not. Once the model is trained and given a text the model will give a toxicity prediction label for each word. Existing methods have been widely using transformers (e.g., BERT+CRF [12], SPAN-BERT [22]) or recurrent neural networks (e.g., BiLSTM [18]) to attain the goal. Some research also experimented with custom loss [59] and data augmentation [55] to boost the performance of toxic span detection.
**Our Method.** Our method is fundamentally different from the existing methods. Instead of considering the toxic span detection as a sequence labeling task, we treat it directly as a generation task. Concretely, the input of our model is the original text that contains the toxic content. We aim to leverage the prompt and the (frozen) LLM to generate text without the toxic span while keeping the rest the same as the input text. Note that, with the prompt, the LLM does not attempt to replace the toxic span in the generated text, rather it generates a, usually, incomplete text that does not have any toxic spans. Then, to detect the toxic span, we run a mapping algorithm to "subtract" the input text from the generated text and consider the rest as the toxic spans (i.e., character-level offsets). Our optimization goal, given the input \(T=\{t\}\) and \(\tilde{T}=\{t\setminus\{S^{t}\}\}\), is summarized in Equation 3.
\[\phi^{*} = \underset{\phi}{\text{arg min}}\quad\mathcal{L}(\tilde{T},f(T, \phi,\theta)) \tag{3}\]
It learns \(M_{\phi}\) (parameterized by \(\phi\)) that nudges the large language model \(\theta\) to remove only toxic spans \(\{S^{t}\}\) from \(X\).
### Task3: Text Detoxification
**Goal.** Text detoxification, as its name suggests, aims to eliminate toxicity from text and generate a detoxified version of the text while preserving the semantic meaning. Different from the previous tasks that only focus on the detection of toxicity (e.g., toxicity classification and toxic span identification), text detoxification addresses the toxic content by proactively rewriting it. An example of toxicity detoxification is shown in Table 1. Formally, for this task, the input is a toxic text \(t\) and our goal is to generate a detoxified version of the text \(\hat{t}\).
**Existing Methods.** Text detoxification can be viewed as a style transfer task. That is, toxicity can be treated as the style of a text. The style transfer methods are applied to rewrite the text with similar semantic meaning without the toxicity style. In previous work [32, 37], both supervised and unsupervised methods are proposed to solve this task in a style transfer manner. Logacheva et al. [32] propose DetoxBART, which fine-tunes the Transformer-based generation model BART [29] on the ParaDetox dataset. Such fine-tuning process makes DetoxBART yield the best performance in terms of detoxification and semantic preservation. The other end-to-end approaches include DualRL [34], Deep Latent Sequence Model (DLSM) [17], Stable Style Transformer (SST) [27], Style Transfer as Paraphrase (STRAP) [24], Paraphrasing GeDi (ParaGeDi) [9], etc.
**Our Method.** The detoxification task is also a generation task. Given the paired dataset (i.e., the toxic text \(T\) and the paraphrased non-toxic counterpart \(\hat{T}\)), our goal is to learn the prompt \(M_{\phi}\) that can better transfer the input text (toxic) into the output text (non-toxic) text while preserving the semantics. The optimization goal is similar to Equation 3 and the only difference is that the label changes from \(\hat{T}\) to \(\hat{T}\) where the former is the texts without toxic spans (incomplete texts) and the later is the detoxified texts (complete texts).
## 4 Datasets and Models
### Datasets
In this paper, we consider eight datasets for the evaluation of the three tasks. Note that, in Task 1 (toxicity classification), for each dataset, we generate a balanced version of it by randomly choosing the same number of samples from the larger category to match the smaller category. We follow the train/test partition of a dataset if they have already been provided. Otherwise, we randomly sample 80% of a dataset as the training dataset and the rest 20% as the testing dataset. Table 2 reports some basic statistics about each dataset. We describe each dataset below.
**HateXplain [35].** It is a benchmark dataset collected from Twitter and Gab for explainable hate speech detection. The dataset is annotated by Amazon Mechanical Turk (MTurk) workers with three labels: hate, offensive, or normal. For our work, we consider both hate and offensive posts as toxic and the rest as non-toxic.
**USElectionHate20 [16].** This dataset is collected from Twitter by selecting tweets that contain election hashtags or politicians' names. The authors manually label a subset of tweets with different stances as well as whether the tweet is hateful/offensive. We consider hateful/offensive tweets as toxic and the rest as non-toxic.
**HateCheck [45].** HateCheck contains a suite of functional tests for hate speech detection models. Each post is labeled by different annotators and we consider the majority votes as the final label of this post.
**SBIC [46].** The Social Bias Inference Corpus (SBIC) is collected from Reddit, Twitter, and fringe Web communities
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Dataset** & **Task** & **\# Train** & **\# Test** \\ \hline
**HateXplain**[35] & 1 & 12,578 & 3,050 \\
**USelectionHate20**[16] \(*\) & 1 & 586 & 118 \\
**HateCheck**[45] & 1 & 1,998 & 484 \\
**SBIC**[46] \(*\) & 1 & 93,346 & 11,000 \\
**MHS**[23] & 1 & 22,700 & 5,762 \\
**ToxicSpan**[39] \(*\) & 2 & 7,888 & 1,991 \\
**Parallel**[11] & 3 & 886 & 222 \\
**ParaDetox**[32] & 3 & 9,551 & 2,388 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Overview of datasets. Note that \(*\) means the dataset provides the train/test partition.
such as Gab, Stormfront, and banned subreddits. The dataset is labeled by MTurk workers. We leverage the v2 version of it for our study and we consider posts labeled offensive as toxic posts and the rest as non-toxic posts.
**MHS [23].** The Measuring Hate Speech (MHS) dataset is collected from comments on social media like YouTube, Twitter, and Reddit. The corpus is labeled by MTurk workers from the US. We consider comments with hate speech score \(\geq\) 0 as toxic and all others as non-toxic.
**ToxicSpan [39].** The ToxicSpan dataset contains \(\sim\)10k English texts filtered from Civil Comments [6] and was formally introduced as SemEval-2021 Task 5 [39]. Each text is reviewed by three to seven raters. Each rater is asked to identify the spans "that constitute anything that is rude, disspectful or unreasonable that would make someone want to leave a conversation" [37]. The lengths of the highlighted spans were decided by the raters.
**Parallel [11].** The Parallel dataset contains 2,279 pairs of (toxic sentence, detoxified sentence). There are 1,108 unique toxic sentences after removing duplicates. Note that for each toxic sentence, the dataset might offer multiple detoxified versions. We only select the first detoxified version to construct the pair.
**ParaDetox [32].** ParaDetox contains 11,939 toxic sentences and 19,766 paraphrased sentences (detoxified sentences). Similar to the Parallel dataset, each toxic sentence might have multiple detoxified versions. We only pick the first detoxified version to construct the pair. The ParaDetox dataset constructed by us has 11,939 pairs in total.
**Remarks.** All the datasets are annotated by human annotators. However, the definition of toxicity might vary across different datasets. For instance, USEelectionHate20 targets hateful tweets against politicians, while SBIC focuses on offensive posts from different Web communities. This may bring challenges for toxicity classifiers such as the Perspective API [4]. On the other hand, our approach diminishes this issue, given that we use a learnable prompt that is tailored for each dataset, effectively capturing the toxic definition of the dataset through the lens of the positive and negative samples in each dataset.
### Models
In this paper, we consider prompt tuning over two families of LLM including GPT2 [43] and T5 [44]. Concretely, we use GPT2-medium, GPT2-large, T5-small, T5-base, and T5-large in our experiments. In Task 1 (Toxicity Classification), the learning rate is set to 0.3, we set the total optimization steps to 2,000 with Adafactor [50] optimizer and the linear learning rate scheduler with 100 warm-up steps. For all models, the effective batch size is set to 32 (batch size of 4/8 with gradient accumulation steps of 8/4 for GPT2-L/Others). We follow the prompt tuning method proposed by Lester et al. [28] in Task 1. In Task 2 (Toxic Span Detection) and Task 3 (Detoxification), we set the training epoch to 5, the initial learning rate to 5e-5, and the optimizer of AdamW [33] with the linear learning rate scheduler. Different from Task 1 (Toxicity Classification), we follow the prompt tuning method proposed by Li and Liang [30] instead as it can achieve better performance in Task 2 and Task 3. We hypothesize that Lester et al. [28] initializes the prompt with embeddings that enumerate the output classes, which makes the method more suitable for the classification task. In contrast, the prompt tuning method proposed by Li and Liang [30] has more tunable parameters than the one proposed by Lester et al. [28]. This method learns transformer activations that are fixed across examples at every network layer, allowing subsequent tokens to attend to this prefix. As such, Li and Liang [30] is a better fit for Task 2 (Toxic Span Detection) and Task 3 (Detoxification).
## 5 Task 1: Toxicity Classification
### Experimental Setup
**Baselines.** Regarding the baselines for Task 1, we consider Google's Perspective API [4] (Perspective), BERT-base trained on toxicity classification corpus [1] (ToxicBERT), and RoBERTa-base trained on toxicity classification corpus [1] (UnRoBERTa). For each baseline, given a text, it provides a toxicity score ranging from 0 to 1. We consider the text with a score larger than 0.5 as toxic otherwise non-toxic. The results with the best threshold (rather than 0.5) are shown in Table 15 in Appendix. Note that for Perspective API, on each dataset, we select the perspective score (e.g., Severe Toxicity) that achieves the best classification result, and report the corresponding performance.
**Datasets.** We use five datasets - HateXplain, USElectionHate20, HateCheck, SBIC, and MHS - to evaluate the baselines and our models. Note that we observe redundant samples on HateXplain, USElectionHate20, and SBIC.v2. However, they are less than 1% and have almost no influence on the final performance based on our initial evaluation.
**Metrics.** We consider accuracy, precision, recall, and \(F_{1}\)-score as the evaluation metrics, which are standard metrics for evaluating the performance of classifiers. Note that we only report the \(F_{1}\)-score on the main paper and put the precision, recall, and accuracy results in Appendix A in Appendix.
### Results
**Overall Performance.** We first show the \(F_{1}\)-score of toxicity classification with toxicity classification in Table 3. The accuracy, precision, and recall are shown in Table 16, Table 17, and Table 18 in the Appendix. We find that, in general, prompt tuning outperforms baselines across different datasets. For instance, on HateXplain, the prompt tuning with GPT2-L achieves 0.731 \(F_{1}\)-score, while the best baseline (Perspective) only achieves 0.703 \(F_{1}\)-score. The statistical test shows that prompt tuning indeed outperforms the best baseline (see Table 14 in Appendix). This indicates that prompt tuning can indeed unleash the power of LLM to perform the toxicity classification task. Also, we observe that a larger LM usually provides a more promising performance on the task, e.g., GPT2-L usually outperforms GPT2-M and T5-B/L is better than T5-S in general. For instance, on HateCheck, GPT2-L achieves 0.892 \(F_{1}\)-score while GPT2-M only has 0.758 \(F_{1}\)-score. This implies that the larger capacity of
LLM would better guide the prompt tuning to achieve better performance.
Comparison to Manual Prompt.As previous work [48, 8] shows that manual prompts can achieve decent performance in the toxicity classification task, we investigate how the performance of prompt tuning compares to manual prompts. We use the below manual prompt following Schick et al. [48] for our comparison study. Note that we also investigate using the descriptions of 6 different scores in Perspective API as the manual prompt and the results show similar trends.
[INPUT] Question: Does the above text contain rude, disrespectful, or unreasonable language? Answer:[MASK] Here, [INPUT] is the placeholder for the text to be classified, "Question: Does the above text contain rude, disrespectful, or unreasonable language? Answer:" is our manual prompt and [MASK] is the classification output by the LLM. The performance is shown in Table 4. We observe that the \(F_{1}\)-score of the manual prompt is substantially lower than the prompt tuning approach (see Table 3). For instance, for the average results, with T5-S, prompt tuning achieves 0.766 \(F_{1}\)-score while manual prompt only reaches 0.165. These results highlight the effectiveness and performance gains when using prompt tuning instead of manual prompts.
can observe that in some cases, the prompt can successfully transfer to another dataset. For instance, the prompts trained on USElectionHate20 can achieve 0.650 \(F_{1}\)-score on HateXplain and 0.733 \(F_{1}\)-score on MHS, which are about 5% lower than the baselines (0.703 accuracy on HateXplain and 0.790 accuracy on MHS according to Table 3). However, the performance is less satisfying in some other cases where the \(F_{1}\)-score is below 0.500. We also notice that the prompt trained on the MHS dataset can better transfer to other datasets. For instance, after training on MHS, the \(F_{1}\)-score is 0.694 on HateXplain and 0.581 on USElectionHate20, which is comparable or even better to the \(F_{1}\)-score provided by the Perspective API (0.703 and 0.506). This can be credited to the fact that MHS covers various kinds of toxicity including insult, humiliation, violence, hate speech, etc. By fine-tuning with the diverse distributed data, the learned prompt is more general and can better transfer to other datasets. On the other hand, prompts learned from dataset like HateXplain is less effective to transfer into other datasets. We suspect this is because these datasets have a relatively narrow definition of toxicity. In general, the prompt learned from a more diverse dataset with different types of toxicities may have a better generalization ability to other datasets. Meanwhile, as we have shown before (see Table 5), the prompts can better fit different downstream datasets with the help of only a small fraction of labeled samples, which further demonstrates the efficacy of prompt learning.
**Comparison with Fine-tuning.** Here we take T5-S on USElectionHate20 as an example. We observe that prompt tuning reaches 0.712 accuracy within 6 minutes, while the best accuracy (evaluated every 200 steps) for fine-tuning the whole model is only 0.619 within 100 minutes. This is because the LLM is trained with a large corpus and can generate informative representations of the inputs. Prompt tuning can guide the model better leverage the representation for the downstream tasks with a small number of parameters, which can adapt faster to new tasks compared to finetuning, especially with fewer training samples.
**Robustness.** Given the misspellings in the training procedure, we do observe that prompt tuning can adapt to the testing posts with misspellings. E.g., on 100 randomly selected toxic posts on HateCheck, there do exist misspelling words like "tr4sh," "4sholes," "Fukc," and "crippl3." And prompt tuning with T5-S can correctly identify them (98% accuracy). We further perturb these 100 evaluation posts by randomly repeating one character of each toxic word several times or adding extra spaces inside the toxic word, e.g., "sluttuttts," and "w h o r e." Note that we leverage such perturbations since we also observe them in the toxic texts and such perturbations are also considered by previous work [19]. We observe that, without further prompt tuning, the evaluation accuracy on these modified 100 posts is still 97%, which remains almost unchanged. This implies that prompt tuning is robust to adversarial perturbation.
**Error Analysis.** Although prompt tuning outperforms other baselines in most cases, wrongly predicted texts still exist (20 in total). We take the USElectionHate20 dataset (with T5-B) as a case study to analyze the wrongly predicted cases. As shown in Table 7, the main reason that causes the wrong prediction is the wrong label, e.g., in the example, we observe some toxicity against Trump, but the text is labeled as non-toxic. Also, we observe that some variations of the slur words and toxic hashtags may cause wrong predictions. Last, prompt tuning is less effective against some texts with implicit toxic content.
**Takeaways.** Our results show that prompt tuning outperforms baselines in the toxicity classification task with sufficient labeled data. Also, the detection performance is still promising with fewer training steps/samples. Another observation is that directly transferring the prompt trained on one dataset into another dataset might be less effective as the two datasets might share different types of toxicity. However, this can be addressed by adding only a small number of labeled samples from the distribution of the testing dataset. Our results suggest that prompt tuning can also serve as an alternative tool to assist the annotation process, especially for
\begin{table}
\begin{tabular}{l|c c c c c} \hline \hline \multirow{2}{*}{**Training Dataset**} & \multicolumn{5}{c}{**Transfer Dataset**} & \\ & **HateXplain** & **USElectionHate20** & **HateCheck** & **SBIC** & **MHS** \\ \hline
**HateXplain** & - & 0.488 & 0.373 & 0.419 & 0.688 \\
**USElectionHate20** & 0.650 & - & 0.472 & 0.485 & 0.733 \\
**HateCheck** & 0.543 & 0.297 & - & 0.534 & 0.579 \\
**SBIC.v2** & 0.638 & 0.404 & 0.646 & - & 0.655 \\
**MHS** & 0.694 & 0.581 & 0.610 & 0.518 & - \\ \hline \hline \end{tabular}
\end{table}
Table 6: \(F_{1}\)-score of Task 1 (Toxicity Classification) when the training dataset is different from the transfer dataset.
Figure 1: \(F_{1}\)-score of Task 1 with different training steps.
the newly emerging toxicity.
## 6 Task 2: Toxic Span Detection
### Experimental Setup
As we observed from Task 1 (Toxicity Classification), T5 models and GPT2 models share similar performance. In the following evaluation, we mainly leverage T5 models as our pre-trained LLMs.
**Baselines.** We consider three baselines, i.e., BiLSTM [18], BERT [12], and SPAN-BERT [22]. Concretely, we follow the default hyper-parameters setting of Pavlopoulos et al. [37]. We train/fine-tune the models for 100 epochs on the training partition of the ToxicSpan dataset and evaluate it on its test partition.
**Datasets.** We use the ToxicSpan dataset to evaluate the baselines and our models.
**Metrics.** We follow previous work [37] and leverage score as the main evaluation metric. Note that the \(F_{1}\)-score in Task 2 is different from Task 1. Concretely, for the \(i\)-th sample, we consider its ground truth span (i.e., the character offsets) as \(S^{i}_{g}\) and the predicted span as \(S^{i}_{p}\). The sample-level precision \(P^{i}\), recall \(P^{i}\), and \(F_{1}\)-score \(F^{i}_{1}\) are defined as the following:
\[P^{i}(S^{i}_{g},S^{i}_{p})=\frac{|S^{i}_{g}\cap S^{i}_{p}|}{|S^{i }_{p}|} \tag{4}\] \[R^{i}(S^{i}_{g},S^{i}_{p})=\frac{|S^{i}_{g}\cap S^{i}_{p}|}{|S^{i }_{g}|}\] (5) \[F^{i}_{1}(S^{i}_{g},S^{i}_{p})=\frac{2\cdot P^{i}(S^{i}_{g},S^{i }_{p})\cdot R^{i}(S^{i}_{g},S^{i}_{p})}{P^{i}(S^{i}_{g},S^{i}_{p})+R^{i}(S^{i}_ {g},S^{i}_{p})} \tag{6}\]
Note that if the ground truth span \(S^{i}_{g}\) and the predicted span \(S^{i}_{p}\) are both empty, we consider \(F^{i}_{1}(S^{i}_{g},S^{i}_{p})=1\) (\(F^{i}_{1}(S^{i}_{g},S^{i}_{p})=0\) if one of them is empty). Then, we average the \(F_{1}\)-score for all samples to obtain a single \(F_{1}\)-score.
Parallel dataset (we used in Task 3) and form a new testing dataset. Given the prompt trained with T5-L on ToxicSpan, we observe that our method can correctly identify the toxic spans on 85% of posts. We then dive deeper into the failed cases and find that most of them belong to Categories 1 and 8 as shown in Table 9. In general, this case study demonstrates that prompt tuning can indeed transfer to out-of-distribution data.
**Comparison with Fine-tuning.** For Task 2, we also compare the performance of prompt tuning with fine-tuning. Taking T5-L model as an example, we observe that, with the same training epochs, prompt tuning yields slightly better performance (0.643 \(F_{1}\)-score) than fine-tuning (0.628 \(F_{1}\)-score) and costs less time. This indicates that prompt tuning can unleash the power of LLM with only limited effort.
**Robustness.** Following the perturbation strategy in Task 1, we perturb 100 randomly selected posts from TSD and compare the performance with the original posts. We observe that prompt tuning reports the same toxic span for 57 perturbed posts. For 38 perturbed posts, prompt tuning failed to detect or can only detect part of the toxic spans. For the rest 5 perturbed posts, prompt tuning can obtain even better toxic spans than their original version. Compared to Task 1, prompt tuning is less robust in Task 2. This can be credited to the lack of perturbed toxic spans in the training dataset, which may be mitigated by introducing perturbation during the training phase as well.
**Error Analysis.** We conduct a case study regarding the wrongly detected spans. Concretely, we randomly select 100 test samples with wrongly predicted spans and manually verify the possible reasons. Then, we categorize the reasons into 9 categories (see Table 9). Note that each test sample is manually verified by three annotators to put into a category with full agreement. We find that a substantial percentage of wrong span predictions in categories 2, 3, 4, and 5 (47%) are caused by the problematic ground truth label. For instance, in category 2, the ground truth span contains both toxic and non-toxic text. Note that the ground truth inconsistency is caused by the fact that the lengths of the toxic spans were decided by the raters [39]. The ToxicSpan dataset accepts character offsets that at least two raters have included each character offset in their spans. Category 2 actually covers the corner cases relating to such human errors/bias when building the ToxicSpan dataset. Nevertheless, our method successfully detects the real toxic span "cowards" from this example. Also, in category 3, the toxic span is not labeled by the ground truth. However, they are accurately detected by our method. We also observe that prompt tuning may fail to identify some ambiguous toxic spans such as the "embarrassment" example shown in category 4 (Table 9). A more interesting case (category 5) shows that our method can dig out the missing toxic span from the text. For instance, the ground truth span only contains "stupid", while our method discovers "idiots" as well. This case demonstrates the potential of prompt tuning to become an effective tool to improve the annotation quality of toxic spans. We also notice that the cases in categories 1, 6, 7, 8, and 9 (53%) are caused (or partially caused) by our method. For category 1, we observe that our method repeats the original sentence without any change. We then diver deeper into those samples and find that they are mainly short sentences or contain less toxic spans, which may lead the prompt to become less sensitive to these cases. For category 6, we observe that our method successfully generates the sentence without toxic spans, but the mapping algorithm fails to provide an exact span area as the ground truth span, e.g., prompt tuning includes the quota into the toxic span as well since it serves as an emphasize to the toxic expression. In category 9, we observe that our method overlooks the ground truth span, but surprisingly detects a new span like the "crap" example. Those wrong cases show that toxic span detection from the view of prompt tuning is not perfect, but prompt tuning shows its great potential in facilitating and correcting the toxic span detection process. For instance, it can serve as an assistant tool for better annotation quality.
**Takeaways.** We observe that prompt tuning can achieve comparable performance with the best conventional method, i.e., SPAN-BERT, but with much less time cost. Also, the performance is relatively stable even with fewer training epochs. This further demonstrates the potential of leveraging prompt tuning to tackle the toxic span detection tasks and provides evidence for better span labeling. We also show that prompt tuning, in some cases, can identify additional toxic spans not labeled by the ground truth (i.e., human annotators).
## 7 Task 3: Detoxification
Different from previous tasks that only focus on toxicity detection, this task aims to detoxify the given text while preserving the corresponding semantic meaning.
### Experimental Setup
**Baselines.** We use the vanilla version of BART [29] and the DetoxBART [32] as the baselines. Note that the DetoxBART is also trained on the ParaDetox dataset for 10,000 epochs according to Logacheva et al. [32].
**Datasets.** We use Parallel and ParaDetox datasets to evaluate the performance of baselines and prompt tuning.
**Metrics.** To quantify the quality of the detoxification, we consider two aspects, i.e., the detoxification effectiveness and the utility of the generated sentences. For detoxification effectiveness, we leverage the Perspective API to quantify the toxicity level change since it offers the best performance among all baselines and is robust on different datasets. Specifically, we first measure the average toxicity score change and then quantify the percentage of texts that has high toxicity score (0.7 or 0.9), following the guidelines of Perspective API.3 Note that we use \(\mathbf{T_{avg}}\), \(\mathbf{T_{0.7}}\), and \(\mathbf{T_{0.9}}\) to denote the average toxicity score of texts, the ratio of texts that have toxicity score over 0.7, and the ratio of texts that has toxicity score over 0.9, respectively. Regarding the utility, we consider five different metrics. We first consider BLEU score as the utility evaluation metric, which is also widely
used in previous work [32, 54]. Then we quantify the semantic preservation by comparing the text embeddings similarity between the original text and the detoxification text. Concretely, we consider two types of embedding following [32], i.e., contextual string embeddings [25] from flairNLP [2], which is denoted as SIM (F), and SIMILE proposed by Wieting et al. [60], which is denoted as SIM (W). We denote the two types of embedding similarities as SIM (F) and SIM (W), respectively. Besides, we also use the token-level perplexity [43] to measure the fluency of the text, where lower perplexity denotes better fluency.
### Results
The detoxification performance on different datasets is shown in Table 10. We observe that DetoxBART performs slightly better in detoxifying the text than prompt tuning. For instance, on ParaDetox, DetoxBART reduces the \(\mathbf{T_{avg}}\), \(\mathbf{T_{0.7}}\), and \(\mathbf{T_{0.9}}\) to 0.180, 0.013 and 0, respectively while prompt tuning on T5-L can reduce them into 0.213, 0.037, and 0.003 respectively. This means that ParaDetox has better detoxification effectiveness than prompt tuning. However, we also observe that the text quality generated with prompt tuning is better than the DetoxBART. For instance, on ParaDetox, compared to DetoxBART, the PT (T5-B) has a higher BLEU score, SIM (W), SIM (F), while attaining a smaller TokenPPL. This indicates the text generated by prompt tuning has better fluency and can better preserve the semantic meaning of the original text. In general, we consider both DetoxBART and prompt tuning as successful methods as they can largely reduce the toxicity level while preserving the semantic meaning and fluency of the original text.
Different Epochs.We then investigate how the training epochs affect the detoxification effectiveness and the model's utility regarding semantic preservation. The results are shown in Figure 4 and Figure 3, respectively. From Figure 4, we have three observations, first, we find that more training epochs lead to better detoxification performance. For instance, on Parallel, prompt tuning on T5-L can reduce the
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category** & **Reason** & **Text Example** & **Percentage (\%)** \\ \hline
1 & Labeled by ground truth & they’re not patriots. they’re vandals, [thieves], and [thieves], they’ve plastered a facade of patriotism over their outrage at being expected to obey the law. & 17 \\ \hline
2 & GT contains both toxic and non-toxic spans. & adn is endorsing, without officially endorsing. [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves,thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves],[thieves], [thieves], [thieves], [thieves], [thieves], [thieves,thieves, [th], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves,thieves], [thieves], [thieves], [thieves], [thieves], [thieves,th], [thieves], [thieves,thieves, [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [th], [thieves], [thieves], [thieves], [thieves], [thieves],[thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves,th], [thieves], [thieves,th], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [thieves], [th], [
**Tavg** to 0.616 with 1 epoch, while decreasing to 0.397 with 5 epochs. Second, prompt tuning on larger models lead to better detoxification performance, e.g., T5-L performs the best while T5-S performs the worst. This is expected as a larger model can represent the data in a more informative way thus better guiding the prompt tuning in the direction of detoxification. Third, in a larger dataset such as Paradox, prompt tuning already achieves good detoxification performance in the early epoch, e.g., the first or second epoch. Our results further exemplify the effectiveness of prompt tuning as the time cost is much less than training the detoxification model like DetoxBART.
Regarding utility, we find that the utility is relatively stable for different models in different epochs. This indicates that those LLMs have good generation ability in general.
**Prompt Transferability.** We then take ParaDetox as the training dataset and Parallel as the testing dataset to investigate the generalizability power of prompt tuning. With T5-B trained on ParaDetox, the \(T_{avg}\), \(T_{0.7}\), and \(T_{0.9}\) on Parallel drop to 0.251, 0.027, and 0.000, respectively, which are even better than the original results shown in Table 10 (0.408, 0.256, and 0.032). One possible reason is that ParaDetox contains a larger number of training data, which better guides the prompt for the detoxification tasks and makes it more transferable to other datasets like Parallel.
**Comparison with Fine-tuning.** For Task 3, we take the T5-L model on Parallel as a case study. We observe that, prompt tuning can reduce the toxicity of posts to a larger extent, e.g., the **Tavg** of prompt tuning is 0.396, while the value is 0.437 for fine-tuning. On the other hand, we find that fine-tuning can generate more fluent sentences, e.g., the BLEU score is 0.795 for fine-tuning, while only 0.754 for prompt tuning. In general, prompt tuning can still be considered as a lightweight plugin to adapt LLMs to new tasks.
**Robustness.** We again follow the perturbation strategy in Task 1 to perturb 100 randomly selected posts from the Parallel dataset. We observe that, for the original version of these 100 posts, prompt tuning (with T5-L) can reduce the \(T_{avg}\), \(T_{0.7}\), and \(T_{0.9}\) from 0.725, 0.590, and 0.130 to 0.357, 0.120, and 0.010, respectively, while the values are 0.402, 0.180, and 0.020 for the perturbed 100 posts, which is close to detoxify the original version. This indicates that prompt tuning is relatively robust in Task 3.
**Case Study.** We then dive deeper into the generated text of the ParaDetox dataset and check them manually. We consider both successful cases (C1 and C2) and failed cases (W1-W5). Table 11 shows the examples of these cases. In most cases, prompt tuning is powerful in reducing the toxicity level of the sentence while preserving its semantic meaning. For example, in C1, our method achieves similar detoxification performance (toxicity score decreases from 0.827 to around 0.163). Also, our method preserves the semantic meaning properly. In C2, we observe that our method can even detoxify the sentence better than the ground truth.
Among the 2,388 text samples, we observe that there are 88 detoxification samples (3.68%) that still have a high toxicity score, i.e., larger than 0.7. We manually check those samples and find that they can be categorized into 5 different wrong categories (W1-W5). For W1 (6/88), we observe that the sentence is hard to be detoxified, and the ground truth sentence is identical to the original sentence. For W2 (52/88), prompt tuning just directly repeats the original sentence without any modification. For W3 (27/88), we observe that prompt tuning indeed preserves the semantic meaning and reduces the toxicity level. We acknowledge that for some implicit toxic content, as shown in the example, it might be harder for the prompt model to detect and eliminate them perfectly. For W4 (1/88), prompt tuning actually provides better semantic preservation compared to the ground truth. For W5 (1/88), we observe that prompt tuning just considers "i jus clicked tht nasty shit" as toxic parts and directly removes them. During the labeling, we notice that there indeed exists a trade-off between detoxification and semantic preservation. However, in most cases, prompt tuning can do well on both aspects (see also Table 10). It indicates that prompt tuning can be a good tool for assisting the detoxification task, e.g., providing possible solutions for the annotators to make their decision. Currently, our current prompt tuning is based on paired datasets, which is similar to machine translation. However, such datasets are usually small. One promising
\begin{table}
\begin{tabular}{l l|c c c|c c c c} \hline \hline
**Dataset** & **Method** & **Tavg**\(\downarrow\) & **T0.7**\(\downarrow\) & **T0.9**\(\downarrow\) & **BLEU**\(\uparrow\) & **SIM (W)**\(\uparrow\) & **SIM (F)**\(\uparrow\) & **TokenPPL**\(\downarrow\) \\ \hline \multirow{8}{*}{**Parallel**} & None & 0.755 & 0.676 & 0.135 & 1.000 & 1.000 & 1.000 & 227.834 \\ & GroundTruth & 0.178 & 0.009 & 0.000 & 0.491 & 0.757 & 0.669 & 550.725 \\ & BART & 0.754 & 0.676 & 0.135 & 0.999 & 0.999 & 0.998 & 227.904 \\ & DetoxBART & 0.242 & 0.036 & 0.000 & 0.708 & 0.879 & 0.843 & 236.654 \\ & PT (T5-S) & 0.573 & 0.463 & 0.077 & 0.835 & 0.927 & 0.939 & 326.696 \\ & PT (T5-B) & 0.408 & 0.256 & 0.032 & 0.770 & 0.898 & 0.909 & 301.597 \\ & PT (T5-L) & 0.396 & 0.329 & 0.031 & 0.754 & 0.881 & 0.889 & 284.861 \\ \hline \multirow{8}{*}{**ParaDetox**} & None & 0.775 & 0.778 & 0.134 & 1.000 & 1.000 & 1.000 & 330.829 \\ & GroundTruth & 0.166 & 0.000 & 0.000 & 0.633 & 0.828 & 0.778 & 393.800 \\ \cline{1-1} & BART & 0.774 & 0.777 & 0.133 & 0.999 & 0.999 & 0.998 & 331.250 \\ \cline{1-1} & DetoxBART & 0.180 & 0.013 & 0.000 & 0.688 & 0.862 & 0.832 & 438.242 \\ \cline{1-1} & PT (T5-S) & 0.253 & 0.081 & 0.007 & 0.760 & 0.910 & 0.905 & 593.442 \\ \cline{1-1} & PT (T5-B) & 0.224 & 0.051 & 0.005 & 0.754 & 0.920 & 0.897 & 499.851 \\ \cline{1-1} & PT (T5-L) & 0.213 & 0.037 & 0.003 & 0.743 & 0.916 & 0.886 & 404.565 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Performance of Task 3. The arrow denotes which direction is for better results.
direction that we aim to explore in our future work is to combine the paired dataset with the unpaired dataset (i.e., it only contains sets of toxic and non-toxic contents but without the pairs) to jointly fine-tune the prompt.
**Takeaways.** We empirically show that prompt tuning can reduce the toxicity level to a large extent and better preserve the semantic meanings. An interesting observation is that the semantic meaning of the original sentence can be properly preserved even with fewer training epochs due to the strong representation ability of the LLM. However, with fewer epochs, the detoxification performance might be less satisfying as the process of toxic to non-toxic contents is more difficult than previous tasks and needs more learning steps to better guide the prompt tuning. The effective detoxification and semantic preserving abilities make prompt tuning a strong competitor to conventional methods in the detoxification task.
## 8 Related Work
**Prompt Learning.** Prompt learning is a new paradigm in natural language processing (NLP) [31]. It allows users to directly specify the task they want in natural language for the pre-trained language model to interpret and complete. This paradigm paves way for using a single LLM as the _universal solver_ for various understanding and generation tasks, such as text classification [47], machine translation [44], semantic parsing [52], question answering [20], etc. To un-leash the full potential, research on prompt learning has been investigating automatically inducing the discrete/continuous prompts [57, 30], multi-prompt learning [42, 20], prompt training, and fine-tuning strategy [13, 41], transferability of prompts [40], etc. Our work is built on top of prompt learning. We conduct the first systematic hateful language study from the prompt tuning perspective.
**Toxicity Classification.** The problem of toxic online content is a longstanding and challenging [5] problem affecting our society. Motivated by the impact that the problem has on both the online and offline world, the research community and the industry devoted substantial resources to developing models to detect toxic content. One of the most used tools for assessing toxicity online is Perspective API [4], a set of machine learning models trained on a human-annotated dataset, released by Google. The Perspective API, given a piece of text, provides a set of scores that correspond to how likely the text is toxic, attacking specific identities, sexually explicit, etc. At the same time, Google released its annotated dataset, which enabled other researchers to develop more models aiming to tackle the problem. One such example is Detoxify [1], which leverages the power of transformer models to detect toxicity in text, across multiple languages.
Davidson et al. [10] highlight that there is a distinction between offensive language and hate speech. Also, the authors release HateSonar, a machine learning model, that identifies whether a piece of text contains offensive language or hate speech. As previous research notes [61], however, the HateSonar classifier performs poorly compared to the Perspective API, when tested on comments left on news articles. Zimmerman et al. [66] highlight that by leveraging deep learning ensembles, we can improve the performance of previous models in detecting hate speech on Twitter. Other work focuses on identifying the targets of toxic content [53, 14], or on identifying specific forms of toxic content such as Antisemitism [62, 36], Islamophobia [58], and Sinophobia [56, 65].
Figure 4: Detoxification effectiveness of Task 3 with different training epochs.
Figure 3: Utility of Task 3 with different training epochs.
All of the above-mentioned efforts in detecting toxic content are based on fine-tuning existing models or developing dedicated classifiers focusing on the specific task of detecting toxic content. Recently, the pre-train and prompt paradigm is becoming increasingly popular, hence the research community started investigating how prompt learning can be leveraged to tackle the problem of toxic content online. In particular, Chiu et al. [8] use OpenAI's GPT-3 language model to investigate the performance of prompt learning in the task of detecting racist or sexist content. They find that by using a pre-defined prompt and a few-shot learning setting, they can identify racist or sexist content with an accuracy of up to 85%, highlighting that prompt learning can play a role in identifying toxic content. Similarly, Schick et al. [48] find that language models can identify toxic content and whether the generated text contains undesirable biases, all using prompt learning techniques. Also, they propose a de-biasing method, which helps the language model generate less biased content. Overall, both works [8, 48] highlight that large language models and prompt learning can detect toxic content with a decent performance. While this previous work is essential, it is limited in the sense that it focuses only on the toxicity classification task and, more importantly, relies on manual pre-defined prompts. In contrast, our work provides a comprehensive evaluation of how large language models and prompt learning can assist in tackling the problem of toxic content by considering multiple tasks (toxicity classification, toxic span detection, and detoxification). Also, we show that by using prompt tuning techniques, instead of pre-defined prompts, we can substantially increase the performance of the language models in the three tasks.
**Toxic Span Detection.** Toxic span detection [39] aims to identify the specific span that makes the sentence to be toxic. Pavlopoulos et al. [37] treat this task as the sequence labeling task to annotate the suspicious span in the sentence. Three models including BiLSTM [18], BERT [12], and SPAN-BERT [22] are considered. We instead formalize this task as a generation task and show that prompt-tuning can achieve comparable performance to the SPAN-BERT but with much less computational time.
**Detoxification.** Detoxification aims to reduce the toxicity level of the sentence while preserving the semantic meaning to the largest extent. It is similar to neural style transfer [21]. Laugier et al. [26] propose a self-supervised method named CAE-T5 to learn the transformation from toxic to civil from the unpaired corpus. Logacheva et al. [32] develop DetoxBART which fine-tunes the BART model on the ParaDetox dataset to achieve better performance. Our work is substantially different from their work as we do not need to fine-tune the model but just the prompt, which is less costly. We notice that conventional methods like DetoxBART can achieve better detoxification performance while prompt tuning can better preserve semantic information.
## 9 Conclusion
In this paper, we performed the first extensive evaluation of using prompt learning with tunable prompts (prompt tuning) to tackle the problem of toxic content. Particularly, we focused on three tasks (toxicity classification, toxic span detection, detoxification) and assessed the performance of prompt tuning and how it compares to state-of-the-art baselines in these tasks. Among other things, we find that prompt tuning can achieve comparable or even better performance compared to the baselines. As shown by our evaluation, integrating prompt tuning into toxic content research can better help to improve the dataset quality and the model utility as the toxicity label (Task 1), predicted toxic span (Task 2), and
\begin{table}
\begin{tabular}{c l l l} \hline \hline
**Category** & **Original** & **Ground Truth** & **Generated** \\ \hline C1 & if assad wasn ’t such a dick then involvement would not be necessary. [0.827] & if assad wasn’t such that bad involvement would not be necessary. [0.163] & if assad wasn ’t such a bad person then involvement would not be necessary. [0.159] \\ \hline C2 & your reading comprehension is more fucked up than a football bat. [0.835] & your reading comprehension is worse than a football bat. [0.575] & your reading comprehension is not good [0.355] \\ \hline W1 & if you hold a knife against the cops you deserve to be shot! [0.827] & if you hold a knife against the cops you deserve to be shot! [0.827] & if you hold a knife against the cops you deserve to be shot! [0.774] \\ \hline W2 & did they bring along a signedx-glossy of boy wonder and castro as a token of their stupidity? & did they bring along a signedx-glossy of boy wonder and castro as a token [0.133] & did they bring along a signedx-glossy of boy wonder and castro as a token of their stupidity? [0.754] \\ \hline W3 & someone should give this idid a slap on the head. [0.939] & someone should give him a lesson. [0.257] & someone should give him a slap on the head. [0.721] \\ \hline W4 & i hate guys with a bitch mouth sit the fuck down nigea [0.985] & I hate bad guys [0.440] & i hate guys with a bad mouth sit down [0.721] \\ \hline W5 & wt!!
detoxified sentence (Task 3) can be used to assist the labeling procedure.
**Limitations.** Naturally, our work has some limitations. First, we use GPT2 and T5 as the LLMs to demonstrate the efficacy of prompt tuning. Our evaluation has demonstrated that prompt tuning can perform well even with these LLMs, and larger models generally perform better (see Table 3). While we acknowledge that conducting experiments with larger models with billions of parameters would be appealing, our hardware capabilities limit such endeavors. Also, we use Perspective API as an indicator to quantify the toxicity level (e.g., on Task 3), which is likely to yield some false positives/false negatives. Nevertheless, detecting toxic content is an open research challenge and the Perspective API is also leveraged by previous work [48, 51], indicating that it is a good proxy for assessing toxic content. Despite these limitations, we believe that our research can pave new ways for the study of toxic content, as researchers with limited computing resources can utilize currently available pre-trained large language models to perform important toxicity-related tasks with acceptable performance.
**Acknowledgment.** We thank the anonymous reviewers and our shepherd for their invaluable comments and feedback that helped us improve our manuscript. This work is partially funded by the European Health and Digital Executive Agency (HADEA) within the project "Understanding the individual host response against Hepatitis D Virus to develop a personalized approach for the management of hepatitis D" (D-Solve) (grant agreement number 101057917).
| オンラインにおける有害コンテンツの拡散は、ユーザーエクスペリエンスと社会全体に悪影響を与える重要な問題です。この問題の重要性と影響を促されて、研究は、人間が annotate し、トレーニングされた機械学習 (ML) モデルを開発することに焦点を当てています。これらの努力は重要ですが、これらのモデルは通常、汎用性が良くなく、新しいトレンドに対応することができません (例: 新しい有害な用語の出現)。現在、オンラインにおける社会問題に対処するためのアプローチは変革しており、特に、GPT-3 や T5 と 같은 大規模なコーパスでトレーニングされ、汎用性の高い大規模言語モデル (LLM) を利用することに焦点を当てています。この仕事では、LLM とヒント学習を使用して有害なコンテンツの問題に対処する方法について調査しています。特に、3 つのタスクを対象にしています。1) 毒性分類、2) 毒性スペーントーク、3) デ |
2306.13880 | Causality and stability analysis for the minimal causal spin
hydrodynamics | We perform the linear analysis of causality and stability for a minimal
extended spin hydrodynamics up to second order of the gradient expansion. The
first order spin hydrodynamics, with a rank-3 spin tensor being antisymmetric
for only the last two indices, are proved to be acausal and unstable. We then
consider the minimal causal spin hydrodynamics up to second order of the
gradient expansion. We derive the necessary causality and stability conditions
for this minimal causal spin hydrodynamics. Interestingly, the satisfaction of
the stability conditions relies on the equations of state for the spin density
and chemical potentials. Moreover, different with the conventional relativistic
dissipative hydrodynamics, the stability of the theory seems to be broken at
the finite wave-vector when the stability conditions are fulfilled at small and
large wave-vector limits. It implies that the behavior in small and large
wave-vector limits may be insufficient to determine the stability conditions
for spin hydrodynamics in linear mode analysis. | Xin-Qing Xie, Dong-Lin Wang, Chen Yang, Shi Pu | 2023-06-24T07:06:44 | http://arxiv.org/abs/2306.13880v3 | # Causality and stability analysis for the minimal causal spin hydrodynamics
###### Abstract
We perform the linear analysis of causality and stability for a minimal extended canonical spin hydrodynamics up to second order of the gradient expansion. The first order canonical spin hydrodynamics are proved to be acausal and unstable. To remove the unstable and acausal modes, we then formulate the minimal causal spin hydrodynamics up to second order of the gradient expansion. We derive that causality and stability conditions for this minimal causal spin hydrodynamics. Interestingly, the satisfaction of the stability condition relies on the equations of state for the spin density and chemical potentials. Moreover, different with the conventional relativistic dissipative hydrodynamics, the stability of the theory seems to be broken at the finite wave-vector when the stability conditions are fulfilled at small and large wave-vector limits. It implies that the linear stability conditions are necessary but may be insufficient.
Introduction
Relativistic heavy ion collisions provide a novel platform to study the spin physics. In non-central relativistic heavy-ion collisions, the quark-gluon plasma (QGP) with large angular momentum perpendicular to the reaction plane is created. Because of the total angular momentum conservation, the averaged spin of final particles produced from QGP are polarized along the direction of the initial orbital angular momentum [1; 2; 3], as known as the global polarization. The measurements of the global polarization for \(\Lambda,\overline{\Lambda}\), and other hyperons [4; 5; 6; 7; 8; 9; 10] can be understood well by various phenomenological models [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. The experimental data also indicates that the QGP generated in non-central relativistic heavy-ion collisions is the most vortical fluid ever observed [4]. STAR [6; 27] and ALICE Collaboration [28] also measured the local polarization of \(\Lambda\) and \(\overline{\Lambda}\) along the beam and out-of-plane directions. Interestingly, the sign of local polarization in theoretical calculations is opposite to that of experimental data [15; 29; 30; 31; 23; 16]. To resolve the disagreement, a great deal of effort has been taken in feed-down effects [32; 33], hadronic interactions [34; 35], relativistic spin hydrodynamics [36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73], statistical models [74; 75; 29], quantum kinetic theory [76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105; 106; 107; 108; 109; 110; 111; 112; 113], effective theories [114; 115; 116], and other phenomenological models [117; 118; 119; 20; 210; 211; 23; 23; 212; 213; 214; 215; 216; 217; 218]. Although there are much important progress [116; 120; 121; 122; 123; 124; 125; 126; 127; 128], the local polarization has not been fully understood. Another important phenomenon related to spin, called the spin alignment of vector mesons proposed by Refs. [1; 2; 3], has drawn a lot of attentions. The spin alignment is characterized by the deviation of \(\rho_{00}\) from \(1/3\), where \(\rho_{00}\) is the \(00\) component of the spin density matrix of vector mesons [129]. A non-vanishing \(\rho_{00}-1/3\) indicates a net spin alignment of vector mesons. The experimental results [130; 131; 132; 133; 134; 135] show that the magnitude of the spin alignment of vector meson is much larger than that caused by vorticity and other conventional effects [2; 136; 137; 138; 139; 20; 214; 215; 216; 217; 218; 219; 22; 230; 231; 232; 233; 234; 235; 236; 237; 238; 239; 240; 241; 242; 243; 244; 244; 245; 246; 247; 248; 25
to construct spin hydrodynamics, such as entropy current analysis [45; 57; 58; 59; 67; 73; 48; 50; 49; 51], quantum kinetic theory [49; 51; 55; 56; 63; 65; 66; 69; 85; 93; 100; 110; 145], holographic duality [52; 53], and effective Lagrangian method [36; 37].
In spite of the substantial efforts, the arbitrariness due to pseudo-gauge transformations in spin hydrodynamics is not fully understood. Through the pseudo-gauge transformations [146; 147], one can obtain new forms of energy momentum tensor and spin tensor without affecting the conservation law. Although such transformations have no impacts on the total conserved charges, they indeed change the values of locally defined quantities, e.g., energy momentum tensor and spin tensor [145; 146; 147; 141]. Thus, different pseudo-gauge transformations give rise to different frameworks of the spin hydrodynamics, e.g., canonical [45; 59], Belinfante [48], Hilgevoord-Wouthuysen (HW) [148; 65], de Groot-van Leeuwen-van Weert (GLW) [43; 149] forms. Which framework is suitable for understanding the experimental data leads to intense discussions [150; 151; 152; 48; 153; 154; 41].
In the canonical spin hydrodynamics, the canonical spin operator fulfills the SO(3) algebra of angular momentum [153], thereby establishing a inherent connection to the conventional spin in quantum mechanics. Moreover, the transformation between the orbital angular momentum and spin in canonical spin hydrodynamics is transparent.
So far, the canonical spin hydrodynamics in the first order of the gradient expansion has been established. Before simulating the canonical spin hydrodynamics, it is necessary to investigate the theory's causality and stability, as is done in conventional hydrodynamics. In fact, the first order conventional relativistic hydrodynamics at Landau frame in the gradient expansion are always acausal and unstable, e.g., see the discussions in Refs. [154; 155; 156; 157]. Therefore, the question whether the first order spin hydrodynamics can be casual or stable arises. Several studies conclude that the canonical spin hydrodynamics up to the first order in the gradient expansion may be acausal and unstable in the linear modes analysis [70; 71]. In the early study [45], the authors have modified the constitutive relations for the antisymmetric part of energy momentum tensor through the equations of motion for the fluid and the stability conditions of this first order theory in the rest frame of fluid seem to be satisfied in the linear modes analysis. Later on, the Ref. [71] shows that this first order theory may be acausal, while Ref. [70] finds the stability condition (which corresponds to Eq. (42) in this work) may not be satisfied.
In this work, we systematically investigate the causality and stability for the canonical
spin hydrodynamics in the linear modes analysis. Our findings indicate that the canonical spin hydrodynamics up to the first order in the gradient expansion is acausal and unstable even when using the replacement mentioned by Ref. [45]. The acausal and unstable modes can usually be removed when extending the theory up to the second order in the gradient expansion. Therefore, we follow the method outlined in the conventional hydrodynamics [156; 157; 158; 159; 160] to formulate the minimal causal spin hydrodynamics. It is sufficient to see whether the causality and stability can be recovered up to the second order in the gradient expansion [156; 157; 158; 159; 160]. We then analyze the causality and stability for this minimal extended theory.
The paper is organized as follows. We first review the first order canonical spin hydrodynamics in Sec. II and show it is acausal and unstable in Sec III. In Sec IV, we formulate the minimal causal spin hydrodynamics following the method outlined in the conventional hydrodynamics. In Sec V, we analyze the causality and stability for the minimal causal spin hydrodynamics in the rest frame and comment the results in moving frames. We summarize this work in Sec. VI.
Throughout this work, we work with the metric \(g_{\mu\nu}=\text{diag}\{+,-,-,-\}\) and \(\Delta_{\mu\nu}=g_{\mu\nu}-u_{\mu}u_{\nu}\). For a rank-2 tensor \(A^{\mu\nu}\), we introduce the short hand notations \(A^{(\mu\nu)}\equiv(A^{\mu\nu}+A^{\nu\mu})/2\), \(A^{[\mu\nu]}\equiv(A^{\mu\nu}-A^{\nu\mu})/2\), and \(A^{<\mu\nu>}\equiv\frac{1}{2}[\Delta^{\mu\alpha}\Delta^{\nu\beta}+\Delta^{\mu \beta}\Delta^{\nu\alpha}]A_{\alpha\beta}-\frac{1}{3}\Delta^{\mu\nu}(\Delta^{ \alpha\beta}A_{\alpha\beta})\).
## II First order canonical spin hydrodynamics
In this section, let us briefly review the first order relativistic spin hydrodynamics. In canonical spin hydrodynamics, we have the conservation equations for energy, momentum, total angular momentum, and particle number, i.e., [45; 48; 50; 58; 59; 67; 161]
\[\partial_{\mu}\Theta^{\mu\nu}=0,\quad\partial_{\lambda}J^{\lambda\mu\nu}=0, \quad\partial_{\mu}j^{\mu}=0, \tag{1}\]
where \(\Theta^{\mu\nu}\) is the energy momentum tensor, \(J^{\lambda\mu\nu}\) is the total angular momentum current, and \(j^{\mu}\) is the current for particle number. Different with conventional relativistic hydrodynamics, the total angular momentum conservation equation in Eq.(1) plays a crucial role to describe the evolution of spin. The total angular momentum current in the canonical form can be written as [45; 48]
\[J^{\lambda\mu\nu} = x^{\mu}\Theta^{\lambda\nu}-x^{\nu}\Theta^{\lambda\mu}+\Sigma^{ \lambda\mu\nu}, \tag{2}\]
where the first two terms corresponds to the conventional orbital angular momentum, and \(\Sigma^{\lambda\mu\nu}\) is usually called as the canonical rank-3 spin tensor. Using Eq.(2), the conservation equation \(\partial_{\lambda}J^{\lambda\mu\nu}=0\) can be rewritten as the spin evolution equation,
\[\partial_{\lambda}\Sigma^{\lambda\mu\nu}\:=\:-2\Theta^{[\mu\nu]}. \tag{3}\]
Eq.(3) implies that the anti-symmetric part of energy momentum tensor \(\Theta^{[\mu\nu]}\) is the source for spin, and the spin can be viewed as a conserved quantity if and only if \(\Theta^{[\mu\nu]}=0\).
After introducing the spin degrees of freedom, the thermodynamic relations in spin hydrodynamics are modified as [45; 48; 50; 58; 59; 67; 161]
\[e+p = Ts+\mu n+\omega_{\mu\nu}S^{\mu\nu}, \tag{4}\] \[de = Tds+\mu dn+\omega_{\mu\nu}dS^{\mu\nu}. \tag{5}\]
where \(e,p,T,s,n,\mu,\omega_{\mu\nu}\), and \(S^{\mu\nu}\) denote energy density, pressure, temperature, entropy density, particle number density, chemical potential, spin chemical potential, and spin density. The spin density is defined as
\[S^{\mu\nu}\equiv u_{\lambda}\Sigma^{\lambda\mu\nu} \tag{6}\]
with the fluid velocity \(u^{\mu}\). Analogy to the relationship between \(\mu\) and \(n\), here we introduce the anti-symmetric spin chemical potential \(\omega_{\mu\nu}\) as the conjugate of \(S^{\mu\nu}\).
In general, the energy momentum tensor and particle current can be decomposed as
\[\Theta^{\mu\nu} = eu^{\mu}u^{\nu}-(p+\Pi)\Delta^{\mu\nu}+2h^{(\mu}u^{\nu)}+\pi^{ \mu\nu}+2q^{[\mu}u^{\nu]}+\phi^{\mu\nu}, \tag{7}\] \[j^{\mu} = nu^{\mu}+\nu^{\mu}, \tag{8}\]
where \(h^{\mu},\nu^{\mu}\), \(\Pi\), and \(\pi^{\mu\nu}\) stand for heat current, particle diffusion, bulk viscous pressure, and shear stress tensor, respectively, and the antisymmetric parts \(2q^{[\mu}u^{\nu]}\) and \(\phi^{\mu\nu}\) are related to the spin effects [45; 48]. As for the rank-3 spin tensor \(\Sigma^{\lambda\mu\nu}\), we have,
\[\Sigma^{\lambda\mu\nu}\:=\:u^{\lambda}S^{\mu\nu}+\Sigma^{\lambda\mu\nu}_{(1)}, \tag{9}\]
where the spin density \(S^{\mu\nu}\) defined in Eq.(6) has six independent degrees of freedom [45; 48].
In this work, we follow the power counting scheme in Refs. [48; 62; 64].
\[S^{\mu\nu}\sim O(1),\ \omega_{\mu\nu}\sim O(\partial),\ \Sigma^{\lambda\mu\nu}_{(1)} \sim O(\partial). \tag{10}\]
The spin density \(S^{\mu\nu}\) is chosen as the leading order in the gradient expansion. It corresponds to the case in which the most of particles in the system are polarized, i.e. the order of \(S^{\mu\nu}\) is considered as the same as the one for the number density \(n\). While in Refs. [45; 59], the authors have chosen a different power counting scheme, \(S^{\mu\nu}\sim O(\partial)\), \(\omega_{\mu\nu}\sim O(\partial)\), \(\Sigma^{\lambda\mu\nu}_{(1)}\sim O(\partial^{2})\).
Following [45; 48], it is straightforward to get the entropy production rate,
\[\partial_{\mu}\mathcal{S}^{\mu}_{\rm can} = (h^{\mu}-\frac{e+p}{n}\nu^{\mu})\left(\partial_{\mu}\frac{1}{T}+ \frac{1}{T}Du_{\mu}\right)+\frac{1}{T}\pi^{\mu\nu}\partial_{\mu}u_{\nu}-\frac{ 1}{T}\Pi(\partial\cdot u) \tag{11}\] \[+\frac{1}{T}\phi^{\mu\nu}(\partial_{\mu}u_{\nu}+2\omega_{\mu\nu} )+\frac{q^{\mu}}{T}\left(T\partial_{\mu}\frac{1}{T}-Du_{\mu}+4\omega_{\mu\nu} u^{\nu}\right)+O(\partial^{3}),\]
where \(\mathcal{S}^{\mu}_{\rm can}\) is the entropy density current. The second law of thermodynamics \(\partial_{\mu}\mathcal{S}^{\mu}_{\rm can}\geq 0\) can give us the first order constitutive relations [45; 48],
\[h^{\mu}-\frac{e+p}{n}\nu^{\mu} = \kappa\Delta^{\mu\nu}\left[\frac{1}{T}\partial_{\nu}T-(u\cdot \partial)u_{\nu}\right], \tag{12}\] \[\pi^{\mu\nu} = 2\eta\partial^{<\mu}u^{\nu>},\] (13) \[\Pi = -\zeta\partial_{\mu}u^{\mu},\] (14) \[q^{\mu} = \lambda\Delta^{\mu\nu}\left[\frac{1}{T}\partial_{\nu}T+(u\cdot \partial)u_{\nu}-4\omega_{\nu\alpha}u^{\alpha}\right],\] (15) \[\phi^{\mu\nu} = 2\gamma_{s}\Delta^{\mu\rho}\Delta^{\nu\sigma}(\partial_{[\rho}u_ {\sigma]}+2\omega_{\rho\sigma}), \tag{16}\]
where the heat conductivity coefficient \(\kappa\), shear viscosity coefficient \(\eta\), and bulk viscosity \(\zeta\) also exist in conventional hydrodynamics, while \(\lambda\) and \(\gamma_{s}\) are new coefficients corresponding to the interchange of spin and orbital angular momentum. The entropy principles also requires that the transport coefficients
\[\kappa,\eta,\zeta,\lambda,\gamma_{s}>0, \tag{17}\]
are positive. Note that, pointed out by Refs. [67; 162], some cross terms between the different dissipative currents may also exist due to the Onsager relation, but here we neglect them for simplicity.
Before ending this section, we would like to comment on the heat flow \(h^{\mu}\). Interestingly, when we set \(\nu^{\mu}=0\) and \(n=0\), we find that one cannot fix the expression for heat current \(h^{\mu}\) in the first order of gradient expansion. By using \(\Delta_{\nu\alpha}\partial_{\mu}\Theta^{\mu\nu}=0\) and Eqs.(4,5), we find that \((\partial_{\mu}\frac{1}{T}+\frac{1}{T}Du_{\mu})\sim O(\partial^{2})\) when \(\nu^{\mu}=0\) and \(n=0\). In that case, the term \(h^{\mu}(\partial_{\mu}\frac{1}{T}+\frac{1}{T}Du_{\mu})\sim O(\partial^{3})\), will be neglected in the entropy production rate (11), i.e., we cannot determine the
expression of \(h^{\mu}\) by the entropy principle there. The similar behavior was also observed in conventional hydrodynamics [163; 164].
## III Unstable and Acausal Modes in the First Order Spin Hydrodynamics
In this section, we analyze the causality and stability for the first order spin hydrodynamics. It is well-known that the conventional relativistic hydrodynamics in Landau frame up to the first order in gradient expansion are always acausal, e.g., see Refs. [154; 155] as the early pioneer works.
In the linear modes analysis, one can consider the perturbations \(\delta X\) to the hydrodynamical quantities \(X\) at the equilibrium. By assuming the \(\delta X\sim\delta\tilde{X}e^{i\omega t-ikx}\) with \(\delta\tilde{X}\) being constant in space-time, one can solve the dispersion relation \(\omega=\omega(k)\) from the conservation equations. In the conventional hydrodynamics, the causality condition is usually given by [165; 166; 167; 157; 163; 168; 169; 170; 164; 165; 166; 167]
\[\lim_{k\rightarrow\infty}\left|\text{Re}\ \frac{\omega}{k}\right|\leq 1, \tag{18}\]
where the condition (18) can also be written as \(\lim_{k\rightarrow\infty}\left|\text{Re}\ \frac{\partial\omega}{\partial k}\right|\leq 1\) in some literature [166; 167; 157]. However, the above condition is insufficient to guarantee the causality. We need an extra condition that, [168]
\[\lim_{k\rightarrow\infty}\left|\frac{\omega}{k}\right|\ \text{is bounded}. \tag{19}\]
As pointed out by the early pioneer work [168], the unbounded \(\lim_{k\rightarrow\infty}\left|\frac{\omega}{k}\right|\) gives the infinite propagating speed of the perturbation, even if the \(\omega\) is pure imaginary. One simple example is the non-relativistic diffusion equation, \(\partial_{t}n-D_{n}\partial_{x}^{2}n=0\) with \(D_{n}\) being the diffusion constant. It is easy to check that its dispersion relation gives \(\omega=iD_{n}k^{2}\), which satisfies condition (18) but does not obey condition (19). Therefore, the perturbation in the non-relativistic diffusion equations has the unlimited propagating speed, i.e. with any compact initial value for \(n(t_{0},x)\), the \(n(t_{0}+\Delta t,x)\) at \(x\rightarrow\infty\) can still get the influence [169]. We emphasize that the conditions (18,19) are necessary but not sufficient to guarantee that the theory is casual [170; 171]. One example is the transverse perturbations of an Eckart fluid with shear viscous tensor, whose dispersion relation satisfies the conditions (18,19),
but the velocity can exceed the speed of light (see Eq. (47) and (48) in Ref. [155] for the perturbation equations and the propagating velocity).
The stability means that the the imaginary part of \(\omega=\omega(k)\) must be positive, i.e.
\[\text{Im }\omega(k)>0. \tag{20}\]
Note that the case of \(\text{Im }\omega=0\) corresponds to the neutral equilibrium, which means the equilibrium state is not unique. In this work, we will not consider such special cases, and we only consider the condition (20) to study the stability of spin hydrodynamics as in Ref. [70].
It is necessary to study the causality and stability for the relativistic spin hydrodynamics in the first order. To see whether the first order spin hydrodynamics can be casual or not, we consider the linear modes analysis to the system, i.e. we take the small perturbations on top of static equilibrium. Following Refs. [154; 155], the static equilibrium background is assumed to be irrotational global equilibrium state. We label the quantities with subscript \((0)\) as those at the global equilibrium state, while we use "\(\delta X\)" to denote the small perturbations of the quantity \(X\), e.g., \(e_{(0)}\) and \(\delta e\) stand for the energy density at the global equilibrium and the small perturbations of energy density, respectively.
From now on, unless specified otherwise, we adopt the Landau frame, and neglect the conserved charge current \(j^{\mu}\).
We now consider the small perturbations on top of static equilibrium. Not all of the perturbations are independent of each other, and we can choose
\[\delta e,\ \delta u^{i},\ \delta S^{\mu\nu}, \tag{21}\]
as independent variables.
The variation of pressure \(\delta p\) and spin chemical potential \(\delta\omega^{\mu\nu}\) can be expressed as functions of \(\delta e\) and \(\delta S^{\mu\nu}\) through
\[\delta p=c_{s}^{2}\delta e,\quad\delta\omega^{0i}=\chi_{b}\delta S^{0i}+\chi_{ e}^{0i}\delta e,\quad\delta\omega^{ij}=\chi_{s}\delta S^{ij}+\chi_{e}^{ij} \delta e, \tag{22}\]
where the speed of sound \(c_{s}\), and \(\chi_{b}\), \(\chi_{s}\),\(\chi_{e}^{\mu\nu}\) are in general the functions of thermodynamic variables. For simplicity, we take \(c_{s}\), \(\chi_{b}\), \(\chi_{s}\),\(\chi_{e}^{\mu\nu}\) as constants in the linear modes analysis. Note that \(\chi_{e}^{\mu\nu}\) comes from the anisotropy of the system. Under the assumption of an irrotational global equilibrium, from Eq. (15) the spin chemical potential vanishes \(\omega_{(0)}^{\mu\nu}=0\). For
simplicity, we further choose \(S^{\mu\nu}_{(0)}=0\). The variation of the temperature \(\delta T\) can be obtained by the thermodynamics relations, with the help of Eqs.(4,5),
\[\delta T=\frac{T_{(0)}}{e_{(0)}+p_{(0)}}\left[\delta p-T_{(0)}S^{\mu\nu}_{(0)} \delta\left(\frac{\omega_{\mu\nu}}{T}\right)\right]=\frac{T_{(0)}c_{s}^{2} \delta e}{e_{(0)}+p_{(0)}}. \tag{23}\]
Next, we consider the variation of the conservation equations \(\partial_{\mu}\delta\Theta^{\mu\nu}=0\) and \(\partial_{\lambda}\delta J^{\lambda\mu\nu}=0\), where the perturbations \(\delta\Theta^{\mu\nu}\) and \(\delta J^{\lambda\mu\nu}\) can be derived from the constitutive relations in Eqs.(2,7,12-16). It is straightforward to obtain the linearized equations for the independent perturbations \(\delta e,\delta\vartheta^{i},\delta S^{\mu\nu}\),
\[0 = (\partial_{0}+\frac{1}{2}\lambda^{\prime}c_{s}^{2}\partial_{i} \partial^{i}+4\lambda\chi_{e}^{0i}\partial_{i})\delta e+(\partial_{i}+\frac{ 1}{2}\lambda^{\prime}\partial_{i}\partial_{0})\delta\vartheta^{i}+D_{b} \partial_{i}\delta S^{0i}, \tag{24}\] \[0 = (4\gamma_{s}\chi_{e}^{ij}\partial_{i}-c_{s}^{2}\partial^{j}- \frac{1}{2}c_{s}^{2}\lambda^{\prime}\partial_{0}\partial^{j}-4\lambda\chi_{e}^ {0j}\partial_{0})\delta e+(\gamma_{\parallel}-\gamma_{\perp}-\gamma^{\prime}) \partial^{j}\partial_{i}\delta\vartheta^{i}\] (25) \[+[\partial_{0}-\frac{1}{2}\lambda^{\prime}\partial_{0}\partial_{0 }+(\gamma_{\perp}+\gamma^{\prime})\partial^{i}\partial_{i}]\delta\vartheta^{j }-D_{b}\partial_{0}\delta S^{0j}+D_{s}\partial_{i}\delta S^{ij},\] \[0 = (\lambda^{\prime}c_{s}^{2}\partial^{i}+8\lambda\chi_{e}^{0i}) \delta e+\lambda^{\prime}\partial_{0}\delta\vartheta^{i}+(2D_{b}-\partial_{0 })\delta S^{0i},\] (26) \[0 = 8\gamma_{s}\chi_{e}^{ij}\delta e+2\gamma^{\prime}\partial^{i} \delta\vartheta^{j}-2\gamma^{\prime}\partial^{j}\delta\vartheta^{i}+(2D_{s}+ \partial_{0})\delta S^{ij}. \tag{27}\]
Here we introduce the following shorthand notations,
\[D_{s} \equiv 4\gamma_{s}\chi_{s},\quad D_{b}\equiv 4\lambda\chi_{b},\quad \delta\vartheta^{i}\equiv(e_{(0)}+p_{(0)})\delta u^{i},\quad\lambda^{\prime} \equiv\frac{2\lambda}{e_{(0)}+p_{(0)}},\] \[\gamma^{\prime} \equiv \frac{\gamma_{s}}{e_{(0)}+p_{(0)}},\quad\gamma_{\perp}\equiv \frac{\eta}{e_{(0)}+p_{(0)}},\quad\gamma_{\parallel}\equiv\frac{\frac{4}{3} \eta+\zeta}{e_{(0)}+p_{(0)}}. \tag{28}\]
In the linear modes analysis, the perturbations are assumed along the \(x\) direction only,
\[\delta e=\delta\tilde{e}e^{i\omega t-ikx},\ \delta\vartheta^{i}=\delta \tilde{\vartheta}^{i}e^{i\omega t-ikx},\ \delta S^{\mu\nu}=\delta\tilde{S}^{\mu\nu}e^{i\omega t-ikx}, \tag{29}\]
where \(\delta\tilde{e}\), \(\delta\tilde{\vartheta}^{i}\), and \(\delta\tilde{S}^{\mu\nu}\) are independent of space and time.
Inserting the perturbations in Eq. (29) into Eqs.(24-27) yields,
\[{\cal M}_{1}\delta\tilde{X}_{1}\:=\:0, \tag{30}\]
where
\[\delta\tilde{X}_{1}\equiv(\delta\tilde{e},\delta\tilde{\vartheta}^{x},\delta \tilde{S}^{0x},\delta\tilde{\vartheta}^{y},\delta\tilde{S}^{0y},\delta\tilde{ S}^{xy},\delta\tilde{\vartheta}^{z},\delta\tilde{S}^{0z},\delta\tilde{S}^{xz}, \delta\tilde{S}^{yz})^{\rm T}, \tag{31}\]
and
\[{\cal M}_{1}\equiv\left(\begin{array}{cccc}M_{1}&0&0&0\\ A_{1}&M_{2}&0&0\\ A_{2}&0&M_{2}&0\\ A_{3}&0&0&M_{3}\end{array}\right), \tag{32}\]
with
\[M_{1} \equiv \left(\begin{array}{ccc}i\omega+\frac{1}{2}\lambda^{\prime}c_{s}^ {2}k^{2}-4ik\lambda\chi_{e}^{0x}&\frac{1}{2}\lambda^{\prime}k\omega-ik&-ikD_{b} \\ \frac{1}{2}\lambda^{\prime}c_{s}^{2}k\omega-ikc_{s}^{2}-4i\omega\lambda\chi_{e }^{0x}&\gamma_{\parallel}k^{2}+i\omega+\frac{1}{2}\lambda^{\prime}\omega^{2}&- i\omega D_{b}\\ ik\lambda^{\prime}c_{s}^{2}+8\lambda\chi_{e}^{0x}&i\omega\lambda^{\prime}&2D_{b}-i \omega\end{array}\right), \tag{33}\] \[M_{2} \equiv \left(\begin{array}{ccc}k^{2}(\gamma_{\perp}+\gamma^{\prime})+ i\omega+\frac{1}{2}\lambda^{\prime}\omega^{2}&-i\omega D_{b}&-ikD_{s}\\ i\omega\lambda^{\prime}&2D_{b}-i\omega&0\\ 2ik\gamma^{\prime}&0&2D_{s}+i\omega\end{array}\right),\] (34) \[M_{3} \equiv 2D_{s}+i\omega. \tag{35}\]
The off-diagonal blocks \(A_{1}\), \(A_{2}\), \(A_{3}\) in the matrix \(\mathcal{M}_{1}\), whose expressions are shown in Appendix A, and are irrelevant to the following discussions. The non-trivial solutions in Eq.(30) requires,
\[0=\det\mathcal{M}_{1}=\det M_{1}\cdot(\det M_{2})^{2}\cdot\det M_{3}. \tag{36}\]
From Eqs.(33-35), we find that Eq.(36) is a polynomial equation for two variables \(\omega\) and \(k\). Solving this equation gives the dispersion relations \(\omega=\omega(k)\).
The \(\det M_{3}=0\) gives a non-hydrodynamic mode,
\[\omega = 2iD_{s}, \tag{37}\]
which corresponds to the spin relaxation [45; 59]. The stability condition (20) requires that \(D_{s}>0\).
The dispersion relation solved from \(\det M_{1}=0\) and \(\det M_{2}=0\) are lengthy and complicated, so here we only discuss the relations in small \(k\) and large \(k\) limits to analyze stability and causality. In the \(k\to 0\) limit, the dispersion relations are
\[\omega = \pm c_{s}k+\frac{i}{2}(\gamma_{\parallel}\mp 4c_{s}\lambda\chi_{e }^{0x}D_{b}^{-1})k^{2}+O(k^{3}), \tag{38}\] \[\omega = (-i\pm\sqrt{4D_{b}\lambda^{\prime}-1})\lambda^{\prime-1}+O(k),\] (39) \[\omega = i\gamma_{\perp}k^{2}+O(k^{3}),\] (40) \[\omega = 2iD_{s}+O(k^{2}). \tag{41}\]
where the dispersion relations (38-39) and (39-41) are solved from \(\det M_{1}=0\) and \(\det M_{2}=0\), respectively. The modes in Eq. (38) and Eq. (40) correspond the sound and shear modes in the conventional hydrodynamics [154; 156; 157; 166], respectively. The stability condition
(20) for the dispersion relation in Eqs.(38-41) gives,
\[D_{s}>0,\ \lambda^{\prime}<0,\ D_{b}<-4c_{s}\lambda\gamma_{\parallel}^{-1}| \chi_{e}^{0x}|\leq 0. \tag{42}\]
However, conditions (42) contradict with the entropy principle in Eq. (17), i.e. \(\lambda^{\prime}=2\lambda/(e_{(0)}+p_{(0)})>0\) defined in Eq.(28) with \(\lambda>0\) and \(e_{(0)}+p_{(0)}>0\).
In the \(k\rightarrow\infty\) limit, the dispersion relations become,
\[\omega = -4iD_{b}\gamma_{\parallel}^{-1}\lambda^{\prime-1}k^{-2}+O(k^{-3}), \tag{43}\] \[\omega = -ic_{s}^{2/3}\gamma_{\parallel}^{1/3}k^{4/3}+O(k),\] (44) \[\omega = (-1)^{1/6}c_{s}^{2/3}\gamma_{\parallel}^{1/3}k^{4/3}+O(k),\] (45) \[\omega = (-1)^{5/6}c_{s}^{2/3}\gamma_{\parallel}^{1/3}k^{4/3}+O(k),\] (46) \[\omega = -2iD_{b}+O(k^{-1}),\] (47) \[\omega = 2iD_{s}\gamma_{\perp}(\gamma^{\prime}+\gamma_{\perp})^{-1}+O(k^ {-1}),\] (48) \[\omega = \pm ik\sqrt{2\lambda^{\prime-1}(\gamma^{\prime}+\gamma_{\perp}) }+O(k^{0}), \tag{49}\]
where first four modes come from \(\det M_{1}=0\), and others can be derived by \(\det M_{2}=0\). Obviously, Eq.(49) contains an unstable mode.
On the other hand, we also find that in Eqs.(44-46) \(|\omega/k|\) is unbounded, which violates the causality condition (19). We also notice that Ref. [71] has also analyzed the causality for the first order spin hydrodynamics in small \(k\) limit.
We find that the first order spin hydrodynamics is acausal and unstable similar as the conventional relativistic hydrodynamics in Landau frame.
Before ending this section, we comment on the condition (42). We notice that the dispersion relations in Refs. [70; 71; 45] are different with ours in Eqs.(37-49). Let us explain what happens here. The energy momentum conservation equation \(\Delta_{\mu\alpha}\partial_{\nu}\Theta^{\mu\nu}=0\), gives the acceleration equations for the fluid velocity,
\[(u\cdot\partial)u^{\mu}=\frac{1}{T}\Delta^{\mu\nu}\partial_{\nu}T+O(\partial^ {2}). \tag{50}\]
In Refs.[70; 71; 45], the authors have replaced \((u\cdot\partial)u^{\mu}\) in \(q^{\mu}\) in Eq.(15) by Eq. (50) and gotten another expression for \(q^{\mu}\)
\[q^{\mu}=\lambda\left(\frac{2\Delta^{\mu\nu}\partial_{\nu}p}{e+p}-4\omega^{\mu \nu}u_{\nu}\right)+O(\partial^{2}). \tag{51}\]
Although \(q^{\mu}\) in Eq.(51) (also in Refs. [70; 71; 45] ) is equivalent to our \(q^{\mu}\) in Eq.(15) up to the first order in gradient expansion, we emphasize that these two \(q^{\mu}\) correspond to different hydrodynamic frames and will lead to different hydrodynamic equations (also see Refs. [163; 172] for the general discussion for these kinds of replacement in relativistic hydrodynamics). Different with our Eqs. (43 - 49), the dispersion relations computed with the \(q^{\mu}\) in Eq. (51) are stable and satisfy causality condition (18) in the rest frame under certain conditions. However, they do not obey the causality condition (19) and the whole theory become acausal, e.g., one mode in Refs. [70; 71; 45] is
\[\omega=i(\gamma^{\prime}+\gamma_{\perp})k^{2}\text{ as }k\rightarrow\infty, \tag{52}\]
breaks the causality condition (19).
We now conclude that the first order spin hydrodynamics at static equilibrium state are unstable and acausal in the rest frame. We do not need to discuss the stability and causality of the first order spin hydrodynamics in moving frames again.
## IV Minimal causal spin hydrodynamics
In the previous section, we have shown that the first order spin hydrodynamics in Landau frame are acausal and unstable. The acausal and unstable theory is not physical, we therefore need to consider the second order the spin hydrodynamics in gradient expansion. In this section we follow the idea of minimal causal extension in conventional hydrodynamics and implement it to the spin hydrodynamics.
Up to now, there are two ways to establish causal hydrodynamics. The first way is to add the second order corrections to the dissipative terms, such as the Muller-Israel-Stewart (MIS) theory [158; 159] or other related second order hydrodynamics. The MIS theory is a famous causal conventional hydrodynamic theory up to \(O(\partial^{2})\) in gradient expansion. Here, we consider a relativistic dissipative hydrodynamics with the bulk viscous pressure \(\Pi\) only as an example to explain why the MIS theory can be casual. The entropy current in MIS theory is assumed to be [173; 159; 174]
\[\mathcal{S}^{\mu}=su^{\mu}-\frac{\mu}{T}\nu^{\mu}+\frac{1}{T}h^{\mu}-\frac{1} {2T}\beta_{0}u^{\mu}\Pi^{2}+..., \tag{53}\]
where the coefficient \(\beta_{0}>0\) and the ellipsis stands for other possible \(O(\partial^{2})\) terms. Then
the second law of thermodynamics \(\partial_{\mu}\mathcal{S}^{\mu}\geq 0\) leads to,
\[\tau_{\Pi}\frac{d}{d\tau}\Pi+\Pi=-\zeta\partial_{\mu}u^{\mu}+..., \tag{54}\]
where \(d/d\tau\equiv u^{\mu}\partial_{\mu}\), and \(\tau_{\Pi}=\zeta\beta_{0}>0\) is defined as the relaxation time for the bulk viscous pressure. If the \(\tau_{\Pi}\to 0\), the hydrodynamic equations reduce to parabolic equations and become acausal. With a finite \(\tau_{\Pi}\), the hydrodynamic equations are hyperbolic and can be causal under certain conditions [156; 157; 156; 175; 176]. In linear modes analysis, the dispersion relations from Eq. (54) satisfy causal conditions (18, 19) when the relaxation time \(\tau_{\Pi}\) is sufficiently large. The second order constitutive equations for shear viscous tensor \(\pi^{\mu\nu}\), heat flow \(h^{\mu}\) and heat current \(\nu^{\mu}\) can be obtained in a similar way. These equations represent evolution equations that incorporate the respective relaxation time [159; 173; 174]. Apart from the MIS theory, many other second order causal conventional hydrodynamic theories, e.g., Baier-Romatschke-Son-Starinets-Stephanov (BRSSS) theory [177] and the Denicol-Niemi-Molnar-Rischke (DNMR) theory [178], have been established. All of them contain the terms proportional to the relaxation times and can be causal and stable under certain conditions [177; 179; 180]. Following these discussion, we can say that the key to recover the causality of the theory is to introduce the terms proportional to relaxation time.
Different with the above second order theories, the Bemfica-Disconzi-Noronha-Kovtun (BDNK) [163; 164; 165; 181; 182; 183] is a first order hydrodynamic theory in general (fluid) frames. It roughly says that one can choose some preferred frames to satisfy the causality and stability conditions. Unfortunately, the commonly used Landau or Eckart frame are not the preferred fluid frames in the BDNK theory. Therefore, we will not discuss the spin hydrodynamics in the BDNK theory in this work. We also notice that recent studies in Ref. [184] discuss the casual spin hydrodynamics in the first order similar to BDNK theory.
In this work, we follow the basic idea in MIS, BRSSS, and DNMR theories to construct a simplified causal spin hydrodynamics. Instead of considering the complete second order spin hydrodynamics, we only analyze the called "minimal" extended second order spin hydrodynamics. Here, the word "minimal" means that we concentrate on the essential terms in the second order of gradient expansion to get a causal theory and neglect the other terms which do not contribute to the dispersion relations in the linear modes analysis. As mentioned below Eq. (54), the key to get the causal theory is to add the terms proportional to the relaxation times similar to \(\tau_{\Pi}d\Pi/d\tau\), in the left hand side of Eq.(54). Following this idea,
the constitutive equations (12-16) in the minimal extended causal spin hydrodynamics can be rewritten as,
\[\tau_{q}\Delta^{\mu\nu}\frac{d}{d\tau}q_{\nu}+q^{\mu} = \lambda(T^{-1}\Delta^{\mu\alpha}\partial_{\alpha}T+Du^{\mu}-4\omega ^{\mu\nu}u_{\nu}), \tag{55}\] \[\tau_{\phi}\Delta^{\mu\alpha}\Delta^{\nu\beta}\frac{d}{d\tau} \phi_{\alpha\beta}+\phi^{\mu\nu} = 2\gamma_{s}\Delta^{\mu\alpha}\Delta^{\nu\beta}(\partial_{[\alpha }u_{\beta]}+2\omega_{\alpha\beta}),\] (56) \[\tau_{\pi}\Delta^{\alpha<\mu}\Delta^{\nu>\beta}\frac{d}{d\tau} \pi_{\alpha\beta}+\pi^{\mu\nu} = 2\eta\partial^{<\mu}u^{\nu>},\] (57) \[\tau_{\Pi}\frac{d}{d\tau}\Pi+\Pi = -\zeta\partial_{\mu}u^{\mu}, \tag{58}\]
where \(\tau_{q},\tau_{\phi},\tau_{\pi}\) and \(\tau_{\Pi}\) are positive relaxation times for \(q^{\mu},\phi^{\mu\nu},\pi^{\mu\nu},\Pi\), respectively. Eqs. (57,58) are the same as those in the conventional hydrodynamics1[156; 157; 156]. Recently, the second order spin hydrodynamics similar to MIS theory has been introduced in Ref. [73]. Our minimal causal spin hydrodynamics can be regarded as a simplified version of it.
Footnote 1: Another kinds of the minimal causal theory is discussed in Ref.[156; 160], in which the extended dissipative terms can not be determined from the entropy principle \(\partial_{\mu}\mathcal{S}^{\mu}\geq 0\).
We also notice that in the Refs. [60], the authors have proposed the same expressions for \(q^{\mu}\) and \(\phi^{\mu\nu}\) as presented in Eqs. (55, 56) for minimal causal spin hydrodynamics.
At last, we comment on the relaxation time \(\tau_{q}\) and \(\tau_{\phi}\). Different with total particle number or total energy of the fluid, the total spin or polarization is not a conserved quantity, i.e., the spin density should and will decay with time. Therefore, the two modes described by Eqs. (39, 41) or Eqs. (47, 48) can also be interpreted as the relaxation modes for the spin density. Similarly, we can interpret that \(\tau_{q}\) and \(\tau_{\phi}\) as the relaxation times for the sources that induce spin generation.
## V Causality and stability analysis for minimal causal spin hydrodynamics
In this section we analyze the causality and stability of the minimal causal spin hydrodynamics. We use the similar notations in Sec. III, i.e., for a physical quantity \(X\), we use \(X_{(0)}\) and \(\delta X\) to denote the \(X\) at the global equilibrium state and the small perturbations of the quantity \(X\), respectively. We adopt the independent perturbations as
\[\delta e,\ \delta u^{i},\ \delta S^{\mu\nu},\ \delta\Pi,\ \delta\pi^{ij}, \tag{59}\]
where \(\delta\pi^{i}_{\ i}=0\) and \(\delta\pi^{ij}=\delta\pi^{ji}\).
We first start from the spin hydrodynamics in the rest frame, i.e., \(u^{\mu}_{(0)}=(1,0)\). The conservation equations \(\partial_{\mu}\delta\Theta^{\mu\nu}=0\) and \(\partial_{\lambda}\delta J^{\lambda\mu\nu}=0\) with the constitutive equations (55 - 58) read,
\[0 = (\lambda^{\prime}c_{s}^{2}\partial^{i}+8\lambda\chi_{e}^{0i}) \delta e+\lambda^{\prime}\partial_{0}\delta\vartheta^{i}+(2D_{b}-\tau_{q} \partial_{0}\partial_{0}-\partial_{0})\delta S^{0i}, \tag{60}\] \[0 = 8\gamma_{s}\chi_{e}^{ij}\delta e+2\gamma^{\prime}(\partial^{i} \delta\vartheta^{j}-\partial^{j}\delta\vartheta^{i})+(\tau_{\phi}\partial_{0} \partial_{0}+\partial_{0}+2D_{s})\delta S^{ij},\] (61) \[0 = \tau_{\pi}\partial_{0}\delta\pi^{ij}+\delta\pi^{ij}-\gamma_{ \perp}(\partial^{i}\delta\vartheta^{j}+\partial^{j}\delta\vartheta^{i}-\frac {2}{3}g^{ij}\partial_{k}\delta\vartheta^{k}),\] (62) \[0 = \tau_{\Pi}\partial_{0}\delta\Pi+\delta\Pi+(\gamma_{\parallel}- \frac{4}{3}\gamma_{\perp})\partial_{i}\delta\vartheta^{i},\] (63) \[0 = \partial_{0}\delta e+\partial_{i}\delta\vartheta^{i}+\frac{1}{2} \partial_{0}\partial_{i}\delta S^{0i},\] (64) \[0 = -c_{s}^{2}\partial^{j}\delta e+\partial_{0}\delta\vartheta^{j}- \partial^{j}\delta\Pi+\partial_{i}\delta\pi^{ij}-\frac{1}{2}\partial_{0} \partial_{0}\delta S^{0j}-\frac{1}{2}\partial_{0}\partial_{i}\delta S^{ij}, \tag{65}\]
where \(\chi_{b},\chi_{e}^{\mu\nu},\chi_{s},D_{s},D_{b},\delta\vartheta^{i},\lambda^{ \prime},\gamma^{\prime},\gamma_{\perp},\gamma_{\parallel}\) are defined in Eqs.(22,28) and we have used the spin evolution equation (3) to replace \(\delta q^{i}\) and \(\delta\phi^{ij}\) by \(\delta S^{\mu\nu}\),
\[\delta q^{i}=\frac{1}{2}\partial_{0}\delta S^{0i},\quad\delta\phi^{ij}=-\frac{ 1}{2}\partial_{0}\delta S^{ij}. \tag{66}\]
### Zero modes in rest frame
Following the conventional hydrodynamics, we consider a fluid with the dissipative terms \(q^{\mu}\) and \(\phi^{\mu\nu}\) only for simplicity, i.e., we remove Eqs. (62, 63) and take \(\delta\Pi=0\) and \(\delta\pi^{ij}=0\) in Eqs. (60, 61, 64, 65). We consider the solutions Eq.(29) and derive
\[\mathcal{M}^{\prime}_{2}\delta\tilde{X}^{\prime}_{2} = 0, \tag{67}\]
where \(\delta\tilde{X}^{\prime}_{2}\) and \(\mathcal{M}^{\prime}_{2}\) are given by
\[\delta\tilde{X}^{\prime}_{2} \equiv (\delta\tilde{e},\delta\tilde{\vartheta}^{x},\delta\tilde{S}^{0x },\delta\tilde{\vartheta}^{y},\delta\tilde{S}^{0y},\delta\tilde{S}^{xy}, \delta\tilde{\vartheta}^{z},\delta\tilde{S}^{0z},\delta\tilde{S}^{xz},\delta \tilde{S}^{yz})^{\rm T}, \tag{68}\]
and
\[\mathcal{M}^{\prime}_{2} \equiv \left(\begin{array}{cccc}M^{\prime}_{4}&0&0&0\\ A^{\prime}_{4}&M^{\prime}_{5}&0&0\\ A^{\prime}_{5}&0&M^{\prime}_{5}&0\\ A^{\prime}_{6}&0&0&M^{\prime}_{6}\end{array}\right), \tag{69}\]
with
\[M_{4}^{\prime} = \left(\begin{array}{ccc}i\omega&-ik&\frac{1}{2}\omega k\\ -ikc_{s}^{2}&i\omega&\frac{1}{2}\omega^{2}\\ \lambda^{\prime}c_{s}^{2}ik+8\lambda\chi_{e}^{0x}&\lambda^{\prime}i\omega&2D_{b }+\tau_{q}\omega^{2}-i\omega\end{array}\right), \tag{70}\] \[M_{5}^{\prime} = \left(\begin{array}{ccc}i\omega&\frac{1}{2}\omega^{2}&-\frac{1} {2}\omega k\\ \lambda^{\prime}i\omega&2D_{b}+\tau_{q}\omega^{2}-i\omega&0\\ 2\gamma^{\prime}ik&0&-\tau_{\phi}\omega^{2}+i\omega+2D_{s}\end{array}\right),\] (71) \[M_{6}^{\prime} = -\tau_{\phi}\omega^{2}+i\omega+2D_{s}. \tag{72}\]
The off-diagonal matrices \(A_{4,5,6}^{\prime}\) are put in Appendix A.
The dispersion relations \(\omega=\omega(k)\) are derived from
\[\det{\cal M}_{2}^{\prime}=\det M_{4}^{\prime}\cdot(\det M_{5}^{ \prime})^{2}\cdot\det M_{6}^{\prime}=0. \tag{73}\]
We find that there exists a zero mode coming from the equation \(\det M_{5}^{\prime}=0\). We will discuss the zero modes at the end of this subsection. Now, let us focus on the nonzero modes. The \(\det M_{6}^{\prime}=0\) gives two non-hydrodynamic modes
\[\omega=\frac{1}{2\tau_{\phi}}(i\pm\sqrt{8D_{s}\tau_{\phi}-1}). \tag{74}\]
From \(\det M_{4}^{\prime}=0\) and \(\det M_{5}^{\prime}=0\), we obtain the dispersion relation in small \(k\) limit,
\[\omega = \pm c_{s}k\mp 2ic_{s}\lambda\chi_{e}^{0x}D_{b}^{-1}k^{2}+O(k^{3}), \tag{75}\] \[\omega = \left[i\pm\sqrt{-4D_{b}(2\tau_{q}-\lambda^{\prime})-1}\right](2 \tau_{q}-\lambda^{\prime})^{-1}+O(k),\] (76) \[\omega = \frac{1}{2\tau_{\phi}}(i\pm\sqrt{8D_{s}\tau_{\phi}-1})+O(k), \tag{77}\]
and, in large \(k\) limit,
\[\omega = \pm k\sqrt{\frac{c_{s}^{2}(3\lambda^{\prime}+2\tau_{q})}{2\tau_{ q}-\lambda^{\prime}}}+\frac{4i\lambda^{\prime}}{(2\tau_{q}-\lambda^{\prime})(2 \tau_{q}+3\lambda^{\prime})}\mp\frac{8\lambda\chi_{e}^{0x}}{c_{s}\sqrt{\left( \lambda^{\prime}-2\tau_{q}\right)(3\lambda^{\prime}+2\tau_{q})}}+O(k^{-1}), \tag{78}\] \[\omega = \frac{i\pm\sqrt{-1-4D_{b}(2\tau_{q}+3\lambda^{\prime})}}{2\tau_{ q}+3\lambda^{\prime}}+O(k^{-1}),\] (79) \[\omega = \pm\sqrt{\frac{2\gamma^{\prime}\tau_{q}}{(2\tau_{q}-\lambda^{ \prime})\tau_{\phi}}}k+i\frac{[\tau_{q}(2\tau_{q}-\lambda^{\prime})+\lambda^{ \prime}\tau_{\phi}]}{2\tau_{q}\tau_{\phi}(2\tau_{q}-\lambda^{\prime})}+O(k^{-1}),\] (80) \[\omega = \frac{i\pm\sqrt{-1-8D_{b}\tau_{q}}}{2\tau_{q}}+O(k^{-1}). \tag{81}\]
The causality conditions (18, 19) requires,
\[0\leq\frac{c_{s}^{2}(3\lambda^{\prime}+2\tau_{q})}{2\tau_{q}-\lambda^{\prime}} \leq 1,\ 0\leq\frac{2\gamma^{\prime}\tau_{q}}{(2\tau_{q}-\lambda^{\prime})\tau_{ \phi}}\leq 1, \tag{82}\]
which implies that the relaxation times \(\tau_{q},\tau_{\phi}\) cannot be arbitrarily small. It is consistent with the discussion in Sec. IV.
The stability condition (20) leads to
\[\tau_{q}>\lambda^{\prime}/2,\ D_{s}>0,\ D_{b}<0,\ \chi_{e}^{0x}=0, \tag{83}\]
where \(\chi_{e}^{0x}=0\) comes from the stability of the sound mode (75). Although the conditions in Eq.(83) are derived from the small \(k\) and large \(k\) limits only, we can implement the Routh-Hurwitz criterion [163; 164; 165; 182; 186; 187] to prove that the conditions (83) are sufficient and necessary for stability, i.e., if (83) are satisfied, then \(\text{Im}\ \omega>0\) for all \(k\). Details for the proof can be found in Appendix B.
At last, let us comment on the zero modes. The zero modes, which gives \(\omega=0\), coming from Eq. (65) with vanishing \(\delta\Pi,\delta\pi^{ij}\). Generally, the zero modes in the linear modes analysis do not mean the perturbations are not decaying with time. It indicates that there exists the nonlinear modes in Eq. (65) with vanishing \(\delta\Pi,\delta\pi^{ij}\). To continue our analysis, we need to set non-vanishing \(\delta\Pi,\delta\pi^{ij}\).
### Causality analysis in the rest frame
Next, we substitute the plane wave solutions Eq.(29) and
\[\delta\Pi=\delta\tilde{\Pi}e^{i\omega t-ikx},\ \delta\pi^{ij}=\delta\tilde{\pi} ^{ij}e^{i\omega t-ikx}, \tag{84}\]
with \(\delta\tilde{\Pi},\delta\tilde{\pi}^{ij}\), being constants, into Eqs.(60-65), and obtain the matrix equation
\[\mathcal{M}_{2}\delta\tilde{X}_{2}\:=\:0, \tag{85}\]
where \(\delta\tilde{X}_{2}\) and \(\mathcal{M}_{2}\) are given by
\[\delta\tilde{X}_{2} \equiv (\delta\tilde{e},\delta\tilde{\vartheta}^{x},\delta\tilde{S}^{0x },\delta\tilde{\Pi},\delta\tilde{\pi}^{xx},\delta\tilde{\vartheta}^{y},\delta \tilde{S}^{0y},\delta\tilde{S}^{xy},\delta\tilde{\pi}^{xy} \tag{86}\] \[,\delta\tilde{\vartheta}^{z},\delta\tilde{S}^{0z},\delta\tilde{ S}^{xz},\delta\tilde{\pi}^{xz},\delta\tilde{S}^{yz},\delta\tilde{\pi}^{yy}, \delta\tilde{\pi}^{yz})^{\text{T}}.\]
\[\mathcal{M}_{2} = \left(\begin{array}{cccc}M_{4}&0&0&0\\ A_{4}&M_{5}&0&0\\ A_{5}&0&M_{5}&0\\ A_{6}&0&0&M_{6}\end{array}\right), \tag{87}\]
with
\[M_{4} = \left(\begin{array}{cccc}i\omega&-ik&\frac{1}{2}\omega k&0&0\\ -ikc_{s}^{2}&i\omega&\frac{1}{2}\omega^{2}&-ik&-ik\\ ik\lambda^{\prime}c_{s}^{2}+8\lambda\chi_{e}^{0x}&i\omega\lambda^{\prime}&2D_{ b}+\tau_{q}\omega^{2}-i\omega&0&0\\ 0&-ik(\gamma_{\parallel}-\frac{4}{3}\gamma_{\perp})&0&i\omega\tau_{\Pi}+1&0 \\ 0&-\frac{4}{3}ik\gamma_{\perp}&0&0&i\omega\tau_{\Pi}+1\end{array}\right), \tag{88}\] \[M_{5} = \left(\begin{array}{cccc}2ik\gamma^{\prime}&0&-\tau_{\phi} \omega^{2}+i\omega+2D_{s}&0\\ i\omega&\frac{1}{2}\omega^{2}&-\frac{1}{2}\omega k&-ik\\ i\omega\lambda^{\prime}&2D_{b}+\tau_{q}\omega^{2}-i\omega&0&0\\ -ik\gamma_{\perp}&0&0&i\omega\tau_{\Pi}+1\end{array}\right),\] (89) \[M_{6} = \left(\begin{array}{cccc}-\tau_{\phi}\omega^{2}+i\omega+2D_{s} &0&0\\ 0&i\omega\tau_{\Pi}+1&0\\ 0&0&i\omega\tau_{\Pi}+1\end{array}\right). \tag{90}\]
The submatrices \(A_{4,5,6}\) in Eq.(87) are shown in Appendix A. If there exist nonzero plane wave solutions, we have
\[0=\det\mathcal{M}_{2}=\det M_{4}\cdot(\det M_{5})^{2}\cdot\det M_{6}. \tag{91}\]
We observe the zero modes in Eq. (65) disappear. It indicates that the current analysis is consistent with the assumption of linear response. The dispersion relations \(\omega=\omega(k)\) are the solutions to the polynomial equation (91).
The \(\det M_{6}=0\) gives,
\[\omega = \frac{i}{\tau_{\pi}}, \tag{92}\] \[\omega = \frac{1}{2\tau_{\phi}}(i\pm\sqrt{8D_{s}\tau_{\phi}-1}), \tag{93}\]
which are non-propagating modes or non-hydrodynamic modes.
In \(k\to 0\) limit, the \(\det M_{4}=0\) and \(\det M_{5}=0\) give
\[\omega = \frac{i}{\tau_{\pi}}+O(k), \tag{94}\] \[\omega = \frac{i}{\tau_{\Pi}}+O(k),\] (95) \[\omega = \pm c_{s}k+\frac{i}{2}(\gamma_{\parallel}\mp 4c_{s}\lambda\chi_{e }^{0x}D_{b}^{-1})k^{2}+O(k^{3}),\] (96) \[\omega = \left[i\pm\sqrt{-4D_{b}(2\tau_{q}-\lambda^{\prime})-1}\right](2 \tau_{q}-\lambda^{\prime})^{-1}+O(k),\] (97) \[\omega = i\gamma_{\perp}k^{2}+O(k^{3}),\] (98) \[\omega = \frac{1}{2\tau_{\phi}}(i\pm\sqrt{8D_{s}\tau_{\phi}-1})+O(k), \tag{99}\]
where Eq.(94) and Eq.(97) are doubly degenerate. In large \(k\) limit, we have
\[\omega = -4iD_{b}\gamma_{\parallel}^{-1}\lambda^{\prime-1}k^{-2}+O(k^{-3}), \tag{100}\] \[\omega = \frac{3i\gamma_{\parallel}}{\tau_{\pi}(3\gamma_{\parallel}-4 \gamma_{\perp})+4\gamma_{\perp}\tau_{\Pi}}+O(k^{-1}),\] (101) \[\omega = c_{1}k+i\frac{c_{2}}{c_{3}}+O(k^{-1}),\] (102) \[\omega = \pm\sqrt{\frac{2\tau_{q}(\gamma^{\prime}\tau_{\pi}+\gamma_{\perp }\tau_{\phi})}{(2\tau_{q}-\lambda^{\prime})\tau_{\pi}\tau_{\phi}}}k+ic_{4}+O(k ^{-1}),\] (103) \[\omega = \frac{i\pm\sqrt{-1-8D_{b}\tau_{q}}}{2\tau_{q}}+O(k^{-1}),\] (104) \[\omega = \frac{i(\gamma^{\prime}+\gamma_{\perp})\pm c_{5}}{2(\gamma^{ \prime}\tau_{\pi}+\gamma_{\perp}\tau_{\phi})}+O(k^{-1}), \tag{105}\]
where \(c_{1}\) is
\[c_{1} = \sqrt{\frac{b_{1}^{1/2}\pm(b_{1}-b_{2})^{1/2}}{6(2\tau_{q}- \lambda^{\prime})\tau_{\pi}\tau_{\Pi}}},\mbox{or}\ -\sqrt{\frac{b_{1}^{1/2}\pm(b_{1}-b_{2})^{1/2}}{6(2\tau_{q}- \lambda^{\prime})\tau_{\pi}\tau_{\Pi}}}, \tag{106}\] \[b_{1} \equiv \{8\gamma_{\perp}\tau_{q}\tau_{\Pi}+\tau_{\pi}[2\tau_{q}(3\gamma _{\parallel}-4\gamma_{\perp})+3\tau_{\Pi}c_{s}^{2}(3\lambda^{\prime}+2\tau_{ q})]\}^{2},\] (107) \[b_{2} \equiv 12c_{s}^{2}\lambda^{\prime}(2\tau_{q}-\lambda^{\prime})\tau_{ \pi}\tau_{\Pi}[\tau_{\pi}(3\gamma_{\parallel}-4\gamma_{\perp})+4\gamma_{\perp }\tau_{\Pi}], \tag{108}\]
and
\[c_{2} = -3c_{1}^{4}[2\tau_{\pi}\tau_{\Pi}+(2\tau_{q}-\lambda^{\prime})( \tau_{\pi}+\tau_{\Pi})]+48c_{1}^{3}\lambda\chi_{e}^{0x}\tau_{\pi}\tau_{\Pi}-3c _{s}^{2}\gamma_{\parallel}\lambda^{\prime} \tag{109}\] \[+c_{1}^{2}\{6\gamma_{\parallel}\tau_{q}+(6\gamma_{\parallel}-8 \gamma_{\perp})\tau_{\pi}+8\gamma_{\perp}\tau_{\Pi}+3c_{s}^{2}[2\tau_{\pi} \tau_{\Pi}+(3\lambda^{\prime}+2\tau_{q})(\tau_{\pi}+\tau_{\Pi})]\}\] \[-8c_{1}\lambda\chi_{e}^{0x}[(3\gamma_{\parallel}-4\gamma_{\perp}) \tau_{\pi}+4\gamma_{\perp}\tau_{\Pi}],\] \[c_{3} = -2c_{s}^{2}\lambda^{\prime}[(3\gamma_{\parallel}-4\gamma_{\perp })\tau_{\pi}+4\gamma_{\perp}\tau_{\Pi}]-18c_{1}^{4}(2\tau_{q}-\lambda^{ \prime})\tau_{\pi}\tau_{\Pi}\]
\[+4c_{1}^{2}[3c_{s}^{2}(3\lambda^{\prime}+2\tau_{q})\tau_{\pi}\tau_{ \Pi}+2(3\gamma_{\parallel}-4\gamma_{\perp})\tau_{q}\tau_{\pi}+8\gamma_{\perp} \tau_{q}\tau_{\Pi}], \tag{110}\] \[c_{4} = \frac{\gamma_{\perp}[\tau_{q}(2\tau_{q}-\lambda^{\prime})+\lambda ^{\prime}\tau_{\pi}]\tau_{\phi}^{2}+\gamma^{\prime}\tau_{\pi}^{2}[\tau_{q}(2 \tau_{q}-\lambda^{\prime})+\lambda^{\prime}\tau_{\phi}]}{2(2\tau_{q}-\lambda^ {\prime})\tau_{q}\tau_{\pi}\tau_{\phi}(\gamma^{\prime}\tau_{\pi}+\gamma_{\perp} \tau_{\phi})},\] (111) \[c_{5} = \sqrt{8D_{s}\gamma_{\perp}(\gamma^{\prime}\tau_{\pi}+\gamma_{ \perp}\tau_{\phi})-(\gamma^{\prime}+\gamma_{\perp})^{2}}. \tag{112}\]
The \(\det M_{4}=0\) gives Eqs.(94-97) and Eqs.(100-102), while \(\det M_{5}=0\) gives Eqs.(94,97-99) and Eqs.(103-105).
Now, let us analyze the causality conditions. From Eqs.(100-105), we find that all modes in minimal causal spin hydrodynamics correspond to finite propagation speed since \(|\omega/k|\) is bounded as \(k\to+\infty\). Imposing Eq.(18) on the propagating modes in Eqs.(102-103), the causality requires,
\[0\leq\frac{b_{1}^{1/2}\pm(b_{1}-b_{2})^{1/2}}{6(2\tau_{q}-\lambda^{\prime}) \tau_{\pi}\tau_{\Pi}}\leq 1\ \text{and}\ 0\leq\frac{2\tau_{q}(\gamma^{\prime}\tau_{\pi}+\gamma_{\perp}\tau_{\phi})}{ (2\tau_{q}-\lambda^{\prime})\tau_{\pi}\tau_{\phi}}\leq 1, \tag{113}\]
which implies that the relaxation times \(\tau_{q},\tau_{\pi},\tau_{\Pi},\tau_{\phi}\) cannot be arbitrarily small and is consistent with the discussion in Sec. IV. We also notice that the causality conditions (113) reduce to Eq. (82) when we take a smooth limit \(\tau_{\pi},\tau_{\Pi},\gamma_{\perp},\gamma_{\parallel}\to 0\).
### Non-trivial stability conditions in rest frame
The requirement of stability is non-trivial. Inserting Eq.(20) into Eqs.(92-105) yields,
\[\tau_{q} > \lambda^{\prime}/2, \tag{114}\] \[D_{s} > 0,\quad D_{b}<-4c_{s}\lambda\gamma_{\parallel}^{-1}|\chi_{e}^{0x }|\leq 0,\] (115) \[b_{1} > b_{2}>0,\quad\frac{c_{2}}{c_{3}}>0. \tag{116}\]
The stability condition \(\lambda^{\prime}<0\) in Eq.(42) for the first order spin hydrodynamics becomes \(\lambda^{\prime}<2\tau_{q}\) in Eq. (114). When the relaxation time \(\tau_{q}\) is sufficiently large, the inequality \(\lambda^{\prime}<2\tau_{q}\) is satisfied, and then the previous unstable modes are removed. We also notice that the conditions (114, 115) agree with Eq. (83) except the \(\chi_{e}^{0x}=0\). The strong constraint \(\chi_{e}^{0x}=0\) is released in this case.
The satisfaction of the stability condition (115) relies on the specific equation of state governing \(S^{\mu\nu}\) and \(\omega^{\mu\nu}\). In Ref.[70], it was found that the stability condition (115) cannot be satisfied if \(\delta S^{\mu\nu}\sim T^{2}\delta\omega^{\mu\nu}\)[62; 64]. In more general cases, we can have
\[u_{\mu}\delta\omega^{\mu\nu} = \chi_{1}u_{\mu}\delta S^{\mu\nu}, \tag{117}\]
\[\Delta^{\mu\alpha}\Delta^{\nu\beta}\delta\omega_{\alpha\beta} = (\chi_{1}+\chi_{2})\Delta^{\mu\alpha}\Delta^{\nu\beta}\delta S_{ \alpha\beta}, \tag{118}\]
where \(\chi_{1,2}\) are susceptibility corresponding to the \(S^{0i}\) and \(S^{ij}\) in the rest frame. In this case, according to the definitions in Eqs.(22,28), the stability condition (115) is satisfied if \(\chi_{2}>-\chi_{1}>0\). Details can be found in Appendix C. Note that the parameters \(\chi_{1}\) and \(\chi_{2}\) strongly depends on the equation of state for \(S^{\mu\nu}\) and \(\omega^{\mu\nu}\). To determine the equation of state, we need the microscopic theories, and we will leave it for the future studies.
Another remarkable observation for the stability conditions is that there exists unstable modes at finite \(k\). Eqs.(114, 115, 116) are the stability conditions in small \(k\) and large \(k\) limits only. We still need to study the \(\mathrm{Im}\omega\) in finite \(k\) region. One analytic method, named the Routh-Hurwitz criterion [163; 164; 165; 182; 186; 187], are usually implemented to study the sign of \(\mathrm{Im}\omega\) in finite \(k\) region. Unfortunately, \(\det\mathcal{M}_{2}\) cannot be reduced to the form that Routh-Hurwitz criterion applies, thus, we analyze the behavior of \(\mathrm{Im}\omega\) numerically instead of the Routh-Hurwitz criterion. For a finite \(k\), we find that \(\mathrm{Im}\omega\) can be negative, even if all the conditions (114, 115, 116) are satisfied. In Fig. 1, we present an example to show that \(\mathrm{Im}\omega\) can be negative for finite \(k\). We choose the parameters as,
\[c_{s}=\frac{1}{\sqrt{3}}, \lambda\chi_{e}^{0x}=\frac{1}{8},\ \tau_{\pi}=4\tau_{\Pi},\ \tau_{\phi}=2\tau_{\Pi},\ \tau_{q}=10\tau_{\Pi},\ \lambda^{\prime}=\frac{1}{2}\tau_{\Pi},\] \[\gamma_{\parallel}=\frac{7}{10}\tau_{\Pi}, \gamma_{\perp}=\frac{1}{2}\tau_{\Pi},\ \gamma^{\prime}=\tau_{\Pi},\ D_{s}=\frac{1}{2\tau_{\Pi}},\ D_{b}=-\frac{1}{2 \tau_{\Pi}}. \tag{119}\]
It is straightforward to verify that the parameters in Eq.(119) satisfy the stability and causality constraints (18, 19, 20). We pick up two modes derived from \(\det M_{4}=0\). We observe that the \(\mathrm{Im}\ \omega\) at both small and large \(k\) limits are positive, while it becomes negative when \(k\tau_{\Pi}\sim 0.5\) and \(k\tau_{\Pi}\sim 10.0\), i.e., the modes are unstable in finite \(k\) region.
We comment on the unstable modes at finite \(k\). The unstable modes in the minimal causal spin hydrodynamics are significantly different with those in the conventional hydrodynamics. As discussed in Refs. [165; 166; 167; 168; 156; 157], the stability conditions obtained in \(k\to 0\) and \(k\to+\infty\) limits are sufficient to ensure the stability at any real \(k\). However, it looks failed in minimal causal spin hydrodynamics. It implies that the conditions (114, 115, 116) are necessary but may not be sufficient. At last, it is still unclear whether the unstable modes at finite \(k\) indicate the fluid becomes unstable or not.
### Causality and stability analysis for extended \(q^{\mu}\) and \(\phi^{\mu\nu}\)
We notice that when \(q^{\mu}\) and \(\phi^{\mu\nu}\) are coupled in the second order constitutive equations the dispersion relation will be modified. Therefore, we extend \(q^{\mu}\) and \(\phi^{\mu\nu}\) in Eqs.(55-56) as follows,
\[\tau_{q}\Delta^{\mu\nu}\frac{d}{d\tau}q_{\nu}+q^{\mu} = \lambda\left(T^{-1}\Delta^{\mu\nu}\partial_{\nu}T+u^{\nu}\partial _{\nu}u^{\mu}-4u_{\nu}\omega^{\mu\nu}\right)+g_{1}\Delta^{\mu\nu}\partial^{ \rho}\phi_{\nu\rho}, \tag{120}\] \[\tau_{\phi}\Delta^{\mu\alpha}\Delta^{\nu\beta}\frac{d}{d\tau} \phi_{\alpha\beta}+\phi^{\mu\nu} = 2\gamma_{s}\left(2\Delta^{\mu\alpha}\Delta^{\nu\beta}\omega_{ \alpha\beta}+\partial_{\perp}^{[\mu}u^{\nu]}\right)+g_{2}\Delta^{\mu\alpha} \Delta^{\nu\beta}\partial_{[\alpha}q_{\beta]}, \tag{121}\]
where \(g_{1,2}\) are new transport coefficients describing the coupling between \(q^{\mu}\) and \(\phi^{\mu\nu}\). Following the same method, Eq.(60-61) become,
\[0 = (\lambda^{\prime}c_{s}^{2}\partial^{i}+8\lambda\chi_{e}^{0i}) \delta e+\lambda^{\prime}\partial_{0}\delta\vartheta^{i}+(2D_{b}-\tau_{q} \partial_{0}\partial_{0}-\partial_{0})\delta S^{0i}-g_{1}\partial_{j}\partial _{0}\delta S^{ij}, \tag{122}\] \[0 = 8\gamma_{s}\chi_{e}^{ij}\delta e+2\gamma^{\prime}(\partial^{i} \delta\vartheta^{j}-\partial^{j}\delta\vartheta^{i})+(\tau_{\phi}\partial_{0} \partial_{0}+\partial_{0}+2D_{s})\delta S^{ij}\] (123) \[+\frac{1}{2}g_{2}\partial^{i}\partial_{0}\delta S^{0j}-\frac{1}{ 2}g_{2}\partial^{j}\partial_{0}\delta S^{0i}.\]
Figure 1: We plot the imaginary parts of \(\omega\tau_{\Pi}\) as a function of \(k\tau_{\Pi}\) in three modes derived from \(\det M_{4}=0\). The parameters are chosen as in Eq. (119), which satisfy the causality and stability conditions Eqs. (18, 19, 20). The solid, dashed and dotted lines stand for three unstable modes.
First, we consider the \(q^{\mu}\) and \(\phi^{\mu\nu}\) only and neglect other dissipative terms for simplicity. In this case, \(M_{5}^{\prime}\) in Eq.(71) reads
\[M_{5}^{\prime} = \left(\begin{array}{ccc}i\omega&\frac{1}{2}\omega^{2}&-\frac{1}{ 2}\omega k\\ \lambda^{\prime}i\omega&2D_{b}+\tau_{q}\omega^{2}-i\omega&g_{1}\omega k\\ 2\gamma^{\prime}ik&-\frac{1}{4}g_{2}\omega k&-\tau_{\phi}\omega^{2}+i\omega+2D _{s}\end{array}\right), \tag{124}\]
while the matrix \(M_{4}^{\prime}\) is the same as before. The dispersion relations in Eq.(80-81) give
\[\omega = \pm\sqrt{\frac{m}{4(2\tau_{q}-\lambda^{\prime})\tau_{\phi}}}k+ \frac{1}{2}i\left(\frac{2}{2\tau_{q}-\lambda^{\prime}}+\frac{1}{\tau_{\phi}}- \frac{8\gamma^{\prime}}{m}\right)+\mathcal{O}(k^{-1}), \tag{125}\] \[\omega = \frac{4\gamma^{\prime}(i\pm\sqrt{-1-D_{b}m\gamma^{\prime-1}})}{ m}+\mathcal{O}(k^{-1}), \tag{126}\]
where
\[m = 2g_{1}g_{2}+8g_{1}\gamma^{\prime}+g_{2}\lambda^{\prime}+8\gamma^{ \prime}\tau_{q}. \tag{127}\]
We notice that the zero modes mentioned in Sec.V.1 cannot be solved by introducing the coupling between \(q^{\mu}\) and \(\phi^{\mu\nu}\).
Imposing Eq.(18) to the propagating modes in Eqs.(102-103), the causality conditions in Eq.(82) becomes
\[0\leq\frac{c_{s}^{2}(3\lambda^{\prime}+2\tau_{q})}{2\tau_{q}- \lambda^{\prime}}\leq 1,\ 0\leq\frac{m}{4(2\tau_{q}-\lambda^{\prime})\tau_{\phi}}\leq 1. \tag{128}\]
Inserting Eq.(20) into these new dispersion relations, Eq.(83) gives
\[\tau_{q}>\lambda^{\prime}/2,\ D_{s}>0,\ D_{b}<0,\ \chi_{e}^{0x}=0,\quad m>8 \gamma^{\prime}\left(\frac{2}{2\tau_{q}-\lambda^{\prime}}+\frac{1}{\tau_{ \phi}}\right)^{-1}. \tag{129}\]
Similarly, we can still implement the Routh-Hurwitz criterion to verify the stability conditions are sufficient and necessary in this case. Details are shown in Appendix (D).
Next, we need to consider all dissipative terms. The Eq.(89) becomes
\[M_{5} = \left(\begin{array}{ccc}2ik\gamma^{\prime}&-\frac{1}{4}g_{2} \omega k&-\tau_{\phi}\omega^{2}+i\omega+2D_{s}&0\\ i\omega&\frac{1}{2}\omega^{2}&-\frac{1}{2}\omega k&-ik\\ i\omega\lambda^{\prime}&2D_{b}+\tau_{q}\omega^{2}-i\omega&g_{1}\omega k&0\\ -ik\gamma_{\perp}&0&0&i\omega\tau_{\Pi}+1\end{array}\right), \tag{130}\]
while the \(M_{4}^{\prime}\) is unaffected. The Eqs.(103-105) becomes,
\[\omega = \pm\sqrt{\frac{f+f^{\prime}}{8(2\tau_{q}-\lambda^{\prime})\tau_{ \pi}\tau_{\phi}}}k+i\frac{f+f^{\prime}}{4(2\tau_{q}-\lambda^{\prime})\tau_{\pi }\tau_{\phi}}c_{6}+\mathcal{O}(k^{-1}), \tag{131}\] \[\omega = \pm\sqrt{\frac{f-f^{\prime}}{8(2\tau_{q}-\lambda^{\prime})\tau_{ \pi}\tau_{\phi}}}k+i\frac{f-f^{\prime}}{4(2\tau_{q}-\lambda^{\prime})\tau_{ \pi}\tau_{\phi}}c_{7}+\mathcal{O}(k^{-1}),\] (132) \[\omega = \pm 4\sqrt{\frac{-D_{b}D_{s}}{g_{1}g_{2}}}k^{-1}+4i\frac{[D_{s} \gamma_{\perp}-D_{b}(\gamma_{\perp}+\gamma^{\prime})]}{g_{1}g_{2}\gamma_{ \perp}}k^{-2}+\mathcal{O}(k^{-3}), \tag{133}\]
where \(m\) is given by Eq. (127) and
\[f = m\tau_{\pi}+8\gamma_{\perp}\tau_{q}\tau_{\phi}, \tag{134}\] \[f^{\prime} = \{-32g_{1}g_{2}\gamma_{\perp}(2\tau_{q}-\lambda^{\prime})\tau_{ \pi}\tau_{\phi}+(m\tau_{\pi}+8\gamma_{\perp}\tau_{q}\tau_{\phi})^{2}\}^{1/2},\] (135) \[d = 4g_{1}^{2}(g_{2}+4\gamma^{\prime})^{2}\tau_{\pi}^{2}+[g_{2} \lambda^{\prime}\tau_{\pi}+8\tau_{q}(\gamma^{\prime}\tau_{\pi}+\gamma_{\perp} \tau_{\phi})]^{2}\] (136) \[+4g_{1}\tau_{\pi}[g_{2}^{2}\lambda^{\prime}\tau_{\pi}+4g_{2} \gamma^{\prime}(\lambda^{\prime}+2\tau_{q})\tau_{\pi}+8g_{2}\gamma_{\perp}( \lambda^{\prime}-\tau_{q})\tau_{\phi}+32\gamma^{\prime}\tau_{q}(\gamma^{ \prime}\tau_{\pi}+\gamma_{\perp}\tau_{\phi})],\] \[c_{6} = -\frac{1}{(f^{\prime 2}+fd^{1/2})}\left[m\tau_{\pi}(2\tau_{q}- \lambda^{\prime})(\tau_{\phi}-\tau_{\pi})\right.\] (137) \[+8\gamma_{\perp}\tau_{q}\tau_{\phi}(\tau_{q}-\tau_{\pi})(\lambda ^{\prime}-2\tau_{\phi})+16\gamma^{\prime}\tau_{\pi}^{2}\tau_{\phi}(2\tau_{q}- \lambda^{\prime})\] \[\left.-f^{\prime}(2\tau_{q}-\lambda^{\prime})(\tau_{\pi}+\tau_{ \phi})+2\tau_{\pi}\tau_{\phi}(-m\tau_{\pi}-8\gamma_{\perp}\lambda^{\prime} \tau_{\phi}+8\gamma^{\prime}\tau_{q}^{2}-f^{\prime})\right],\] \[c_{7} = -\frac{c_{72}}{c_{71}},\] (138) \[c_{71} = -4g_{1}^{2}(g_{2}+4\gamma^{\prime})^{2}\tau_{\pi}^{2}+[g_{2} \lambda^{\prime}\tau_{\pi}+8\tau_{q}(\gamma^{\prime}\tau_{\pi}+\gamma_{\perp} \tau_{\phi})](-g_{2}\lambda^{\prime}\tau_{\pi}-8\gamma^{\prime}\tau_{q}\tau_ {\pi}\] (139) \[-8\gamma_{\perp}\tau_{q}\tau_{\phi}+d^{1/2})-2g_{1}\tau_{\pi}\{2g _{2}^{2}\lambda^{\prime}\tau_{\pi}+g_{2}[8\gamma^{\prime}(\lambda^{\prime}+2 \tau_{q})\tau_{\pi}+16\gamma_{\perp}\lambda^{\prime}\tau_{\phi}\] \[-16\gamma_{\perp}\tau_{q}\tau_{\phi}-d^{1/2}]-4\gamma^{\prime}(16 \gamma^{\prime}\tau_{q}\tau_{\pi}+16\gamma_{\perp}\tau_{q}\tau_{\phi}-d^{1/2})\},\] \[c_{72} = -f^{\prime}(2\tau_{q}-\lambda^{\prime})(\tau_{\pi}+\tau_{\phi})+ m\tau_{\pi}(2\tau_{q}-\lambda^{\prime})(\tau_{\pi}-\tau_{\phi})-16\gamma^{ \prime}\tau_{\pi}^{2}\tau_{\phi}(2\tau_{q}-\lambda^{\prime})\] (140) \[-8\gamma_{\perp}\tau_{q}\tau_{\phi}(2\tau_{q}-\lambda^{\prime})( \tau_{\pi}-\tau_{\phi})+\tau_{\pi}\tau_{\phi}(-2f^{\prime}+2m\tau_{\pi}-16 \gamma_{\perp}\tau_{q}\tau_{\phi}+16\gamma_{\perp}\lambda^{\prime}\tau_{ \phi}).\]
From these new dispersion relations, we obtain causality conditions,
\[0\leq\frac{b_{1}^{1/2}\pm(b_{1}-b_{2})^{1/2}}{6(2\tau_{q}-\lambda^{\prime}) \tau_{\pi}\tau_{\Pi}}\leq 1\ \mbox{and}\ 0\leq\frac{f+f^{\prime}}{8(2\tau_{q}-\lambda^{\prime})\tau_{\pi}\tau_{\phi}} \leq 1, \tag{141}\]
which reproduces Eq.(113) when \(g_{1},g_{2}\to 0\).
The stability conditions in Eq.(83) becomes,
\[\tau_{q}-\frac{\lambda^{\prime}}{2} > 0, \tag{142}\] \[D_{s}>0,\quad-4c_{s}\lambda\gamma_{||}^{-1}|\chi_{e}^{0x}|-D_{b} > 0, \tag{143}\]
\[b_{1}>b_{2}>0,\quad\frac{c_{2}}{c_{3}} > 0, \tag{144}\] \[g_{1}g_{2}>0,\quad f>0,\quad f^{\prime} > 0,\] (145) \[\mathrm{Rec}_{6}>0,\quad\mathrm{Rec}_{7} > 0. \tag{146}\]
Unfortunately, we find that the extended \(q^{\mu}\) and \(\phi^{\mu\nu}\) cannot remove the unstable modes at finite \(k\) coming from \(\mathrm{det}M_{4}=0\). We choose the parameters satisfying the causality conditions (141) and stability conditions (142 - 146), and consider the influence on dispersion relations of \(g_{1},g_{2}\). For simplicity, we choose the parameters as the same as in Eq. (119) with \((g_{1}/\tau_{\Pi},g_{2}/\tau_{\Pi})=(0.0,0.0),(2.0,0.1),(6.0,0.1),(6.0,0.05)\). We find that one modes from \(\mathrm{det}\,M_{5}=0\) becomes unstable at finite \(k\) with \((g_{1}/\tau_{\Pi},g_{2}/\tau_{\Pi})=(6.0,0.1),(6.0,0.05)\) as shown in Fig. 2.
As a brief summary, the extended \(q^{\mu}\) and \(\phi^{\mu\nu}\) can modify the causality and stability
conditions, but cannot remove the zero modes when we turn off other dissipative effects. The unstable modes at finite \(k\) also cannot be cured by the extended \(q^{\mu}\) and \(\phi^{\mu\nu}\).
### Causality and stability in moving frames
Let us briefly discuss the causality and stability of the minimal causal spin hydrodynamics in moving frames.
For the causality in a moving frame, we refer to the studies in Ref. [163]. The authors in Ref. [163] has studied the dispersion relations at large \(k\) limit in moving frames and demonstrate that the system is causal in moving frames if it is causal in the rest frame. Thus, the minimal causal spin hydrodynamics is causal in moving frames when the causality condition (113) in the rest frame are satisfied.
For the stability, it has also been proved that if a causal theory is unstable in the rest frame, then it is also unstable in moving frames (also see Theorem 2 of Ref.[188]). We now apply this theorem to the minimal causal spin hydrodynamics. If the equation of state gives \(\delta\omega^{\mu\nu}=\chi_{1}\delta S^{\mu\nu}\) with constant \(\chi_{1}\), the minimal causal spin hydrodynamics will be unstable in moving frames since it has unstable modes in the rest frame. For more general cases, the stability of the theory in both moving frames and the rest frame depends on the equation of state for \(S^{\mu\nu}\) and \(\omega^{\mu\nu}\).
In summary, the minimal causal spin hydrodynamics is causal in any reference frame when Eq.(113) is fulfilled. Hence, we have solved the problem of acausality by introducing the minimal causal spin hydrodynamics. However, the stability of minimal causal spin hydrodynamics remains unclear. Our findings indicate that the validity of the stability condition (115) is highly contingent upon the equation of state governing spin density and spin chemical potential. Moreover, we also find that the stability conditions (114, 115, 116) obtained at \(k\to 0\) and \(k\rightarrow+\infty\) are necessary but not sufficient.
## VI Conclusion
In this work, we investigate the causality and stability of canonical spin hydrodynamics in the linear modes analysis.
In linear modes analysis, we consider perturbations to the spin hydrodynamics near the
static equilibrium. We obtain the dispersion relations \(\omega=\omega(k)\) and analyze the all possible modes. The results show the stability condition (42) cannot be fulfilled. Moreover, the value of \(|\omega/k|\) in Eqs. (44-46) is unbounded, which violates the causality condition (19). In Refs.[70; 71; 45], the expression of \(q^{\mu}\) are modified by using the equation of motion for the fluid. We emphasize that the first order spin hydrodynamics in Refs.[70; 71; 45] are still acausal since one mode shown in Eq. (52) breaks the causality condition (19). We conclude that the canonical spin hydrodynamics in the first order of gradient expansion are acausal and unstable.
We then follow the basic idea in MIS, BRSSS, and DNMR theories to construct minimal causal spin hydrodynamics. The constitutive equations (12-16) in a minimal extended causal spin hydrodynamics are replaced by Eqs. (55-58). One can view it as a natural extension of the first order spin hydrodynamics or a simplified version of the complete second order spin hydrodynamics [73]. We investigate the causality and stability for this minimal causal spin hydrodynamics. We analyze the causality and stability for dissipative fluids with \(q^{\mu}\) and \(\phi^{\mu\nu}\) only and find that the zero modes in the linear modes analysis. It indicates that there exists the nonlinear modes. Therefore, we consider dissipative spin fluids with shear viscous tensor and bulk viscous pressure.
For causality, we find that the modes with infinite speed disappear and all modes are causal in the rest frame if the conditions in Eq.(113) are fulfilled. Following the statement in Ref. [163], we comment that the minimal causal spin hydrodynamics are causal in any reference frame when the conditions (113) are fulfilled.
For the stability, although we obtain the stability conditions in Eqs.(114, 115, 116) from the constraints in the \(k\to 0\) and \(k\rightarrow+\infty\) limits, the stability of the theory in both moving frames and the rest frame remains unclear. Two kinds of problems can lead to instabilities. The first one is related to stability condition (115). Interestingly, we prove that the coefficients \(D_{s},D_{b}\) do not obey the stability condition (115) if the equation of state \(S^{\mu\nu}\sim T^{2}\omega^{\mu\nu}\) is adopted. In more general cases, the fulfillment of the stability condition (115) hinges on the specific equations of state. One has to assess the condition (115) on a case-by-case basis. Surprisingly, different with the conventional hydrodynamics, we find that the stability condition (20) breaks at finite \(k\) as shown in Fig. 1. It implies that the conditions (114, 115, 116) are necessary but may not be sufficient.
We also considered the extended \(q^{\mu}\) and \(\phi^{\mu\nu}\), in which the \(q^{\mu}\) and \(\phi^{\mu\nu}\) are coupled in the
second order constitutive equations. The causality and stability conditions are modified in this case. However, in dissipative fluids with \(q^{\mu}\) and \(\phi^{\mu\nu}\) only the zero modes cannot be removed. The unstable modes at finite wavelength are still there.
We conclude that the canonical spin hydrodynamics in the first order of gradient expansion are always acausal and unstable. The minimal causal extension of spin hydrodynamics makes the theory be causal. However, the linear stability of the minimal causal spin hydrodynamics remain unclear. The studies beyond the linear modes analysis may provide us a better and clear answer to the problem of stability.
###### Acknowledgements.
We thank Francesco Becattini, Matteo Buzzegoli, Asaad Daher, Xu-Guang Huang, Jin Hu, Masoud Shokri and David Wagner for helpful discussion during the 7th International Conference on Chirality, Vorticity and Magnetic Field in Heavy Ion Collisions. This work is supported in part by the National Key Research and Development Program of China under Contract No. 2022YFA1605500. This work is partly supported by National Nature Science Foundation of China (NSFC) under Grants No. 12075235 and 12135011.
## Appendix A Off-diagonal submatrices in Eqs.(32, 69, 87)
In this appendix, we list all the off-diagonal submatrices introduced in Eqs.(32,69,87):
\[A_{1}\equiv\left(\begin{array}{ccc}-4i(\omega\lambda\chi_{e}^{0y}+k\gamma_{ s}\chi_{e}^{xy})&0&0\\ 8\lambda\chi_{e}^{0y}&0&0\\ 8\gamma_{s}\chi_{e}^{xy}&0&0\end{array}\right),\ A_{2}\equiv\left(\begin{array} []{ccc}-4i(\omega\lambda\chi_{e}^{0z}+k\gamma_{s}\chi_{e}^{xz})&0&0\\ 8\lambda\chi_{e}^{0z}&0&0\\ 8\gamma_{s}\chi_{e}^{xz}&0&0\end{array}\right), \tag{10}\]
\[A_{3}=A_{6}^{\prime}=\left(\begin{array}{ccc}8\gamma_{s}\chi_{e}^{yz},\ 0,\ 0,\ 0 \end{array}\right),\ A_{4}^{\prime}\equiv\left(\begin{array}{ccc}0&0&0\\ 8\lambda\chi_{e}^{0y}&0&0\\ 8\gamma_{s}\chi_{e}^{xy}&0&0\end{array}\right),\ A_{5}^{\prime}\equiv\left( \begin{array}{ccc}0&0&0\\ 8\lambda\chi_{e}^{0z}&0&0\\ 8\gamma_{s}\chi_{e}^{xz}&0&0\end{array}\right), \tag{11}\]
\[A_{4}=\left(\begin{array}{cccc}8\gamma_{s}\chi_{e}^{xy}&0&0&0&0\\ 0&0&0&0&0\\ 8\lambda\chi_{e}^{0y}&0&0&0&0\\ 0&0&0&0&0\\ \end{array}\right),\ A_{5}=\left(\begin{array}{cccc}8\gamma_{s}\chi_{e}^{xz}&0 &0&0&0\\ 0&0&0&0&0\\ 8\lambda\chi_{e}^{0z}&0&0&0&0\\ 0&0&0&0&0\\ \end{array}\right),\ A_{6}=\left(\begin{array}{cccc}2\gamma_{s}\chi_{e}^{yz} &0&0&0&0\\ 0&\frac{2}{3}ik\gamma_{\perp}&0&0&0\\ 0&0&0&0&0\\ \end{array}\right). \tag{101}\]
Appendix B Discussion on the stability conditions in fluids with \(q^{\mu}\) and \(\phi^{\mu\nu}\) only
In this appendix, we discuss the stability conditions (83) for fluids with \(q^{\mu}\) and \(\phi^{\mu\nu}\) only (see Sec. V.1).
As mentioned, we derive the stability condition (83) from the linear modes analysis in small and large \(k\) limits only. Now, we implement the Routh-Hurwitz criterion [163; 164; 165; 182; 186] to prove that the conditions (83) hold for all real \(k\).
We only need to prove that the nonzero modes derived from \(\det M_{4}^{\prime}=0\) and \(\det M_{5}^{\prime}=0\) satisfy \(\mathrm{Im}\ \omega>0\) for all \(k\). First, we discuss the modes coming from the \(\det M_{4}^{\prime}=0\). The \(\det M_{4}^{\prime}=0\) gives
\[a_{0}\omega^{4}-ia_{1}\omega^{3}-a_{2}\omega^{2}+ia_{3}\omega+a_{4}=0, \tag{102}\]
with
\[a_{0} = \frac{1}{2}(2\tau_{q}-\lambda^{\prime}),\] \[a_{1} = 1,\] \[a_{2} = \frac{1}{2}c_{s}^{2}k^{2}(3\lambda^{\prime}+2\tau_{q})-2D_{b},\] \[a_{3} = c_{s}^{2}k^{2},\] \[a_{4} = -2c_{s}^{2}D_{b}k^{2}. \tag{103}\]
We redefine \(\omega=-i\Delta\) and rewrite Eq.(102) as,
\[a_{0}\Delta^{4}+a_{1}\Delta^{3}+a_{2}\Delta^{2}+a_{3}\Delta+a_{4}=0. \tag{104}\]
Notice that the coefficients \(a_{0,1,2,3,4}\) are pure real. According to the Routh-Hurwitz criterion [163; 164; 165; 182; 186; 187], the stability condition (20), i.e., \(\mathrm{Im}\omega>0\) or \(\mathrm{Re}\Delta<0\), is fulfilled for all nonzero \(k\) if and only if
\[a_{i} > 0,\] \[a_{1}a_{2}a_{3}-a_{1}^{2}a_{4}-a_{0}a_{3}^{2} > 0. \tag{105}\]
When the conditions in Eq.(83) are fulfilled, the first inequality \(a_{i}>0\) are automatically satisfied. The second inequality can be expressed as \(\lambda^{\prime}=2\lambda/[e_{(0)}+p_{(0)}]>0\), which has already been guaranteed by entropy principle (17). Thus the modes derived from \(\det M_{4}^{\prime}=0\) are stable for all \(k\) if condition (83) is satisfied.
Second, we consider the nonzero modes derived from \(\det M_{5}^{\prime}=0\). The \(\det M_{5}^{\prime}=0\) gives \(\omega=0\) or
\[a_{0}^{\prime}\omega^{4}-ia_{1}^{\prime}\omega^{3}-a_{2}^{\prime}\omega^{2}+ia _{3}^{\prime}\omega+a_{4}^{\prime}=0, \tag{105}\]
where
\[a_{0}^{\prime} = \frac{1}{2}\tau_{\phi}(2\tau_{q}-\lambda^{\prime}),\] \[a_{1}^{\prime} = \tau_{\phi}+\frac{1}{2}(2\tau_{q}-\lambda^{\prime}),\] \[a_{2}^{\prime} = 1+D_{s}(2\tau_{q}-\lambda^{\prime})+k^{2}\gamma^{\prime}\tau_{q }-2D_{b}\tau_{\phi},\] \[a_{3}^{\prime} = \gamma^{\prime}k^{2}+2D_{s}-2D_{b},\] \[a_{4}^{\prime} = -4D_{b}D_{s}-2D_{b}\gamma^{\prime}k^{2}. \tag{106}\]
Similarly, the Routh-Hurwitz criterion provides the necessary and sufficient conditions for \(\mathrm{Im}\ \omega>0\) in Eq.(105),
\[a_{i}^{\prime} > 0, \tag{107}\] \[a_{1}^{\prime}a_{2}^{\prime}a_{3}^{\prime}-a_{1}^{\prime 2}a_{4}^{ \prime}-a_{0}^{\prime}a_{3}^{\prime 2} > 0. \tag{108}\]
Each \(a_{i}^{\prime}>0\) does not give new constraints for stability. We now show that the second inequality holds for all \(k\) if the conditions in Eq.(83) are fulfilled. Define a new function \(F(D_{b},D_{s},k)\),
\[F(D_{b},D_{s},k) \equiv a_{1}^{\prime}a_{2}^{\prime}a_{3}^{\prime}-a_{1}^{\prime 2}a_{4}^ {\prime}-a_{0}^{\prime}a_{3}^{\prime 2} \tag{109}\] \[= 4\tau_{\phi}^{2}D_{b}^{2}+\frac{1}{2}[8D_{s}(2\tau_{q}-\lambda^{ \prime})\tau_{\phi}+G(k)]D_{b}+H(D_{s},k),\]
with
\[G(k) \equiv -(2+k^{2}\gamma^{\prime}\lambda^{\prime})(2\tau_{q}-\lambda^{ \prime})-2[2+k^{2}\gamma^{\prime}(3\lambda^{\prime}-4\tau_{q})]\tau_{\phi}, \tag{110}\] \[H(D_{s},k) \equiv \frac{1}{2}(2D_{s}+k^{2}\gamma^{\prime})(2\tau_{q}-\lambda^{ \prime})[1+D_{s}(2\tau_{q}-\lambda^{\prime})+k^{2}\gamma^{\prime}\tau_{q}]\] (111) \[+\frac{1}{2}(2D_{s}+k^{2}\gamma^{\prime})(2+k^{2}\gamma^{\prime} \lambda^{\prime})\tau_{\phi}.\]
Since \(\tau_{q}>\lambda^{\prime}/2\) in Eq.(83), we have \(H(D_{s},k)>0\) for any \(k\) and any \(D_{s}>0\).
Then, we discuss two cases. When
\[8D_{s}(2\tau_{q}-\lambda^{\prime})\tau_{\phi}+G(k)\leq 0, \tag{92}\]
we find \(F(D_{b},D_{s},k)>0\) for any \(D_{b}<0\). In another case, \(8D_{s}(2\tau_{q}-\lambda^{\prime})\tau_{\phi}+G(k)>0\), i.e.,
\[D_{s}>\frac{-G(k)}{8(2\tau_{q}-\lambda^{\prime})\tau_{\phi}}, \tag{93}\]
for each fixed \(D_{s}>0\) and \(k\), the function \(F(D_{b},D_{s},k)\) gets its minimal value
\[F(D_{b},D_{s},k) \geq F(D_{b},D_{s},k)|_{D_{b}=-[8D_{s}(2\tau_{q}-\lambda^{\prime}) \tau_{\phi}+G(k)]/(16\tau_{\phi}^{2})} \tag{94}\] \[= \frac{1}{64\tau_{\phi}^{2}}(2+k^{2}\gamma^{\prime}\lambda^{\prime })(\lambda^{\prime}-2\tau_{q}-2\tau_{\phi})^{2}\] \[\times[16\tau_{\phi}D_{s}-2-k^{2}\gamma^{\prime}(\lambda^{\prime }-8\tau_{\phi})].\]
at
\[D_{b}=-[8D_{s}(2\tau_{q}-\lambda^{\prime})\tau_{\phi}+G(k)]/(16\tau_{\phi}^{2}). \tag{95}\]
Substituting Eq.(93) into Eq.(94) leads to
\[F(D_{b},D_{s},k)\;\geq\;\frac{(2+k^{2}\gamma^{\prime}\lambda^{\prime})^{2}( \lambda^{\prime}-2\tau_{q}-2\tau_{\phi})^{2}(2\tau_{q}-\lambda^{\prime}+4\tau _{\phi})}{64(2\tau_{q}-\lambda^{\prime})\tau_{\phi}^{2}}>0, \tag{96}\]
where we have used \(\tau_{q}>\lambda^{\prime}/2\) in Eq.(83). Thus, the nonzero modes derived from \(\det M_{5}^{\prime}=0\) are stable for all \(k\) if the conditions in Eq.(83) are fulfilled.
Therefore, the conditions (83) are sufficient and necessary for the stability of fluids with \(q^{\mu}\) and \(\phi^{\mu\nu}\) only.
## Appendix C Discussions on the stability conditions (115)
Here, we discuss the stability conditions (115), i.e., \(D_{s}>0\), \(D_{b}<0\).
Let us consider an isotropic fluid at equilibrium, i.e., we assume that there are not preferred directions induced by spin and external fields. In this case, the variation of spin chemical potential is
\[\delta\omega^{\mu\nu}=\chi^{\mu\nu\alpha\beta}\delta S_{\alpha\beta}+\chi^{\mu \nu}_{e}\delta e, \tag{97}\]
with a rank-4 tensor \(\chi^{\mu\nu\alpha\beta}\) and rank-2 tensor \(\chi^{\mu\nu}_{e}\). We find that \(\chi^{\mu\nu\alpha\beta}\) satisfies \(\chi^{\mu\nu\alpha\beta}=-\chi^{\nu\mu\alpha\beta}=-\chi^{\mu\nu\beta\alpha}\).
In an irrotational isotropic background fluid without any external fields, any rank-\(n\) tensor can only be constructed by \(u^{\mu},g^{\mu\nu},\partial^{\mu},\epsilon^{\mu\nu\alpha\beta}\). Back to rank-4 tensor \(\chi^{\mu\nu\alpha\beta}\), in the linear modes analysis, we do not need to consider the part in \(\chi^{\mu\nu\alpha\beta}\) proportional to spacetime derivatives \(\partial^{\mu}\) since those terms in \(\chi^{\mu\nu\alpha\beta}\delta S_{\alpha\beta}\) becomes nonlinear and will be dropped. While the tensor \(\epsilon^{\mu\nu\alpha\beta}\) violates the reflection symmetry and cannot be used there. According to the anti-symmetric properties of \(\chi^{\mu\nu\alpha\beta}\), the only possible expression is
\[\chi^{\mu\nu\alpha\beta}=\frac{\chi_{1}}{2}(g^{\mu\alpha}g^{\nu\beta}-g^{\mu \beta}g^{\nu\alpha})+\frac{\chi_{2}}{2}(\Delta^{\mu\alpha}\Delta^{\nu\beta}- \Delta^{\mu\beta}\Delta^{\nu\alpha}), \tag{100}\]
where \(\chi_{1}\) and \(\chi_{2}\) are scalars.
Substituting Eq.(100) into Eq.(101), we obtain
\[\delta\omega^{\mu\nu}=\chi_{1}\delta S^{\mu\nu}+\chi_{2}\Delta^{\mu\alpha} \Delta^{\nu\beta}\delta S_{\alpha\beta}. \tag{101}\]
One can also write it as
\[u_{\mu}\delta\omega^{\mu\nu} = \chi_{1}u_{\mu}\delta S^{\mu\nu}, \tag{102}\] \[\Delta^{\mu\alpha}\Delta^{\nu\beta}\delta\omega_{\alpha\beta} = (\chi_{1}+\chi_{2})\Delta^{\mu\alpha}\Delta^{\nu\beta}\delta S_{ \alpha\beta}. \tag{103}\]
From the definitions in Eqs.(22,28), we then have
\[D_{s}=4\gamma_{s}(\chi_{1}+\chi_{2}),\ D_{b}=4\lambda\chi_{1}. \tag{104}\]
Since \(\gamma_{s}>0,\lambda>0\), the stability condition (115), \(D_{s}>0,D_{b}<0\), is equivalent to
\[\chi_{2}>-\chi_{1}>0. \tag{105}\]
The equation of state used in our previous works [62; 64] corresponds to \(\chi_{2}=0\) (see Eq.(17) of Ref. [62]) and Eq.(38) of Ref. [64]). In that case, Eq.(105) cannot be satisfied and there exists unstable modes, although the analytic solutions in Refs. [62; 64] do not rely on it. For general cases where \(\chi_{2}\neq 0\), whether the stability condition (115) \(D_{s}>0,D_{b}<0\) is satisfied depends on \(\chi_{1},\chi_{2}\), which relates with the equation of state for \(S^{\mu\nu}\) and \(\omega^{\mu\nu}\). To determine the value of \(\chi_{1},\chi_{2}\), further investigations should be done from the microscopic theory.
Appendix D Discussion on the stability conditions for the case with extended \(q^{\mu}\) and \(\phi^{\mu\nu}\)
As discussed in Appendix (B), we consider the nonzero modes derived from \(\det M^{\prime}_{5}=0\). The \(\det M^{\prime}_{5}=0\) gives \(\omega=0\) or
\[a^{\prime}_{0}\omega^{4}-ia^{\prime}_{1}\omega^{3}-a^{\prime}_{2}\omega^{2}+ia^ {\prime}_{3}\omega+a^{\prime}_{4}=0, \tag{105}\]
where
\[a^{\prime}_{0} = \frac{1}{2}\tau_{\phi}(2\tau_{q}-\lambda^{\prime}),\] \[a^{\prime}_{1} = \tau_{\phi}+\frac{1}{2}(2\tau_{q}-\lambda^{\prime}),\] \[a^{\prime}_{2} = 1+D_{s}(2\tau_{q}-\lambda^{\prime})+\frac{1}{8}k^{2}m-2D_{b}\tau _{\phi},\] \[a^{\prime}_{3} = \gamma^{\prime}k^{2}+2D_{s}-2D_{b},\] \[a^{\prime}_{4} = -4D_{b}D_{s}-2D_{b}\gamma^{\prime}k^{2}. \tag{106}\]
Similarly, the necessary and sufficient conditions for Im \(\omega>0\) in Eq.(105) are
\[a^{\prime}_{i} > 0, \tag{107}\] \[a^{\prime}_{1}a^{\prime}_{2}a^{\prime}_{3}-a^{\prime 2}_{1}a^{ \prime}_{4}-a^{\prime}_{0}a^{\prime 2}_{3} > 0. \tag{108}\]
The first conditions are automatically satisfied when we have the constraints for stability. Then we need to analyze whether Eq.(108) is satisfies under the existing constraints. Define a function \(F(D_{b},D_{s},k)\),
\[F(D_{b},D_{s},k) \equiv a^{\prime}_{1}a^{\prime}_{2}a^{\prime}_{3}-a^{\prime 2}_{1}a^{ \prime}_{4}-a^{\prime}_{0}a^{\prime 2}_{3} \tag{109}\] \[= F_{a}D_{b}^{2}+F_{b}D_{b}+F_{c},\]
where
\[F_{a} \equiv 4\tau_{\phi}^{2},\] \[F_{b} \equiv \left[\frac{1}{2}k^{2}\gamma^{\prime}(2\tau_{q}-\lambda^{\prime} )+(4D_{s}+3k^{2}\gamma^{\prime})\tau_{\phi}\right](2\tau_{q}-\lambda^{\prime} )-\frac{1}{8}(mk^{2}+8)(2\tau_{\phi}+2\tau_{q}-\lambda^{\prime}),\] \[F_{c} \equiv \frac{1}{16}(2D_{s}+k^{2}\gamma^{\prime})\left\{8D_{s}(2\tau_{q} -\lambda^{\prime})^{2}+(2\tau_{q}-\lambda^{\prime})[8+k^{2}(m-8\gamma^{ \prime}\tau_{\phi})]+2(8+k^{2}m)\tau_{\phi}\right\} \tag{110}\] \[> \frac{1}{2}(2D_{s}+k^{2}\gamma^{\prime})\{2\tau_{\phi}+(2\tau_{q} -\lambda^{\prime})[1+D_{s}(2\tau_{q}-\lambda^{\prime})]\}>0.\]
When \(F_{b}<0\), i.e.,
\[D_{s}\;<\;\frac{(mk^{2}+8)(2\tau_{\phi}+2\tau_{q}-\lambda^{\prime})}{32(2\tau_{q}- \lambda^{\prime})\tau_{\phi}}-\frac{k^{2}\gamma^{\prime}(2\tau_{q}-\lambda^{ \prime})}{8\tau_{\phi}}-\frac{3}{4}k^{2}\gamma^{\prime}, \tag{107}\]
we get
\[F(D_{b},D_{s},k)>F(0,D_{s},k)=F_{c}>0. \tag{108}\]
In another case, \(F_{b}\geq 0\), i.e.,
\[D_{s}\;\geq\;\frac{(mk^{2}+8)(2\tau_{\phi}+2\tau_{q}-\lambda^{\prime})}{32(2 \tau_{q}-\lambda^{\prime})\tau_{\phi}}-\frac{k^{2}\gamma^{\prime}(2\tau_{q}- \lambda^{\prime})}{8\tau_{\phi}}-\frac{3}{4}k^{2}\gamma^{\prime}, \tag{109}\]
the function has its minimal value
\[F(D_{b},D_{s},k)_{\rm min} = F(D_{b},D_{s},k)|_{D_{b}=-F_{b}/(2F_{a})} \tag{110}\] \[= -\frac{(2\tau_{\phi}+2\tau_{q}-\lambda^{\prime})^{2}}{1024\tau_ {\phi}^{2}}\left\{8+k^{2}\left[m-4\gamma^{\prime}\left(2\tau_{q}-\lambda^{ \prime}\right)\right]\right\}\] \[\times\left\{8+k^{2}\left[m-4\gamma^{\prime}\left(2\tau_{q}- \lambda^{\prime}\right)\right]-32k^{2}\tau_{\phi}(\gamma^{\prime}+2D_{s})\right\}\] \[\geq \frac{\left\{8+k^{2}\left[m-4\gamma^{\prime}\left(2\tau_{q}- \lambda^{\prime}\right)\right]\right\}^{2}\left(2\tau_{\phi}+2\tau_{q}- \lambda^{\prime}\right)^{3}}{1024\tau_{\phi}^{2}(2\tau_{q}-\lambda^{\prime}) }>0,\]
at
\[D_{b} = -\frac{F_{b}}{2F_{a}}, \tag{111}\] \[D_{s} = \frac{(mk^{2}+8)(2\tau_{\phi}+2\tau_{q}-\lambda^{\prime})}{32(2 \tau_{q}-\lambda^{\prime})\tau_{\phi}}-\frac{k^{2}\gamma^{\prime}(2\tau_{q}- \lambda^{\prime})}{8\tau_{\phi}}-\frac{3}{4}k^{2}\gamma^{\prime}. \tag{112}\]
Therefore, the nonzero modes are stable for all \(k\) if the stability condition (129) is satisfied.
| 因果関係と安定性の線形解析を、最小拡張スピン水素動力学に適用し、勾配拡張の2次の項まで行います。1次のスピン水素動力学は、ランク3スピンテンソルが最後の2つの索引のみを反転させることで、因果関係と不安定となります。そこで、最小因果性スピン水素動力学を勾配拡張の2次の項まで適用し、この最小因果性スピン水素動力学の因果関係と安定性を導き出します。興味深いことに、安定性の条件は、スピン密度の状態方程式と化学ポテンシャルによって満たされます。また、従来のrelativistic dissipative hydrodynamicsとは異なり、安定性の理論は、安定条件が満たされる場合でも、有限の波vectorで破られます。これは、小さな波vectorと大きい波vectorの限界において安定条件が満たされる場合、スピン水素動力学の安定 |
2302.01973 | Measuring The Impact Of Programming Language Distribution | Current benchmarks for evaluating neural code models focus on only a small
subset of programming languages, excluding many popular languages such as Go or
Rust. To ameliorate this issue, we present the BabelCode framework for
execution-based evaluation of any benchmark in any language. BabelCode enables
new investigations into the qualitative performance of models' memory, runtime,
and individual test case results. Additionally, we present a new code
translation dataset called Translating Python Programming Puzzles (TP3) from
the Python Programming Puzzles (Schuster et al. 2021) benchmark that involves
translating expert-level python functions to any language. With both BabelCode
and the TP3 benchmark, we investigate if balancing the distributions of 14
languages in a training dataset improves a large language model's performance
on low-resource languages. Training a model on a balanced corpus results in, on
average, 12.34% higher $pass@k$ across all tasks and languages compared to the
baseline. We find that this strategy achieves 66.48% better $pass@k$ on
low-resource languages at the cost of only a 12.94% decrease to high-resource
languages. In our three translation tasks, this strategy yields, on average,
30.77% better low-resource $pass@k$ while having 19.58% worse high-resource
$pass@k$. | Gabriel Orlanski, Kefan Xiao, Xavier Garcia, Jeffrey Hui, Joshua Howland, Jonathan Malmaud, Jacob Austin, Rishabh Singh, Michele Catasta | 2023-02-03T19:47:22 | http://arxiv.org/abs/2302.01973v3 | # Measuring The Impact Of Programming Language Distribution
###### Abstract
Current benchmarks for evaluating neural code models focus on only a small subset of programming languages, excluding many popular languages such as Go or Rust. To ameliorate this issue, we present the BabelCode framework for execution-based evaluation of any benchmark in any language. BabelCode enables new investigations into the qualitative performance of models' memory, runtime, and individual test case results. Additionally, we present a new code translation dataset called Translating Python Programming Puzzles (TP3) from the Python Programming Puzzles (Schuster et al., 2021) benchmark that involves translating expert-level python functions to any language. With both BabelCode and the TP3 benchmark, we investigate if balancing the distributions of 14 languages in a training dataset improves a large language model's performance on low-resource languages. Training a model on a balanced corpus results in, on average, 12.34% higher \(pass@k\) across all tasks and languages compared to the baseline. We find that this strategy achieves 66.48% better \(pass@k\) on low-resource languages at the cost of only a 12.94% decrease to high-resource languages. In our three translation tasks, this strategy yields, on average, 30.77% better low-resource \(pass@k\) while having 19.58% worse high-resource \(pass@k\).1
Footnote 1: [https://github.com/google-research/babelcode](https://github.com/google-research/babelcode)
## 1 Introduction
In the 2022 StackOverflow Developer Survey, Rust was the 14th most popular programming language despite not ranking in the survey taken five years prior. However, the 13th most popular language, Go, has nearly doubled Rust's number of StackOverflow questions in this time frame. Further, despite their similar popularity, Go has nearly 350% more source code available (Kocetkov et al., 2022). These disparities highlight the problem that many popular programming languages are starkly low-resource, especially compared to the most popular languages.
Despite their impressive generative capabilities, especially in code, Large Language Models' (LLM) are adversely impacted by this language resource imbalance. Thus, developers will likely find minimal utility from LLMs if they are not using the extremely popular languages. It is therefore imperative to investigate how to mitigate the discrepancy between a language's popularity and the amount of data available for it. Prior works focusing on code generation (Ahmad et al., 2021) and multilingual natural language processing (Arivazhagan et al., 2019; Conneau et al., 2019) use temperature-based strategies to balance the training languages. Such a strategy duplicates extremely low-resource languages thousands of times, which has been shown to significantly reduce performance (Allamanis, 2019).
Beyond the the language balancing strategy, evaluating code LLMs in a multi-lingual setting presents significant challenges. Existing datasets are either mono-lingual (Chen et al., 2021; Austin et al., 2021; Lai et al., 2022) or limited to only a subset of popular programming languages (Roziere et al., 2020). Each problem in these datasets, which we henceforth refer to as a _benchmark_, contains an input, and a canonical solution along with the test-cases for checking correctness. Creating a new benchmark for each language of interest would require insurmountable engineering and monetary costs. To address both of these problems, we present the BabelCode framework for execution-based evaluation of _any benchmark_ in _any language_ and use it to investigate the impact of programming language distribution on code generation and translation.
BabelCode is open-sourced, has an extensive test suite, and supports evaluating four benchmarks in 14 languages. It is designed specifically to enable future research directions such as the evaluation of custom data-structures. BabelCode allows investigation of novel research directions through the measurement of memory and runtime usage for a given
prediction, as well as the outcomes of individual test cases. Furthermore, we can use BabelCode to build multi-lingual executiuon based benchmarks from existing mono-lingual datasets. We demonstrate this functionality by creating a new dataset called Translating Python Programming Puzzles (TP3) from the Python Programming Puzzles (Schuster et al., 2021) benchmark, where the objective is to translate expert-level python programs to other languages. The source programs for TP3 are the hand-crafted verification functions for each problem in P3. As the authors hand-wrote each function, they are significantly more complex than the current state-of-the-art code translation benchmarks, such as Transcoder (Roziere et al., 2020), for which code LLMs are already achieving highly impressive results.
Our presented framework is closely related to the concurrent work of MBXP (Athiwaratkun et al., 2022) and Multi-PLE(Cassano et al., 2022). While MBXP is quite similar to BabelCode, it is not open-sourced and requires that the input benchmarks be in Python. Multi-PLE is open-sourced, but only supports generation tasks and contains significant errors in multiple languages. BabelCode addresses these issues through an extensive test suite that ensures that the code generated is correct, and that crucial functionality, such as data structure equivalence, works when executed.
With the BabelCode framework, we investigate remedies to the problems of programming language imbalance. We utilize the Unimax algorithm (Anonymous, 2023) to limit the maximum number of times to duplicate a language's data to a constant \(N\). We then train 1B, 2B, and 4B parameter decoder-only models on both the natural and Unimax \(N\) distributions. We utilize the UL2 (Tay et al., 2022) and causal language modeling training objective. We find that models trained on the balanced dataset significantly outperform the baseline models on low-resource languages across all tasks. Further, we find that the resulting performance drop on high-resource languages is mitigated by increasing the model size.
This paper makes the following key contributions:
* We propose and release BabelCode, a new execution-based evaluation framework that allows for multilingual evaluation of code generation and translation capabilities of code language models. It also supports the easy addition of new benchmark tasks and execution-based metrics.
* We show that the code language models trained on the natural distributions of GitHub source code have poor performance on low-resource languages in both generation and translation tasks.
* We propose a new data balancing strategy for programming languages to improve performance on low-resource languages. We demonstrate that the resulting models outperform the baseline models across all tasks by an average of 12.34% \(pass@k\) for all languages, with a further improvement of 39.70% \(pass@k\) to low-resource languages.
* We find that the average improvements on low-resource languages from training on balanced data do not scale with model size. But scaling model sizes significantly helps the average \(pass@k\) loss compared to the baselines on high-resource languages going from a loss of 39.70% with the 1B model to a loss of 2.47% with the 4B model.
## 2 The BabelCode Framework
BabelCode enables the evaluation of a collection of problems, each consisting of a prompt and a set of test cases, in any language through four stages: 1) represent each test case in our domain specific language (DSL) defined in Figure 2, 2) use this generic form to generate the test cases in the target language from the input and output values, 3) use a Jinja2 template to generate a testing script in the target language, and 4) execute the target script through the command line. This is done autonomously, requiring minimal human
Figure 1: Overview of this work’s contributions.
intervention. We provide an overview of how an example problem is translated in Figure 7.
### Framework Design
BabelCode shares many design similarities to the concurrent work from Athiwaratkun et al. (2022). Specifically, we follow the same approach to inferring argument and return types. We follow the respective documentation and tutorials for each language to determine which native types to use. We also use these docs to determine the docstring formatting and naming convention. These mappings are used to generate unit and integration tests for each language automatically. They ensure that each language's implementation is syntactically correct and that, when executed, the equality comparison is correct.
**DSL Representations:** Using a DSL in the first phase, we do not force the inputs to be Python, thus enabling more flexibility to represent more generic tasks. For example, given the inputs from two test cases: {"a": [[1],[],[80]]} and {"a": []}, we only represent the _types_ in our generic DSL. Thus, the resulting type string for this input is map<string;list<integer>>. We do not represent the actual values in the generic form as we can easily translate literals across languages. This allows users to create a dataset from any language by requiring that they only represent the types of the inputs and outputs in this generic form. The language agnostic nature of the DSL enables future extensions of BabelCode to incorporate complex inputs and outputs such as custom data-structures. For example, the representation of a node class in a BST could be BSTNode<integer;integer>.
**Equality Checking:** We support floating point equivalence to a precision of \(\epsilon=1\mathrm{e}{-6}\) for floats and \(\epsilon=1\mathrm{e}{-9}\) for doubles. To determine if a given value is a float or a double, we count the number of digits after the decimal place. We apply this same logic to int and long by counting the total number of digits. Languages such as C# do not, by default, support deep equivalence of data structures. In such cases, we serialize the objects to JSON and check that the resulting strings are equal. Otherwise, we use the language built-in deep equality functionality.
**Test Statement Execution:** We opt to print the result of each test case (i.e. TEST-0...PASSED) to the standard output in a parseable format across all languages. Along with try-catch blocks, this allows the evaluation of _every_ test case for a given problem. This allows finer analysis of individual programs when compared to using assert statements as it identifies if specific corner cases fail.
**Prompt Translation:** As Wang et al. (2022) showed, LLMs are sensitive to the input prompts for code generation. Therefore BabelCode supports prompt translation and construction for multiple different problem formulations. We replace the names of languages, such as Python, with the target language. We use the language-specific naming convention to properly format the signature in the best practice style. If an argument uses a reserved keyword, we append arg to its name so that it retains the same meaning but will no longer conflict. We replace Python-specific terms with their equivalent names in the target language. For tasks formulated as code-completion, we support formatting the problem description as a native docstring. We do _not_ translate the import statements in the header. Instead, we exclude the headers from all languages to provide a language-agnostic format.
### Differences To Prior Works
We summarize the high-level differences between BabelCode and prior works in Table 1. The **MBXP** framework from Athiwaratkun et al. (2022) is the most similar to our
\begin{table}
\begin{tabular}{l|c c c c c c c c} \hline \hline & Open & \# & NL2C & C2C & Mem. \& Test & Indiv. Test & Lang. Agnostic \\ Name & Sourced & Lang. & Support & Support & Time Metrics & Suite & Case Results & Datasets \\ \hline MultiPL-E & \(\checkmark\) & 18 & \(\checkmark\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\ MBXP & \(\times\) & 10 & \(\checkmark\) & \(\checkmark\) & \(\times\) & \(\times\) & \(\checkmark\) & \(\times\) \\ BabelCode & \(\checkmark\) & 14 & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Differences between BabelCode and prior works. NL2C is natural language to code, while C2C is code to code datasets. BabelCode has an extensive test-suite that automatically tests each language’s implementation and correctness when executed.
Figure 2: BabelCode’s domain specific language for representing the input and output types of a question. Prior works require that the source dataset be written in Python, while our DSL removes this restriction and allows users to create datasets in _any_ language. This enables seamless additions of new languages while simplifying future expansions to features such as custom data structures.
work as discussed in subsection 2.1. Similar to BabelCode, MBXP does have individual test-case results; however, it uses assert statements and thus can only determine the first test-case that fails. MBXP does use language experts to review the generated code's quality and discuss the validation it supports to ensure that generated code parses and/or compiles for its respective language. BabelCode also has this functionality but, additionally, it ensures correctness through a test suite that covers the execution of generated code. We provide scripts to allow validating that source solutions to a dataset pass the generated code. For languages that do not have a solution in the respective dataset, we generate "mock" predictions that return the expected output type. This allows us to ensure that generated code is correct in _all_ supported languages even if no solution exists.
The **MultiPL-E** framework from Cassano et al. (2022) supports 18 languages compared to BabelCode's 16. However, we support four datasets, while MultiPL-E only currently has support for two datasets. In addition, BabelCode also supports fine-grained evaluation metrics for memory, running time, and individual test cases. Our extensive test suite and validation scripts have also exposed many language-specific idiosyncrasies that naive methods of translation fail to handle. For example, in Julia, any "\(\hat{s}\)" will be treated as string interpolation, even if it is in a docstring. Thus, in the majority of cases, these must be escaped. We automatically rename variables that use reserved keywords. In languages such as C#, the \(==\) operator checks equivalence by _reference_ instead of _value_. Besides corner cases, our DSL and templates allow us to effectively implement proper floating point equivalence for problems that return a float. Finally, in many languages, MultiPL-E uses types that are _not_ considered best practice, such as in Scala, where it relies on the Java types ArrayList instead of the native List.
## 3 Low-Resource Code Language Models
Because the data availability can vary greatly by programming language, we can consider the goal of building a multilingual code model as a data-imbalanced multi-task learning problem. Previous work in the multilingual natural language community (Conneau et al., 2019; Arivazhagan et al., 2019) and in the program synthesis space (Ahmad et al., 2021) have used sampling strategies relying on temperature-scaling. In this work, we use the Unimax (Anonymous, 2023) strategy to address this imbalance. The Unimax algorithm assumes that we are given a budget of how many examples we plan to consume during training and a maximum number of times, \(N\), any single example can be duplicated in the training corpus. Then, we separate the data into buckets by programming language and add \(N\) epochs of each of the lowest-resource languages until we can safely distribute the remaining budget across all the remaining languages without exceeding \(N\) epochs over any one of these remaining languages. This will allow us to control the number of epochs \(N\) we perform over the low-resource languages to minimize overfitting while allowing fair distribution of the compute budget to the remaining high-resource languages. We will ablate the choice of \(N\) in our experiments.
## 4 Experimental Setup
### Models
To understand the impact of training decoder-only models on the different programming language distributions, we train models in 3 sizes: 1B, 2B, and 4B. For each of these sizes, we train 5 different models on each distribution: Natural and Unimax \(N\), where \(N\in\{1,2,3,4\}\). The parameters and training differences are listed in Table 2. We follow Chowdhery et al. (2022) for all other architecture choices. Every model has a context window of 2048 and is trained identically with the same vocabulary described in subsection 4.3. We use a base learning rate of 0.01 and a constant warmup with a step inverse decay. The number of warmup steps is kept to 10% of the total training steps per model. The total number of training steps is 38000, 77000, 190000 for the 1B, 2B, and 4B models, respectively. We use the Adafactor optimizer (Shazeer and Stern, 2018) and a batch size of 256. We prepend [code] to the beginning and add the tag [eod] to the end of each file from our training data. Finally, we use the T5X and SeqIO (Roberts et al., 2022) frameworks. We use the UL2 (Tay et al., 2022) objective with an additional causal language modeling objective as described in Appendix A.
### Training Data
Our curated source code corpus was obtained by collecting publicly available code data on the web using a custom code data collection system. We apply a similar license filter as Kocetkov et al. (2022) to remove any files with non-permissible licenses, use simple heuristics to filter out low-quality code and apply near-deduplication to obtain our corpus of high quality, permissive source code. After preprocessing, we select 14 programming languages by their
Figure 3: Different distributions for Unimax with different budgets.
file extensions according to the mapping used by GitHub's Linguist library3 to segment the dataset by language. To calculate the number of examples per language, we use SeqIO's caching feature and take the number of examples after post-processing (Roberts et al., 2022). We list the percentages of all examples and file extensions used per language in Appendix B. With these numbers, we consider the top 7 languages to be **high-resource**(HR): Java, Python, C++, PHP, TypeScript, JavaScript, and Go. We further consider the bottom 7 languages to be **low-resource**(LR): Dart, Lua, Rust, C#, R, Julia, and Haskell.
Footnote 3: [https://github.com/github/linguist/](https://github.com/github/linguist/)
### Vocabulary
The original PaLM (Chowdhery et al., 2022) vocabulary focuses on multilingual natural language. In contrast, we trained our SentencePiece (Kudo and Richardson, 2018) vocabulary with 64k tokens from the training data directly. Each programming language is uniformly sampled to build the vocabulary. In previous works, such as Chen et al. (2021), a list of tokens that consists of a different number of whitespace is manually added to represent code more efficiently. In our work, we rely on the SentencePiece model to learn the whitespace tokens by allowing extra whitespace tokens and whitespace-only tokens. In the end, the model can represent up to 12 whitespaces into one token. In addition, numbers are split into individual tokens.
### Benchmarks
BabelCode currently supports 4 datasets. To allow the translation of any dataset to any language, we modify each benchmark as well as remove problems that were incompatible. These changes are described in Appendix C. For HumanEval (Chen et al., 2021), MBPP (Austin et al., 2021), and Transcoder (Roziere et al., 2020), we add the prefix **BabelCode- (BC)** to indicate that we are using the BabelCode specific version. Further, for Transcoder, we use the same version as in Chowdhery et al. (2022). **BC-HumanEval (BC-HE)** has 161 out of the original 164 HumanEval questions. **BC-MBPP** has 855 of the original 999 questions. **BC-Transcoder (BC-TC)** has 524 of the original 560 questions.
We additionally introduce a new dataset called **Translating Python Programming Puzzles (TP3)**. We take the verification functions from the questions in the original Python Programming Puzzles dataset (Schuster et al., 2021) to create this dataset. These functions are hand-crafted by the authors and are used to check if an answer satisfies the constraints of the puzzle. These puzzles range in difficulty from basic character checking to competitive programming problems. Thus, each verification function is written by an expert python programmer and requires a significant understanding of programming to translate. In total, there are 370 python functions to translate. Examples from TP3 can be found in subsection C.4.
### Evaluation
For BC-HumanEval, we follow Chen et al. (2021) and generate 200 programs per problem. Further, we use a zero-shot prompt described in subsection D.1. We use the built-in docstring translation of BabelCode. We generate 50 programs per problem on our three translation tasks and use the prompts described in subsection D.2. We consider these prompts zero-shot as we do not provide any additional examples. However, we provide the translated signature without the docstring in the prompt. We do not consider this to be data leakage as it is trivial to translate signatures with libraries such as Treesitter4.
Footnote 4: [https://tree-sitter.github.io/tree-sitter/](https://tree-sitter.github.io/tree-sitter/)
For every dataset, we use \(T=0.8\), \(top_{p}=0.95\), and do not use \(top_{k}\). We use the \(pass@k\) estimator (Chen et al., 2021) to measure the performance. We use \(k=100\) and \(k=25\) for generation and translation, respectively.
## 5 Results
### Baseline Models
We report the baseline results for our trained models and PaLM-Coder in Figure 4. On BC-HumanEval, we find that the 2B model has a better \(pass@100\) than that of PaLM-Coder 8B on all but C# and Python. On average, the BC-2B model trained on the natural distribution of GitHub data has average improvements of 48.17% compared to PaLM-Coder 8B despite having a quarter of the number of parameters and training on 6.4B fewer code tokens. Further, we find that the 4B model outperforms PaLM-Coder 62B on 6 of the 14 languages evaluated. This likely results from the 4B model seeing over 53B tokens more than what PaLM-Coder
\begin{table}
\begin{tabular}{c|c|c|c|c} \hline & \# of & & & Train \\ Model & Layers & Heads & \(d_{model}\) & Tokens(B) \\ \hline BC 1B & 16 & 8 & 8192 & 20.2 \\ BC 2B & 24 & 16 & 10240 & 40.4 \\ BC 4B & 26 & 16 & 14336 & 100 \\ \hline PC 8B & 32 & 16 & 4096 & 46.8 \\ PC 62B & 64 & 32 & 8192 & 46.8 \\ \end{tabular}
\end{table}
Table 2: Hyperparameters for models trained (BC) compared with those used to train PaLM-Coder(PC). For PaLM-Coder, we report the number of code tokens trained on. Each BC model is trained on each of the naturally occurring distributions of the GitHub data and each of the distributions is detailed in section 3 where \(N\in\{1,2,3,4\}\)
62B did. Another likely factor in this discrepancy is that the data PaLM-Coder was fine-tuned on included all languages on GitHub in contrast to our filtered training dataset.
We also observe that performance on languages do not scale with respect to their resource level nor the model's size. C#, Dart, Julia, and Haskell have significantly higher gains when scaling to 4B model size when compared to the other languages. While this may be due to the increased number of training tokens, it is not consistent across all LR languages as the increase in performance for R and Lua when scaling from 1B to 2B is similar to that when scaling from 2B to 4B. Instead, this result is likely due to better transfer from languages such as Java, Python, and C++.
The importance of scale for multi-lingual code models is further demonstrated by the results of the translation tasks. We find that in BC-TP3, the 1B and 2B models' performance is similar. However, the most significant gains are from scaling up to 4B where it beats PaLM-Coder 8B on all but three languages in this zero-shot translation. We do make note, though, that while we do not provide any examples for in-context learning, we do provide the signature in the target language during generation. This finding is less pronounced in BC-Transcoder as the scaling observed in all languages is more akin to that seen in BC-HumanEval.
### Impact of Balancing Programming Languages
Figure 5 shows the mean \(pass@k\) scores of different models trained on each of the 5 distributions for each of the 4 datasets. As expected, the natural distribution is optimal if the focus is solely HR languages as the performance losses when training on Unimax balanced data are 15.47%, 14.00%, and 9.35% for the 1B, 2B, and 4B models, respectively. However, for any LR language, Unimax is clearly better given that there is an average \(pass@100\) improvement on these languages of 111.85%, 68.38%, and 19.22% for the 1B, 2B, and 4B size models, respectively. For generation tasks, we find that \(N=3\) is optimal with respect to the difference between performance gained on LR and performance lost on HR languages. On the 1B, 2B, and 4B models, the ones trained on the Unimax 3 dataset had differences of 130.17%, 87.80%, and 36.00%, respectively.
We observe similar scaling trends on TP3, as training on a Unimax distribution yielded average \(pass@25\) improvements to LR languages of 124.45% for the 1B model, 64.51% for the 2B model, and 51.29% for the 4B model when compared to the same sized models trained on the natural distribution. Unlike BC-HumanEval, training the 4B on Unimax Distributions yielded _better_ average HR performance with an increase of 6.80%. As shown in Figure 6, training a 4B model on the Unimax 2 distribution had a mean \(pass@25\) improvement of 71.59% in LR languages and an improvement of 20.31% on HR languages when compared to the natural distribution. Training on other Unimax distributions does not see as large of improvements. For the 4B model, we find mean LR improvements of 42.39%, 52.91%, and 38.26% when trained on the Unimax 1, 3, and 4 distributions, respectively. This indicates that for TP3, at least, balancing the training data for each language improves translation capabilities. However, less Python data
Figure 4: Comparison of the models trained with PaLM-Coder models. For each dataset, we use Chen et al. (2021)\(pass@k\) estimator with \(n=2*k\). We then generate \(n\) samples per problem with \(T=0.8\). Full results can be found in Appendix E. Languages in the X-Axis are sorted from high to low resource. HS is Haskell, JS is JavaScript, Py is Python, and TS is TypeScript.
adversely affects understanding the source code necessary to translate it properly.
When evaluated on BC-Transcoder, we find that LR performance _increased_ with size. When the source language is C++, training on the Unimax distributions yielded an average \(pass@25\) improvements of 7.57%, 6.76%, and 11.80% for the 1B, 2B, and 4B models, respectively. Translating Python to other languages followed this trend with an average change of -26.04%, 15.1%, and 22.47% for the 1B, 2B, and 4B models, respectively. On BC-Transcoder, we find similar benefits when translating from Python to other languages, although the performance on higher resource languages is significantly worse. When translating from C++ to other languages, we find that training both a 1B and 2B model on the UM 4 distribution improves performance on 5 of the 7 LR languages. For 4B sized models, the UM 2 distribution is optimal as LR performance increased by an average of 20.47% when compared to training on the natural distribution. As the source code of BC-Transcoder focuses on language-agnostic algorithm implementations, this scaling trend is most likely due to the importance of a surface-level understanding of the target language. Further, the fact that this trend does not appear for BC-HumanEval or TP3 indicates that neither model size nor duplication of language data enables the model to have a deep understanding of these low-resource languages.
### Qualitative Effects Of Language Balance
We find that, as is expected, decreasing the number of tokens for a language negatively impacts its performance on that language. To compare the overall effects of language balancing at each size, we focus on the Unimax 1 and Unimax 2 distributions as they represent the largest change in proportions of HR languages when compared to the Nat
Figure 5: Effects of scale on the average \(pass@k\) of the high and low resource languages for each of four datasets. Full tabulated results are located in Appendix E.
Figure 6: Mean relative difference of \(pass@k\) for each of the models trained on the different Unimax distributions compared to the \(pass@k\) of the same sized model trained on the Natural distribution. The X-Axis is the language sorted from high to low resource. HS is Haskell and Py is Python. The percent changes for each delta for HR languages are shown in Table 12 and Table 13 for LR languages.
ural distribution. Figure 8 shows that on BC-HumanEval, training on either UM 1 or UM 2 will cause the model to generate fewer correct solutions than when the model is trained on the Natural distribution with respect to HR languages. However, this is _not_ due to those models generating more programs with either compilation or run-time errors as the raw average increase is only 0.40 and 1.15 for the models trained on the Unimax 1 and Unimax 2 respectively. Rather, we find that the largest decrease is in the mean % test cases passed per problem. Training on the Unimax 1 and Unimax 2 distributions results in 5.50% and 9.09% fewer test cases respectively when compared to the model trained on the natural distribution.
On LR languages, the Unimax 1 distribution yielded the best improvements compared to the other distributions. Specifically, the programs generated by the model trained on the Natural distribution passed, on average, 5.13% of the test cases per problem. In comparison, 9.53% and 10.48% of average test cases per problem were solved by the models trained on the Unimax 1 and Unimax 2 distributions. The less than 1% improvement when going from Unimax 1 to Unimax 2 suggests that, for generation tasks, multi-lingual models of code benefit the most from seeing unique data.
In our translation task of TP3, we observe consistent improvements in the mean number of test cases passed for both HR and LR languages. For the former, we observe an average improvement of 2.58% and 3.06% compared to the Natural distribution for the UM 1 and 2 distributions respectively. On LR languages, we find average improvements of 3.40% and 4.99% over the Natural distribution for the UM 1 and UM 2 distributions respectively. These results, along with the performance improvements discussed in subsection 5.2, indicate that translation tasks benefit highly from uniformly balanced languages. This is, likely, due to the task formulation where natural language understanding is not necessary. Higher resource languages are more likely to contain diverse natural language and code pairs due to the language's popularity.
Thus, performance on NL2Code tasks, such as BC-HumanEval, depends on the unique samples of code and doc-strings in the training corpus. Translation, on the other hand, does not have this constraint. Rather, it appears that uniformly balancing languages is the optimal strategy for this task.
## 6 Related Works
**Code Evaluation** Existing code benchmarks have primarily focused on surface matching evaluation (Lu et al., 2021; Yin et al., 2018; Wang et al., 2022b; Husain et al., 2019). Recent works have introduced new execution-based benchmarks for both generation (Austin et al., 2021; Hendrycks et al., 2021; Chen et al., 2021; Lai et al., 2022) and repair (Yasunaga and Liang, 2021) tasks, however, these have been limited to only Python. Additional works have introduced generation (Li et al., 2022) and translation (Roziere et al., 2020) tasks in multiple-languages, but are limited to only C++, Java, and Python. We acknowledge concurrent works by Cassano et al. (2022) and Athiwaratkun et al. (2022) on translating HumanEval and MBPP into multiple programming languages. As we note in subsection 2.2, BabelCode supports deeper analysis on a wider range of tasks while including significant methods for ensuring correctness.
**Code LLMs** Recent years has seen significant interest in LLMs for code. CodeBERT (Feng et al., 2020) is the first work to train an encoder only model on code. CodeT5 (Wang et al., 2021), PLBART (Ahmad et al., 2021), and additional works (Clement et al., 2020; Orlanski and Gittens, 2021; Chakraborty et al., 2022) examine training encoder-decoder models on code. Similar to this work, Ahmad et al. (2021) investigate difference data balancing strategies for pre-training. Our work differs in that we focus on balancing many programming languages in pre-training data. AlphaCode (Li et al., 2022), Codex (Chen et al., 2021), PaLM (Chowdhery et al., 2022), and other works (Nijkamp et al., 2022; Fried et al., 2022; Allal et al., 2023; Christopoulou et al., 2022) have shown that decoder-only code language models achieve exceptional performance on a wide range of tasks. Additional works have investigated different training strategies (Roziere et al., 2020; Bavarian et al., 2022) and different pre-training data (Roziere et al., 2021; Orlanski et al., 2022; Austin et al., 2021).
**Language Balancing** Choosing a proper sampling distribution from a mixture of datasets of various size is a difficult problems. Initial attempts at studying this in the multilingual natural language processing literature relied on temperature-based approaches (Conneau et al., 2019; Arivazhagan et al., 2019). These approaches oversample the low-resource tasks and downsample the high-resource ones. Other works have adopted more dynamic approaches, adapting the sampling rates in an online fashion during training (Wang et al., 2020).
## 7 Conclusion
We proposed the BabelCode framework for multi-lingual execution-based evaluation and a new strategy for balancing programming language distributions. We highlight the ease of creating new benchmarks with BabelCode by proposing the Translating Python Programming Puzzles. Our experiments demonstrate that adjusting how much we oversample low-resource languages and downsample high-resource languages greatly improves low-resource performance with minimal impact to to the performance of high-resource languages in tasks involving either a single or multiple programming language. By open-sourcing BabelCode, future
work can investigate improved balancing strategies along with new multi-lingual programming language questions.
## Acknowledgements
We thank Michael Janner, Owen Lewis, Alex Polozov, Uros Popovic, Deviet Roy, Tal Schuster, and Charles Sutton for their helpful discussions and feedback on the paper.
| ```
現在、ニューラルコードモデルを評価するためのベンチマークは、プログラミング言語のほんの一部に焦点を当てています。GoやRustのような人気言語を排除しています。この問題に対処するため、私たちは、 BabelCodeフレームワークを提供しています。BabelCodeは、任意の言語でベンチマークを評価するための実行ベースの評価を行います。BabelCodeは、モデルのメモリ、実行時間、個別テストケースの結果の質的な性能に関する新しい調査に役立ちます。さらに、Pythonプログラムパズル (Schuster et al. 2021) のベンチマークから、Pythonプログラムパズル (TP3) という新しいコード翻訳データセットを提案します。TP3は、専門的な Python 関数をどの言語にも翻訳するものです。BabelCodeと TP3ベンチマークを用いて、トレーニングデータの言語分布のバランスが、低リソース言語の巨大言語モデルの性能にどのように影響を与えるかを調査しました。バランスの |
2306.02755 | Quantum operations with the time axis in a superposed direction | In the quantum theory, it has been shown that one can see if a process has
the time reversal symmetry by applying the matrix transposition and examining
if it remains physical. However, recent discoveries regarding the indefinite
causal order of quantum processes suggest that there may be other, more general
symmetry transformations of time besides the complete reversal. In this work,
we introduce an expanded concept of matrix transposition, the generalized
transposition, that takes into account general bipartite unitary
transformations of a quantum operation's future and past Hilbert spaces,
allowing for making the time axis definitely lie in a superposed direction,
which generalizes the previously studied `indefinite direction of time', i.e.,
superposition of the forward and the backward time evolution. This framework
may have applications in approaches that treat time and space equally like
quantum gravity, where the spatio-temporal structure is explained to emerge
from quantum mechanics. We apply this generalized transposition to investigate
a continuous generalization of perfect tensors, a dynamic version of tracing
out a subsystem, and the compatibility of multiple time axes in bipartite
quantum interactions. Notably, we demonstrate that when a bipartite interaction
is consistent with more distinct local temporal axes, there is a reduced
allowance for information exchange between the two parties in order to prevent
causality violations. | Seok Hyung Lie, M. S. Kim | 2023-06-05T10:20:59 | http://arxiv.org/abs/2306.02755v3 | # Quantum operations with the time axis in a superposed direction
###### Abstract
In the quantum theory, it has been shown that one can see if a process has the time reversal symmetry by applying the matrix transposition and examining if it remains physical. However, recent discoveries regarding the indefinite causal order of quantum processes suggest that there may be other, more general symmetry transformations of time besides the complete reversal. In this work, we introduce an expanded concept of matrix transposition, the generalized transposition, that takes into account general bipartite unitary transformations of a quantum operation's future and past Hilbert spaces, allowing for making the time axis definitely lie in a superposed direction, which generalizes the previously studied 'indefinite direction of time', i.e., superposition of the forward and the backward time evolution. This framework may have applications in approaches that treat time and space equally like quantum gravity, where the spatio-temporal structure is explained to emerge from quantum mechanics. We apply this generalized transposition to investigate a continuous generalization of perfect tensors, a dynamic version of tracing out a subsystem, and the compatibility of multiple time axes in bipartite quantum interactions. Notably, we demonstrate that when a bipartite interaction is consistent with more distinct local temporal axes, there is a reduced allowance for information exchange between the two parties in order to prevent causality violations.
## 1 Introduction
The arrow of time has been one of the central topics of physics. Although the unique direction of time propagation from the past to the future is so natural for us living in the macroscopic and classical world, the fundamental laws of nature governing the microscopic world appear to be symmetric with respect to the reversal of time direction. There are some attempts to explain the emergence of the arrow time with thermodynamics argument in the classical realm.
Recently, Chiribella and Liu studied the time reversal symmetry of quantum processes [1]. In Ref. [1], it is shown that the most appropriate mathematical representation of the input-output reversion is the matrix transposition, and the quantum processes that are consistent with both directions of time propagation correspond to unital quantum channels. However, the input-output reversion may not be the most general symmetry transformation of the temporal structure of quantum processes considering recent developments in indefinite causal structures of quantum processes [2].
Especially, in some approaches to quantum theory of spatio-temporal structure of the universe like the quantum gravity theory, the spacetime is also treated on the same footing with other quantum objects. In such theories, the existence of the unique flow of time is not assumed. Some approaches explain that the time emerges only after some subspace of the whole Hilbert space of the universe is identified as a 'clock' to provide quantized time parameter [3]. In this picture, there is no immediate reason to expect that there is a unique well-defined direction of time obeyed by every quantum system in the universe, as there is an ambiguity in the choice of a clock system, known as the clock ambiguity [4, 5, 6]. In other words, when interpreted as quantum systems, the distinction between future and past systems is not so clear, and the partition between them need not be unique.
These observations suggest the possibility of altering the direction of temporal direction, not just within a given axis -forward and backward or their superpositions as considered in Ref. [1, 2]- but also through the
transformation of the direction of temporal axis _itself_. In this work, we develop a generalization of the approach of Chiribella and Liu [1] by introducing the _generalized transposition_, which generalizes the conventional matrix transposition and study its applications and implications in various contexts such as tensor network picture of quantum events, perfect tensors and information exchange within bipartite quantum interactions.
### Notation
Without loss of generality, we sometimes identify the Hilbert space \(H_{X}\) corresponding to a quantum system \(X\) with the system itself and use the same symbol \(X\) to denote both. We will denote the dimension of \(X\) by \(|X|\). For any system \(X\), \(X^{\otimes n}\) represents the tensor product of \(n\) copies of \(X\), and when we need to refer to one copy of it, we denote it by \(X^{\prime},X^{\prime\prime}\) etc. In other words, \(X^{\prime}\) is a copy of \(X\) with the same dimension, i.e., \(|X|=|X^{\prime}|\). When there are many systems, all the systems other than \(X\) are denoted by \(\bar{X}\). However, the trivial Hilbert space will be identified with the field of complex numbers and will be denoted by \(\mathds{C}\). The identity operator on system \(X\) is denoted by \(\mathds{1}_{X}\) and the maximally mixed state is denoted by \(\pi_{X}=|X|^{-1}\mathds{1}_{X}\). The space of all bounded operators acting on system \(X\) is denoted by \(\mathfrak{B}(X)\), the real space of all Hermitian matrices on system \(X\) by \(\mathfrak{H}(X)\). The set of all unitary operators in \(\mathfrak{B}(X)\) is denoted by \(\mathfrak{U}(X)\). For any \(M\in\mathfrak{B}(X)\), we let \(\text{Ad}_{M}\) be \(\text{Ad}_{M}(K):=MKM^{\dagger}\). For any matrix \(M\), \(M^{T}\) is its transpose with respect to some fixed basis, and for any \(M\in\mathfrak{B}(X\otimes Y)\), the partial transpose on system \(X\) is denoted by \(M^{T_{X}}\). The Schatten \(p\)-norm of an operator \(X\) is defined as \(\left\|X\right\|_{p}:=\operatorname{Tr}\bigl{[}(X^{\dagger}X)^{p/2}\bigr{]}^{ 1/p}=\{\sum_{i}(s_{i}(X))^{p}\}^{1/p}\) where \(s_{i}(X)\) is the \(i\)-th largest singular value of \(X\). The (Uhlmann) fidelity between two quantum states is defined as \(F(\rho,\sigma):=\left\|\sqrt{\rho}\sqrt{\sigma}\right\|_{1}^{2}\).
The space of all linear maps from \(\mathfrak{B}(X)\) to \(\mathfrak{B}(Y)\) is denoted by \(\mathfrak{L}(X,Y)=\mathfrak{B}(\mathfrak{B}(X),\mathfrak{B}(Y))\) and we will use the shorthand notation \(\mathfrak{L}(X):=\mathfrak{L}(X,X)\). The set of all quantum states on system \(X\) by \(\mathfrak{S}(X)\) and the set of all quantum channels (completely positive and trace-preserving linear maps) from system \(X\) to \(Y\) by \(\mathfrak{C}(X,Y)\) with \(\mathfrak{C}(X):=\mathfrak{C}(X,X)\). Similarly we denote the set of all quantum subchannels (completely positive trace non-increasing linear maps) by \(\tilde{\mathfrak{C}}(X,Y)\) and \(\tilde{\mathfrak{C}}(X):=\tilde{\mathfrak{C}}(X,X)\). We denote the identity map on system \(X\) by \(\text{id}_{X}\). For any completely positive map \(\mathcal{N}=\sum_{i}\text{Ad}_{K_{i}}\), we define its transpose as \(\mathcal{N}^{T}:=\sum_{i}\text{Ad}_{K_{i}^{T}}\).
\(J_{XX^{\prime}}^{\mathcal{N}}\) is the Choi matrix of \(\mathcal{N}\in\mathfrak{L}(X)\) defined as \(J_{XX^{\prime}}^{\mathcal{N}}:=\mathcal{N}_{X}(\phi_{XX^{\prime}}^{+})\) where \(\phi_{XX^{\prime}}^{+}=\left|\phi^{+}\right\rangle\!\!\left\langle\phi^{+} \right|_{XX^{\prime}}\) is a maximally entangled state with \(\left|\phi^{+}\right\rangle_{XX^{\prime}}=\left|X|^{-1/2}\sum_{i}\left|ii\right\rangle _{XX^{\prime}}\). The mapping \(J:\mathfrak{L}(X)\rightarrow\mathfrak{B}(X\otimes X^{\prime})\) defined as \(J(\mathcal{M}):=J_{XX^{\prime}}^{\mathcal{M}}\) itself is called the Choi-Jamiolkowski isomorphism [7, 8]. Unnormalized state \(\sum_{i}\left|ii\right\rangle_{XX^{\prime}}\) will be denoted by \(\left|\Gamma\right\rangle_{XX^{\prime}}\). We call a linear map from \(\mathfrak{L}(X)\) to \(\mathfrak{L}(Y)\) a _supermap_ from \(X\) to \(Y\) and denote the space of supermaps from \(X\) to \(Y\) by \(\mathfrak{S}\mathfrak{L}(X,Y)\) and let \(\mathfrak{S}\mathfrak{L}(X):=\mathfrak{S}\mathfrak{L}(X,X)\). Supermaps preserving quantum channels even when it only acts on a part of multipartite quantum channels are called _superchannel_[2, 9, 10, 11, 12, 13, 14] and the set of all superchannels from \(X\) to \(Y\) is denoted by \(\mathfrak{S}\mathfrak{C}(X,Y)\) and we let \(\mathfrak{S}\mathfrak{C}(X):=\mathfrak{S}\mathfrak{C}(X,X)\). We say a superchannel \(\Omega\in\mathfrak{S}\mathfrak{C}(X)\) is _superunitary_ if there are \(U_{0}\) and \(U_{1}\) in \(\mathfrak{U}(X)\) such that \(\Omega(\mathcal{N})=\text{Ad}_{U_{1}}\circ\mathcal{N}\circ\text{Ad}_{U_{0}}\) for all \(\mathcal{N}\in\mathfrak{L}(X)\).
We define the 'Choi map' \(\mathbb{J}[\Theta]\in\mathfrak{L}(X\otimes X^{\prime},Y\otimes Y^{\prime})\) of supermap \(\Theta\in\mathfrak{S}\mathfrak{L}(X,Y)\) in such a way that the following diagram is commutative:
(1)
Similarly, we define the inverse of the Choi map \(\mathbb{J}^{-1}[\mathcal{N}]\in\mathfrak{S}\mathfrak{L}(X,Y)\) of a linear map \(\mathcal{N}\in\mathfrak{L}(X\otimes X^{\prime},Y\otimes Y^{\prime})\) in such a way that the following diagram commutes:
(2)
## 2 Generalized transpose
Imagine that an experimenter observes a quantum system evolving with the passage of time. The process may appear to have well-defined input and output systems for the experimenter. However, how can one be sure that the quantum system experiences the same passage of time with the classical experimenter outside of the system? This seemingly obvious question is actually highly nontrivial considering the fact that time is not a universal parameter shared by all the systems but a quantity that should be observed with a physical mean as one can see from the difficulty in constructing a satisfactory quantum clock [15, 16]. The possibility of superposition of multiple time evolutions has been studied since decades ago [17]. Especially, with the recent development of indefinite causal structure of quantum systems [18], it is evident that there are no a priori reasons to assume that a quantum process has a unique temporal axis.
Nevertheless, if an experimenter can prescribe a valid description of a given quantum process, e.g., a completely positive trace preserving (CPTP) map, or, a quantum channel, then we can conclude that at least one temporal structure, that is, the one the experimenter follows, is compatible with the given quantum process. However, by no means that should be the unique temporal structure compatible with the process. A quantum process connects input and output systems, but the distinction between them is made from the perspective of the experimenter; There could be other partitionings of the input-output joint system into an alternative input-output system pair (See FIG. 1.) One could consider it a new type of symmetry a quantum process could have. Then, a natural question follows: How can one describe the corresponding symmetry transformation? (See Sec. 3.1 for extended discussion on the necessity of studying temporally indefinite quantum states.)
In this work, we construct such a symmetry transformation by generalizing the matrix transposition. The input-output inversion of quantum operation is given as the transpose operation \(M\mapsto M^{T}\)[1], which can be understood as the rotation by \(180^{\circ}\) in tensor diagram, i.e.
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The unitary operator is the complex generalization of the orthogonal matrix, hence it generalizes the action of rotating to complex Hilbert spaces. First, we observe that we can stretch and curve the wires in (3) to transform it into
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for any \(\sigma\in\mathfrak{B}(A^{\prime})\) and \(\mathcal{N}\in\mathfrak{L}(A)\). For the sake of brevity, we will sometimes use the notation \(\mathcal{N}^{T[W]}:=\mathfrak{T}[W](\mathcal{N})\). This seemingly complicated definition of superchannel \(\mathfrak{T}[W]\) is given in this way so that \((\text{Ad}_{M})^{T[W]}=\text{Ad}_{M^{T[W]}}\). From this, one can easily see that if
\[\mathcal{N}(\rho)=\sum_{n}c_{n}K_{n}\rho K_{n}^{\dagger}, \tag{12}\]
with complex numbers \(c_{n}\in\mathds{C}\), then
\[\mathcal{N}^{T[W]}(\rho)=\sum_{n}c_{n}K_{n}^{T[W]}\rho K_{n}^{T[W]\dagger}. \tag{13}\]
One important distinction should be made at this point. Although they share the same mathematical form, the generalized transposition defined here is for quantum processes, not quantum states. Transposition acting on density matrices is important for testing NPT entanglement [15, 19], but does not necessarily have the operational meaning as the reversal of input-output systems of a quantum process.
Given our tool for describing symmetry transformations of temporal structures in quantum processes, we can now define the compatibility of a quantum process with multiple temporal structures using generalized transposition. It is a direct generalization of _bidirectional operations_ corresponding to the conventional transposition considered in Ref. [1].
**Definition 1**.: A quantum channel \(\mathcal{N}\) is compatible with a generalized transposition \(T[W]\) when \(\mathcal{N}^{T[W]}\) is also a channel.
As closed quantum systems evolve with time via unitary operations, it is considered that unitary operations are basic building blocks of time evolution of quantum systems. We immediately get the following result on the generalized transposition of unitary operations by simply observing that \(\operatorname{Tr}\circ\text{Ad}_{U^{T[W]}}=\operatorname{Tr}\) is equivalent to \(U^{T[W]\dagger}U^{T[W]}=\mathds{1}\).
**Proposition 2**.: If a unitary operation \(\mathcal{U}\) is compatible with \(T[W]\), then \(\mathcal{U}^{T[W]}\) is also a unitary operation.
Formally it is obviously possible to generalize the generalized transpose even further by letting the unitary operation \(W\) to be a general quantum channel, but we focus on unitary cases in this work. It is mainly because allowing for irreversible quantum operations seems to go against the interpretation of the generalized transpose as a coordinate transformation of future and past Hilbert spaces, not an active joint evolution of future and past systems, albeit a probabilistic implementation through quantum teleportation is possible as it is for the transposition [1].
One subclass of generalized transpositions of special interest is that of _unital generalized transpositions_. A bipartite unitary operator \(W\) has the maximally entangled state \(\sum_{i}\left|i\right\rangle\left|i\right\rangle\) as an eigenvector with eigenvalue 1 if and only if \(T[W]\) is unital, since for such \(W,\mathds{1}^{T[W]}=\mathds{1}\). Note that a generalized transposition \(T[W]\) is unital if and only if it is trace-preserving, i.e. \(\operatorname{Tr}M^{T[W]}=\operatorname{Tr}M\) for all \(M\). For example, every fractional transposition, including the usual transposition, is trace-preserving and unital. Since unital generalized transpositions preserve the identity operator, they have the operational meaning as a transformation that preserves 'no event', which is desirable for a transformation of time axis. Imagine that there exists a film that recorded no event happening at all; it is natural to expect that playing it forward, backward, or even in quantumly weird direction of time makes no difference.
One possible problem of the definition of generalized transposition is that it may be too general to represent the transformation of tensor product decomposition because there are multiple bipartite unitary operators \(W\in\mathfrak{U}(AB)\) that preserve the tensor product structure of the Hilbert space. We can observe that the nonlocal properties of \(W\) in the definition of generalized transposition \(T[W]\) correspond to the properties of \(T[W]\) as a transformation of temporal structure, as \(W\) can be interpreted as a bipartite interaction between "future" and "past" systems.
Therefore we consider the equivalence class of bipartite unitary operators that are similar through local unitary operators, i.e., \(\langle W\rangle=\{(u_{1}\otimes u_{2})W(v_{1}\otimes v_{2}):u_{1},v_{1}\in \mathfrak{U}(A),u_{2},v_{2}\in\mathfrak{U}(B)\}\), so that every unitary operator in the same class has the same nonlocal properties. Note that every operator in the same equivalence
class transforms the tensor product structure in the same way. This leaves the problem of choosing a good representative from each equivalence class, and from its desirable properties, we hope to choose a bipartite unitary operator that induces a unital generalized transposition. When is it possible?
We say that a bipartite unitary operator _preserves maximal entanglement (ME)_ when it maps at least one maximally entangled state to a maximally entangled state. This definition when combined with the definition of the equivalence class of locally similar bipartite unitary operators yields the following result.
**Proposition 3**.: There is a unital generalized transposition \(T[V]\) with \(V\in\langle W\rangle\) if and only if \(W\) preserves ME.
Notably, every two-qubit unitary operator preserves ME [20, 21] as there are always at least four maximally entangled states that form an orthonormal basis remaining maximally entangled after the action of the unitary operator. Hence, it is conjectured that every bipartite unitary operator preserves ME, even in higher dimensions [22, 23]. This conjecture can be compactly stated with the generalized transposition.
**Conjecture 4** (Ubb, [22, 23]).: For every \(W\in\mathfrak{U}(AA^{\prime}),\) there exists at least one pair \((U,V)\) of unitary operators in \(\mathfrak{U}(A)\) such that
\[U^{T[W]}=V. \tag{14}\]
Especially, there is a numerical evidence of this conjecture that there is an iterative algorithm that finds a sequence of pairs of quantum states that converge to a pair of maximally entangled states related by a given bipartite unitary operator [23]. If Conjecture 4 is true, then we can always pick a representative that yields the unital generalized transposition from each equivalence class of locally similar bipartite unitary operators. It is equivalent to that the only nontrivial effect of generalized transposition to a transformation comes from its unital part, and all the other effects can be understood as unitary operation applied before and after the transformation in question.
This conjecture, when limited to the class of controlled unitary operators, is equivalent to the following problem.
**Conjecture 5** (Ubb-Cu (Controlled unitary)).: For every set of \(d\) unitary operators \(\{U_{i}\}\) on \(d\)-dimensional Hilbert space \(\mathcal{A}\), there is an orthonormal basis \(\{\ket{\psi_{i}}\}\) of \(\mathcal{A}\) such that \(\{U_{i}\ket{\psi_{i}}\}\) is also an orthonormal basis of \(\mathcal{A}\).
One can see that this conjecture is equivalent to the UBB conjecture for controlled unitary operators of the form \(\sum_{i}\ket{i}\!\bra{i}\otimes U_{i}\) from the fact that arbitrary maximally entangled pure state must have an expression of the form of \(d^{-1/2}\sum_{i}\ket{i}\otimes\ket{\psi_{i}}\) for some orthonormal basis \(\{\ket{\psi_{i}}\}\). Namely, after the action of the unitary operator, the state is transformed into \(d^{-1/2}\sum_{i}\ket{i}\otimes U_{i}\ket{\psi_{i}}\), thus \(\{U_{i}\ket{\psi_{i}}\}\) must be an orthonormal basis, too.
When expressed in this form, it is evident that the UBB-CU conjecture is also equivalent to its classical counterpart. In other words, when it is promised that a random index \(i\) will be picked and accordingly the unitary operator \(U_{i}\) will be applied to the quantum system \(A\) which contains the memory of the index value \(i\), it is natural to conjecture that there exists a deterministic process, represented by a unitary process \(\ket{i}\mapsto\ket{\psi_{i}}\), that prepares a quantum state \(\ket{\psi_{i}}\) that retains the memory of the index \(i\) after the action of \(U_{i}\). The UBB-CU conjecture supposes that exactly such a process always exists for any set of \(\{U_{i}\}\).
One simple example is the case of the generalized transposition corresponding to the CNOT gate, i.e., \(W=\ket{0}\!\bra{0}\otimes\mathds{1}+\ket{1}\!\bra{1}\otimes X\). The Hadamard gate \(H=\ket{+}\!\bra{0}+\ket{-}\!\bra{1}\) is compatible with \(T[W]\), as one can see from
\[H^{T[W]}=XH, \tag{15}\]
where \(X\) is the Pauli-X operators. One can unitalize \(T[W]\) by substituting \(W\) with \(W^{\prime}:=W(\mathds{1}\otimes X^{1/2}H)\), so that \(\mathds{1}^{T[W^{\prime}]}=\mathds{1}\).
We remark that even if not every bipartite unitary operator preserves ME, in light of Proposition 2, we could argue that only generalized transpositions corresponding to those preserve ME are relevant for the temporal structure of quantum processes. It is because if a \(W\) does not preserve ME, then no two unitary operators are
related to each other via the corresponding generalized transposition \(T[W]\). However, if one includes the non-unitary quantum channels into the picture, then it is no a priori clear if there are no pairs of quantum channels related by a non-unital generalized transposition. We leave this problem as an open problem.
Note that the generalized transposition is basis-dependent as the conventional transposition is a basis-dependent operation. There are two layers of basis dependency for input and output systems, i.e, the choice of basis \(\{|i\rangle\}\) and \(\{|j\rangle\}\) in (8). One could interpret Choosing the unital representation locally similar to a given generalized transposition is eliminating one such basis dependency by equalizing the input and output bases.
Just as the transposition can be applied to a part of multipartite operators to define the partial transpose operation, the generalized transposition can also be applied to a part of multipartite operator. If \(M\in\mathfrak{B}(AB)\), then, for arbitrary \(W\in\mathfrak{U}(B^{\otimes 2})\), the partial generalized transposition \(T_{B}[W]\) is defined as \(T_{B}[W]:=\mathsf{id}_{A}\otimes T[W]\)
\[\tikzfig{T_{B}[W]} \tag{16}\]
Using the partial generalized transposition, we can examine the compatibility of bipartite unitary operators with multiple directions of time of a local system. Assume again that two systems \(A\) and \(B\) interact through a bipartite unitary operator \(V\in\mathfrak{U}(AB)\). This assumption alone has a couple of implications. It assumes that there are two subsystems \(A\) and \(B\) that can be localized and identified, which stays so even after the interaction. Also, it also implies that \(A\) and \(B\) appear to share the same time axis during their interaction. However, this need not be the unique description of the direction of time for each system. For example, \(B\) might also appear to evolve in the direction given by a generalized transposition \(T_{B}[W]\) in time from the perspective of \(A\). In this case, for interaction \(V\) to be consistent with the new flow of time as well, its generalized transpose \(V^{T_{B}[W]}\) also should be unitary. The same argument can be applied to general quantum channels, so we give the following definition of compatibility.
**Definition 6**.: A quantum channel \(\mathcal{N}\in\mathfrak{C}(AB)\) is compatible with a generalized transposition \(T_{B}[W]\) on \(B\) when \(\mathcal{N}^{T_{B}[W]}:=(\mathsf{i}\mathfrak{d}_{A}\otimes\mathfrak{T}_{B}[W] )(\mathcal{N})\) is also a channel.
For the case of conventional matrix transposition \(T\), bipartite unitary operators compatible with \(T\) on a subsystem is called to be t-dual or catalytic. (See Sec. 3.4 for more information.)
Finally, we examine the relation between the compatibility of a quantum process with multiple directions of time and that of its causally neighbouring processes. When seen from a broader perspective, no interactions happen isolated as they are embedded in the network of events (e.g., see FIG. 3). For example, at the very least, experimenter prepares an input state and measures the output state of a given quantum channel.
We can model the ambient quantum processes as a quantum superchannel since they map a quantum channel to another quantum channel. Therefore, when we examine the consistency of causalities, it is natural to also require the physicality of the causality of the ambient superchannel. If a quantum channel \(\Phi\in\mathfrak{C}(A)\) embedded in a superchannel \(\mathfrak{F}\) is compatible with a generalized transposition \(\mathfrak{T}[W]\), then, for this generalized transposition of \(\Phi\) to be consistent with \(\mathfrak{F}\) as well, we require that \(\mathfrak{F}\circ\mathfrak{T}[W^{\dagger}]\) is also a superchannel, because (See FIG. 2.)
\[\mathfrak{F}\circ\mathfrak{T}[W^{\dagger}]\left(\mathfrak{T}[W](\Phi)\right) =\mathfrak{F}(\Phi), \tag{17}\]
Figure 2: Compatibility of superchannel with the generalized transposition of its input channel.
and because every superchannel with one input register is guaranteed to be physically implementable with a pre-process and a post-process [9]. In other words, if one tries to re-interpret a given event in a different decomposition of the spacetime, then the events surrounding it must be consistent as well in that decomposition.
This observation severely restricts which state can be fed into a multipartite unitary operator with multiple compatible temporal axes, as the following Proposition shows.
**Proposition 7**.: A state preparation superchannel given as \(\mathfrak{P}^{\sigma}(\mathcal{N}):=\mathcal{N}(\sigma)\) is compatible with a generalized transposition \(T[W]\) of its input channel, i.e., \(\mathfrak{P}^{\sigma}\circ\mathfrak{T}[W^{\dagger}]\) is a superchannel, if and only if there exists a quantum state \(\tau\) such that
\[W(\mathds{1}_{A}\otimes\sigma_{A^{\prime}}^{T})=(\mathds{1}_{A}\otimes\tau_{ A^{\prime}}^{T})W. \tag{18}\]
Note that (18) implies that \(\tau\) and \(\sigma\) are unitarily similar. Proof can be found in Appendix. From Proposition 7, it follows that the preparation of the maximally mixed state is always compatible with an arbitrary generalized transposition of its input channel. (See Sec.3.3 for a related discussion on factorizable maps.) Especially, for the case of time inversion, corresponding to the transposition, \(W\) is the swapping gate and Proposition 7 implies that \(\sigma=\mathds{1}/\operatorname{Tr}[\mathds{1}]\), hence we have the following Corollary.
**Corollary 8**.: The only state preparation compatible with the transposition is the preparation of the maximally mixed state.
This result matches with our intuition that no knowledge can propagate backward in time, as feeding a non-maximally mixed states into a quantum process compatible with inverse evolution can lead to retrocausality. See Sec. 3.4 for the discussion on the constraint Proposition 7 imposes on the information exchange between two quantum systems through a bipartite quantum channel when there are multiple compatible local temporal directions.
## 3 Discussion
### Events in spacetime as a tensor network
In this Section, we delve into a more detailed discussion on spacetime regions with indefinite causal orders and how the generalized transposition can be used in the context. In conventional quantum mechanics, the quantum state of a quantum system \(|\psi(t)\rangle\) at time \(t\) can be written as
Figure 3: Suppose that the dynamics of a set of quantum systems is given as a tensor network. Without presupposing a spacetime structure, it may not be possible to assign the unique temporal axis of the evolution in the tensor network.
\[|\psi(t)\rangle=\left(\prod_{t>t^{\prime}\geq 0}U_{t^{\prime}}\right)|\psi(0) \rangle\,, \tag{19}\]
where each \(U_{t^{\prime}}\) is the unitary operator describing the time evolution from time \(t^{\prime}\) to time \(t^{\prime}+1\). Moreover, each \(U_{t^{\prime}}\) also can be decomposed into interaction between many subsystems located at \(x\), e.g. \(U_{t^{\prime}}=\bigotimes_{x}U_{(x,t^{\prime})}\). The dynamics \(|\psi(t)\rangle\) went through can be depicted as a tensor network resembling FIG 3 where each box is \(U_{(x,t^{\prime})}\). Therefore, once the set of unitary operators \(\{U_{(x,t)}\}\) and their connectivity are given, the dynamics of a set of quantum systems is completely decided. In other words, we can consider the dynamics of quantum systems _a net of events_ composed of unitary operator which can be interpreted as a tensor. This approach shares the same spirit with the approach known as the _event-universe_[24, 25] understanding the universe as a tree of events except that 'events' are unitary evolution, or quantum channels, in this work.
However, what if we do not assume that there is a spacetime with the familiar spatio-temporal structure? What if the existence of the universal axis of time is not given as an additional data outside of the Hilbert space of the universe? There are approaches to quantum gravity in which they treat time, which is often treated as a parameter, on the same footing with space. One of the purposes of theses approaches is to recover time as an emergent entity from quantum theory without supposing its familiar properties as the temporal parameter. Notable examples include those of Cotler [26, 27], Castellani [28, 29] and Dias [30].
We can consider the following time-neutral model of the Hilbert space of the spacetime. Suppose that there exists a 'pre-spacetime' structure \(\mathcal{S}\), which parametrizes different regions of the to-be spacetime that is not necessarily having familiar spatio-temporal properties. We suppose that the Hilbert space \(\mathcal{H}\) of a part of or the whole universe can be decomposed into smaller Hilbert spaces \(\mathcal{H}_{s}\) corresponding to each point \(s\) in \(\mathcal{S}\)
\[\mathcal{H}_{\mathcal{S}}=\bigotimes_{s\in\mathcal{S}}\mathcal{H}_{s}. \tag{20}\]
One possible model of pre-spacetime \(\mathcal{S}\) is the set of points \((\mathbf{x},t)\) in the familiar spacetime before the spatio-temporal structure is assigned. In some literature, the Hilbert space of a quantum system including its behavior through the passage of time is called the _history Hilbert space_[27, 31, 32], hence we will use the same nomenclature for \(\mathcal{H}_{\mathcal{S}}\) in (20). However, since we are yet to allocate the temporal parametric role to parameter \(t\) in \((\mathbf{x},t)\), we stick to the temporally neutral notation \(s\) for each region in \(\mathcal{S}\). Also, for the sake of simplicity, we suppose that the structure of \(\mathcal{S}\) is discrete by interpreting each \(s\) as a region in spacetime rather than a point. Working with continuous tensor product requires the full kit of Fahri-Gutmann path integral formalism [33], which goes beyond the scope of the current work. Generalization to the continuous regime is an interesting future work.
Assuming the existence of a net of events is not conceptually more demanding than other approaches [34, 35] that assume pre-existing Hamiltonian attached to each Hilbert space. It is immediate from the fact that assuming that a Hamiltonian \(H\) governs a quantum system is mathematically equivalent to assuming that the dynamics of the quantum system is described by the unitary operator \(U:=\exp\{-iHt/\hbar\}\). Nevertheless, in this work, we deem the picture of unitary operators and the corresponding quantum channels constituting the history of the universe is conceptually more clear than the picture of Hamiltonians living outside of Hilbert space yet inducing dynamics of quantum systems indirectly.
Additionally, we will work on a plausible assumption that every \(\mathcal{H}_{s}\) is finite-dimensional and isomorphic with each other. Assuming isomorphic structure amounts to assuming a sort of translational symmetry of the pre-spacetime \(\mathcal{S}\) which is usually done in cosmology. Also, there are good reasons to assume that the Hilbert space of the universe is locally finite-dimensional based on the arguments such as the finiteness of the entropy of black holes [36].
Once the history Hilbert space is defined, we assume that an event of the universe is given as a tensor \(X\in\bigotimes_{s\in\mathcal{S}^{\prime}}\mathcal{H}_{s}\) with some \(\mathcal{S}^{\prime}\subseteq\mathcal{S}\) connecting different regions of the pre-spacetime \(\mathcal{S}\). However, we are more familiar with the interpretation of an event as an operator transforms a state into another, hence, whenever we can divide the support Hilbert space of \(X\) (we will sometime call this the history Hilbert space of \(X\)) into two disjoint same-dimensional regions \(\mathcal{S}_{0}\) and \(\mathcal{S}_{1}\), we tentatively interpret \(X\) as an operator between these regions
as follows.
\[X:\bigotimes_{s\in\mathcal{S}_{0}}\mathcal{H}_{s}\to\bigotimes_{s\in\mathcal{S}_{1 }}\mathcal{H}_{s}. \tag{21}\]
This kind of identifying tensor with operator is often done in the field of quantum scrambler [37]. However, it may not be possible to interpret \(X\) as a time-evolution from \(\bigotimes_{s\in\mathcal{S}_{0}}\mathcal{H}_{s}\) to \(\bigotimes_{s\in\mathcal{S}_{1}}\mathcal{H}_{s}\), as the input and output systems can be arbitrarily chosen since \(X\) was given as a tensor in \(\bigotimes_{s\in\mathcal{S}_{0}}\bigcup_{\mathcal{S}_{1}}\mathcal{H}_{s}\) and the operator interpretation was rather arbitrary. At this stage, we only have a set of Hilbert spaces corresponding to each distinguishable points in the 'pre-spacetime' whose physical meaning is still unclear and a set of rather abstract 'events' given as tensors on the Hilbert spaces.
Now, we hope to study the spatio-temporal structure that emerges from the given network of events. This approach is particularly motivated by the recent results including the HaPPY code [38], where the correspondence between the tensor network and spacetime emerged is explored. Especially, it was shown that the correspondence with tensor network can be extended from that of space-like time slices to the whole spacetime, recently [39].
For the case of pre-spacetime \(\mathcal{S}\) without a presupposed temporal structure, there need not be a unique axis of time across the whole tensor network. As an example, one can imagine the following interaction \(X\) between four regions \(s_{i}\) in \(\mathcal{S}\).
\[\begin{array}{c}s_{2}\\ s_{3}\end{array}\]
\(s_{0}\)\(s_{1}\).
One could consider \(X\) as a tensor in \(\bigotimes_{i=0}^{3}\mathcal{H}_{s_{i}}\). However, without a presupposed temporal parameter, there is no _a priori_ reason to suppose that there is a causal order between the regions, as it is evident from that the Hilbert space \(\bigotimes_{i=0}^{3}\mathcal{H}_{s_{i}}\) has infinitely many ways to be decomposed into the tensor product of input and output spaces.
In other words, by changing the basis, the tensor \(X\) as a vector in \(\bigotimes_{i=0}^{3}\mathcal{H}_{s_{i}}\) can appear as \(UX\) with some unitary operator \(U\) on \(\bigotimes_{i=0}^{3}\mathcal{H}_{s_{i}}\). This unitary operation can nontrivially affect the operator interpretation of \(X\) (say, from \(s_{0}s_{1}\) to \(s_{2}s_{3}\)), but the description of the effect can be complicated when one is working with the operator interpretation. In the next Section, we will develop a language that can concisely describe this type transformation of operators.
To recover the temporal structure, we adopt the approach taken in the studies of quantum causal models [40] which assumes that quantum dynamics is fundamentally unitary even when the causal relationship between events is unclear. Say, if \(X\) is a unitary operator from \(s_{0}s_{1}\) to \(s_{2}s_{3}\) in (22), then one could give an interpretation to \(X\) as the time evolution from the temporally preceding regions \(s_{0}s_{1}\) to the temporally succeeding regions \(s_{2}s_{3}\). Hence, although there may not exist a unique global time axis in \(\mathcal{S}\), there still could be a "local time axis", given in terms of the decomposition of its supporting Hilbert space into "future" and "past" subsystems, allowing us to interpret each local interaction as a unitary evolution along some direction in \(\mathcal{S}\).
Of course, the inverse of a unitary operator is still unitary, there is no canonical distinction between "future" and "past" yet, which should be made by other means, e.g. the second law of thermodynamics. Therefore, the terms like axis or direction of time should be understood with this symmetry in consideration. It is somehow similar to that both \(\ket{\psi}\) and \(-\ket{\psi}\) represent the same quantum state; only the relative difference between the orientations of axes is important, as we will see later.
This type of process of finding natural decomposition of the whole Hilbert space into distinguishable'subsystems' from the given dynamics is called quantum mereology [35]. The decomposition studied in Ref. [35] was the bipartition into "system" and "environment" and the criterion was the emergence of quasiclassical behavior. Moreover, there are approaches to explain the emergence of spatial structure from the Hilbert space structure and the given dynamics alone [41, 42, 36, 43]. The goal of this work is in a similar vein, but the interest of this work is more focused on the emergence of temporal structure in microscopic dynamics, and for doing so we identify the decomposition of the history Hilbert space into future and past systems, which allows the unitary time evolution description of each interaction tensor.
We will call a tensor (or an operator that can be interpreted as a tensor) \(Y\in\mathcal{K}\) a _dynamics tensor_ if there exists some tensor product decomposition \(\mathcal{K}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}\) with \(\mathcal{K}_{1}\approx\mathcal{K}_{2}\) such that \(Y\) is unitary when understood as an operator from \(\mathcal{K}_{1}\) to \(\mathcal{K}_{2}\). In other words, \(Y\) is a dynamics tensor if it is possible to interpret it as a time
evolution with respect to some spacetime structure. The following result shows that no special type of tensor network is necessary to represent a dynamics on a pre-spacetime as long as each tensor is properly normalized.
**Theorem 9**.: Every operator \(X\in\mathfrak{B}(\mathcal{K})\) is proportional to a dynamics tensor. Especially, every \(X\) with \(\|X\|_{2}=|\mathcal{K}|^{1/2}\) is a dynamics tensor.
See Appendix for omitted proofs including that of Theorem 9. Treating unitarity as a guideline for assigning temporal order to the pre-spacetime \(\mathcal{S}\) may help explaining the emergence of temporal structure of spacetime from its tensor network structure, but still there remain some ambiguities. Especially, if \(X\) is a _perfect tensor_ considered in Ref. [38], then any bipartition of \(s_{0}s_{1}s_{2}s_{3}\) yields a unitary operator, hence it is compatible with any temporal direction direction across the diagram in (22). Thus, unitarity alone may not yield the unique direction of time.
As we discussed about the unital generalized transpositions and their probable correspondence with temporal axis transformation, now we consider the restriction of the definition of dynamical tensor where the transformation is restricted to unital generalized transpositions. We say that an operator \(X\in\mathfrak{B}(\mathcal{K})\) is a _proper dynamics tensor_ if there exists a unital generalized transposition \(T[W]\) with some \(W\in\mathfrak{U}(\mathcal{K}^{\otimes 2})\) such that \(X^{T[W]}\) is unitary.
**Theorem 10**.: Every operator \(X\in\mathfrak{B}(\mathcal{K})\) is proportional to a proper dynamics tensor if \(|\mathcal{K}|\) is even. Especially, every \(X\) with \(\|X\|_{2}=|\mathcal{K}|^{1/2}\) is a proper dynamics tensor.
Theorem 10 implies that when each subsystem is assumed to be even-dimensional, arbitrary tensor network with properly normalized tensors can be understood as a 'net of events' that each constituting tensor can bee seen as a unitary evolution operator after a 'rotation of time axis' represented by a unital generalized transposition.
This result lessens the weight of the assumption to justify the approach'spacetime as a tensor network'. According to this viewpoint, there may not be a single universal axis of time in the universe, but each subsystem could experience time as it hops around the given tensor network of dynamics tensors to whatever direction that yields the unitary evolution interpretation for the adjacent dynamics tensor. As there is no unique time axis, each subsystem may 'clash' while hopping around the tensor network in different directions. However, this does not necessarily mean that the model is ill-defined or that there is a contradiction, since by satisfying a set of conditions, the interaction between quantum systems with multiple relative configuration of axes of time can be consistent. We will discuss about those conditions in Sec. 3.4.
We remark that we do not claim that we found a canonical way to explain the emergence of the unique arrow of time from any tensor network structure of (pre-)spacetime. But we highlight the fact that there could be multiple quantum systems with multiple compatible directions of time in temporally neutral approaches to quantum gravity, and that the generalized transposition provides a mathematical tool to deal with the symmetry transformation of that structure.
This approach bears some similarity with time-symmetric operational formulation of quantum theory such as Oreshkov and Cerf [44, 45] where quantum theory is analyzed without presupposing background spatio-temporal structure. The main difference is that in this work we accept the existence of well-defined local direction of time by interpreting each event as a unitary evolution.
### Generalized perfect tensors
If we were to pursue the approach where a tensor network of events could model the structure of spacetime, then it is natural to expect that the constituting tensors are covariant under a certain class of generalized transposes as we expect physical laws at each point space are 'isotropic' in some sense if there is no _ad hoc_ temporal axis determined beforehand.
In this Section, we consider continuous generalizations of the class of tensors with particular symmetry known as _perfect tensors_. A perfect tensor \(X\) in \(\bigotimes_{i=1}^{n}\mathcal{H}_{i}\) is a tensor which is unitary for any bipartition \(A,B\) of \(\{1,2,\cdots,n\}\) into input and output nodes with \(|A|=|B|\). This definition can be re-expressed in terms of generalized transpositions. We say that \(X\) is a perfect tensor, when \(X\) is understood as a matrix for some
fixed choice of input and output nodes mentioned above, if \(X^{T[P]}\) with arbitrary permutation \(P\) of \(n\) systems is unitary.
Hence, when understood as a tensor describing dynamics in spacetime as in Section 3.1, one can say that a perfect tensor has an 'isotropic' spatio-temporal structure to some extent in the sense that even after arbitrary permutation of its nodes it stays unitary. Thus, if one intends to explain the emergence of the spacetime structure from the quantum nature of the universe, then one might think that it is desirable to assume that each dynamics tensor is perfect not to a presupposed temporal structure.
However, as we discussed in Section 3.1, even if we assume that the history Hilbert space of the universe is given as \(\mathcal{H}=\bigotimes_{s\in\mathcal{S}}\mathcal{H}_{s}\), one can always express \(\mathcal{H}\) with another tensor product structure with some \(\mathcal{Z}\), i.e. \(\mathcal{H}=\bigotimes_{z\in\mathcal{Z}}\mathcal{H}_{z}\). In this regard, the definition of perfect tensor has a shortcoming that it only considers permutation of tensor components of an already fixed tensor product structure of the Hilbert space. This type of property is similar to basis-dependent properties of vectors in the sense it depends on the choice of tensor product structure.
If we were to examine the isotropy of a tensor with respect to arbitrary unitary transformation of Hilbert space, then we must define the following object we will call a _totally perfect tensor_. A tensor \(X\in\bigotimes_{i}\mathcal{H}_{i}\otimes\bigotimes_{i}\mathcal{K}_{i}\) is totally perfect if (when understood as an operator in \(\mathfrak{B}(\bigotimes_{i}\mathcal{H}_{i},\bigotimes_{i}\mathcal{K}_{i})\)) \(X^{T[W]}\) is unitary for any unitary operator \(W\in\mathfrak{U}(\bigotimes_{i}\mathcal{K}_{i}\otimes\bigotimes_{i}\mathcal{H }_{i})\). Are there totally perfect tensors and do they form isotropic building blocks of spacetime? Or is it the case that there are no such things and the symmetry of choosing regions of spacetime is necessarily broken to some extent? We show that it is the latter.
**Theorem 11**.: There are no totally perfect tensors.
Proof.: Suppose that \(M\) is totally perfect. Then, by letting \(W=V(M^{\dagger}\otimes\mathds{1})\) for an arbitrary unitary operator \(V\), we have \(M^{T[W]}=\mathds{1}^{T[V]}\). Now, by choosing \(V\) such that \(\mathds{1}^{T[V]}\) is non-unitary, we get the desired result. One such example is the generalized controlled-CNOT gate given as \(V=\sum_{i}S^{-i}\otimes|i\rangle\!\langle i|\), where \(S=\sum_{n}|n\oplus 1\rangle\!\langle n|\) where \(\oplus\) is the modular summation operation. For such \(V\), we have \(\mathds{1}^{T[V]}=|0\rangle\left(\sum_{i}\left\langle i\right|\right)\), which is rank-\(1\), hence obviously non-unitary.
One can understand Theorem 11 as the converse result of Theorem 9. Theorem 11 shows that there is no tensor that is unitary with respect to arbitrary tensor decomposition of the ambient Hilbert space. It implies that every dynamics tensor has a set of preferred (or disfavored, to be more accurate) decomposition of the history Hilbert space, albeit it could still have some remaining ambiguity. This observation helps explaining the emergence of definite pre-spacetime structure as Theorem 11 implies that not every superposition of points in the pre-spacetime can be interpreted as a legitimate subsystem participating in physical interactions.
A naturally following question is if there are operators (tensors) that remain unitary under smaller classes of generalized transpositions. This question is natural since one might guess that there are no totally perfect tensors because the set of all generalized transpositions is too large for an operator to remain unitary after the action of them. We first consider the class of unital generalized transpositions, as they have a good property of preserving zero event. We call an operator \(M\) a _properly perfect tensor_ if \(M^{T[W]}\) is unitary whenever the generalized transposition is unital, i.e. \(\mathds{1}^{T[W]}=\mathds{1}\). We will call the operators of the form \(\alpha\mathds{1}\) with some complex number \(\alpha\) scalar operators.
**Proposition 12**.: There are no properly perfect tensors other than scalar operators.
Proof.: Assume that \(M\in\mathfrak{B}(A)\) is a properly perfect tensor that is not a scalar operator, i.e., \(M\neq\alpha\mathds{1}\) for any complex number \(\alpha\). Note that \(M\) is unitary by definition. Let us decompose \(M\) into the trace part and the traceless part. In other words, there exists a traceless operator \(S\) with \(\|S\|_{2}=|A|^{1/2}\) that allows for the following form of expansion of \(M\)
\[M=\cos\theta\ \mathds{1}_{A}+\sin\theta S, \tag{23}\]
for some real value \(\theta\) such that \(\sin\theta\neq 0\). (Further inspection reveals that \(S\) should be unitary, too.) As \(S\) is traceless, we can see that \(\left|\phi^{+}\right\rangle_{AA^{\prime}}\) and \(\left(S\otimes\mathds{1}_{S^{\prime}}\right)\left|\phi^{+}\right\rangle_{AA^{ \prime}}\) are orthogonal. This means that one can construct a unitary operator \(W\) on \(AA^{\prime}\) that maps \(\left|\phi^{+}\right\rangle_{AA^{\prime}}\) to itself and \(\left(S\otimes\mathds{1}_{S^{\prime}}\right)\left|\phi^{+}\right\rangle_{AA^{ \prime}}\) to
\((X\otimes\mathds{1}_{S^{\prime}})\left|\phi^{+}\right\rangle_{AA^{\prime}}\), where \(X=\sum_{i}|i\oplus 1(\text{mod }|A|)\rangle\!\langle i|\) is the generalized Pauli \(X\) operator. Note that \(X\) is also traceless. For such \(W\), \(T[W]\) is unital, hence we have
\[N:=M^{T[W]}=\cos\theta\ \mathds{1}_{A}+\sin\theta X. \tag{24}\]
This operator cannot be unitary since \(N^{\dagger}N=\mathds{1}_{A}+\sin\theta\cos\theta(X+X^{\dagger})\) and the second term does not vanish. It contradicts the assumption that \(\mathcal{M}\) is a properly perfect tensor, hence the desired result follows.
Proposition 12 implies that non-scalar dynamics tensors not only have disfavored spacetime structures as a whole, but also have disfavored temporal structures, if we accept the correspondence between transformations of time axis and unital generalized transpositions.
Nevertheless, as we have seen before, there are unitary operators that are also symmetric, i.e. \(U=U^{T}\) in every dimension. (Note that the direct sum of such operators is also symmetric and unitary.) They are also invariant under fractional transpositions, hence they are all unitary after arbitrary fractional transposition. Hence, we will call an operator \(M\) that remain unitary after every fractional transposition, i.e. \(M^{T(\theta)}\) is unitary for every \(\theta\), a _rotationally perfect tensor_ and summarize the observation given above as follows. Although we still lack a complete geometric interpretation of the fractional transposition, one can imagine that the rotationally perfect tensors as tensors that have a time evolution interpretation along with any axis in the space-time plane.
**Proposition 13**.: There are rotationally perfect tensors in every dimension.
### Supertrace and Factorizalbe Maps
In this Section, we introduce a mathematical tool related to the generalized transposition for modelling the loss of _dynamical quantum information_ processes. Quantum superchannels are transformations that map quantum channels to quantum channels. Its formal similarity with quantum channels enabled many results about quantum channels to be translated over to quantum superchannels, but not every component of static quantum information processes has been translated into the language of the dynamical setting.
One of such component is information loss. In the static setting, the loss of quantum information is modelled with the (partial) trace operation, and the causality of quantum operation is also formulated in terms of the trace operation. However, to the best of our knowledge, there is no analogue of trace operation for quantum channels, although it is naturally possible that one loses all the information about input and output of a quantum channel.
We propose the _supertrace_ as the superchannel counterpart of the trace operation, denoted by \(\mathfrak{T}\mathfrak{r}\). The supertrace is defined in such a way that the following diagram is commutative:
(25)
In other words, \(\mathfrak{T}\mathfrak{r}=\mathbf{J}^{-1}[\mathrm{Tr}]=J^{-1}\circ\mathrm{Tr}\circ J\). Here, we slightly abused the notations by identifying isomorphic trivial Hilbert spaces \(\mathsf{C}^{*}\approx\mathds{C}\approx\mathfrak{L}(\mathds{C})\approx \mathfrak{B}(\mathds{C}\otimes\mathds{C})\) and letting \(J:\mathfrak{L}(\mathds{C})\rightarrow\mathfrak{B}(\mathds{C}\otimes\mathds{C})\) be identified with \(\mathrm{id}_{\mathds{C}}\). Equivalently, \(\mathfrak{T}\mathfrak{r}[\mathcal{M}]:=\mathrm{Tr}\!\left[J_{XX^{\prime}}^{ \mathcal{M}}\right]=\mathrm{Tr}\!\left[\mathcal{M}(\pi_{X})\right]\) for all \(\mathcal{M}\in\mathfrak{L}(X)\). Similarly to partial trace, \(\mathfrak{T}\mathfrak{r}_{X}\) is a shorthand expression of \(\mathfrak{T}\mathfrak{r}_{X}\otimes\mathfrak{i}\mathfrak{o}_{\tilde{X}}\). It is consistent with the definition of map reduction of Ref. [46] where it was defined only for semicausal maps.
Note that the supertrace lacks a few tracial properties such as cyclicity when applied naively, i.e., \(\mathfrak{T}\mathfrak{r}[\mathcal{A}\circ\mathcal{B}]\neq\mathfrak{T}\mathfrak{ r}[\mathcal{B}\circ\mathcal{A}]\) in general on \(\mathfrak{L}(X)\). However, it generalizes the operational aspect of trace as the discarding action. For example, every quantum channel \(\mathcal{N}\) is normalized in supertrace, i.e., \(\mathfrak{T}\mathfrak{r}[\mathcal{N}]=1\). Not only that, if some linear map \(\mathcal{M}\) is a quantum channel in _some_ configuration, i.e., if \(\mathcal{M}^{T[W]}\) is a quantum channel for some \(W\), then it is still normalized, \(\mathfrak{T}\mathfrak{r}[\mathcal{M}]=1\). We leave a remark that it is unrelated to the supertrace \(\mathsf{Str}\) frequently used in the field of supersymmetry [47] or the supertrace \(\hat{\mathrm{Tr}}\) defined as an operator on endomorphism spaces [48].
We leave a remark that the normalization condition of quantum processes in Oreshkov and Cerf's generalized process theoretic approach to quantum theory without predefined time [44, 45] can be compactly expressed with supertrace. A quantum operation in their formalism is a CP map \(\mathcal{M}\) that has unit supertrace: \(\mathfrak{Tr}[\mathcal{M}]=1\).
The supertrace yields another way of marginalizing multipartite quantum channels. In other words, we can apply the supertrace to a local system of a bipartite channel to get the quantum channel on the other quantum system. It has advantage over the other definition of marginalization of quantum channel that requires the bipartite channel to be no-signalling [49] that the supertrace can be applied to any bipartite channel.
Oftentimes, quantum channels are referred to as deterministic quantum processes in the sense it preserves the trace of input state so that the transformation from the input state to the output state is guaranteed. However, a critical review of its implementation is needed to examine if it can be realized truly deterministically. The Stinespring dilation theorem [50] implies that for every quantum channel \(\mathcal{N}\in\mathfrak{C}(X)\), there exists an 'ancillary system' \(Y\) and a unitary operation \(\mathcal{U}\in\mathfrak{U}\mathfrak{O}(XY)\) such that, for every \(\rho\in\mathfrak{B}(X)\),
\[\mathcal{N}(\rho)=\mathrm{Tr}_{Y}\,\mathcal{U}(\rho_{X}\otimes|0\rangle\! \langle 0|_{Y}), \tag{26}\]
for some \(|0\rangle\) in \(Y\). We have to note that, unless \(\mathcal{N}\) is a unitary operation, \(Y\) is not a 1-dimensional system that only admits \(|0\rangle\), but a nontrivial quantum system that can be in some state other than \(|0\rangle\). Thus, one role of system \(Y\) is providing working space so that information of \(X\) can move around to produce the wanted outcome. Another role is providing _purity_. System \(Y\) is prepared in a pure state initially, so that the entropy of \(X\) can be disposed of. However, how is this pure state \(|0\rangle\) prepared? One might claim that the initialization map \(\mathcal{I}\in\mathcal{C}(Y)\) given as
\[\mathcal{I}(\sigma)=\mathrm{Tr}[\sigma]\,|0\rangle\!\langle 0|\,, \tag{27}\]
can prepare the pure state \(|0\rangle\), but any Stinespring dilation of \(\mathcal{I}\) itself, for example,
\[\mathcal{I}(\sigma)=\mathrm{Tr}_{Z}\,F(\sigma_{Y}\otimes|0\rangle\!\langle 0 |_{Z})F^{\dagger}, \tag{28}\]
with the swapping operator \(F\) requires yet another pure ancilla state \(|0\rangle_{Z}\), so the problem will be repeated _ad infinitum_. Indeed, Landauer's principle asserts that initializing a quantum system inevitably produces heat [51]; entropy can only be displaced, not destroyed, under reversible evolution. The generated heat consumes the purity of the heat absorbent and we again confront the problem of initializing the absorbent.
Another potential solution is preparing the pure state by measuring the ancilla system and choosing the wanted measurement outcomes. However, an operation that can be realized only when some probabilistic measurement outcome happens cannot be deterministic. We also interpret that not just pure states, but any non-maximally mixed quantum state indicates partial knowledge on the given quantum system.
Therefore, by the same argument, every quantum channel that can be deterministically implemented must be possible to realize with a maximally mixed ancilla system, i.e.,
\[\mathcal{N}(\rho)=\mathrm{Tr}_{Y}\,\mathcal{U}(\rho_{X}\otimes\pi_{Y}). \tag{29}\]
Quantum maps of this form are known as _(exactly) factorizable maps_. From the Stinespring dilation of (29), we can see that any factorizable map \(\mathcal{M}\) has the following simple expression in terms of supertrace,
\[\mathcal{M}=\mathfrak{T}_{Y}\mathcal{U}, \tag{30}\]
with some unitary operation \(\mathcal{U}\in\mathfrak{U}\mathfrak{O}(XY)\). This expression is surprisingly similar to purification of quantum states, i.e. a mixed state \(\rho_{A}\) can be always purified with some environment system \(B\) and purification \(|\psi\rangle_{AB}\) such that
\[\rho_{A}=\mathrm{Tr}_{B}\,|\psi\rangle\!\langle\psi|_{AB}\,. \tag{31}\]
Therefore, we will call \(\mathcal{U}\) in (30) the _purification_ of factorizable map \(\mathcal{M}\). See Appendix E for discussion of general factorizable maps and their relation to general \(C^{*}\)-algebras.
By the operational meaning of factorizable maps given here, we can appreciate the physical significance of the mathematical result that not every unital map is factorizable [52]. That is, unital maps that are not factorizable require nondeterministic preparation of ancilla systems.
However, the formal similarity of purifications of factorizable maps and quantum states has limitations. For instance, the Schrodinger-HJW theorem [53, 54] does not hold for purification of factorizable maps when we try to generalize it straightforwardly.
**Proposition 14**.: For some factorizable map \(\mathcal{N}\in\mathfrak{C}(A)\) and two purifications \(\mathcal{U}\) and \(\mathcal{V}\) of \(\mathcal{N}\) on \(AB\), there exists no superunitary operation \(\Upsilon\in\mathfrak{S}\mathfrak{L}(B)\) such that
\[\mathcal{U}=(\mathfrak{i}\mathfrak{o}_{A}\otimes\Upsilon_{B})(\mathcal{V}). \tag{32}\]
Proof.: Consider two ways of implementing the completely depolarizing map
\[\mathcal{C}(\rho)=\pi\operatorname{Tr}[\rho]. \tag{33}\]
The first method is simply swapping the maximally mixed state with the input state and the second is catalytically depolarizing the input state using a randomness source. The unitary operation of the latter is catalytic; its partial transpose is still unitary and this property does not change after local superunitary. However, the former is not catalytic; the partial transpose of the swapping gate is no longer unitary. Therefore, they cannot be superunitarily similar.
Nevertheless, we have the following result immediate from the Schrodinger-HJW theorem.
**Proposition 15**.: For any factorizable map \(\mathcal{N}\in\mathfrak{C}(A)\) with two purifications \(\mathcal{U}=\operatorname{Ad}_{U}\) and \(\mathcal{V}=\operatorname{Ad}_{V}\) of \(\mathcal{N}\) on \(AB\), there exists a unitary operator \(W\in\mathfrak{U}(B^{\otimes 2})\) so that they are related by a generalized transposition \(T[W]\), i.e.,
\[\mathcal{U}=(\mathfrak{i}\mathfrak{o}_{A}\otimes\mathfrak{T}_{B}[W])( \mathcal{V}). \tag{34}\]
One possible interpretation of Proposition 15 is that losing dynamical quantum information is not just losing two set of data, namely the input and the output information of a given process, in a fixed temporal structure. Losing dynamical quantum information is symmetric with respect to the transformation of the spatio-temporal structure modelled by generalized transpositions, and it is natural in the sense that there is no _a priori_ reason to believe that a quantum system that you have no information at all is governed by the same flow of time with you and the bipartite unitary operator \(W\) redirects the temporal progress of the lost system. Indeed, applying a generalized transposition followed by the supertrace is same with applying the supertrace alone, i.e.,
\[\mathfrak{T}\mathfrak{r}\circ\mathfrak{T}[W]=\mathfrak{T}\mathfrak{r}. \tag{35}\]
We remark that (35) bears a striking similarity with the definition of causality for quantum processes,
\[\operatorname{Tr}_{Y}\circ\mathcal{E}=\operatorname{Tr}_{X}. \tag{36}\]
Indeed \(\mathfrak{T}[W]\) in (35) simply changing your spacetime-coordinate system for a quantum system about which you have no information at all should not affect all the other processes, hence it expresses a sort of 'logical causality'.
### Compatibility of state preparation
In this Section, we show that consistency of causal structure is deeply related to flow of information through multipartite interaction, which is greatly important in the study of quantum scrambler, as it was demonstrated in the task of quantum hacking [55]. As the most evident example, Corollary 8 practically allows no information exchange between two systems compatible with opposite temporal directions. The impossibility of preparing a system compatible with opposite direction of time in a specific state of your choice is evident from the fact that such an action will lead to signaling to the past from the perspective of the inverted system. For example, if a qubit appears to propagate back to the past, then the ability of preparing it in either \(|0\rangle\) or \(|1\rangle\) is equivalent to the ability of forcing its measurement outcome to be either \(\langle 0|\) or \(\langle 1|\) on demand in the opposite temporal flow, which leads to retrocausality from that perspective.
In fact, in the quantum setting, bipartite unitary operators \(U\) on \(AB\) with the unitary partial transpose \(U^{T_{B}}\) are known as _catalytic_ unitary operators [56, 57, 58] or _T-dual_ unitaries [59]. They allow information exchange between two systems only in the form of randomization given as a unital map [60, 61], hence no information leaks to a system initially prepared in the maximally mixed state, as it cannot be randomized more. This is the very reason why these unitary operators can be used for catalysis of quantum randomness [56, 57, 58, 62, 63].
In other words, if you are interacting with a quantum system that is compatible with two opposite temporal flows, then non of your information leaks to it from your perspective. It hints that the more the effect of the given generalized transposition deviates from the usual flow of time, the less information can be leaked through the given interaction. Henceforth, we can ask the following interesting question: System \(A\), a quantum system on your side, is going to interact with another system \(B\). You have no knowledge about the interaction between \(A\) and \(B\) other than it is also compatible with another spatio-temporal structure of \(B\) modelled by a generalized transposition \(T_{B}[W]\) on \(B\). Does this condition constrain the maximum amount of information that can be leaked from \(A\) to \(B\)?
For this purpose, let us define the (geometric) non-swappability of bipartite quantum channel \(\mathcal{N}\in\mathfrak{C}(AB)\) with \(|A|=|B|\)
\[\Xi(\mathcal{N}):=\frac{1}{2}\min_{\mathcal{C}_{A}\mathcal{C}_{B}}\| \mathcal{N}-(\mathcal{C}_{A}\otimes\mathcal{C}_{B})\circ\text{Ad}_{F}\|_{ \diamond}, \tag{37}\]
where \(\mathcal{C}_{A}\) and \(\mathcal{C}_{B}\) are local unitary operators. In other words, \(\Xi(\mathcal{N})\) is the diamond norm distance between \(\mathcal{N}\) and the set of swapping unitary operations up to local unitary. In this sense, one say say that \(\Xi(\text{Ad}_{W})\) is the measure of how close global behavior of \(T[W]\) is to the usual transposition.
We also define the _geometric capacity_\(C_{G}(\mathcal{N})\) of quantum channel \(\mathcal{N}\) as
\[C_{G}(\mathcal{N}):=\frac{1}{2}\min_{\tau}\|\mathcal{N}-\mathcal{E}_{\tau}\|_ {\diamond}, \tag{38}\]
where \(\mathcal{E}_{\tau}(\rho):=\tau\operatorname{Tr}[\rho]\) from \(A\) to \(B\) is the initialization map. In other words, \(C_{G}(\mathcal{N})\) measures the distance between \(\mathcal{N}\) and the closest initialization map. State initialization maps completely destroy the information of the input system. Thus, we can say that the farther away a channel is from initialization maps, the more information it preserves.
Hence, when \(\cdot\otimes\sigma\) is understood as a quantum channel that attaches an ancilla system in state \(\sigma\), we can interpret
\[\mathcal{L}_{B(A}(\mathcal{M}|\sigma):=C_{G}(\operatorname{Tr}_{A}[\mathcal{ M}(\,\cdot\otimes\sigma)]), \tag{39}\]
as the measure of information leakage from \(A\) to \(B\) for any bipartite channel \(\mathcal{M}\in\mathfrak{C}(AB)\) when the system \(B\) is initially prepared in the state \(\sigma\).
**Proposition 16**.: Let \(\mathcal{U}\) be a bipartite quantum operations on \(AB\) compatible with a generalized transposition \(T_{B}[W]\) on \(B\), i.e. \(\mathcal{V}:=\mathcal{U}^{T[W^{\dagger}]}\) is also a quantum operation with \(\mathcal{W}=\text{Ad}_{W}\). Then, information leakage from \(A\) to \(B\) through \(\mathcal{U}\) is limited by the non-swappability of \(\mathcal{W}\), i.e.,
\[\max_{\sigma}\mathcal{L}_{B(A}(\mathcal{U}|\sigma)\leq\Xi(\mathcal{W}), \tag{40}\]
where the maximization is over quantum states \(\sigma\) that are compatible with \(T_{B}[W]\).
Proof is given in Appendix. One could understand that Proposition 16 provides the robustness of Corollary 8. For example, if \(\mathcal{W}\) is the swapping operation corresponding to the ordinary transposition, the right hand side of (40) vanishes, so there could be no information leakage from \(A\) to \(B\) through \(\mathcal{U}\). Even when \(T[W]\) is slightly different from the ordinary transposition, the information leakage is also very small.
On the other extreme, we examine how highly information leaking interactions restrict the form of compatible partial generalized transposition. We can measure the information destruction by quantum channel \(\mathcal{N}\) with the minimum sine metric between the Choi matrices of \(\mathcal{N}\) and an arbitrary unitary operation. Here, the _sine metric_\(d_{S}(\rho,\sigma)\)[64, 65] between quantum states \(\rho\) and \(\sigma\) is given as
\[d_{S}(\rho,\sigma)=\sqrt{1-F(\rho,\sigma)}. \tag{41}\]
Therefore, our (geometric) measure of _information destruction_ in \(\mathcal{N}=\sum_{i}\text{Ad}_{N_{i}}\in\mathfrak{C}(A)\) can be expressed as
\[D_{S}(\mathcal{N}):=\sqrt{1-|A|^{-2}\max_{Y\in\mathfrak{U}(A)}\sum_{i}|\text{ Tr}[YN_{i}]|^{2}}. \tag{42}\]
As a special case, \(D_{S}(\mathcal{N})\) vanishes if and only if \(\mathcal{N}\) is a unitary operation that never destroys input information. By using this measure, we can define a geometric measure of _information non-leakage_ of bipartite channel \(\mathcal{M}\) given as
\[\mathcal{K}_{B\langle A}(\mathcal{M}|\sigma):=D_{S}(\text{Tr}_{A}[\mathcal{M} (\cdot\otimes\sigma)]). \tag{43}\]
Similarly, we can define the following sine metric-based measure of _non-catalyticity_ of bipartite unitary operations,
\[\mathcal{D}_{B\langle A}(\mathcal{M}|\sigma):=\min_{\xi\in(B)}d_{S}(\text{Tr} _{A}[\mathcal{M}(\phi^{+}_{AA^{\prime}}\otimes\sigma^{T}_{B})],\pi_{A^{\prime }}\otimes\xi_{B}). \tag{44}\]
It is a non-catalyticity measure for bipartite unitary operations since, when \(\mathcal{N}=\text{Ad}_{Y}\) is a unitary operation, \(\mathcal{D}_{B\langle A}(\mathcal{N}|\sigma)=0\) if and only if \(Y\) is a catalytic unitary operator [56, 57]. After preparing these definitions, we can introduce another approximate result on the relation between compatible generalized partial transpositions and information leakage of bipartite quantum channels.
**Proposition 17**.: Let \(\mathcal{U}\) be a bipartite unitary operations on \(AB\) compatible with a generalized transposition \(T_{B}[W]\) on \(B\), i.e. \(\mathcal{V}:=\mathcal{U}^{T[W\uparrow]}\) is also a quantum operation with \(\mathcal{W}=\text{Ad}_{W}\). Then, non-catalyticity of \(\mathcal{W}\) is limited by the information non-leakage by \(\mathcal{U}\), i.e.,
\[\mathcal{D}_{B^{\prime}\langle B}(\mathcal{W}|\sigma)\leq 2\mathcal{K}_{B \langle A}(\mathcal{U}|\sigma), \tag{45}\]
for all \(\sigma\) that are compatible with \(T_{B}[W]\).
Proof can be found in Appendix. For example, if \(\mathcal{U}\) leaks all of the information of input \(A\) to output \(B\) for a certain initial state \(\sigma_{B}\) of \(B\) so that \(\mathcal{K}_{B\langle A}(\mathcal{U}|\sigma)=0\) for some \(\sigma\), then any compatible generalized partial transposition \(T_{B}[\mathcal{W}]\) must be catalytic, i.e. \(\mathcal{W}^{T_{B}}\) is still unitary.
## Acknowledgements
SHL thanks Varun Narasimhachar for insightful discussions. This work was supported by National Research Foundation of Korea grants funded by the Korea government (Grants No. 2019M3E4A1080074, No. 2020R1A2C1008609 and No. 2020K2A9A1A06102946) via the Institute of Applied Physics at Seoul National University and by the Ministry of Science and ICT, Korea, under the ITRC (Information Technology Research Center) support program (IITP-2020-0-01606) supervised by the IITP (Institute of Information & Communications Technology Planning & Evaluation) and the quantum computing technology development program of the National Research Foundation of Korea(NRF) funded by the Korean government (Ministry of Science and ICT (MSIT)) (Grants No.2021M3H3A103657312). SHL was also supported by the start-up grant of the Nanyang Assistant Professorship of Nanyang Technological University, Singapore.
## Appendix A Proof of Theorem 1
Proof.: Note that \(Y\in\mathcal{K}_{1}\otimes\mathcal{K}_{2}\) being a dynamics tensor is equivalent to the existence of some \(W\in\mathfrak{U}(\mathcal{K})\) such that \(Y^{T[W]}=\mathds{1}_{\mathcal{K}_{1}}\) when \(Y\) is understood as an operator in \(\mathfrak{B}(\mathcal{K}_{1},\mathcal{K}_{2})\). For any two unit vectors in the same Hilbert space, there exists a unitary operator that transforms one to another. Therefore, when \(X\) is interpreted as an operator in \((\mathcal{K})\), there exists a unitary operator \(W\in\mathfrak{U}(\mathcal{K}\otimes\mathcal{K})\) that transforms \(\|X\|_{2}^{-1}\sum_{i}X\left|i\right>\otimes\left|i\right>\) to \(\left|\phi^{+}\right>\). For such \(W\), we have \(X^{T[W]}=|\mathcal{K}|^{-1/2}\|X\|_{2}\mathds{1}_{\mathcal{K}}\)
Proof of Theorem 10
Proof.: Suppose that \(|\mathcal{K}|\) is even. Arbitrary operator \(Y\in\mathfrak{B}(\mathcal{K})\) has an expansion of the form of \(Y=a\mathds{1}_{\mathcal{K}}+J\) with a traceless operator \(J\). Without loss of generality, we assume that \(a\) is real. Note that \(\mathds{1}_{\mathcal{K}}\) and \(J\) are orthogonal to each other. Hence, there exists a unitary operator \(W\) on \(\mathcal{K}^{\otimes 2}\) that preserves \(\left|\phi^{+}_{\mathcal{K}^{\otimes 2}}\right\rangle\) but transforms \((J\otimes\mathds{1}_{\mathcal{K}})\left|\phi^{+}_{\mathcal{K}^{\otimes 2}}\right\rangle\) to \((J^{\prime}\otimes\mathds{1}_{\mathcal{K}})\left|\phi^{+}_{\mathcal{K}^{ \otimes 2}}\right\rangle\) where \(J^{\prime}\) is an arbitrary traceless unitary operator such that \(J^{\prime\dagger}=-J^{\prime}\), which always exists when \(|\mathcal{K}|\) is even. An example is direct sum of \(i\sigma_{Y}\) where \(\sigma_{Y}\) is the \(2\times 2\) Pauli Y operator. Then, \(Y^{T[W]}\) is proportional to a unitary operator as \(Y^{T[W]\dagger}Y^{T[W]}=|a|^{2}\mathds{1}_{\mathcal{K}}+a(J^{\prime}+J^{ \prime\dagger})+J^{\prime\dagger}J=(|a|^{2}+1)\mathds{1}_{\mathcal{K}}\).
## Appendix C Proof of Theorem 7
Proof.: For a state preparation superchannel given as \(\mathfrak{P}^{\sigma}(\mathcal{N}):=\mathcal{N}(\sigma)\) to be compatible with a generalized transposition \(T[W]\) of its input channel, equivalently for \(\mathfrak{P}^{\sigma}\circ\mathfrak{T}[W^{\dagger}]\) to be a superchannel, it must be possible to decompose it into
\[\mathfrak{P}^{\sigma}\circ\mathfrak{T}[W^{\dagger}](\mathcal{L})=\operatorname {Tr}_{A^{\prime}}[\operatorname{Ad}_{Q}\circ(\mathcal{L}_{A}\otimes\mathds{1 }_{A^{\prime}})(\tau_{AA^{\prime}})], \tag{46}\]
for any \(\mathcal{L}\in\mathfrak{L}(A)\), where \(Q\in\mathfrak{U}(AA^{\prime})\) and \(\tau_{AA^{\prime}}\) is a pure quantum state on \(AA^{\prime}\)[9]. By applying \(\operatorname{Tr}_{A}\) on the both hands of (46) and taking the adjoint (as a map on \(\mathcal{L}(AA^{\prime})\)) and taking the matrix transposition (as a matrix in \(\mathcal{L}(AA^{\prime})\)), we get that from the arbitrariness of \(\mathcal{L}\)
\[W(\mathds{1}_{A}\otimes\sigma_{A^{\prime}}^{T})W^{\dagger}=(\mathds{1}_{A} \otimes\tau_{A^{\prime}}^{T}). \tag{47}\]
It follows that \(\mathds{1}_{A}\otimes\tau_{A^{\prime}}\) and \(\mathds{1}_{A}\otimes\sigma_{A^{\prime}}\), thus in turn \(\tau_{A}\) and \(\sigma_{A}\) have the same spectrum, therefore they are unitarily similar.
Conversely, if (47) holds for some \(\tau_{A}\), then we can set \(Q=W\) and \(\tau_{AA^{\prime}}=(\operatorname{id}_{A^{\prime}}\otimes\operatorname{Ad}_{ \sqrt{\tau_{A}}})(\Gamma_{AA^{\prime}})\) to express \(\mathfrak{P}^{\sigma}\circ\mathfrak{T}[W^{\dagger}]\) as a superchannel form as in (46).
## Appendix D Proof of Proposition 16
Proof.: We can observe that, by using the definition of generalized transposition (8), \(\operatorname{Tr}_{A}\circ\mathcal{U}\circ\mathcal{A}_{\sigma}(\rho)\) can be expressed as for any \(\rho\),
\[\operatorname{Tr}_{AB^{\prime}}[(\mathds{1}_{AB}\otimes\sigma_{B^{\prime}}^{T })\mathcal{W}_{BB^{\prime}}\circ\mathcal{V}_{AB}(\rho_{A}\otimes\Gamma_{BB^{ \prime}})]. \tag{48}\]
However, due to the compatibility of preparing \(\sigma\) with \(T[W]\), by Proposition 7 (Note that \(W\) and \(W^{\dagger}\) are switched in this proof), for the preparation superchannel inputting \(\sigma\) to \(\mathcal{U}\) to be compatible with the transformation \(\mathcal{U}\mapsto\mathfrak{T}[W^{\dagger}](\mathcal{U})\), there must exist a quantum state \(\bar{\sigma}\) that is unitarily similar to \(\sigma^{T}\) and satisfies
\[(\mathds{1}_{B}\otimes\sigma_{B^{\prime}}^{T})W=W(\mathds{1}_{B}\otimes\bar{ \sigma}_{B^{\prime}}). \tag{49}\]
Hence, for a purification \(\psi_{BB^{\prime}}=(\operatorname{id}_{B}\otimes\operatorname{Ad}_{\bar{ \sigma}})\) of \(\bar{\sigma}_{B^{\prime}}\), we have that (48) equals
\[\operatorname{Tr}_{AB^{\prime}}\circ\mathcal{W}_{BB^{\prime}}\circ\mathcal{V}_ {AB}(\rho_{A}\otimes\psi_{BB^{\prime}}). \tag{50}\]
Now, we let \(\mathcal{M}\in\mathfrak{C}(B^{\prime},B)\) be a channel that achieves
\[F_{B(B^{\prime}}(\mathcal{W})=\frac{1}{2}\|\operatorname{Tr}_{B^{\prime}}\circ \mathcal{W}-\operatorname{Tr}_{B}\otimes\mathcal{M}\|_{\diamond}. \tag{51}\]
By the submultiplicative property of the diamond norm, as \(\|\operatorname{Tr}_{A}\circ\mathcal{V}_{AB}\circ\mathcal{A}_{\psi}\|_{\diamond}=1\), we have
\[\frac{1}{2}\|\operatorname{Tr}_{A}\circ\mathcal{U}\circ\mathcal{A}_{\sigma}- \mathcal{E}_{\mathcal{M}(\bar{\sigma}_{B^{\prime}})}\|_{\diamond}\leq F_{B(B^{ \prime}}(\mathcal{W}). \tag{52}\]
Here, we used the fact that
\[(\operatorname{Tr}_{AB}\otimes\mathcal{M})\mathcal{V}_{AB}(\rho_{A}\otimes\psi_{ BB^{\prime}})=\mathcal{M}(\tilde{\sigma}_{B^{\prime}})\operatorname{Tr}\rho, \tag{53}\]
and the definition of \(\mathcal{E}_{\tau}\). After the minimization over \(\tau\), (40) follows as the choice of \(\sigma\) was arbitrary.
### Proof of Proposition 17
Proof.: For simplicity, we first assume that \(\sigma_{B}=\pi_{B}.\) There exist a unitary operator \(Y\) on \(B\) and, by Uhlmann's theorem, a pure quantum state \(\eta_{AB^{\prime}}\) such that \(d_{S}(\eta_{AB^{\prime}}\otimes J^{\mathrm{Adj}}_{A^{\prime}B},J^{\mathcal{U} }_{ABA^{\prime}B^{\prime}})=\mathcal{K}_{B\langle A}(\mathcal{U}|\pi_{B})\). By the monotonicity of fidelity under partial trace, after tracing out \(AB\), we have \(d_{S}(\pi_{A^{\prime}}\otimes\eta_{B^{\prime}},\pi_{A^{\prime}}\otimes\pi_{B^ {\prime}})=d_{S}(\eta_{B^{\prime}},\pi_{B^{\prime}})\leq\mathcal{K}_{B\langle A }(\mathcal{U}|\pi_{B})\). Again, by Uhlmann's theorem, there exists a unitary operator \(Z\) such that \(d_{S}(\eta_{AB^{\prime}},J^{\mathrm{Ad}_{Z}}_{AB^{\prime}})=d_{S}(\eta_{AB^{ \prime}}\otimes J^{\mathrm{Adj}},J^{\mathrm{Ad}_{Z}}_{AB^{\prime}}\otimes J^{ \mathrm{Adj}})\leq\mathcal{K}_{B\langle A}(\mathcal{U}|\pi_{B})\). Therefore, by the triangle inequality [65], we get that \(d_{S}(J^{\mathrm{Ad}_{Z}}\otimes J^{\mathrm{Adj}},J^{\mathcal{U}})\) is upper bounded by
\[d_{S}(\eta_{AB^{\prime}}\otimes J^{\mathrm{Adj}},J^{\mathcal{U}})+d_{S}(\eta_ {AB^{\prime}}\otimes J^{\mathrm{Adj}},J^{\mathrm{Ad}_{Z}}\otimes J^{\mathrm{ Adj}}), \tag{54}\]
where the subscripts for the Choi matrices are omitted. Both terms are upper bounded by \(\mathcal{K}_{B\langle A}(\mathcal{U}|\pi_{B})\). Again, by the monotonicity of fidelity, by applying \(\operatorname{Tr}_{AB}\circ\mathcal{W}_{BB^{\prime}}\) to both \(J^{\mathrm{Ad}_{Z}}\otimes J^{\mathrm{Adj}}\) and \(J^{\mathcal{U}}\), we get
\[d_{S}(\operatorname{Tr}_{B}[\mathcal{W}_{BB^{\prime}}(\phi^{+}_{A^{\prime}B} \otimes\pi_{B^{\prime}})],\pi_{A^{\prime}B^{\prime}})\leq 2\mathcal{K}_{B \langle A}(\mathcal{U}|\pi_{B}). \tag{55}\]
Now, observe that the left hand side is \(\mathcal{D}_{B\langle B^{\prime}}(\mathcal{W}|\pi_{B^{\prime}})\).
For general \(\sigma_{B}\), the proof is more or less similar except for that \(J^{\mathcal{U}}_{ABA^{\prime}B^{\prime}}\) is replaced with \(\zeta_{ABA^{\prime}B^{\prime}}:=(\operatorname{id}_{A^{\prime}B^{\prime}} \otimes\mathcal{U}_{AB})(\phi^{+}_{AA^{\prime}}\otimes\sigma_{BB^{\prime}})\), where \(\sigma_{BB^{\prime}}:=\operatorname{Ad}_{\sqrt{\sigma_{B}}}(\phi^{+}_{BB^{ \prime}})\) is a purification of the given \(\sigma_{B}\). Then there exist \(\eta_{AB^{\prime}}\) and some unitary \(Y\) such that \(d_{S}(\eta_{AB^{\prime}}\otimes J^{\mathrm{Adj}}_{A^{\prime}B^{\prime}},\zeta _{ABA^{\prime}B^{\prime}})=\mathcal{K}_{B\langle A}(\mathcal{U}|\sigma_{B})\) and \(d_{S}(\eta_{B^{\prime}},\sigma_{B^{\prime}})\leq\mathcal{K}_{B\langle A}( \mathcal{U}|\sigma_{B})\). Note that \(\sigma_{B^{\prime}}=\sigma_{B}^{T}\). As we did for \(\sigma_{B}=\pi_{B}\) case, we apply \(\mathcal{W}\) on \(BB^{\prime}\) and trace out \(AB\) to both \(\zeta_{ABA^{\prime}B^{\prime}}\) and \(J^{\mathrm{Adj}}_{A^{\prime}B}\otimes\eta_{AB^{\prime}}\). By using the compatibility condition (18) and that \(\mathcal{U}^{T[W]}=\mathcal{V}\), we get that the former turns into \(\pi_{A^{\prime}}\otimes\tau_{B^{\prime}}^{T}\) for some \(\tau_{B}\) and the latter is mapped to \(\operatorname{Tr}_{B}[\mathcal{W}(J^{\mathrm{Adj}}_{A^{\prime}B}\otimes\eta_ {B^{\prime}})]\). Note that the sine metric between \(\operatorname{Tr}_{B}[\mathcal{W}(J^{\mathrm{Adj}}_{A^{\prime}B}\otimes\eta_ {B^{\prime}})]\) and \(\operatorname{Tr}_{B}[\mathcal{W}(J^{\mathrm{Adj}}_{A^{\prime}B}\otimes\sigma_ {B^{\prime}})]\) is upper bounded by \(d_{S}(\eta_{B^{\prime}},\sigma_{B^{\prime}})\leq\mathcal{K}_{B\langle A}( \mathcal{U}|\sigma_{B})\). Since \(d_{S}(\pi_{A^{\prime}}\otimes\tau_{B^{\prime}},\operatorname{Tr}_{B}[ \mathcal{W}](J^{\mathrm{Adj}}_{A^{\prime}B}\otimes\sigma_{B^{\prime}}))\) bounds \(\mathcal{D}_{S}(\mathcal{W}|\sigma)\) from above, by using the triangle inequality of the sine metric, we get the wanted result.
## Appendix E Factorizable maps with general \(C^{*}\)-algebra
In contrast to the fact that postselecting a certain measurement outcome of a system whose state can be changed when affected but has not yet been examined before is not deterministic, we claim that generating a state with randomness which cannot be altered afterwards is deterministically implementable. This is because, in that case, we assume that there is no room for the change of the ancilla system, so no measurement is required to identify its state, and we are not selecting a certain subset of states but using the whole probabilistically mixed state.
Hence, we can also deterministically implement the aforementioned exactly factorizable maps conditioned on the classical register. This more general set of quantum maps are known as _factorizable_ maps and can be expressed as follows with some probability distribution \(\{p_{i}\}\),
Since \(|Y|^{-1}\sum_{i}p_{i}\operatorname{Tr}_{Y_{i}}\) is a tracial state of \(C^{*}\)-algebra \(\bigoplus_{i}\mathfrak{B}(Y_{i})\) for every probability distribution \(\{p_{i}\}\), by just naming it \(\operatorname{Tr}_{Y}:=\sum_{i}p_{i}\operatorname{Tr}_{Y_{i}}\) so that \(\mathfrak{T}_{Y}:=\sum_{i}p_{i}\mathfrak{T}_{Y_{i}}\), we again recover the purification expression for an arbitrary factorizable map (30).
\[\mathcal{N}(\rho)=\sum_{i}p_{i}\operatorname{Tr}_{Y_{i}}\mathcal{U}_{i}(\rho_{X} \otimes\pi_{Y_{i}}). \tag{56}\]
Here, \(Y\) is decomposed into superselection sectors \(Y=\bigoplus_{i}Y_{i}\) with the orthogonal projector \(\Pi_{i}\) onto each subspace \(Y_{i}\), and \(\mathcal{U}_{i}\in\mathfrak{U}\mathfrak{O}(X,Y_{i})\). Note that this is a simpler expression for finite dimensional \(Y\), but the concept of factorizable maps can be also defined on infinite dimensional systems. (See [66, 67] for more information.) We focus on finite dimensional cases in this work for simplicity. | 量子論において、行列転置を適用して時間反転の対称性を検証することができると示されている。しかし、量子プロセスにおける不定因果性の発見から、時間における一般の対称性変換の可能性が示唆される。この研究では、量子演算の過去と現在Hilbert空間の一般化された転置という概念を導入する。これは、時間軸を完全に逆転しているのではなく、相対的な方向に置くことを可能にする。これは、これまで研究された「不定方向の時間」の一般化である、時間軸の重ね合わせを表す。このフレームワークは、時間と空間を量子力学のように等しく扱う量子重力など、時間と空間を統合するアプローチに適用できる。私たちは、この一般化された転置を、完璧なテンソル、サブシステムをトレースする動的バージョンなど、時間軸の多様性と相対的な相互作用を研究する。特に、二対性相互 |
2307.15100 | Singlet magnetism in intermetallic UGa$_2$ unveiled by inelastic x-ray
scattering | Using high resolution tender-x-ray resonant inelastic scattering and
hard-x-ray non-resonant inelastic scattering beyond the dipole limit we were
able to detect electronic excitations in intermetallic UGa$_2$ that are highly
atomic in nature. Analysis of the spectral lineshape reveals that the local
$5f^2$ configuration characterizes the correlated nature of this ferromagnet.
The orientation and directional dependence of the spectra indicate that the
ground state is made of the $\Gamma_1$ singlet and/or $\Gamma_6$ doublet
symmetry. With the ordered moment in the $ab$ plane, we infer that the
magnetism originates from the higher lying $\Gamma_6$ doublet being mixed with
the $\Gamma_1$ singlet due to inter-site exchange, qualifying UGa$_2$ to be a
true quantum magnet. The ability to observe atomic excitations is crucial to
resolve the on-going debate about the degree of localization versus itineracy
in U intermetallics. | Andrea Marino, Martin Sundermann, Denise S. Christovam, Andrea Amorese, Chun-Fu Chang, Paulius Dolmantas, Ayman H. Said, Hlynur Grrtarsson, Bernhard Keimer, Maurits W. Haverkort, Alexander V. Andreev, Ladilav Havela, Peter Thalmeier, Liu Hao Tjeng, Andrea Severing | 2023-07-27T16:22:18 | http://arxiv.org/abs/2307.15100v1 | # Singlet magnetism in intermetallic UGa\({}_{2}\) unveiled by inelastic x-ray scattering
###### Abstract
Using high resolution tender-x-ray resonant inelastic scattering and hard-x-ray non-resonant inelastic scattering beyond the dipole limit we were able to detect electronic excitations in intermetallic UGa\({}_{2}\) that are highly atomic in nature. Analysis of the spectral lineshape reveals that the local \(5f^{2}\) configuration characterizes the correlated nature of this ferromagnet. The orientation and directional dependence of the spectra indicate that the ground state is made of the \(\Gamma_{1}\) singlet and/or \(\Gamma_{6}\) doublet symmetry. With the ordered moment in the \(ab\) plane, we infer that the magnetism originates from the higher lying \(\Gamma_{6}\) doublet being mixed with the \(\Gamma_{1}\) singlet due to inter-site exchange, qualifying UGa\({}_{2}\) to be a true quantum magnet. The ability to observe atomic excitations is crucial to resolve the on-going debate about the degree of localization versus itineracy in U intermetallics.
+
Footnote †: The Netherlands
+
Footnote †: The Netherlands
## I Introduction
Actinide intermetallics show a wealth of fascinating phenomena that includes heavy-fermion behavior, hidden order or unconventional magnetism, unconventional superconductivity, the combination of ferromagnetism and superconductivity [1; 2; 3], orbital multicomponent [4], or spin-triplet superconductivity [5; 6; 7; 8] with unusual topological properties [9]. It is generally understood that those complex emergent properties originate from the intricate interplay of band formation and correlations involving the \(5f\) electrons. It is, however, far from clear how to describe quantitatively the electronic structure of these systems, for example, whether an itinerant approach [2; 3] or an embedded impurity model which includes explicitly the local degrees of freedom [10; 11; 12] would be the better starting point. The main problem is that many intermetallic uranium compounds, perhaps with the exception of UPd\({}_{3}\)[13; 14], do not exhibit excitations in their inelastic neutron scattering data. It is therefore challenging to understand if remnants of atomic-like states are at all present in these compounds, let alone to pinpoint which multiplet and/or crystal-field state is actually occupied.
Here we investigate UGa\({}_{2}\) as a representative case for many metallic U compounds in which the relative importance between itinerancy and localization is at issue in explaining the physical properties. UGa\({}_{2}\) crystallizes in the hexagonal AlB\({}_{2}\) structure (space group P6/_mmm_) [15], with the U-U distances well above the Hill limit [16], and orders ferromagnetically below \(T_{c}\) = 125 K with a small orthorhombic distortion [17]. The moments are aligned in the \(ab\)-plane along the \(a\) crystallographic direction. The size of the uranium moment as determined by neutron diffraction [18] and magnetization [19; 20; 21] measurements amounts to about 3 \(\mu_{B}\), quite a high value as compared to other magnetically ordered uranium intermetallics, and suggests a more localized nature of the \(5f\) states in this binary. Inelastic neutron scattering, however, did not find crystal-field excitations; only magnons below 10 meV were observed [22]. Attempts to explain the magnetism have been based on local \(f^{2}\) and \(f^{3}\) charge configurations [23; 24; 25], and on approaches that include itinerancy [24; 26]. De Haas - van Alphen [20] and photoemission [27; 28] experiments indicate that UGa\({}_{2}\) is neither localized nor itinerant. Spectroscopically, photoemission experiments are also not conclusive: core level data on bulk samples were interpreted as indicative for the localized nature of the \(5f\) states, based on the satellite structure of the U \(4f\) core level that looks very different from that of itinerant UB\({}_{2}\)[27; 29]. On the other hand, data on UGa\({}_{2}\) films [30] seem to support itinerancy, based on the fact that the satellites appear at different energy positions than in prototypical UPd\({}_{3}\).
The observation of multiplets would provide direct evidence of the presence of atomic-like states. Furthermore, multiplets are a unique fingerprint of the configuration that determines the symmetry. Here resonant inelastic x-ray scattering (RIXS) is the ideal method because it covers a wide range of energy transfers. Already in
2006 Kvashnina _et al._ and Burotin _et al._ used RIXS at the U \(O\)-edge (\(\approx 100\) eV) to distinguish valence states in semiconducting UO\({}_{2}\) and other U and actinide oxides [31; 32; 33], and Wray _et al._ and Liu _et al._ reported excitations in the \(O\)-edge RIXS spectra of intermatallic U compounds [34; 35]. However, the signal to background ratio of these \(f\)-\(f\) excitations is very small at the U \(O\)-edge because of the strong elastic tail in the extreme ultraviolet. A recent publication of soft RIXS data at the U \(N_{4}\) edge (\(\approx 778\) eV), also of UO\({}_{2}\)[36], encouraged some of the authors of this manuscript to repeat the \(N_{4}\)-edge experiment with the same experimental set-up for the intermetallic large moment antiferromagnet UNi\({}_{2}\)Si\({}_{2}\). The result was discouraging, with absolutely no inelastic intensity observed [37]. Another trial at the N\({}_{5}\)-edge (\(\approx 736\) eV) of the hidden order compound URu\({}_{2}\)Si\({}_{2}\) gave the same negative result [38]. Kvashnina _et al._, on the other hand, reported tender x-ray RIXS experiments at the U \(M_{4}\) (\(\approx 3730\) eV) and \(M_{5}\) (\(\approx 3552\) eV) edge with a resolution of 1 eV for UO\({}_{2}\), and for the two intermetallic compounds UPd\({}_{3}\) and URu\({}_{2}\)Si\({}_{2}\). Distinct excitations are observed at about 3 - 7 eV (valence band into unoccupied 5\(f\) states) and 18-20 eV (U 5\(f\) to U 6\(p\)), both at the \(M_{4}\) and \(M_{5}\) edge [39]. These data show that the realization of _high-resolution tender_ RIXS at the U \(M\)-edges is the most promising direction to aim at, not only because of the expected stronger signal, but also because the tender x-ray regime does not require cleaving; it would even allow the confinement of samples. The latter would be a great advantage when performing experiments on U and especially actinide containing samples.
Here we utilize this new spectroscopic tool, namely _high-resolution tender_ RIXS at the U \(M_{5}\) edge to tackle the origin of the magnetism in UGa\({}_{2}\). With tender RIXS, we were able to detect pronounced atomic multiplet states in the intra-valence band excitation spectrum of UGa\({}_{2}\). We also present hard x-ray core-level non-resonant inelastic scattering data (NIXS, also known as x-ray Raman) in the beyond-dipole limit at the U \(O_{4,5}\)-edge, confirming the RIXS result. Also in the high energy NIXS spectrum we observed states that are highly atomic in nature. Our analysis ultimately indicates that UGa\({}_{2}\) is a singlet ferromagnet.
## II Methods
### High resolution tender RIXS at U \(M_{5}\)-edge
In a U \(M_{4,5}\)-edge RIXS experiment, see Fig. 1 (a), a 3\(d\) core electron is excited from the 3\(d^{10}5f^{n}\) ground state \(|g\rangle\) into the 5\(f\) shell by the absorption of incoming photons at the \(M_{5}\) (3552 eV) or \(M_{4}\) (3730 eV) edge, leading to 3\(d_{5/2}^{9}5f^{n+1}\) or 3\(d_{3/2}^{9}5f^{n+1}\) intermediate states \(|i\rangle\), respectively. The subsequent de-excitation of the 3\(d\) core can be into the ground state (elastic peak), into an excited state of the same local charge configuration (\(ff\) excitations, phonons, magnons) [40; 41; 42; 43], or into an excited state of a different charge configuration (charge transfer excitations) [44; 45; 46].
Fig. 1 (b) depicts the experimental geometry where the scattering angle is set at 2\(\Theta=90^{\circ}\) to minimize the elastic intensity. The UGa\({}_{2}\) samples used for the experiments were grown with the Czochralski method [21] and their surface is the \(ab\) plane, i.e. it has the [001] orientation.
High resolution tender RIXS was performed at the Max-Planck RIXS end station (IRIXS) of the P01 beamline at Petra III/DESY in Hamburg. The instrument is unique, since it allows to perform RIXS experiments with tender x-rays (2.5 - 3.5 keV) and good resolution [47]. For example, a resolution of 100 meV can be achieved at the Ru \(L_{3}\)-edge at 2840 eV. The IRIXS beamline uses the hard x-ray set-up [47]. For the U \(M_{5}\) edge at 3550 eV a diced quartz wave (112), pressed and glued on a concave Si lens has been used as analyzer crystal [48]. The instrument 150 meV Gaussian response function at the U \(M_{5}\) edge is estimated by measuring a carbon tape. The experiment was performed with horizontal polar
Figure 1: (a) RIXS process at the U \(M_{4,5}\) edge for a 5\(f^{n}\) ground state. (b) Scattering geometry of the RIXS experiment. The scattering angle is kept at 2\(\Theta=90^{\circ}\). (c) Fluorescence-yield \(M_{5}\) XAS spectrum of UGa\({}_{2}\). Large dots mark the photon energies where RIXS is measured (\(h\nu_{1}\), \(h\nu_{2}\), and \(h\nu_{3}\)). (d) \(M_{5}\) RIXS spectrum of UGa\({}_{2}\) acquired at \(h\nu_{2}\) and \(\theta=45^{\circ}\).
ization of the the incident photons, a scattering angle \(2\Theta=90^{\circ}\) to minimize elastic intensity and sample angles of \(\theta=20^{\circ},45^{\circ}\) and \(80^{\circ}\) (see Fig. 1(b)). Temperature was kept at \(35\,\mathrm{K}\).
### NIXS with hard x-rays at U \(O_{4,5}\)-edge
NIXS with hard x-rays (\(10\,\mathrm{keV}\)) and large momentum transfer is dominated by higher-than-dipole transitions [49; 50], which are more excitonic in contrast to the dipole contribution [51; 52; 53; 54]. The direction of the momentum transfer \(\vec{q}\) in NIXS plays an analogous role as the electric field vector \(\vec{E}\) in XAS and is sensitive to the symmetry of the crystal-field ground state.
The experiments are performed at the Max-Planck NIXS end stations of the P01 beamline at Petra III/DESY in Hamburg. A sketch and description of the NIXS experimental setup is shown in Fig. 2 of [55]. \(10\,\mathrm{keV}\) photons are used. The average scattering angle \(2\Theta\) at which the analyzers are positioned is \(\approx\!150^{\circ}\), thus yielding a momentum transfer of \(|\vec{q}|\)=(9.6\(\pm\)0.1) A\({}^{-1}\) at elastic scattering. An instrumental energy resolution of about \(0.8\,\mathrm{eV}\) FWHM is achieved. The sample is kept in a vacuum cryostat at \(T=5\,\mathrm{K}\). The O\({}_{4,5}\) edge of U is measured with momentum transfer \(\vec{q}\) parallel to the \(a\) and \(c\) crystallographic directions.
## III Results
Fig. 1 (c) shows the experimental U \(M_{5}\)-edge x-ray absorption (XAS) spectrum of UGa\({}_{2}\). The large dots mark the photon energies used in this RIXS study, \(E_{\mathrm{res}}\)-\(4\,\mathrm{eV}\) (\(h\nu_{1}\)), \(E_{\mathrm{res}}\) (\(h\nu_{2}\)), and \(E_{\mathrm{res}}\)+\(4\,\mathrm{eV}\) (\(h\nu_{3}\)) with \(E_{\mathrm{res}}\) = \(3552\,\mathrm{eV}\). In Fig. 1 (d) the RIXS spectrum of UGa\({}_{2}\) is displayed for a wide energy range up to \(8\,\mathrm{eV}\) energy transfer taken at the \(M_{5}\) resonance (\(h\nu_{2}\)) with the sample angle of \(\theta=45^{\circ}\). The spectrum exhibits sharp peaks below \(2\,\mathrm{eV}\) that are on top of a broad feature that arises most likely from charge transfer excitations. The sharp peaks are very typical of local atomic-like excitations.
Fig. 2 shows a close-up of the first \(2\,\mathrm{eV}\) of RIXS spectra that were measured with different incident energies, \(h\nu_{1}\), \(h\nu_{2}\) and \(h\nu_{3}\). The data are normalized to the peak at \(1.05\,\mathrm{eV}\). The intensities of the peaks vary considerably with the incoming photon energy so that three inelastic excitations at \(0.44\), \(0.70\) and \(1.05\,\mathrm{eV}\) can be identified. We assign these to intermultiplet \(ff\) transitions since the energies are too high for magnons and crystal-field excitations.
Full atomic multiplet calculations assuming a \(5f^{2}\) and alternatively a \(5f^{3}\) configuration were carried out to simulate the spectra. For this, the _Quanty_ code [56] was used with the atomic values of the the spin-orbit constant and the \(5f\)-\(5f\) and \(3d\)-\(5f\) Slater integrals from the Atomic Structure Code by Robert D. Cowan [57] as input parameters, whereby the spin orbit constant was reduced by \(10\%\) and the Slater integrals by \(45\%\) in order to take configuration interaction effects and covalence into account [58; 59; 60]. These are typical reduction factors for uranium compounds [36]. A crystal-field (CF) potential was always considered. CF parameters were taken from fits to the magnetic susceptibility or magnetization, for the \(5f^{2}\) from Ref. [25] and for the \(5f^{3}\) configuration from Ref. [20; 23], or constructed to test different CF ground state wave functions. Furthermore, a Lorentzian broadening of about \(6\,\mathrm{eV}\) in the intermediate state was used, based on the width of the \(M_{5}\) XAS spectrum, and a Gaussian broadening of \(150\,\mathrm{meV}\) to account for the experimental resolution.
Fig. 2 also shows the simulation for the \(5f^{2}\) and for the \(5f^{3}\) configuration. The calculation based on the \(5f^{2}\) reproduces the experimental data in terms of peak positions as well as variation of the peak intensities with incident energy. The vertical lines represent the histogram of the multiplet states and provide a straightforward labeling of the peaks. The \(5f^{3}\) simulation, on the other hand, does not reproduce the experimental data. It turns out that no matter how the reduction factors are tuned, an agreement cannot be achieved for \(5f^{3}\) (see Appendix VII.1). Hence we conclude, the atomic-like states in UGa\({}_{2}\) are given by the \(5f^{2}\) configuration.
Next we determine the CF symmetry of the ground state. Here we ignore the slight orthorhombic distortion below \(T_{c}\)[17] since it is only a very small magnetostriction correction to the hexagonal crystal-field analysis. In \(D_{6h}\) the hexagonal CF splits the nine-fold degenerate \(J\!=\!4\) Hund's rule ground state of the U \(5f^{2}\) configuration into
Figure 2: \(M_{5}\) RIXS spectra of UGa\({}_{2}\) acquired at \(h\nu_{1}\), \(h\nu_{2}\), and \(h\nu_{3}\) with a sample angle \(\theta=45^{\circ}\) as compared to simulations using an \(5f^{2}\) and \(5f^{3}\) configuration.
three singlets and three doublets. These be written in the \(J_{z}\) representation as:
\[\Gamma_{1} =|0\rangle \tag{1}\] \[\Gamma_{3} =\frac{1}{\sqrt{2}}|+3\rangle+\frac{1}{\sqrt{2}}|-3\rangle\] (2) \[\Gamma_{4} =\frac{1}{\sqrt{2}}|+3\rangle-\frac{1}{\sqrt{2}}|-3\rangle\] (3) \[\Gamma_{5}^{1} =\sin\phi|\pm 4\rangle+\cos\phi|\mp 2\rangle\] (4) \[\Gamma_{5}^{2} =\cos\phi|\pm 4\rangle-\sin\phi|\mp 2\rangle\] (5) \[\Gamma_{6} =|\pm 1\rangle \tag{6}\]
Although the CF splitting is below the resolution limit of the present RIXS experiment, it is possible to obtain information about the ground state symmetry by measuring the orientation dependence of the scattering signal [61]. The two panels of Fig. 3(a) show the RIXS spectra for two incident energies (\(h\nu_{1}\) and \(h\nu_{3}\)), whereupon each energy was measured for the three sample angles \(\theta=80^{\circ},45^{\circ}\) and \(20^{\circ}\). The \(\theta\) rotation is in the [001]-[-210] plane, see Fig. 1(b). The intensities are again normalized to the peak at 1.05 eV energy transfer. A pronounced orientation dependence can be seen in the spectra for \(h\nu_{1}\), that has almost disappeared for \(h\nu_{3}\).
Again the data are compared to the full multiplet calculations. In Fig. 3(b) we start with the calculations using the \(5f^{2}\) crystal field parameters of Richter \(et\)\(al.\)[25]. With this set of parameters, the ground state is the \(\Gamma_{1}\), the first excited state the \(\Gamma_{6}\) at 3.3 meV, the second excited state the \(\Gamma_{5}^{1}\) with \(\sin\phi\) = 0.81 at 5.9 meV, and all other states at 13 meV or higher. The calculations were performed for T = 35 K and include a molecular field \(I_{e}\) of 1.78 meV as will be explained later. We observe that the calculations reproduce the experiment well: the simulation captures the strong orientation dependence for \(h\nu_{1}\) in the correct sequence and its decrease for \(h\nu_{3}\).
To understand whether a different order of CF states would also be able to reproduce the experiment, we calculate the spectra for different CF ground states. To this end, we tuned slightly the CF parameters such that the desired CF state becomes the ground state and then we carried out the calculation for T = 0 K. The results are displayed in Fig. 3(c). We observe that a \(\Gamma_{1}\) ground state, or a \(\Gamma_{6}\), or also a \(\Gamma_{5}^{2}\) with \(\cos\phi\) \(\approx\) 1 have the correct trend in the orientation dependence for \(h\nu_{1}\) and its reduction at \(h\nu_{3}\). A \(\Gamma_{3}\), \(\Gamma_{4}\), or \(\Gamma_{5}^{1}\) as ground state, on the other hand, produces an orientation dependence that is opposite to the experiment so that these three states can be excluded. The NIXS experiment below will show that also the \(\Gamma_{5}^{2}\) cannot be the ground state. We thus conclude that the good simulation is based on the strong orientation dependence provided by the \(\Gamma_{1}\) or the \(\Gamma_{6}\) low lying states, which gets counteracted at 35 K by the Boltzmann occupation of a higher lying state with opposite orientation dependence, such as the \(\Gamma_{5}^{1}\).
Fig. 4(a) shows the \(O_{4,5}\) edge NIXS data of UGa\({}_{2}\) at 5 K for \(\vec{q}\,||\,c\) and \(\vec{q}\,||\,a\), revealing a strong directional dependence. In Fig. 4(b) the (pseudo) isotropic U \(O_{4,5}\) NIXS spectrum, constructed from \(I_{iso}=(I_{\mathbf{q}||c}+2I_{\mathbf{q}||a})/3\), is displayed and compared to atomic simulations without considering the crystal-field Hamiltonian. The Slater integrals for the \(5f\)-\(5f\) and \(5d\)-\(5f\) Coulomb interactions are reduced by about 40% with respect to their atomic values. The value of the momentum transfer in the simulation is set to \(|\vec{q}|\,=\,11.1\) A\(\cdot\)1 in order to account for the U \(5f\) radial wave function in the solid being different from the calculated atomic value. An arctangent type of background is added to account for the edge jump. A Gaussian broadening of 0.8 eV and a Lorentzian broadening of 1.3 eV account for instrumental resolution and lifetime effects, respectively. The simulations are performed both for an \(5f^{2}\) and \(5f^{3}\) configuration and also here only the \(f^{2}\) simulation reproduces the experimental lineshape, whereas the \(f^{3}\) does not (see Appendix VII.2). This finding is fully consistent with the RIXS results.
Footnote 1: The \(5f^{2}\) configuration is shown in Fig. 4(b).
Focussing now on the directional dependence, we cal
Figure 3: (a) Experimental RIXS spectrum of UGa\({}_{2}\) acquired at \(h\nu_{1}\) and \(h\nu_{3}\) with sample angles \(\theta=80^{\circ},45^{\circ}\) and \(20^{\circ}\). (b) Simulated 35 K RIXS spectra for an \(5f^{2}\) configuration with the crystal-field parameters of Richter \(et\)\(al.\)[25] and a molecular field \(I_{e}\) of 1.78 meV. (c) Simulated 0 K RIXS spectra for each of the six crystal-field states of the \(5f^{2}\).
culate the spectra for each of the six possible crystal-field states. The results are displayed in Fig. 4(c). Comparison with experiment immediately excludes the \(\Gamma_{3}\) and \(\Gamma_{4}\) singlets, as well as the \(\Gamma_{5}\) doublets for any range of the parameter \(\phi\) (see equations above). For the \(\Gamma_{5}^{(1,2)}\) doublets only the extreme cases of \(\phi=0^{o}\) are shown, since the spectra for all other \(\phi\) values fall between these two extremes. The \(\Gamma_{1}\) singlet and the \(\Gamma_{6}\) doublet, on the other hand, show the same directional dependence as the experiment, thus confirming the RIXS results.
## IV Discussion
The above NIXS and RIXS results find that the \(5f^{2}\) configuration dominates the local electronic structure of UGa\({}_{2}\) and that the symmetry of the CF ground state is either given by the \(\Gamma_{1}\) singlet and/or \(\Gamma_{6}\) doublet. However, we can further exclude the \(\Gamma_{6}\) doublet as ground state because it would yield an ordered moment along \(c\) and not in the \(ab\)-plane. Hence, the \(\Gamma_{1}\) singlet state must be the lowest one in energy. Yet, the \(\Gamma_{6}\) is also a necessary ingredient for the magnetism in UGa\({}_{2}\) as we will discuss in the following.
In a conventional local moment magnet the non-vanishing temperature independent moments are present at each lattice site and then order spontaneously at the transition temperature creating a self-consistent molecular field. This is basically a classical concept modified only by the influence of semiclassical quantum fluctuations which reduce the size of the ordered moment by a modest amount. A \(\Gamma_{1}\) ground state would not carry a local moment so that the semiclassical picture of magnetic order does not apply, it rather must be classified as a true quantum magnet where the creation of the local moments and their ordering appears spontaneously at \(T_{c}\). This mechanism of induced magnetic order is caused by the non-diagonal mixing of \(\Gamma_{1}\) with excited \(\Gamma_{6}\) states due to the effective inter-site exchange coupling that forms the true ground state superposition below the ordering temperature. Induced quantum magnetism in singlet ground state systems has been explored in \(d^{4}\) transition metal [62, 63] or \(4f^{2}\) Pr materials (see Ref. [64] and references therein). In these cases the presence of multiplets is clear. Singlet magnetism is however rarely recognized in U compounds [64, 65, 66, 67, 68], where pinpointing the U \(5f^{2}\) configuration is already challenging (see also Appendix VII.4).
Looking at the simple structure of CF states, we realize that indeed \(\Gamma_{6}\) is the only possible excited state that has non-vanishing mixing matrix elements \(\langle\Gamma_{1}|J_{x}|\Gamma_{6}\rangle\) for the in-plane total angular momentum operators (not for \(J_{z}\)) so that there can be no coupling to any other state when we restrict to the Hilbert space of the ground state multiplet (\(J\!=\!4\)). This explains naturally that the ordered moment must lie in the hexagonal plane and at the same time the anisotropy of the paramagnetic susceptibility. For the induced moment mechanism of magnetic order to work, i.e. to produce a finite ordering temperature, the effective exchange \(I_{e}\) must surpass a critical value. Here \(I_{e}\) is the Fourier transform \(I(\mathbf{q})\) of the inter-site coupling \(I_{ij}\) at the ordering vector \(\mathbf{q}\) where \(I(\mathbf{q})\) is at its maximum. In a singlet ground state system with a \(\Gamma_{1}\!\cdot\!\Gamma_{6}\) splitting energy \(\Delta\) a spontaneous induced moment can only appear when the control parameter \(\xi\) is larger than \(1\), with \(\xi\!=\!2\alpha^{2}\,I_{e}/\Delta\)[64, 66] and \(\alpha^{2}\!=\!\sum_{\sigma}|\langle\Gamma_{1}|J_{x}|\Gamma_{6,\sigma}\rangle| ^{2}\), where \(\sigma\) is the degeneracy index of the \(\Gamma_{6}\) states, with numerical value \(\alpha\!=\!3.1\). The saturation moment at zero temperature then is given by \(m_{0}/(g_{J}\mu_{B})\!=\!\langle J_{x}\rangle_{0}\!=\!\alpha\xi^{-1}(\xi^{2} -1)^{1/2}\) (\(g_{J}=0.8\)) which vanishes when approaching the critical value from above \(\xi\to 1^{+}\), and becomes equal to \(\langle J_{x}\rangle_{0}=\alpha\), that of a quasi-degenerate \(\Gamma_{1}\!\cdot\!\Gamma_{6}\) system, when \(\xi\gg 1\), i.e. where the effective exchange strongly dominates over the splitting. In UGa\({}_{2}\) the moment of \(3\,\mu_{B}\) / U is close to the latter case of the exchange dominated regime. See also Appendix VII.4.
The thermal occupation of higher levels has to be considered for the determination of the temperature dependence of \(\langle J_{x}\rangle_{T}\) and \(T_{c}\) as a function of the exchange \(I_{e}\) and CF splitting \(\Delta\). This can be done within our full multiplet calculation (including all angular momentum multiplets \(J\) and their respective CF multiplets) by solving iteratively the selfconsistency equation \(\langle J_{x,y}\rangle_{T}=\sum_{n}p_{n}\langle n|J_{x,y}|n\rangle\) where \(E_{n}(\langle J_{x,y}\rangle_{T})\) and \(|n\rangle(\langle J_{x,y}\rangle_{T})\) are the eigenenergies and eigenstates in the presence of the molecular field \(I_{e}\langle J_{x,y}\rangle_{T}\) for the given values of multiplet model parameters. Here \(p_{n}\!=\!Z^{-1}\exp(-E_{n}/T)\) with \(Z=\sum_{m}\exp(-E_{m}/T)\) are the thermal level occupations.
Figure 4: (a) NIXS spectra for \(\vec{q}\!\parallel\!a\) and \(\vec{q}\!\parallel\!c\) together with the simulation using the crystal field parameters of Richter \(et\,al.\)[25] and a molecular field \(I_{e}\) of \(1.78\,\)meV. (b) (Pseudo)isotropic data and full multiplet simulations without crystal-field for the U \(f^{2}\) and U \(f^{3}\) configurations. (d) Simulated NIXS spectra for each of the six crystal-field states of the \(f^{2}\) together with the corresponding charge densities.
The saturation moment \(m_{0}/\mu_{B}=g_{J}\langle J_{x}\rangle_{0}\) may then be plotted as function of the splitting \(\Delta\) and exchange \(I_{e}\) as shown in Fig. 5. Here the CF parameters from Ref. [25] (apart from the off-diagonal \(A_{6}^{6}\)) are scaled to modify the splitting \(\Delta\). It would be interesting to measure \(\Delta\) using Raman spectroscopy [69]. For \(\Delta\) = 3.3 meV and \(I_{e}\) = 1.78 meV, corresponding to \(m_{0}(\Delta,I_{e})\) \(\approx\) 3 \(\mu_{B}\) (one point on the white line in Fig. 5), we obtain \(T_{c}\) \(\approx\) 125 K, in agreement with experiment. For completeness, we finally examine the impact of the self consistent molecular field with \(I_{e}\) = 1.78 meV on the RIXS (see Fig. 3 (b)) and NIXS (see Fig. 4 (a)) spectra. Here the CEF scheme giving \(\Delta\) = 3.3 meV and the Boltzmann population of excited states is also considered. We see that the molecular field has little impact on the spectra, since, although mixed, the \(\Gamma_{1}\) and \(\Gamma_{6}\) show very similar lineshapes by themselves.
## V Conclusion
In summary, with tender RIXS at the U \(M_{5}\) edge and hard x-ray NIXS at the U \(O_{4,5}\)-edge we have unveiled the U \(5f^{2}\) multiplets inUGa\({}_{2}\) and shown that the magnetism is determined by the U \(5f^{2}\) configuration with a \(\Gamma_{1}\) singlet ground state and a \(\Gamma_{6}\) doublet nearby. UGa\({}_{2}\), therefore, classifies as a quantum magnet. The origin of the induced magnetic order is due to the non-diagonal mixing of \(\Gamma_{1}\) with excited \(\Gamma_{6}\) states due to the effective inter-site exchange coupling below \(T_{c}\).
## VI Acknowledgements
All authors thank C. Geibel and A.C. Lawson for fruitful discussions, and acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. A.S. and A.A. benefited from support of the German Research Foundation (DFG), Project No. 387555779. L.H. and A.V.A. benefited from support of the Czech Science Foundation, Project No. 21-09766S.
## VII Appendix
### \(f^{3}\) RIXS simulation with different values of the reduction factors
Fig. 6 shows simulated \(f^{3}\) RIXS spectra with \(5f-5f\) and Slater integral reduction to 60%, 80% and 100%. The crystal-field is not included in the calculations.
Figure 6: Ionic \(f^{3}\) simulated RIXS for different values of the \(5f\)-\(5f\) reduction factors.
Figure 5: Magnetic moment \(m_{0}\) at zero temperature as a function of the \(\Gamma_{1}-\Gamma_{6}\) splitting \(\Delta\) and the effective exchange \(I_{e}\). The white line denotes the contour with \(m_{0}=3\,\mu_{B}\)
### Crystal-field parameters and ground state symmetry
Table 7 summarizes the crystal-field parameters and the corresponding ground state symmetries used in the RIXS calculations. For the \(\Gamma_{5}\) states the relative \(J_{z}=|\pm 4\rangle\) and \(J_{z}=|\mp 2\rangle\) contribution is specified. The parameters giving a \(\Gamma_{1}\) ground state are taken rom Ref. [25].
### Basics of induced moment magnetism
In this work the magnetism of UGa\({}_{2}\) is interpreted in terms of a localized model consisting of \(5f\) CEF states for \(J=4\). The on-site exchange interaction (resulting from Anderson-type on-site hybridisation and Coulomb repulsion) between conduction and f-electron is assumed to have been eliminated leading to an effective RKKY-interaction \(I_{ij}\) between \(5f\) states on different sites \(i,j\). \(I_{e}\) is the Fourier transform \(I(\mathbf{q})\) of the inter-site coupling \(I_{ij}\) at the ordering vector \(\mathbf{q}\) where \(I(\mathbf{q})\) is at its maximum. Restricting to FM with \(\mathbf{q}=0\) and to nearest neighbor terms only, the effective interaction is given by \(I_{e}=zI_{nn}\) (z=coordination number). If the 5f ground state were degenerate and carried an effective moment (i.e. having nonzero matrix elements of \(\mathbf{J}\) within the multiplet) a quasiclassical ferromagnetic order would appear for any size of \(I_{e}\) where moments are simply aligned at a temperature \(T_{C}\sim I_{e}\). Here, however, the lowest 5f states are nonmagnetic singlet \(\Gamma_{1}\) ground state and \(\Gamma_{6}\) doublet excited state at energy \(\Delta\). Due to their absent moments the FM order in UGa\({}_{2}\) can only appear through a more subtle mechanism called 'induced order'. This mechanism is well established for several 4f Pr and 5f U compounds with nonmagnetic low lying CEF states as in the present case. We refer to previous Refs. [66; 64; 67; 68; 69; 70; 71; 72; 73; 74] for the detailed discussion of the subject. Although the \(\Gamma_{1},\Gamma_{6}\) states do not carry a moment there are _nondiagonal_ matrix elements \(\alpha/\sqrt{2}=\langle\Gamma_{1}|J_{x}|\Gamma_{6\sigma}\rangle\) (\(\sigma=1,2\)) of in-plane dipolar moment \(J_{x}\) (and similar for \(J_{y}\)) connecting them accross the CEF gap \(\Delta\). This means that n.n. inter-site interaction terms like \(I_{ij}J_{x}(i)J_{x}(j)\) are able to mix the excited state \(\Gamma_{6}\) into the noninteracting ground state \(\Gamma_{1}\) and form spontaneously a new magnetic ground state at each site which is a superposition \(|\Gamma_{1}^{\prime}\rangle=u|\Gamma_{1}\rangle+v|\Gamma_{6}\rangle\) (and similar for the excited state). In this way the ground state moment appearance and its ordering happens simultaneously. The size of the ordered moment is then \(\langle J_{x}\rangle=2uv\alpha(n_{1}^{\prime}-n_{6}^{\prime})\) where \(n_{1,6}^{\prime}\) denote the thermal occupations of the CEF states which also depend on \(\langle J_{x}\rangle\). This represents a molecular field equation for the induced moment \(\langle J_{x}\rangle\). When temperature is lowered the occupation difference increases which may lead to a nonzero induced moment, provided the prefactor in the above equation is sufficiently large. This can be evaluated as a condition for the control parameter \(\xi=2\alpha^{2}I_{e}/\Delta>1\) to achieve a finite T\({}_{C}\) and a saturation moment at \(T=0\) given by \(\langle J_{x}\rangle_{0}=\alpha\xi^{-1}(\xi^{2}-1)^{\frac{1}{2}}\). At zero temperature varying \(\xi\) across the quantum critical point (QCP) \(\xi=1\) we obtain a quantum phase transition from the paramagnetic (\(\xi<1\)) to the (ferro-) magnetic (\(\xi>1\)) state. In particular close to the QCP the induced moment quantum magnetism shows anomalous dependence of small saturation moment and low ordering temperature on the control parameter and is quite different from the quasiclassical magnetism where the influence of quantum fluctuations on moment and transition temperature is moderate.
Figure 7: Ionic \(f^{3}\) simulated isotropic NIXS for different values of the \(5f\)-\(5f\) and \(5d\)-\(5f\) reduction factors. | ```
高解像度のTender-x-ray resonant inelastic scatteringとhard-x-ray non-resonant inelastic scatteringを適用することで、電子の励起を intermetallic UGa$_2$ には非常に原子的な性質を持つものを検出することができました。スペクトルラインの形状の解析により、この Ferromagnet の性質を特徴づけるローカル$5f^2$ 構成が確認されました。スペクトルの方向と方向性依存性から、 ground state は $\Gamma_1$ singlet と/または $\Gamma_6$ doublet の双重性で構成されていることが推定されました。$ab$ 平面上での秩序化されたモーメントに基づいて、磁気は、サイト間交換によって $\Gamma_6$ doublet が $\Gamma_1$ singlet と混合しているため、UGa$_2$ が真の量子磁石であることを示唆しています。原子励起を観測できる能力は、U |
2307.01175 | Patient-centric health data sovereignty: an approach using Proxy
re-encryption | The exponential growth in the digitisation of services implies the handling
and storage of large volumes of data. Businesses and services see data sharing
and crossing as an opportunity to improve and produce new business
opportunities. The health sector is one area where this proves to be true,
enabling better and more innovative treatments. Notwithstanding, this raises
concerns regarding personal data being treated and processed. In this paper, we
present a patient-centric platform for the secure sharing of health records by
shifting the control over the data to the patient, therefore, providing a step
further towards data sovereignty. Data sharing is performed only with the
consent of the patient, allowing it to revoke access at any given time.
Furthermore, we also provide a break-glass approach, resorting to Proxy
Re-encryption (PRE) and the concept of a centralised trusted entity that
possesses instant access to patients' medical records. Lastly, an analysis is
made to assess the performance of the platform's key operations, and the impact
that a PRE scheme has on those operations. | Bruno Rodrigues, Ivone Amorim, Ivan Costa, Alexandra Mendes | 2023-07-03T17:39:02 | http://arxiv.org/abs/2307.01175v1 | # Patient-centric health data sovereignty: an approach using Proxy re-encryption+
###### Abstract
The exponential growth in the digitisation of services implies the handling and storage of large volumes of data. Businesses and services see data sharing and crossing as an opportunity to improve and produce new business opportunities. The health sector is one area where this proves to be true, enabling better and more innovative treatments. Notwithstanding, this raises concerns regarding personal data being treated and processed. In this paper, we present a patient-centric platform for the secure sharing of health records by shifting the control over the data to the patient, therefore, providing a step further towards data sovereignty. Data sharing is performed only with the consent of the patient, allowing it to revoke access at any given time. Furthermore, we also provide a _break-glass_ approach, resorting to Proxy Re-encryption (PRE) and the concept of a centralised trusted entity that possesses instant access to patients' medical records. Lastly, an analysis is made to assess the performance of the platform's key operations, and the impact that a PRE scheme has on those operations.
Keywords:data-sovereignty cryptography PRE- access delegation-e-health PHR
## 1 Introduction
The ever growing digitisation of services that we use daily, as well as the increasing interest in data crossing and sharing to improve processes, services, and achieve new business opportunities, raises concerns regarding how data is handled and processed. In the healthcare sector, data sharing is not only beneficial,
but also needed to provide the best care possible to the patients. However, this data is also highly sensitive, which requires special care. Several governmental measures have already been taken to improve and standardise the way in which data is shared, such as the European Data Governance Act [1], GDPR4, and, more specifically in personal health information, HIPAA5 and HITECH6. These directives instigate a user-centric paradigm, granting individuals sovereignty over their data.
Footnote 4: [https://data.europa.eu/eli/reg/2016/679/oj](https://data.europa.eu/eli/reg/2016/679/oj)
Footnote 5: [https://www.cdc.gov/phlp/publications/topic/hipaa.html](https://www.cdc.gov/phlp/publications/topic/hipaa.html)
Footnote 6: [https://www.hipaajournal.com/what-is-the-hitech-act/](https://www.hipaajournal.com/what-is-the-hitech-act/)
Several approaches have been proposed for ensuring security and privacy in e-health systems. Conventional encryption techniques like AES and ECC are commonly used [5]. However, these techniques become problematic when data needs to be shared among multiple entities due to redundancy and computational burden [6]. Attribute-Based Encryption (ABE) is another solution [6], but it has its own complexities and limitations, such as managing attribute-based keys and overriding policies in emergencies [7]. ABE also lacks the fine-grained access control necessary for a patient-centric sovereign approach.
Proxy Re-encryption (PRE) is a cryptographic solution for secure data sharing without prior knowledge of the recipient. Unlike ABE, it does not rely on policies or attributes. PRE converts a ciphertext to a recipient's key without revealing the plaintext to the intermediary entity. It is particularly useful in semi-trusted cloud environments [17]. In e-health, PRE has already been used to securely share medical records [20, 23, 19, 26], including in emergency scenarios [19]. However, challenges remain in terms of revocability, computational effort, and safeguarding emergencies [26]. Existing solutions for emergency scenarios are limited and rely on assumptions that may impact efficiency and reliability.
In this context, it is necessary to develop a platform that addresses the aforementioned concerns. This includes enabling more control over the data by the patient while ensuring the safety of that data, even in semi-trusted environments. This contributes to the collaborative aspect of e-health and thus enables better treatments and advancements in the health sector.
In this paper, we present a platform that leverages PRE to enhance health data sharing. Umbral's PRE [16] is used as the foundation for re-encryption processes, through which we achieve unidirectionality and non-interactivity, ensuring secure re-encryption from the patient to the data user (e.g., practitioners or health centres) without requiring the data user's private key. This approach centres on the expressed opinion of the patient to authorise data sharing, eliminating the need for prior identification of authorised parties -- a drawback identified in previous solutions. Additionally, our platform offers revocability options, such as time-based access limits and patient-initiated access revocation. Importantly, the revocation of accesses does not require changes to the encrypted healthcare database, distinguishing our platform from the ones that rely on identity and attribute-based PRE schemes.
Furthermore, in the context of healthcare, it is crucial to ensure data sharing in emergency situations when explicit patient consent may not be possible. Our platform addresses this challenge by incorporating a trusted entity for data access when patient authorisation is infeasible.
In summary, our main contributions are:
* A patient-centric platform, that empowers patients with sovereign control over their health data, enabling granular access control and facilitating the sharing of health records only with explicit consent.
* Robust data protection using Umbral's PRE, ensuring secure and encrypted health data sharing without compromising the data user's private key.
* A robust access revocation mechanism that enables time-based access limits and supports manual revocation by the patient at any time and with immediate effect.
* A break-glass mechanism to ensure seamless emergency data access.
The remainder of this paper is organised as follows. Section 2 introduces basic concepts and definitions, as well as the classification and properties of PRE schemes. Furthermore, an analysis is made concerning the framework on which the access delegation mechanism is based. Section 3 presents the current picture of the PRE and the advancements regarding break-glass scenarios. Section 4 details the proposed solution and its implementation. Section 5 is concerned with the performance test, respective results and discussion. Lastly, Section 6 presents the conclusions and future work.
## 2 Proxy Re-encryption
PRE is a cryptographic technique that enables a third-party entity, named proxy, to delegate access to encrypted data, without being able to infer the plaintext content of that data. This is achieved by transforming a ciphertext encrypted under one key into a ciphertext encrypted under a different key.
### Syntax and basic definitions
Since PRE can be seen as a way to delegate decryption rights to a party, it is possible to categorise the different entities according to the delegation relation they possess with each other. The _delegator_ is the entity that owns the data and delegates decryption rights. The _proxy_ is the intermediary entity in the delegation process, which uses a re-encryption key (PRK) to transform the ciphertext encrypted under the delegator's public key into a ciphertext that can be decrypted only by using the delegatee's private key. Finally, the _delegatee_ is the entity that accesses the information through delegation of decryption rights by the delegator.
Definition 1 (Pre): A PRE scheme can be defined based on five different algorithms:
* _KeyGen_ -- _On input of a security parameter \(n\), the key generation algorithm KeyGen outputs a public/private key pair (\(pk_{A}\), \(sk_{A}\)) for a given user A.
* _ReKey_ -- _On input of a public/private key pair (\(pk_{A}\), \(sk_{A}\)) for user A and a public/private key pair (\(pk_{B}\), \(sk_{B}\)) for user B, a PRK \(rk_{A\to B}\) is computed.
* _Encrypt_ -- _Given the input of a public key \(pk_{A}\) and a message \(m\in M\), the encryption algorithm outputs a ciphertext \(c_{A}\in C_{1}\).
* _ReEncrypt_ -- _On input of a ciphertext \(c_{A}\in C_{1}\) and a PRK \(rk_{A\to B}\), the re-encryption algorithm ReEncrypt transforms a ciphertext \(c_{A}\in C_{1}\) into a ciphertext \(c_{B}\in C_{2}\).
* _Decrypt_ -- _Given a private key \(sk_{A}\) from user A and a ciphertext \(c_{A}\in C_{S}\) (\(S\in\{1,2\}\)) from user A, the same executes the decryption algorithm and outputs the original message \(m\in M\).
According to Qin et al.[18], a PRE scheme can be classified based on its abilities. For example, regarding its directionality, we say that the scheme is _unidirectional_ if it enables the delegator's ciphertext to be re-encrypted into the delegatee's ciphertext but not vice versa. Otherwise, we call it _bidirectional_. The multi-use/single-use classification focuses on the number of times the PRK can be used to re-encrypt data. In _multi-use_ schemes, the PRK can be utilised to perform several re-encryptions. In the case of a _single-use_ scheme, the PRK can only be used to perform a single transformation. Interactivity dictates whether the re-encryption is computed using just the public key from the delegatee (_non-interactive_ scheme) or both the public and private keys (_interactive_ scheme). Depending on the scenario of utilisation, some properties may be more desirable than others.
Other authors classify PRE schemes according to their way of functioning [9; 10]. For example, an Identity-Based PRE (IB-PRE) scheme derives public keys from identity attributes (e.g. email). The messages are encrypted using an identity string from the delegatee. Attribute-Based PRE (AB-PRE) schemes allow transforming a ciphertext defined by a set of attributes or access policies into another ciphertext with a different set of attributes.
### Umbral's PRE scheme
The Umbral PRE scheme is, in its essence, a threshold PRE scheme. This scheme features an Elliptic Curve Integrated Encryption Scheme (EICS-KEM) inspired in [2] and proposes several improvements over the original PRE scheme proposed by [4], namely unidirectionality, non-interactivity, and verifiability. It relies on the concept of semi-trusted proxies, also known as _n_svulas. Being a threshold PRE scheme, it splits the PRK according to shares. The _threshold_ portion of the scheme dictates the minimum number of those shares required to decrypt the information.
Splitting the PRK across multiple proxies brings some benefits namely eliminating a single point of failure, in case of a malfunction or compromise of one of the proxies the PRK is still safeguarded.
The re-encryption processes in our platform are supported by pyUmbral [15], a Python-based implementation of Umbral.
Fig. 1 presents an overview of the key processes and data flows involved in the Umbral PRE scheme. This system beholds seven main processes: _Encapsulation_, _Encryption_, _Generate PRK fragments_, _Re-encapsulation_, _Decapsulation_, and _Decryption_. These processes are supported by three major cryptographic methods: Key Encapsulation Mechanism (KEM), Data Encapsulation Mechanism (DEM), and Shamir Secret Sharing (SSS) [21].
The first step in this process is _Encapsulation_. This is achieved through the use of a Key Encapsulation Mechanism (KEM), in this case, a loosely inspired implementation of ECIES-KEM introduced by [22]. The KEM is fed with Alice's public key \(pk_{A}\) and outputs a symmetric key \(K\), and a _capsule_.
With the capsule and the symmetric key \(K\), the _Encryption_ process is performed using a Data encapsulation mechanism (DEM) which uses Authenticated Encryption with Additional Data (AEAD). This outputs a ciphertext encrypted with the symmetric key.
When the data is encrypted and stored in the cloud, in order for the access delegation to occur, there is a need to generate a PRK. This is performed by the _Generate PRK fragments_ process resorting to the notions present in Shamir Secret Sharing (SSS), Alice's private key and signing key \(signk_{A}\), and Bob's public key \(pk_{B}\). This enables the generation of the PRK fragments or _kFrags_. The number of fragments is defined by the number of shares.
The _kFrags_ are stored by the proxy for further use in the _Re-encapsulation_ process. This process is responsible for generating the _cFrags_ which enables Bob to gain access to the file at a later stage. To generate the _cFrags_ just the capsule and the _kFrags_ are needed. This is due to the fact that this PRE scheme performs the re-encryption over the capsule.
Figure 1: Procedural overview of pyUmbral PRE scheme
Lastly, once Bob wants to retrieve a file, the _Decapsulation_ process needs to happen. This process resorts to SSS in order to reconstruct the symmetric key \(k\). To do so, Alice's public key, Alice's verifying key \(vk_{A}\) for signature verification of the _cFrags_, Bob's private key \(sk_{B}\), and the _capsule_ are needed. Through the use of a Key Derivation Function within the KEM, it is possible to derive the symmetric key \(K\) which together with the ciphertext is passed to the DEM. The DEM performs the _Decryption_ process and outputs the plaintext content of the file that Bob can now use.
## 3 Related work
The notion of PRE made its first appearance in 1998 when Blaze et al. [4] introduced the concept of bidirectional and multi-use PRE. Several works have been published since then with new PRE schemes providing new functionalities and relying on different mathematical assumptions. For example, both [8] and [11] proposed a unidirectional, single-use PRE scheme, but the first relies on threshold PKE, while the second is based on lattice-hardness problems. In 2015, [14] also proposed a unidirectional and single-use PRE scheme, which can be classified as attribute-based. Later, in 2017, [16] presented a unidirectional, non-interactive, and verifiable PRE scheme which is threshold-based.
In the context of healthcare data sharing, PRE has also been widely explored. In fact, several works address security, privacy, and confidentiality when it comes to the design and implementation of e-health systems. However, there is still a lack of development concerning safeguarding emergency scenarios in the context of e-health systems [26]. Works that address this kind of scenario in its design, refer to this as break-glass approaches. In 2017, [3] proposed a framework for the secure sharing of Personal Health Records (PHRs) that relies on attribute-based PRE and which addresses emergency scenarios. The break-glass capabilities are provided with ABE. In this scheme, the emergency department attribute is always appended to the policy that encrypts the patient PHR, thus providing instant access to the entity from the moment the same is uploaded. The problem with this approach, and in general with ABE approaches, is that they present some caveats, namely key management and resorting to other mechanisms in break-glass approaches. This is due to the fact that emergency normally means an exception to a policy and, thus, overriding that same policy might be a hefty task in some implementations. In 2019, [25] also proposed an approach that is based on an attribute-based PRE, and provided self-adaptive access control, meaning that the system can automatically adapt to normal or emergency situations. However, their break-glass mechanism resorts to a _password-based_ paradigm. This approach raises some concerns, namely in the assumption that the individual that stores the password, has the necessary means to ensure its secrecy. More recently, in 2022, [13] proposed a system for IoT sensors combining PRE and PKE with equality test, permitting searches under different public keys and secure data sharing. However, it does not discuss emergency situations. In the same year, [20], proposed a non-interactive, multi-use, certifiateless PRE
for sharing health data in a cloud environment. Even though their approach gives full control to the data owner, it has two important drawbacks, namely it is interactive and does not propose a break-glass mechanism. Also in 2022, [24] published a secure data sharing and authorised Searchable framework for e-healthcare systems. This framework lies on a conditional and unidirectional PRE scheme with keyword search. It is also idealised for managing sensible data from medical wearable devices. This platform has some disadvantages namely regarding the PRK generation performance. Also, this work does not address emergency situations. Finally, in 2022, [12] propose a framework which is also based on attribute-based PRE that features break-glass capabilities. However, it leaves open the possible solution for revocability. That being said, there is a need to develop a solution that can cope with all the aforementioned concerns and that contributes to a more reliable and robust break-glass approach.
## 4 Patient-centric health data sovereignty
In this section, we introduce the envisioned solution for a patient-centric platform that enables health data sovereignty through PRE. The subsequent section presents the architecture of the solution, followed by a description of the processes involved in the key operations for access delegation.
### Proposed Solution
The proposed solution consists of four main nodes: the client, the resource server, the proxy server, and the authorization server, as depicted in Figure 2.
The client node hosts the client-side application developed with Next.js7. This client node communicates with the server nodes via Representation State Transfer (REST) and the Hypertext Transfer Protocol (HTTP). The business logic is divided between the resource and proxy server nodes. The resource server is based on the FastAPI framework8 running in a Python environment. This server is trusted by the data delegator and it is responsible for assisting the client-side operations, namely feeding the data the client node needs to display the information to the user. The resource server node also performs some core operations such as the initial encryption and final decryption of the Electronic Health Record (EHR) stored in the database server node hosted in a cloud environment (MongoDB9) as well as the management of delegation requests (accept or decline). Some other complementary operations are also performed such as the generation of the PRK which is stored afterwards by the proxy server node, and the signature verification of the PRK fragments and capsule fragments.
Footnote 7: [https://nextjs.org/](https://nextjs.org/)
Footnote 8: [https://fastapi.tiangolo.com/](https://fastapi.tiangolo.com/)
Footnote 9: [https://www.mongodb.com/](https://www.mongodb.com/)
The proxy server is solely responsible for the process of EHR delegation, being used for the re-encryption of the capsules and the storage of the PRK.
The authorisation server is responsible for performing the authentication of the different users of the platform as well as the issuing and claims verification of the authorisation tokens. These tokens are subsequently used to consume the APIs provided by the resource and proxy server nodes. This node is also associated with two persistence nodes. An in-memory database (REDIS10 instance) for persisting and performing the lookup of the refresh tokens and a MongoDB instance for storing general purpose user information such as name, email, password, public and verifying keys and roles.
Footnote 10: [https://redis.com/](https://redis.com/)
Figure 2: Deployment diagram of the idealised architecture
### Authentication/Authorisation
The authorisation is performed by resorting to JSON Web Tokens (JWT) which are signed using HMAC SHA256. This ensures the tokens can not be tampered with or changed, thus enabling secure transmissions between parties.
The authentication flow comprises a traditional email/password authentication, where each user needs to provide a valid email and password. In case of successful authentication, a pair of tokens are issued (access/refresh token) containing some claims needed to support the client-side application. These claims follow the standards and restrictions defined in Request for Comments 751911. Besides the pair of tokens, a Cross-site Request Forgery token is also sent for further protection in requests that require cookies. The refresh token is also sent in a cookie configured with the _secure_ and _httpOnly_ flags to ensure it is only transmitted through HTTPS and not available to JavaScript in case of a Cross-site Scripting vulnerability in the client-side application.
Footnote 11: [https://datatracker.ietf.org/doc/html/rfc7519#section-4](https://datatracker.ietf.org/doc/html/rfc7519#section-4)
Since JWT tokens are self-contained, there is no natural way of revoking them. In order to tackle this problem anti-theft mitigation techniques were implemented: _refresh token rotation_ and _token reuse detection_.
### Access delegation scenario
Access delegation is the core problem tackled in this work. The next sections dissect the access delegation flow from the moment the file is uploaded by the patient to the moment the plaintext content is retrieved by the healthcare provider. For demonstration purposes, the step-by-step process between two entities, Alice (delegator) and Bob (delegatee), is presented.
**Upload of an EHR** The access delegation starts with the upload of an EHR by Alice. When Alice uploads a new EHR, which can be a Portable Document Format (PDF) or an image, the resource server encrypts the file using a symmetric key resulting from the encapsulation process and stores it together with the capsule, resulting from the _encapsulation_ process, and an associated _userId_.
Another process that is also performed in this step and further detailed in Section 4.4 is the safeguarding of emergency situations. Besides the persistence of the file in the database, a PRK is also generated in order to provide access to the predefined trusted entity. This ensures that the trusted party possesses the means to access the file from the moment it is uploaded and that no extra input from the user is needed in this regard. This PRK is sent to the proxy for subsequent use.
**Bob requests access to an EHR** When Bob wants to access Alice's uploaded EHR, he needs to formalise his intentions by issuing a share request to the resource server containing the EHR's _resourceId_. In this step, the system checks if Bob is the owner of the EHR. This prevents a user from performing a share request to itself, something that violates the business rules of the platform.
Once this validation is performed, and provided with the _resourceId_, the resource server generates a share request that includes the _resourceId_, the _delegatorId_ and the _delegateeId_, as well as a _status_ that is set to pending by default.
**Alice answers the share request** Now that Bob asked Alice for access to the EHR, Alice is now capable of answering the share request. Depending on Alice's answer, the execution flow might have two outcomes:
**Accept scenario** -- In case of an acceptance, Alice needs to generate the PRK required to re-encrypt the capsule and further enable Bob to have access to the plaintext content of the EHR. To achieve such a feat, Alice requires his secret key along with his signing key pair, needed to verify the signature of the _kFrags_ and _cFrags_ at a later stage, as well as Bob's public key. Notice that just the public key is needed, due to the non-interactivity property of this PRE scheme. Lastly, since the underlying scheme of the access delegation mechanism is a threshold PRE scheme, there is also the need to provide a _threshold_ which defines the minimum number of shares needed to decrypt the capsule and the number of _shares_ which dictates the number of outputted PRK fragments. This last aforementioned operation outputs the _kFrags_, which are sent to the proxy along with a _shareId_ binding the PRK to a specific share request. Both attributes are persisted by the proxy for further use once Bob retrieves the EHR.
The share request operation ends with the status update of the share request, which is defined as accepted, together with an arbitrary expiration date defined by Alice. This expiration date is optional, being possible to share an EHR indefinitely or temporarily, in which case the share request is automatically revoked through a cron job once that date is transposed. This ensures the time-based access delegation aspect that this work contributes to.
**Decline scenario** -- In case Alice declines the share request the status is updated accordingly and no other action is performed.
**Bob retrieves the EHR** Now that Alice explicitly delegated access to the EHR, Bob is now capable of retrieving it. To do so, Bob performs a request to the resource server, which requires Bob's secret key and the _resourceId_, which uniquely identifies the EHR. A file ownership verification is also performed since the decryption steps are different for a delegator and a delegatee, where the former does not have the need to re-encrypt the _capsule_.
As stated previously, ownership trails different paths regarding execution flow. With that said, the following can happen whether the user is or not a data owner.
**Data owner** -- In case the user that requests the file is a data owner, a hybrid encryption approach is used, thus no re-encryption takes place.
**Not a data owner** -- If the user is not the data owner, meaning they are a delegatee, a collaborative operation between the resource and proxy servers is required to take place. For this specific scenario, Bob needs to ask the proxy to re-encrypt the capsule using the previously generated PRK. To that purpose, the resource server retrieves the EHR details and sends the capsule to the proxy server. The proxy, equipped with the capsule and the PRK fragments _kFrags_, performs the _re-encapsulation_ process outputting the _cFrags_. These _cFrags_ are
sent back to the resource server, which validates their signature through Alice's verifying key. Once the capsule fragments are validated, Bob decrypts the file by opening the capsule. This last step encompasses Bob's private key, Alice's verifying key and the verified _cFrags_. With the plaintext content of the EHR, Bob is now capable of accessing the information.
**Some important remarks** to highlight are that the secret key used in the sharing process is never shared with the intermediary entity or proxy, making it semi-trusted. Additionally, the proxy only stores the PRK, which alone does not grant it the capability to decrypt the file. Furthermore, even if the stored information such as the capsule, PRK, and ciphertext were to be leaked from the database, the safety and integrity of the EHRs would still be preserved, as they are not sufficient for decrypting the EHRs.
### Break-glass approach
Safeguarding emergency scenarios is of paramount importance in a health-related platform. Therefore, we adopted amd approach that features a central trustworthy entity responsible for managing the authorisation in emergency scenarios. This trustworthy entity is seen as a government entity that is responsible for managing such issues and has full access to the files submitted in the platform.
The implementation is similar to what is described in Section 4.3 regarding Alice accepting the share request. However, in this case, there is no explicit acceptance of the share request. When an EHR is uploaded, the trusted entity user is retrieved from the database and a PRK is generated. An accepted share request is automatically created for the trusted entity, which links the PRK to the share request between the patient and the trusted entity.
Regarding the process of retrieving the EHR, it follows a similar procedure as depicted in Section 4.3. Just like in a regular file retrieval, since the share request is automatically accepted and the proxy possesses the PRK, the trusted entity requests the proxy to re-encrypt the capsules, enabling the final decryption to take place.
This approach vastly reduces the dependency on external actors, increasing the reliability and availability of the idealised break-glass approach. Having a dedicated entity for this purpose enables instant and swift access to the information if needed.
## 5 Performance analysis
In this section, we present the performance tests conducted to evaluate our platform. Given the common concerns of limited hardware infrastructures and sub-optimal conditions in governmental adoption cases, it is important to assess the responsiveness of the key operations offered by the platform. Our main goal is to quantitatively analyze the performance of the most computationally intensive operations and assess the impact of the PRE scheme. As there are no specific regulations, indications, or suggestions regarding performance for this type of
platform, our tests are purely quantitative and based on known factors and conditions.
The performance tests were carried out on a deployed version of the platform, hosted in Microsoft Azure using a Free F1 tier running Linux and Python 3.10. While these specifications may be basic, they are sufficient to simulate a sub-optimal environment. In real-world scenarios, it is common for governments to have financial restrictions, making it likely that the platform would be deployed on infrastructure with modest specifications. The tests were conducted using Apache JMeter as the tool of choice.
In the rest of this section, we present the results related to the three most crucial operations of the platform and which involve the use of PRE: file upload, accepting a share request, and file retrieval. Additionally, a brief analysis of the results is also presented.
**File upload** The performance tests depicted in this section aim to evaluate how the different file sizes impact the upload performance of files.
Since the size of EHRs depends on various factors, such as the patient's medical history, the image resolution of the machines used for exams, and the content of the file itself, determining an average file size becomes challenging. Therefore, we conducted our experiments using two different file sizes: 1MB and 10MB.
Figure 3 illustrates the results obtained from a series of twenty runs performed for each file size.
It can be observed that a tenfold increase in file size reflected an average increase of 2715 _milliseconds_(ms) when comparing file sizes of 1MB and 10MB respectively. The former took an average of 1154 _ms_ and the latter an average of 3870 _ms_.
Figure 3: Performance Tests - File Size Uploads Bar Chart
Despite a time of almost four _seconds_, and considering this is not an ideal response time for a REST API, it should be taken into account the complexity of the operations performed. Since this is not a critical operation when it comes to performance, these values are acceptable.
**Accepting a share request** The acceptance of a share request is a key operation in the platform described in this paper. Although its performance does not possess a high impact on the efficiency of the platform, it does provide valuable information regarding the PRE process. In this operation, the PRK is generated and sent to the proxy for persistence purposes. Notice that, in this case, there was no need to perform the tests for both file sizes since the PRK generation only depends on cryptographic keys.
Regarding the results of these tests, the average time obtained in 20 runs was 869 _ms_. This quick response was expected since the generation of the PRK fragments is a relatively simple operation that depends on the cryptographic key from both ends, the signature, and the number of shares. Additionally, there was not a significant variation among the twenty runs that were performed. This is supported by the low standard deviation of just 188 _ms_.
**File retrieval** This set of tests aims to assess the impact of file sizes and the use of PRE on a file retrieval scenario. The tests were conducted for both regular decryption and PRE decryption. To evaluate the impact of file sizes, the tests were performed for both 1MB and 10MB file sizes.
Moving on to the obtained results (Fig.4), a 1MB file took an average of 903 _ms_ to be retrieved while the 10MB one took an average of 2529 _ms_. Regarding file retrieval with PRE, the 1MB file took an average of 1245 _ms_ and 2877 _ms_ for the 10MB file.
We have also evaluated the impact of re-encryption on file retrieval operations (Fig.5) by directly measuring the difference between regular decryption and PRE for each file size. This resulted in an average difference of 342 _ms_ for the 1MB file and 348 _ms_ for the other one.
The results of our tests indicate that there is a similar average difference between regular and PRE decryption for both file sizes. This similarity can be attributed to the fact that the re-encryption process only affects the capsule,
not the actual file. Since the sizes of the capsule and cryptographic keys are similar in both scenarios, it is expected that the results would be similar as well. The file size does not significantly impact the re-encryption of the capsule, but rather affects the overhead associated with fetching the file from the database and delivering it in the response.
Regarding the obtained results, they were deemed satisfactory since most operations do not possess restrictive requirements when it comes to performance.
Regarding more critical operations such as file retrieval, considering the computational effort and infrastructure complexity required to ensure full correctness with the underlying threshold PRE scheme, the results were deemed satisfactory. It is important to note that these tests were conducted in a shared infrastructure with modest specifications. Thus, it was not possible to control the current workload of the servers during the tests, which may have impacted negatively the aforementioned results
## 6 Conclusion
In this paper, we present a PRE-based platform for the secure sharing of e-health, considering a sovereign approach focused on the patient. This approach is achieved by ensuring that the patient's data is only shared with their explicit consent. Furthermore, it also enables robust revocability by the patient, without requiring updates on the encrypted EHR database, further contributing to a user-centric approach. Non-interactivity is also a key characteristic of our platform, which does not require sharing user's private key for the re-encryption process to occur. Another key achievement of our work is the proposed break-glass mechanism. Since some implementations fall short in terms of revocability, and only a few contemplate PRE in emergency scenarios, our solution uses a central trusted entity to which the proxy delegates access from the moment the EHR is uploaded to the platform. This eliminates the need to trust external actors in the system, increasing reliability and allowing swift access to the information in critical situations. There are other key characteristics of our platform worth highlighting. Firstly, it uses symmetric encryption to encrypt the EHR, which is faster than PKE. Secondly, the re-encryption process is performed over the capsule, which tends to have a much smaller size compared to a PHR. The tests that were conducted and our results show that the most demanding task is the upload of the EHR, as expected, because it requires the encapsulation process to occur and the encryption of the EHR. However, the re-encryption process does not show a significant increase when the size of the uploaded files increases. This is because the re-encryption does not involve the EHR. Our platform provides a solution to the sharing of medical data that incorporates key functionalities not covered together in previous literature, such as unidirectionality, non-interactivity, revocability, and a mechanism to deal with emergency situations. This solution contributes to the collaborative aspect of e-health and enables better, and more informed treatments supported by the increased exchange of information between providers.
Regarding future work, it would be beneficial to extend the architecture to accommodate multiple proxies instead of using just one. This could be achieved by utilising a blockchain network where the proxies work together to re-encrypt the capsules, thus enabling all the benefits that a threshold-based scheme has to offer. Furthermore, additional tests could be performed using different environments and network conditions to cover more use case scenarios.
| サービスのデジタル化による指数関数的な成長は、大量のデータの処理と保管を意味します。企業やサービスは、データ共有や渡すことを機会として、改善と新しいビジネス機会を創造する見込みです。このことは、特に医療分野において、より良いかつ革新的な治療が可能となることを証明しています。しかしながら、個人データの処理や扱いに関連する懸念が生まれています。この論文では、患者中心のプラットフォームを提案し、患者にデータを管理権を移譲することで、医療記録の共有を安全に行うことを目的としています。これにより、データ所有権を強化するステップが進むと期待されています。データ共有は、患者に同意を得て行われ、いつでもアクセスを解除することができます。さらに、緊急対応として、プロキシ再暗号化 (PRE) と、患者医療記録に即時アクセスできる中央の信頼できるエンティティの概念を採用しています。最後に、プラットフォーム |
2303.13086 | Controlled Lagrangians and Stabilization of Euler--Poincaré Equations
with Symmetry Breaking Nonholonomic Constraints | We extend the method of Controlled Lagrangians to nonholonomic
Euler--Poincar\'e equations with advected parameters, specifically to those
mechanical systems on Lie groups whose symmetry is broken not only by a
potential force but also by nonholonomic constraints. We introduce
advected-parameter-dependent quasivelocities in order to systematically
eliminate the Lagrange multipliers in the nonholonomic Euler--Poincar\'e
equations. The quasivelocities facilitate the method of Controlled Lagrangians
for these systems, and lead to matching conditions that are similar to those by
Bloch, Leonard, and Marsden for the standard holonomic Euler--Poincar\'e
equation. Our motivating example is what we call the pendulum skate, a simple
model of a figure skater developed by Gzenda and Putkaradze. We show that the
upright spinning of the pendulum skate is stable under certain conditions,
whereas the upright sliding equilibrium is always unstable. Using the matching
condition, we derive a control law to stabilize the sliding equilibrium. | Jorge S. Garcia, Tomoki Ohsawa | 2023-03-23T07:55:25 | http://arxiv.org/abs/2303.13086v2 | Controlled Lagrangians and Stabilization of Euler-Poincare Equations with Symmetry Breaking Nonholonomic Constraints
###### Abstract.
We extend the method of Controlled Lagrangians to nonholonomic Euler-Poincare equations with advected parameters, specifically to those mechanical systems on Lie groups whose symmetry is broken not only by a potential force but also by nonholonomic constraints. We introduce advected-parameter-dependent quasivelocities in order to systematically eliminate the Lagrange multipliers in the nonholonomic Euler-Poincare equations. The quasivelocities facilitate the method of Controlled Lagrangians for these systems, and lead to matching conditions that are similar to those by Bloch, Leonard, and Marsden for the standard Euler-Poincare equation. Our motivating example is what we call the pendulum state, a simple model of a figure skater developed by Gzenda and Putkaradze. We show that the upright spinning of the pendulum skate is stable under certain conditions, whereas the upright sliding equilibrium is always unstable. Using the matching condition, we derive a control law to stabilize the sliding equilibrium.
Key words and phrases:Stabilization; controlled Lagrangians; nonholonomic Euler-Poincare; broken symmetry; semidirect product 2020 Mathematics Subject Classification: 34H15, 37J60, 70E17, 70G45, 70F25, 70G65, 93D05, 93D15
## 1. Introduction
### Motivating Example: Pendulum Skate
Consider what we call the _pendulum skate_ shown in Figure 1. It is a simple model for a figure skater developed and analyzed by Gzenda and Putkaradze [13], and consists of a skate--sliding without friction on the surface--with a pendulum rigidly attached to it.
Following [13] (see also Section 2.1 below), the configuration space is the semidirect product Lie group \(\mathsf{SE}(3):=\mathsf{SO}(3)\ltimes\mathbb{R}^{3}\), or the matrix group
\[\mathsf{SE}(3)=\bigg{\{}(R,\mathbf{x}):=\begin{bmatrix}R&\mathbf{x}\\ \mathbf{0}^{T}&1\end{bmatrix}\ |\ R\in\mathsf{SO}(3),\,\mathbf{x}\in\mathbb{R}^{3} \bigg{\}}. \tag{1}\]
The gravity breaks the \(\mathsf{SE}(3)\)-symmetry of the pendulum skate, just as in the well-known example of the heavy top in the semidirect product theory of mechanics [8, 15, 17, 18]. Hence Gzenda and Putkaradze [13] used the unit vector \(\mathbf{\Gamma}\)--the vertical upward direction seen from the body frame as depicted in Figure 1 as in the heavy top equations--as an advected parameter vector, and derived its equations of motion as nonholonomic Euler-Poincare equations with advected parameters on \(\mathfrak{se}(3)\times(\mathbb{R}^{3})^{*}\), where \(\mathfrak{se}(3)\) is the Lie algebra of \(\mathsf{SE}(3)\) and the dual space \((\mathbb{R}^{3})^{*}\) of \(\mathbb{R}^{3}\) is for advected parameters \(\mathbf{\Gamma}\) that take care of the broken \(\mathsf{SE}(3)\)-symmetry.
It turns out that it is not just the gravity that breaks the symmetry. The constraints imposed by the rink--including the nonholonomic constraint that the skate cannot slide sideways--also break the symmetry as well. We shall treat this in detail later, but here is an intuitive explanation of why it breaks the symmetry: When we start off with the configuration space \(\mathsf{SE}(3)\) without the
gravity nor the rink, we have an ambient space without any preferred direction or orientation--hence the \(\mathsf{SE}(3)\)-symmetry. However, by introducing the rink to the setting, we effectively introduce a special direction--the unit normal vector to the rink--to the ambient space, thereby breaking the \(\mathsf{SE}(3)\)-symmetry. This is just like how the gravity breaks the symmetry by introducing the special "vertical" direction to the ambient space that would be otherwise uniform in any direction.
The main motivation for this work is to stabilize nonholonomic mechanical systems on Lie groups with such broken symmetry. We shall show in Section 5 that, for the pendulum skate, the upright spinning and upright sliding motions--ubiquitous in figure skating--are equilibria of the nonholonomic Euler-Poincare equations derived in [13]. As the intuition suggests, the spinning equilibrium is stable only under certain conditions, whereas the sliding equilibrium is always unstable.
Motivated by the problem of finding a control law to stabilize the sliding equilibrium, we would like to extend the method of Controlled Lagrangians of Bloch et al. [3] to the Euler-Poincare equations with symmetry-breaking nonholonomic constraints. Particularly, our main goal is to build on the nonholonomic Euler-Poincare theory of Schneider [20] (see also Holm [14, Section 12.3]) to derive matching conditions for the Controlled Lagrangians applied to such systems.
The pendulum skate indeed necessitates a slight generalization of [20] because it does not fit into the most general setting of [20, Section 2.1 and Theorem 1]. We note that Gay-Balmaz and Yoshimura [12] made such a generalization in a more abstract and general Dirac structure setting. Our focus is rather on first having a concrete expression for the nonholonomic Euler-Poincare equations, particularly on a systematic way to eliminate the Lagrange multipliers in the equations of motion arising from the constraints.
Developing a general method of controlled Lagrangians for nonholonomic systems is challenging, particularly because of the Lagrange multipliers. Indeed, extensions of the method of Controlled Lagrangians to nonholonomic systems are limited to a very special class of Lagrange-d'Alembert equations; see, e.g., Zenkov et al. [24, 25, 26, 28]. To our knowledge, an extension to nonholonomic Euler-Poincare equations has been done by Schneider [20] only for the so-called Chaplygin top.
### Main Results and Outline
We build on the work of Schneider [20] to develop matching conditions for mechanical systems on Lie group \(\mathsf{S}\) with symmetry-breaking nonholonomic constraints.
Figure 1. What we call the pendulum skate here is a simple model for a figure skater developed by Gzenda and Putkaradze [13]. The unit vector \(\boldsymbol{\Gamma}\)—the vertical upward direction _seen from the body frame_\(\{\mathbf{E}_{i}\}_{i=1}^{3}\)—is the advected parameter here.
Particularly, we assume: (i) The left \(\mathsf{S}\)-invariance of the Lagrangian is broken, but can be recovered using an advected parameter \(\Gamma\); (ii) the nonholonomic constraints are not left-invariant either, but this broken symmetry is also recovered by using the same \(\Gamma\). We shall explain the details of this setting in Section 2 using the pendulum skate as an example.
In Section 3, we formulate the reduced Lagrange-d'Alembert principle and derive the nonholonomic Euler-Poincare equations with broken symmetry as Proposition 5, giving a generalization of the General Theorem of Schneider [20, Theorem 1]. However, as mentioned earlier, it is a special case of the Dirac reduction for nonholonomic systems by Gay-Balmaz and Yoshimura [12], and is included here only for completeness.
The ideas that nonholonomic constraints may be symmetry-breaking and that the broken symmetry may be recovered using an advected parameter are not new; they were discussed in [20] as well as Tai [21], Burkhardt and Burdick [6], and Burkhardt [7]. Our treatment is more systematic as in [12] and applies to more general constraints than those of [20].
The above result leads to the notion of \(\Gamma\)-dependent quasivelocities in Section 4. Quasivelocities have been often used in nonholonomic mechanics; see, e.g., Ball et al. [2], Bloch et al. [4], Zenkov [23], Zenkov et al. [27] and references therein. However, to our knowledge, the idea of advected-parameter-dependent quasivelocities is new. Those \(\Gamma\)-dependent quasivelocities help us eliminate the Lagrange multipliers in the nonholonomic Euler-Poincare equations derived in Proposition 5.
In Section 5, we apply the result from Section 4 to the pendulum skate, find a family of equilibria including the upright spinning and upright sliding ones, and analyze their stability.
In Section 6, we show that the nonholonomic Euler-Poincare equations written in the \(\Gamma\)-dependent quasivelocities help us extend the Controlled Lagrangians of Bloch et al. [3]--developed for the _standard_ Euler-Poincare equation--to our nonholonomic setting. Indeed, the derivation and expressions of the resulting matching conditions in Proposition 13 are almost the same as those in [3] thanks to the formulation using the quasivelocities. The result also generalizes the simpler and ad-hoc matching of [20] for the Chaplygin top and our own for the pendulum skate in [11].
Finally, in Section 7, we apply the Controlled Lagrangian to find a feedback control to stabilize the sliding equilibrium, and illustrate the result in a numerical simulation.
## 2. Broken Symmetry in Lagrangian and Nonholonomic Constraints
In this section, we would like to use the pendulum skate of Gzenda and Putkaradze [13] to describe the basic ideas behind mechanical systems on Lie groups with symmetry-breaking nonholonomic constraints.
### Pendulum Skate
Let \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}\) and \(\{\mathbf{E}_{1},\mathbf{E}_{2},\mathbf{E}_{3}\}\) be the spatial and body frames, respectively, where the body frame is aligned with the principal axes of inertia; particularly \(\mathbf{E}_{1}\) is aligned with the edge of the blade as shown in Figure 1. The two frames are related by the rotation matrix \(R(t)\in\mathsf{SO}(3)\) whose column vectors represent the body frame viewed in the spatial one at time \(t\).
The origin of the body frame is the blade-ice contact point, and has position vector \(\mathbf{x}(t)=(x_{1}(t),x_{2}(t),0)\) at time \(t\) in the spatial frame. However, we shall treat the position \(\mathbf{x}\) of the contact point as a vector in \(\mathbb{R}^{3}\) and impose \(x_{3}=0\) as a constraint. Hence, as mentioned earlier, the configuration space is the semidirect product Lie group \(\mathsf{SE}(3):=\mathsf{SO}(3)\ltimes\mathbb{R}^{3}\) from (1), where the multiplication rule is given by
\[(R_{0},\mathbf{x}_{0})\cdot(R,\mathbf{x})=(R_{0}R,R_{0}\mathbf{x}+\mathbf{x} _{0}).\]
Let us find the Lagrangian of the system. If \(t\mapsto s(t)=(R(t),\mathbf{x}(t))\) is the dynamics of the system in \(\mathsf{SE}(3)\), then
\[s^{-1}\dot{s}=\begin{bmatrix}R^{T}\dot{R}&R^{T}\dot{\mathbf{x}}\\ \mathbf{0}^{T}&0\end{bmatrix}=:\begin{bmatrix}\hat{\Omega}&\mathbf{Y}\\ \mathbf{0}^{T}&0\end{bmatrix}=:(\hat{\Omega},\mathbf{Y})\in\mathfrak{se}(3),\]
where \(\hat{\Omega}:=R^{T}\dot{R}\) is the body angular velocity; \(\mathbf{Y}:=R^{T}\dot{\mathbf{x}}\) is the velocity of the blade-ice contact point seen from the body frame.
Suppose that the center of mass is located at \(l\mathbf{E}_{3}\) in the body frame as shown in Figure 1, and hence, in the spatial frame, is located at \(\mathbf{x}_{\mathrm{cm}}(t):=\mathbf{x}(t)+lR(t)\mathbf{E}_{3}\). Then, the Lagrangian \(L_{\mathbf{e}_{3}}\colon T\mathsf{SE}(3)\to\mathbb{R}\) is given by:
\[L_{\mathbf{e}_{3}}\Big{(}R,\mathbf{x},\dot{R},\dot{\mathbf{x}} \Big{)} :=\frac{1}{2}\operatorname{tr}\Big{(}\dot{R}\dot{\mathbb{I}}\dot {R}^{T}\Big{)}+\frac{m}{2}\|\dot{\mathbf{x}}_{\mathrm{cm}}\|^{2}-m\mathsf{g} \mathbf{e}_{3}^{T}\mathbf{x}_{\mathrm{cm}}\] \[=\frac{1}{2}\operatorname{tr}\Big{(}R^{T}\dot{R}\dot{\mathbb{I}} \dot{R}^{T}R\Big{)}+\frac{m}{2}\Big{\|}\dot{\mathbf{x}}+\dot{R}l\mathbf{E}_{3} \Big{\|}^{2}-m\mathsf{g}l\mathbf{e}_{3}^{T}R\mathbf{E}_{3}\] \[=\frac{1}{2}\operatorname{tr}\Big{(}\hat{\Omega}\mathbb{J}\hat{ \Omega}^{T}\Big{)}+\frac{m}{2}\Big{\|}\mathbf{Y}+l\hat{\Omega}\mathbf{E}_{3} \Big{\|}^{2}-m\mathsf{g}l(R^{T}\mathbf{e}_{3})^{T}\mathbf{E}_{3},\]
where \(m\) is the total mass, \(\mathsf{g}\) is the gravitational acceleration, \(\|\cdot\|\) is the Euclidean norm, and \(\mathbb{J}\) is the inertia matrix.
We also identify \(\mathbf{\Omega}=(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathbb{R}^{3}\) with \(\hat{\Omega}\in\mathfrak{so}(3)\) via the hat map [16, SS5.3]:
\[\widehat{\widehat{\widehat{\phantom{\widehat{\phantom{\widehat{\phantom{ \widehat{\phantom{\widehat{\phantom
as well as its induced representation \((s,\gamma)\mapsto s\gamma\) on \(X^{*}\), i.e.,
\[\mathsf{S}\times X^{*}\to X^{*};\qquad(s,\gamma)\mapsto s\gamma\quad\text{such that}\quad\langle s\gamma,\mathsf{x}\rangle=\big{\langle}\gamma,s^{-1}\mathsf{x} \big{\rangle}\quad\forall\mathsf{x}\in X. \tag{4}\]
We assume that we can recover the (left) \(\mathsf{S}\)-symmetry as follows: For every \(s_{0},s\in\mathsf{S}\) and every \(\gamma\in X^{*}\),
\[L(s_{0}s,s_{0}\dot{s},s_{0}\gamma)=L(s,\dot{s},\gamma).\]
As a result, we may define the reduced Lagrangian as follows:
\[\ell\colon\mathsf{s}\times X^{*}\to\mathbb{R};\qquad\ell(\xi,\Gamma):=L(e,\xi,\Gamma), \tag{5}\]
where \(\mathsf{s}\) is the Lie algebra of \(\mathsf{S}\).
**Example 1** (Lagrangian of pendulum skate [13]).: For the pendulum skate from Section 2.1, we define the extended Lagrangian \(L\colon T^{*}\mathsf{SE}(3)\times(\mathbb{R}^{3})^{*}\to\mathbb{R}\) as follows:
\[L\Big{(}R,\mathbf{x},\dot{R},\dot{\mathbf{x}},\boldsymbol{\gamma}\Big{)}= \frac{1}{2}\boldsymbol{\Omega}^{T}\mathbb{I}\boldsymbol{\Omega}+\frac{m}{2} \|\mathbf{Y}+l\boldsymbol{\Omega}\times\mathbf{E}_{3}\|^{2}-m\text{gl}(R^{T} \boldsymbol{\gamma})^{T}\mathbf{E}_{3}\]
so that \(L\Big{(}R,\mathbf{x},\dot{R},\dot{\mathbf{x}},\mathbf{e}_{3}\Big{)}=L_{ \mathbf{e}_{3}}\Big{(}R,\mathbf{x},\dot{R},\dot{\mathbf{x}}\Big{)}\).
We also define an \(\mathsf{SE}(3)\)-representation on \(\mathbb{R}^{3}\) by setting \((R,\mathbf{x})\mathbf{y}:=R\mathbf{y}\). Then, identifying \((\mathbb{R}^{3})^{*}\) with \(\mathbb{R}^{3}\) via the dot product, we have
\[((R,\mathbf{x})\boldsymbol{\gamma})\cdot\mathbf{y}=\boldsymbol{\gamma}\cdot \big{(}(R,\mathbf{x})^{-1}\mathbf{y}\big{)}=\boldsymbol{\gamma}\cdot\big{(}R^ {-1}\mathbf{y}\big{)}=(R\boldsymbol{\gamma})\cdot\mathbf{y}.\]
Therefore, we have
\[\mathsf{SE}(3)\times(\mathbb{R}^{3})^{*}\to(\mathbb{R}^{3})^{*};\qquad((R, \mathbf{x}),\boldsymbol{\gamma})\mapsto R\boldsymbol{\gamma} \tag{6}\]
using the standard matrix-vector multiplication.
Then we see that the \(\mathsf{SE}(3)\)-symmetry is recovered: For every \((R_{0},\mathbf{x}_{0}),(R,\mathbf{x})\in\mathsf{SE}(3)\) and every \(\boldsymbol{\gamma}\in\mathbb{R}^{3}\),
\[L\Big{(}R_{0}R,R_{0}\mathbf{x}+\mathbf{x}_{0},R_{0}\dot{R},R_{0}\dot{\mathbf{ x}},R_{0}\boldsymbol{\gamma}\Big{)}=L\Big{(}R,\mathbf{x},\dot{R},\dot{\mathbf{x}}, \boldsymbol{\gamma}\Big{)}.\]
Therefore, we may define the reduced Lagrangian
\[\begin{split}&\ell\colon\mathfrak{s}\mathfrak{s}(3)\times(\mathbb{ R}^{3})^{*}\cong\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to \mathbb{R};\\ &\ell(\boldsymbol{\Omega},\mathbf{Y},\boldsymbol{\Gamma}):=\frac{ 1}{2}\boldsymbol{\Omega}^{T}\mathbb{I}\boldsymbol{\Omega}+\frac{m}{2}\| \mathbf{Y}+l\boldsymbol{\Omega}\times\mathbf{E}_{3}\|^{2}-m\text{gl}\boldsymbol {\Gamma}^{T}\mathbf{E}_{3},\end{split} \tag{7}\]
which agrees with [13, Eq. (1)]. Since \(\gamma_{0}=\mathbf{e}_{3}\) for our Lagrangian, the advected parameter is \(\boldsymbol{\Gamma}=R^{T}\gamma_{0}=R^{T}\mathbf{e}_{3}\), which gives the vertical upward direction (essentially the direction of gravity) seen from the body frame; see Figure 1.
_Remark 2_ (Intuition behind symmetry recovery).: Here is an intuitive interpretation of how the symmetry recovery works in Example 1 above. When we talked about the broken \(\mathsf{SE}(3)\)-symmetry in (2), the vertical upward direction \(\mathbf{e}_{3}\) (essentially the direction of gravity) was fixed, and the \(\mathsf{SE}(3)\)-action \((R_{0},\mathbf{x}_{0})\cdot(R,\mathbf{x})=(R_{0}R,R_{0}\mathbf{x}+\mathbf{x}_ {0})\) rotates the pendulum skate by \(R_{0}\) and translates it by \(\mathbf{x}_{0}\) but _left the direction of the gravity unchanged_, resulting in a system whose direction of gravity is different from the original configuration \((R,\mathbf{x})\); hence the symmetry is broken. On the other hand, when we introduced \(\boldsymbol{\gamma}\) as a new variable in the Lagrangian above and let \(\mathsf{SE}(3)\) act on \(\gamma\) as defined in (6), we _co-rotated the direction of the gravity along with the pendulum skate_, resulting in a system with the same relative direction of gravity as the original one; hence the whole system--now involving the variable direction of gravity--possesses the \(\mathsf{SE}(3)\)-symmetry.
### Broken Symmetry in Nonholonomic Constraints
Following Gay-Balmaz and Yoshimura [12, Section 4.2], we assume that the system is subject to nonholonomic constraints with a fixed parameter \(\gamma_{0}\in X^{*}\), that is, the constraint is defined by a corank \(r\) smooth distribution \(\mathcal{D}_{\gamma_{0}}\subset T\mathsf{S}\) so that the dynamics \(t\mapsto s(t)\in\mathsf{S}\) satisfies \(\dot{s}(t)\in\mathcal{D}_{\gamma_{0}}(s(t))\). In other words, we have one-forms \(\{\Psi^{a}_{\gamma_{0}}\}_{a=1}^{r}\) on \(\mathsf{S}\) such that the annihilator \(\mathcal{D}^{\circ}_{\gamma_{0}}(s)=\operatorname{span}\{\Psi^{a}_{\gamma_{0}}( s)\}_{a=1}^{r}\), i.e., \(\dot{s}(t)\in\ker\Psi^{a}_{\gamma_{0}}(s(t))\) for any \(a\in\{1,\ldots,r\}\).
We assume that \(\mathcal{D}_{\gamma_{0}}\) is _not_\(\mathsf{S}\)-invariant, i.e.,
\[T_{s}\mathsf{L}_{s_{0}}(\mathcal{D}_{\gamma_{0}}(s))\neq\mathcal{D}_{\gamma_{ 0}}(s_{0}s)\text{ for some }s,s_{0}\in\mathsf{S},\]
or in terms of the one-forms,
\[(\mathsf{L}^{*}_{s_{0}}\Psi^{a}_{\gamma_{0}})(s)\neq\Psi^{a}_{\gamma_{0}}(s) \text{ for some }s,s_{0}\in\mathsf{S}\text{ and some }a\in\{1,\ldots,r\}.\]
We also assume that we may recover the \(\mathsf{S}\)-symmetry in a similar way as above: Defining
\[\mathcal{D}\colon\mathsf{S}\times X^{*}\to T\mathsf{S}\quad\text{with} \quad\mathcal{D}(s,\gamma)\subset T_{s}\mathsf{S}\quad\text{and}\quad\mathcal{ D}(s,\gamma_{0})=\mathcal{D}_{\gamma_{0}}(s)\quad\forall s\in\mathsf{S},\]
we have the \(\mathsf{S}\)-invariance in the following sense:
\[T_{s}\mathsf{L}_{s_{0}}(\mathcal{D}(s,\gamma))=\mathcal{D}(s_{0}s,s_{0} \gamma)\quad\forall s,s_{0}\in\mathsf{S}\quad\forall\gamma\in X^{*}.\]
In other words, we have the following \(r\) one-forms on \(\mathsf{S}\) with a parameter in \(X^{*}\):
\[\Psi^{a}\colon\mathsf{S}\times X^{*}\to T^{*}\mathsf{S}\quad\text{with}\quad \Psi^{a}(s,\gamma)\subset T^{*}_{s}\mathsf{S}\quad\forall s\in\mathsf{S}\quad \forall\gamma\in X^{*}\quad\forall a\in\{1,\ldots,r\}\]
satisfying \(\mathcal{D}^{\circ}(s,\gamma)=\operatorname{span}\{\Psi^{a}(s,\gamma)\}_{a=1} ^{r}\) and
\[(\mathsf{L}^{*}_{s_{0}}\Psi^{a})(\,\cdot\,,s_{0}\gamma)=\Psi^{a}(\,\cdot\,, \gamma)\quad\forall\gamma\in X^{*}\quad\forall a\in\{1,\ldots,r\}. \tag{8}\]
We may then define the following parameter-dependent subspaces in \(\mathsf{s}\) and parameter-dependent elements in \(\mathsf{s}^{*}\):
\[\mathfrak{d}(\Gamma):=\mathcal{D}(e,\Gamma)\subset\mathsf{s},\qquad\psi^{a}( \Gamma):=\Psi^{a}(e,\Gamma)\in\mathsf{s}^{*}\quad\forall a\in\{1,\ldots,r\} \tag{9}\]
so that the annihilator \(\mathfrak{d}^{\circ}(\Gamma)=\operatorname{span}\{\psi^{a}(\Gamma)\}_{a=1}^{r }\subset\mathsf{s}^{*}\).
**Example 3** (No-side-sliding constraint of pendulum skate).: The skate blade moves without friction, but with a constraint that prohibits motions perpendicular to its edge. This means that the spatial velocity \(\dot{\mathbf{x}}\) has no components in the direction perpendicular to the plane spanned by \(\{R\mathbf{E}_{1},\mathbf{e}_{3}\}\), i.e., \(\langle\dot{\mathbf{x}},R\mathbf{E}_{1}\times\mathbf{e}_{3}\rangle=0\) as shown in Figure 2. In other words, setting \(\Psi_{\mathbf{e}_{3}}(R,\mathbf{x})=(R\mathbf{E}_{1}\times\mathbf{e}_{3})^{T} \mathbf{d}\mathbf{x}\), one may write the constraint as \(\dot{\mathbf{x}}\in\ker\Psi_{\mathbf{e}_{3}}\). It is easy to see that the \(\mathsf{SE}(3)\)-symmetry is broken: If we take \((R_{0},\mathbf{x}_{0})\in\mathsf{SE}(3)\) with \(R_{0}^{T}\mathbf{e}_{3}\neq\mathbf{e}_{3}\), then
\[(\mathsf{L}^{*}_{(R_{0},\mathbf{x}_{0})}\Psi_{\mathbf{e}_{3}})(R,\mathbf{x}) =((R_{0}R\mathbf{E}_{1})\times\mathbf{e}_{3})^{T}\mathbf{d}(R_{0} \mathbf{x})\] \[=\big{(}(R\mathbf{E}_{1})\times(R_{0}^{T}\mathbf{e}_{3})\big{)}^{ T}\mathbf{d}\mathbf{x}\] \[\neq\Psi_{\mathbf{e}_{3}}(R,\mathbf{x})\]
in general, although \(\mathsf{L}^{*}_{(R_{0},\mathbf{x}_{0})}\Psi_{\mathbf{e}_{3}}=\Psi_{\mathbf{e}_{ 3}}\) if \(R_{0}^{T}\mathbf{e}_{3}=\mathbf{e}_{3}\).
One notices that this is the same type of broken symmetry as in the Lagrangian from Example 1 caused by the gravity. This suggests the same remedy applied to the Lagrangian would recover the \(\mathsf{SE}(3)\)-symmetry of the constraint too. Indeed, define
\[\Psi\colon\mathsf{SE}(3)\times(\mathbb{R}^{3})^{*}\to T^{*}\mathsf{SE}(3); \qquad\Psi((R,\mathbf{x}),\boldsymbol{\gamma}):=((R\mathbf{E}_{1})\times \boldsymbol{\gamma})^{T}\mathbf{d}\mathbf{x}\]
so that \(\Psi((R,\mathbf{x}),\mathbf{e}_{3})=\Psi_{\mathbf{e}_{3}}(R,\mathbf{x})\). Then we see that, for every \((R_{0},\mathbf{x}_{0}),(R,\mathbf{x})\in\mathsf{SE}(3)\) and every \(\boldsymbol{\gamma}\in\mathbb{R}^{3}\),
\[(\mathsf{L}^{*}_{(R_{0},\mathbf{x}_{0})}\Psi)((R,\mathbf{x}),R_{0 }\boldsymbol{\gamma}) =((R_{0}R\mathbf{E}_{1})\times(R_{0}\boldsymbol{\gamma}))^{T} \mathbf{d}(R_{0}\mathbf{x})\] \[=((R\mathbf{E}_{1})\times\boldsymbol{\gamma})^{T}\mathbf{d} \mathbf{x}\] \[=\Psi((R,\mathbf{x}),\boldsymbol{\gamma}).\]
Therefore, we define the \(\boldsymbol{\Gamma}\)-dependent element
\[\psi^{3}(\boldsymbol{\Gamma}):=\Psi((I,\mathbf{0}),\boldsymbol{\Gamma})=( \mathbf{0},\mathbf{E}_{1}\times\boldsymbol{\Gamma})\in\mathfrak{se}(3)^{*} \cong\mathbb{R}^{3}\times\mathbb{R}^{3},\]
where we used the superscript \(3\) because there are two other constraints as we shall see in Example 8.
_Remark 4_.: Why does the same advected parameter \(\boldsymbol{\Gamma}\)--which we used to recover the broken symmetry _due to the gravity_--work in order to recover the broken symmetry _due to the nonholonomic constraint_ as well? The nonholonomic constraint is characterized by how one introduces the risk into the system. In this problem, we introduced the risk as a horizontal plane--perpendicular to the direction of gravity. So \(\boldsymbol{\Gamma}\) gives _both_ the direction of gravity _and_ the orientation of the risk seen from the body frame.
## 3. Nonholonomic Euler-Poincare Equation with Advected Parameters
This section gives a review of the reduced Lagrange-d'Alembert principle for mechanical systems with broken symmetry. Our result gives a slight generalization of the General Theorem of Schneider [20, Theorem 1]. However, as mentioned earlier, ours is a special case of Gay-Balmaz and Yoshimura [12, Theorem 4.3] as well, and so we include this result only for completeness.
### Reduced Lagrange-d'Alembert Principle with Broken Symmetry
We would like to find the reduced equations of motion exploiting the recovered symmetry. For our setting, one needs to turn the variational principle of Holm et al. [15, Theorem 3.1] (see also Cendra et al. [8, Theorem 1.1]) into a Lagrange-d'Alembert-type incorporating the symmetry-breaking nonholonomic constraints. We note that this is done in Schneider [20, Theorem 1] with a semidirect product Lie group \(\mathsf{S}=\mathsf{G}\ltimes V\) assuming a special class of nonholonomic constraints.
Figure 2. No side sliding.
Before describing our version of the reduced Lagrange-d'Alembert principle with broken symmetry, let us introduce some notation used in the result to follow. For any curve \(t\mapsto s(t)\in\mathsf{S}\), we define \(t\mapsto\xi(t):=s(t)^{-1}\dot{s}(t)\in\mathsf{s}\); conversely, given \(t\mapsto\xi(t)\in\mathsf{s}\), we define \(t\mapsto s(t)\in\mathsf{S}\) via \(\dot{s}(t)=s(t)\xi(t)\) with the left translation. Similarly, for any (infinitesimal) variation \(t\mapsto\delta s(t)\in T_{s(t)}\mathsf{S}\) of the curve \(t\mapsto s(t)\), we define \(t\mapsto\eta(t):=s(t)^{-1}\delta s(t)\) and conversely as well.
We shall also use the Lie algebra representation
\[\mathsf{s}\times X\to X;\qquad(\xi,\mathsf{x})\mapsto\left.\frac{d}{d\varepsilon }\exp(\varepsilon\xi)\mathsf{x}\right|_{\varepsilon=0}=:\xi\mathsf{x},\]
and
\[\mathsf{s}\times X^{*}\to X^{*};\qquad(\xi,\gamma)\mapsto\xi\gamma\quad\text{ so that}\quad\langle\xi\gamma,\mathsf{x}\rangle=-\langle\gamma,\xi\mathsf{x}\rangle\quad\forall \mathsf{x}\in X, \tag{10}\]
as well as the momentum map \(\mathbf{K}\colon X\times X^{*}\to\mathsf{s}^{*}\) defined by
\[\langle\mathbf{K}(\mathsf{x},\Gamma),\xi\rangle=\langle\Gamma,\xi\mathsf{x} \rangle=-\langle\xi\Gamma,\mathsf{x}\rangle\quad\forall\xi\in\mathsf{s}, \tag{11}\]
where \(\xi\mathsf{x}\) is defined in the similar way as in (10) using (3).
Also, for any smooth function \(f\colon E\to\mathbb{R}\) on a real vector space \(E\), let us define its functional derivative \(\delta f/\delta x\in E^{*}\) at \(x\in E\) such that, for any \(\delta x\in E\), under the natural dual pairing \(\langle\,\cdot\,,\,\cdot\,\rangle\colon E^{*}\times E\to\mathbb{R}\),
\[\left.\left\langle\frac{\delta f}{\delta x},\delta x\right\rangle=\left.\frac {d}{d\varepsilon}f(x+\varepsilon\delta x)\right|_{\varepsilon=0}.\]
**Proposition 5** (Reduced Lagrange-d'Alembert Principle with Broken Symmetry).: _Let \(\gamma_{0}\in X^{*}\) be fixed, and suppose that the Lagrangian \(L_{\gamma_{0}}\colon T\mathsf{S}\to\mathbb{R}\) and the distribution \(\mathcal{D}_{\gamma_{0}}\subset T\mathsf{S}\) defining nonholonomic constraints satisfy the assumptions on symmetry recovery from Sections 2.2 and 2.3._
_Then the following are equivalent:_
1. _The curve_ \(t\mapsto s(t)\in\mathsf{S}\) _with the constraint_ \(\dot{s}(t)\in\mathcal{D}_{\gamma_{0}}(s(t))\) _satisfies the Lagrange-d'Alembert principle_ \[\delta\int_{t_{0}}^{t_{1}}L_{\gamma_{0}}(s(t),\dot{s}(t))\,dt=0\] _subject to_ \(\delta s(t_{0})=\delta s(t_{1})=0\) _and_ \(\delta s(t)\in\mathcal{D}_{\gamma_{0}}(s(t))\) _for any_ \(t\in(t_{0},t_{1})\)_._
2. _The curve_ \(t\mapsto\xi(t)\in\mathsf{s}\) _along with_ \[t\mapsto\Gamma(t):=s(t)^{-1}\gamma_{0}\in X^{*}\] (12) _and with the constraint_ \[\xi(t)\in\mathfrak{d}(\Gamma(t))\iff\langle\psi^{a}(\Gamma(t)),\xi(t)\rangle =0\quad\forall t\in[t_{0},t_{1}]\quad\forall a\in\{1,\ldots,r\}\] (13) _satisfies the following reduced Lagrange-d'Alembert principle in terms of the reduced Lagrangian_ \(\ell\colon\mathsf{s}\times X^{*}\to\mathbb{R}\) _defined in (_5_):_ \[\delta\int_{t_{0}}^{t_{1}}\ell(\xi(t),\Gamma(t))\,dt=0,\] (14) _where variations of_ \(t\mapsto(\xi(t),\Gamma(t))\) _are subject to the constraints_ \[\delta\xi=\dot{\eta}+\mathrm{ad}_{\xi}\,\eta,\qquad\delta\Gamma=-\eta\,\Gamma\] (15) _for every curve_ \(t\mapsto\eta(t)\in\mathsf{s}\) _satisfying_ \[\delta\eta(t_{0})=\delta\eta(t_{1})=0,\qquad\langle\psi^{a}(\Gamma(t)),\eta(t) \rangle=0\quad\forall t\in[t_{0},t_{1}]\quad\forall a\in\{1,\ldots,r\}.\] (16)
_._
3. _The curve_ \(t\mapsto(\xi(t),\Gamma(t))\) _with_ \(\xi(t)\in\mathfrak{d}(\Gamma(t))\) _satisfies the nonholonomic Euler-Poincare equations with advected parameter_ \(\Gamma\)_:_ \[\frac{d}{dt}\bigg{(}\frac{\delta\ell}{\delta\xi}\bigg{)} =\mathrm{ad}_{\xi}^{\ast}\,\frac{\delta\ell}{\delta\xi}+\mathbf{K} \bigg{(}\frac{\delta\ell}{\delta\Gamma},\Gamma\bigg{)}+\lambda_{a}\psi^{a}( \Gamma),\] (17a) \[\dot{\Gamma} =-\xi\Gamma\] (17b) _with_ \(\Gamma(t_{0})=s(t_{0})^{-1}\gamma_{0}\)_, where_ \(\{\lambda_{a}\}_{a=1}^{r}\) _are Lagrange multipliers._
Proof.: Let us first show the equivalence between (i) and (ii). First, let us show that the two action integrals are equal. Indeed, using the definition (5) of \(\ell\),
\[L_{\gamma_{0}}(s,\dot{s})=L(s,\dot{s},\gamma_{0})=L\big{(}e,s^{-1}\dot{s},s^{- 1}\gamma_{0}\big{)}=\ell(\xi,\Gamma).\]
It is a standard result in the Euler-Poincare theory that all variations of \(t\mapsto s(t)\) with fixed endpoints induce and are induced by variations of \(t\mapsto\xi(t)\) of the form \(\delta\xi=\dot{\eta}+\mathrm{ad}_{\xi}\,\eta\) with \(\eta\) vanishing at the endpoint; see, e.g., [15, Theorem 3.1 and Lemma 3.2]. Also, \(\delta\Gamma=-\eta\Gamma\) easily follows from the definitions (10) and (12). Finally, one can show the equivalence between the nonholonomic constraints on \(\delta s\) and \(\eta\) as follows: For every \(a\in\{1,\ldots,r\}\),
\[\delta s\in\mathcal{D}_{\gamma_{0}}(s)\iff 0 =\langle\Psi^{a}(s,\gamma_{0}),\delta s\rangle\] \[=\big{\langle}\Psi^{a}(s,\gamma_{0}),ss^{-1}\delta s\big{\rangle}\] \[=\langle(L_{s}^{\ast}\Psi^{a})(e,\gamma_{0}),\eta\rangle\] \[=\big{\langle}\Psi^{a}(e,s^{-1}\gamma_{0}),\eta\big{\rangle}\quad( \because(8))\] \[=\langle\psi^{a}(\Gamma),\eta\rangle\quad(\because(9)\text{ and definition of }\Gamma).\]
It remains to show the equivalence between (ii) and (iii). As is done in the proof of [15, Theorem 3.1],
\[\delta\int_{t_{0}}^{t_{1}}\ell(\xi(t),\Gamma(t))\,dt =\int_{t_{0}}^{t_{1}}\bigg{(}\bigg{\langle}\frac{\delta\ell}{ \delta\xi},\delta\xi\bigg{\rangle}+\bigg{\langle}\delta\Gamma,\frac{\delta \ell}{\delta\Gamma}\bigg{\rangle}\bigg{)}dt\] \[=\int_{t_{0}}^{t_{1}}\bigg{(}\bigg{\langle}\frac{\delta\ell}{ \delta\xi},\dot{\eta}+\mathrm{ad}_{\xi}\,\eta\bigg{\rangle}-\bigg{\langle}\eta \Gamma,\frac{\delta\ell}{\delta\Gamma}\bigg{\rangle}\bigg{)}dt\] \[=-\int_{t_{0}}^{t_{1}}\bigg{\langle}\frac{d}{dt}\bigg{(}\frac{ \delta\ell}{\delta\xi}\bigg{)}-\mathrm{ad}_{\xi}^{\ast}\,\frac{\delta\ell}{ \delta\xi}-\mathbf{K}\bigg{(}\frac{\delta\ell}{\delta\Gamma},\Gamma\bigg{)}, \eta\bigg{\rangle}dt.\]
Therefore, the reduced Lagrange-d'Alembert principle (14) with the nonholonomic constraints (16) yields (17a), whereas taking the time derivative of (12) yields (17b). Conversely, it is clear that (17a) implies (14) with the constraints imposed in (ii), and integrating (17b) with the initial condition on \(\Gamma\) given in (iii) yields (12).
**Example 6** (General Theorem of Schneider [20, Theorem 1]).: Let \(\mathsf{G}\) be a Lie group, \(V\) be a vector space, and construct the semidirect product Lie group \(\mathsf{S}:=\mathsf{G}\ltimes V\) under the multiplication
\[s_{1}\cdot s_{2}=(g_{1},x_{1})\cdot(g_{2},x_{2})=(g_{1}g_{2},g_{1}x_{2}+x_{1}),\]
where \(\mathsf{G}\times V\to V;(g,x)\mapsto gx\) is a representation. In what follows, we shall use the other induced \(\mathsf{G}\)- and \(\mathfrak{g}\)-representations on \(V\) and \(V^{\ast}\) defined in the same way we did for \(X\) and \(X^{\ast}\) above, as well as the associated momentum map \(\mathbf{J}\colon V\times V^{\ast}\to\mathfrak{g}^{\ast}\) defined in the same way as \(\mathbf{K}\) was in (11). Indeed, it is also assumed in [20] that \(X=V\) and the \(\mathsf{S}\)-representation on \(X^{\ast}\) is defined as \((g,x)\gamma:=g\gamma\) using the \(\mathsf{G}\)-representation on \(V^{\ast}\) induced by the above \(\mathsf{G}\)-representation on \(V\)
Hence, writing \(\xi=(\Omega,Y)\in\mathfrak{s}=\mathfrak{g}\ltimes V\), it follows that \(\xi\gamma=(\Omega,Y)\gamma=\Omega\gamma\) as well. It is then straightforward to see that, for every \((\Omega_{i},Y_{i})\in\mathfrak{s}=\mathfrak{g}\ltimes V\) with \(i=1,2\),
\[\operatorname{ad}_{(\Omega_{1},Y_{1})}(\Omega_{2},Y_{2}):=[(\Omega_{1},Y_{1}), (\Omega_{2},Y_{2})]=(\operatorname{ad}_{\Omega_{1}}\Omega_{2},\,\Omega_{1}Y_{2 }-\Omega_{2}Y_{1}),\]
What correspond to \(\xi,\eta\in\mathfrak{s}=\mathfrak{g}\ltimes V\) in this example are
\[(\Omega,Y) :=(g,x)^{-1}(\dot{g},\dot{x})=\big{(}g^{-1}\dot{g},g^{-1}\dot{x} \big{)},\] \[(\Sigma,Z) :=(g,x)^{-1}(\delta g,\delta x)=\big{(}g^{-1}\delta g,g^{-1} \delta x\big{)},\]
respectively. Then the constraint on \(\xi\) in (15) becomes
\[(\delta\Omega,\delta Y)=(\dot{\Omega},\dot{Y})+(\operatorname{ad}_{\Omega} \Sigma,\,\Omega Z-\Sigma Y), \tag{18}\]
In [20], the nonholonomic constraints are assumed to be in the form
\[\langle\Psi((g,x),\gamma),(\dot{g},\dot{x})\rangle=\dot{x}-\bar{\Psi}((g,x), \gamma)\dot{g}=0\quad\text{with}\quad\bar{\Psi}((g,x),\gamma)\colon T_{g} \mathsf{G}\to V\]
with the following (left) \(\mathsf{S}\)-invariance:
\[\Psi((g_{0},x_{0})\cdot(g,x),(g_{0},x_{0})\gamma)=\Psi((g,x),\gamma)\quad \forall(g_{0},x_{0}),(g,x)\in\mathsf{S}\quad\forall\gamma\in X^{*}.\]
Hence we have
\[\langle\psi(\Gamma),(\Omega,Y)\rangle:=Y-\bar{\psi}(\Gamma)\Omega\quad\text{ with}\quad\bar{\psi}(\Gamma):=\bar{\Psi}(e,\Gamma),\]
and thus the nonholonomic constraints (13) and (16) applied to \((\Omega,Y)\) and \((\Sigma,Z)\) yield \(Y=\bar{\psi}(\Gamma)\Omega\) and \(Z=\bar{\psi}(\Gamma)\Sigma\). Substituting these expressions into the second equation in (18) yields
\[\delta Y=\frac{d}{dt}\big{(}\bar{\psi}(\Gamma)\Omega\big{)}+\Omega\,\bar{\psi }(\Gamma)\,\Sigma-\Sigma\,\bar{\psi}(\Gamma)\,\Omega,\]
which was in (2) of the General Theorem of Schneider [20, Theorem 1].
**Example 7** (The Veselova system [22]).: Let \(\mathsf{S}=\mathsf{G}\) (not a semidirect product). We still assume that the same broken _left_ symmetry and the recovery of _left_ symmetry for the Lagrangian \(L_{\gamma_{0}}\) described in Section 2.2, but with \(X=\mathfrak{g}\) so that \(\gamma_{0}\in\mathfrak{g}^{*}\) and the adjoint representation \(g\gamma:=\operatorname{Ad}_{g^{-1}}^{*}\gamma\) on \(X^{*}=\mathfrak{g}^{*}\). We also assume that the constraint distribution \(\mathcal{D}\subset T\mathsf{G}\) is corank 1, and is _right_\(\mathsf{G}\)-invariant:
\[T_{g}\mathsf{R}_{g_{0}}(\mathcal{D}(g))=\mathcal{D}(g_{0}g)\quad\forall g,g_{0 }\in\mathsf{G},\]
where \(\mathsf{R}\colon\mathsf{G}\to\mathsf{G}\) is the right translation. So we may rewrite the constraint \(\dot{g}\in\mathcal{D}(g)\) in terms of \(\omega:=T_{g}\mathsf{R}_{g^{-1}}(\dot{g})=:\dot{g}g^{-1}\), i.e., the spatial angular velocity in the rigid body setting:
\[\omega\in\mathfrak{d}:=\mathcal{D}(e).\]
We then further assume that \(\mathfrak{d}=\ker\gamma_{0}\) with the same parameter \(\gamma_{0}\in\mathfrak{g}^{*}\) for the Lagrangian, so that
\[\omega\in\mathfrak{d}=\ker\gamma_{0}\iff\langle\gamma_{0},\omega\rangle=0.\]
This is an example of the so-called nonholonomic LR systems [9, 10, 22].
In short, the right invariant constraint is breaking the left invariance of the system. Indeed, noting that
\[\omega=\dot{g}g^{-1}=gg^{-1}\dot{g}g^{-1}=\operatorname{Ad}_{g}\Omega\quad \text{with}\quad\Omega:=g^{-1}\dot{g},\]
we may rewrite the above constraint as follows: Setting \(\Gamma=g^{-1}\gamma_{0}=\operatorname{Ad}_{g}^{*}\gamma_{0}\),
\[\langle\gamma_{0},\operatorname{Ad}_{g}\Omega\rangle=\big{\langle} \operatorname{Ad}_{g}^{*}\gamma_{0},\Omega\big{\rangle}=\big{\langle}g^{-1} \gamma_{0},\Omega\big{\rangle}=0\iff\langle\psi(\Gamma),\Omega\rangle=0\quad \text{with}\quad\psi(\Gamma):=\Gamma,\]
Particularly, with \(\mathsf{G}=\mathsf{SO}(3)\) as in [22], we have \(X^{*}=\mathfrak{so}(3)^{*}\), and so under the standard identification \(\mathfrak{so}(3)\cong\mathfrak{so}(3)^{*}\cong\mathbb{R}^{3}\), we have
\[\operatorname{ad}_{\boldsymbol{\Omega}}^{*}\boldsymbol{\Pi}=\boldsymbol{\Pi} \times\boldsymbol{\Omega},\qquad\mathbf{K}(\mathbf{y},\boldsymbol{\Gamma})= \mathbf{y}\times\boldsymbol{\Gamma},\qquad\boldsymbol{\Omega}\boldsymbol{ \Gamma}=\boldsymbol{\Omega}\times\boldsymbol{\Gamma},\]
and hence (17) gives
\[\frac{d}{dt}\bigg{(}\frac{\partial\ell}{\partial\boldsymbol{\Omega}}\bigg{)}= \frac{\partial\ell}{\partial\boldsymbol{\Omega}}\times\boldsymbol{\Omega}+ \frac{\partial\ell}{\partial\boldsymbol{\Gamma}}\times\boldsymbol{\Gamma}+ \lambda\boldsymbol{\Gamma},\qquad\dot{\boldsymbol{\Gamma}}=\boldsymbol{ \Gamma}\times\boldsymbol{\Omega}\]
with constraint \(\boldsymbol{\Gamma}\cdot\boldsymbol{\Omega}=0\). This gives the Veselova system [22] with the standard kinetic minus potential form of the Lagrangian.
**Example 8** (Pendulum skate [13]).: As discussed in Section 2.1, \(\mathsf{S}=\mathsf{SE}(3)=\mathsf{SO}(3)\ltimes\mathbb{R}^{3}\) here. The system is subject to two more constraints in addition to the no-side-sliding constraint from Example 3 (see [13]). The following constraints are actually holonomic as the derivations to follow suggest. However, we shall impose them as constraints on \(\mathfrak{s}\) for the Euler-Poincare formalism.
* _Pitch constancy_: The blade does not rock back and forth. Specifically, the direction \(R\mathbf{E}_{1}\) of the blade in the spatial frame is perpendicular to \(\mathbf{e}_{3}\) (see Figure 2): \[0=(R\mathbf{E}_{1})^{T}\mathbf{e}_{3}=(R\mathbf{E}_{1})^{T}R\boldsymbol{ \Gamma}=\mathbf{E}_{1}^{T}\boldsymbol{\Gamma}=\Gamma_{1}.\] (19) Taking the time derivative and using (17b) (which is \(\dot{\boldsymbol{\Gamma}}=\boldsymbol{\Gamma}\times\boldsymbol{\Omega}\) here) \[0=\mathbf{E}_{1}^{T}\dot{\boldsymbol{\Gamma}}=\mathbf{E}_{1}^{T}(\boldsymbol{ \Gamma}\times\boldsymbol{\Omega})=(\mathbf{E}_{1}\times\boldsymbol{\Gamma})^ {T}\boldsymbol{\Omega},\] giving \[\psi^{1}(\boldsymbol{\Gamma}):=(\mathbf{E}_{1}\times\boldsymbol{\Gamma}, \boldsymbol{0})\in\mathfrak{se}(3)^{*}\cong\mathbb{R}^{3}\times\mathbb{R}^{3}.\]
* _Continuous contact_: The skate blade is in permanent contact with the plane of the ice, i.e., \(x_{3}=\mathbf{e}_{3}^{T}\mathbf{x}=0\). Taking the time derivative, \[0=\mathbf{e}_{3}^{T}\dot{\mathbf{x}}=(R^{T}\mathbf{e}_{3})^{T}R^{T}\dot{ \mathbf{x}}=\boldsymbol{\Gamma}^{T}\mathbf{Y},\] giving \[\psi^{2}(\boldsymbol{\Gamma}):=(\boldsymbol{0},\boldsymbol{\Gamma})\in \mathfrak{se}(3)^{*}\cong\mathbb{R}^{3}\times\mathbb{R}^{3}.\]
Combining the above two constraints with the no-side-sliding constraint from Example 3, we have
\[\mathfrak{d}^{\circ}(\boldsymbol{\Gamma})=\operatorname{span}\{\psi^{a}( \boldsymbol{\Gamma})\}_{a=1}^{3}\subset\mathfrak{se}(3)^{*}.\]
We also have, for every \((\boldsymbol{\Omega},\mathbf{Y})\in\mathfrak{se}(3)\cong\mathbb{R}^{3}\times \mathbb{R}^{3}\) and every \((\boldsymbol{\Pi},\mathbf{P})\in\mathfrak{se}(3)^{*}\cong\mathbb{R}^{3}\times \mathbb{R}^{3}\),
\[\operatorname{ad}_{(\boldsymbol{\Omega},\mathbf{Y})}^{*}(\boldsymbol{\Pi}, \mathbf{P})=(\boldsymbol{\Pi}\times\boldsymbol{\Omega}+\mathbf{P}\times \mathbf{Y},\,\mathbf{P}\times\boldsymbol{\Omega}),\]
and for every \((\mathbf{y},\boldsymbol{\Gamma})\in X\times X^{*}\cong\mathbb{R}^{3}\times \mathbb{R}^{3}\),
\[\mathbf{K}(\mathbf{y},\boldsymbol{\Gamma})=(\mathbf{y}\times\boldsymbol{ \Gamma},\boldsymbol{0}). \tag{20}\]
Hence the nonholonomic Euler-Poincare equations with advected parameter (17) give
\[\frac{d}{dt}\bigg{(}\frac{\partial\ell}{\partial\boldsymbol{\Omega }}\bigg{)} =\frac{\partial\ell}{\partial\boldsymbol{\Omega}}\times\boldsymbol{ \Omega}+\frac{\partial\ell}{\partial\boldsymbol{\Gamma}}\times\mathbf{Y}+ \frac{\partial\ell}{\partial\boldsymbol{\Gamma}}\times\boldsymbol{\Gamma}+ \lambda_{1}(\mathbf{E}_{1}\times\boldsymbol{\Gamma}), \tag{21}\] \[\frac{d}{dt}\bigg{(}\frac{\partial\ell}{\partial\boldsymbol{Y}} \bigg{)} =\frac{\partial\ell}{\partial\boldsymbol{\Gamma}}\times\boldsymbol{ \Omega}+\lambda_{2}\boldsymbol{\Gamma}+\lambda_{3}(\mathbf{E}_{1}\times \boldsymbol{\Gamma}),\] \[\dot{\boldsymbol{\Gamma}} =\boldsymbol{\Gamma}\times\boldsymbol{\Omega},\]
which is Eq. (8) of [13], and also gives Eq. (9) of [13] with the reduced Lagrangian (7).
## 4. Eliminating Lagrange Multipliers
One may algebraically find concrete expressions for the the Lagrange multipliers in the nonholonomic Euler-Poincare equation (17). However, they tend to be quite complicated, even with a rather simple Veselova system as shown in Fedorov and Jovanovic [9, Eq. (4.3)]. Such a complication is detrimental when applying the method of Controlled Lagrangians, because it is difficult to "match" two equations if their structures are unclear. In this section, we introduce \(\Gamma\)-dependent quasivelocities and systematically eliminate the Lagrange multipliers in (17).
### \(\Gamma\)-dependent Hamel Basis
We decompose the Lie algebra \(\mathfrak{s}\) into the \(\Gamma\)-dependent constraints subspace \(\mathfrak{d}(\Gamma)\) and its complement: Setting \(n:=\dim\mathfrak{s}\),
\[\mathfrak{s}=\mathfrak{d}(\Gamma)\oplus\mathfrak{v}(\Gamma)\quad\text{with} \quad\mathfrak{d}(\Gamma)=\operatorname{span}\{\mathcal{E}_{\alpha}(\Gamma) \}_{\alpha=1}^{n-r},\quad\mathfrak{v}(\Gamma)=\operatorname{span}\{\mathcal{E }_{a}(\Gamma)\}_{a=1}^{r}.\]
Note that we are using Greek indices for \(\mathfrak{d}\), whereas \(a,b,c,\dots\) for \(\mathfrak{v}\) and \(i,j,k,\dots\) for the entire \(\mathfrak{s}\).
So we may use Einstein's summation convention to write \(\xi\in\mathfrak{s}\) as
\[\xi=v^{i}(\Gamma)\mathcal{E}_{i}(\Gamma)=v^{\alpha}(\Gamma)\mathcal{E}_{ \alpha}(\Gamma)+v^{a}(\Gamma)\mathcal{E}_{a}(\Gamma),\]
and refer to \(\{v^{i}(\Gamma)\}_{i=1}^{n}\) as the (_\(\Gamma\)-dependent_) _quasvelocities_. For brevity, we shall often drop the \(\Gamma\)-dependence in what follows. The advantage of this constraint-adapted Hamel basis is the following:
\[\xi\in\mathfrak{d}\iff v^{a}=0,\]
where it is implied on the right-hand side that the equality holds for every \(a\in\{1,\dots,r\}\). Hence we may simply drop some of the coordinates to take the constraints into account.
Given a standard (\(\Gamma\)-independent) basis \(\{E_{i}\}_{i=1}^{n}\) for \(\mathfrak{s}\), we may write \(\{\mathcal{E}_{i}(\Gamma)\}_{i=1}^{n}\) as
\[\mathcal{E}_{j}(\Gamma)=\mathcal{E}_{j}^{i}(\Gamma)E_{i}\iff E_{i}=(\mathcal{ E}^{-1})_{i}^{j}(\Gamma)\,\mathcal{E}_{j}(\Gamma), \tag{22}\]
where we abuse the notation as follows: \(\mathcal{E}\) is the \(n\times n\) matrix whose columns are \(\{\mathcal{E}_{j}\}_{j=1}^{n}\) so that \(\mathcal{E}_{j}^{i}\) denotes the \((i,j)\)-entry of \(\mathcal{E}\) as well as the \(i\)-th component of \(\mathcal{E}_{j}\) with respect to the standard basis \(\{E_{i}\}_{i=1}^{n}\).
Suppose that we have the structure constants \(\{c_{ij}^{k}\}_{1\leq i,j,k\leq n}\) for \(\mathfrak{s}\) with respect to the standard basis \(\{E_{i}\}_{i=1}^{n}\), i.e.,
\[[E_{i},E_{j}]=c_{ij}^{k}E_{k}.\]
Then the structure constants with respect to \(\{\mathcal{E}_{j}\}_{j=1}^{n}\) are also \(\Gamma\)-dependent:
\[[\mathcal{E}_{i}(\Gamma),\mathcal{E}_{j}(\Gamma)]=\mathcal{C}_{ij}^{k}(\Gamma )\,\mathcal{E}_{k}(\Gamma)\quad\text{with}\quad\mathcal{C}_{ij}^{k}(\Gamma): =\mathcal{E}_{i}^{l}(\Gamma)\,\mathcal{E}_{j}^{m}(\Gamma)\,c_{lm}^{p}\,( \mathcal{E}^{-1})_{p}^{k}(\Gamma). \tag{23}\]
### Eliminating the Lagrange Multipliers
Let us eliminate the Lagrange multipliers and find a concrete coordinate expression for (17a) in terms of the \(\Gamma\)-dependent quasivelocities.
Imposing the constraint \(\xi\in\mathfrak{d}\) and using the \(\Gamma\)-dependent basis \(\{\mathcal{E}_{i}\}_{i=1}^{n}\) for \(\mathfrak{s}\), we write
\[\mu:=\left.\frac{\delta\ell}{\delta\xi}\right|_{\mathrm{c}},\qquad p_{i}:= \langle\mu,\mathcal{E}_{i}\rangle, \tag{24}\]
where the subscript \(\,(\,\cdot\,)\rvert_{\mathrm{c}}\) indicates that the constraint \(\xi\in\mathfrak{d}(\Gamma)\) is applied _after_ computing what is inside \((\,\cdot\,)\). As a result, \(\{p_{i}\}_{i=1}^{n}\) are written in terms of \(\{v^{\alpha}\}_{\alpha=1}^{n-r}\).
Let \(\{\mathbf{e}_{i}\}_{i=1}^{\dim X}\) be a basis for \(X\) and \(\{\mathbf{e}_{*}^{i}\}_{i=1}^{\dim X}\) be its dual basis for \(X^{*}\), and write
\[\Gamma=\Gamma_{i}\,\mathbf{e}_{*}^{i},\qquad\xi\Gamma=\kappa_{ij}^{k}\,\xi^{j }\,\Gamma_{k}\,\mathbf{e}_{*}^{i}\]
using constants \(\kappa_{ij}^{k}\) that are determined by the \(\mathfrak{s}\)-representation (10) on \(X^{*}\). Then we have:
**Theorem 9**.: _The nonholonomic Euler-Poincare equations (17) are written in coordinates as_
\[\dot{p}_{\alpha} =-\mathcal{B}^{i}_{\alpha\beta}(\Gamma)\,p_{i}\,v^{\beta}+K^{k}_{ \alpha j}\left.\frac{\partial\ell}{\partial\Gamma_{j}}\right|_{\mathrm{c}} \Gamma_{k}, \tag{25a}\] \[\dot{\Gamma}_{i} =-\varkappa^{k}_{i\beta}(\Gamma)\,v^{\beta}\,\Gamma_{k}, \tag{25b}\]
_where_
\[K^{k}_{ij} :=\Big{\langle}\mathbf{K}\Big{(}\mathbf{e}_{j},\mathbf{e}^{k}_{ \star}\Big{)},\mathcal{E}_{i}\Big{\rangle},\qquad\varkappa^{k}_{i\beta}( \Gamma):=\kappa^{k}_{ij}\,\mathcal{E}^{j}_{\beta}(\Gamma), \tag{26}\] \[\mathcal{B}^{i}_{\alpha\beta}(\Gamma) :=\mathcal{C}^{i}_{\alpha\beta}(\Gamma)+\mathcal{F}^{i}_{\alpha \beta}(\Gamma)\quad\text{with}\quad\mathcal{F}^{i}_{\alpha\beta}(\Gamma):=D^{ j}\mathcal{E}^{l}_{\alpha}(\Gamma)\,(\mathcal{E}^{-1})^{i}_{l}(\Gamma)\, \varkappa^{k}_{j\beta}(\Gamma)\,\Gamma_{k}, \tag{27}\]
_and \(D^{j}\mathcal{E}^{i}_{\alpha}\) stands for \(\partial\mathcal{E}^{i}_{\alpha}/\partial\Gamma_{j}\)._
_Remark 10_.: As explained above, \(\{p_{i}\}_{i=1}^{n}\) are written in terms of \(\{v^{\alpha}\}_{\alpha=1}^{n-r}\), and so (25) gives the closed set of equations for \(\{v^{\alpha}\}_{\alpha=1}^{n-r}\) and \(\Gamma\).
Proof.: It is straightforward to see that (25b) follows from (17b): Since \(\xi^{j}=v^{\beta}\,\mathcal{E}^{j}_{\beta}\) for every \(\xi\in\mathfrak{d}\), using the definitions of \(\kappa\) and \(\varkappa\) from above,
\[\dot{\Gamma}_{i}=-(\xi\Gamma)_{i}=-\kappa^{k}_{ij}\,\xi^{j}\,\Gamma_{k}=- \varkappa^{k}_{i\beta}(\Gamma)\,v^{\beta}\,\Gamma_{k}.\]
It remains to derive (25a) from (17a). Imposing the constraint \(\xi\in\mathfrak{d}\) to (17a) and using \(\mu\) defined in (24), we have
\[\dot{\mu}=\mathrm{ad}^{*}_{\xi}\,\mu+\mathbf{K}\bigg{(}\left.\frac{\delta\ell} {\delta\Gamma}\right|_{\mathrm{c}},\Gamma\bigg{)}+\lambda_{a}\psi^{a}(\Gamma).\]
Then, taking the paring of both sides with \(\mathcal{E}_{\alpha}\), we have
\[\left\langle\dot{\mu},\mathcal{E}_{\alpha}\right\rangle=\left\langle\mathrm{ ad}^{*}_{\xi}\,\mu,\mathcal{E}_{\alpha}\right\rangle+\left\langle\mathbf{K} \bigg{(}\left.\frac{\delta\ell}{\delta\Gamma}\right|_{\mathrm{c}},\Gamma\bigg{)},\mathcal{E}_{\alpha}\right\rangle+\underbrace{\left\langle\lambda_{a}\psi^{a},\mathcal{E}_{\alpha}\right\rangle}_{0}.\]
The left-hand side becomes
\[\left\langle\dot{\mu},\mathcal{E}_{\alpha}\right\rangle =\frac{d}{dt}\left\langle\mu,\mathcal{E}_{\alpha}\right\rangle- \left\langle\mu,\dot{\mathcal{E}}_{\alpha}\right\rangle\] \[=\dot{p}_{\alpha}-\left\langle\mu,\dot{\mathcal{E}}_{\alpha}\right\rangle\] \[=\dot{p}_{\alpha}-\left\langle\mu,D\mathcal{E}_{\alpha}(\Gamma) \dot{\Gamma}\right\rangle\] \[=\dot{p}_{\alpha}+\left\langle\mu,D\mathcal{E}_{\alpha}(\Gamma) \xi\Gamma\right\rangle,\]
but then
\[D\mathcal{E}_{\alpha}(\Gamma)\xi\Gamma =\frac{\partial\mathcal{E}^{l}_{\alpha}}{\partial\Gamma_{j}}(\xi \Gamma)_{j}E_{l}\] \[=D^{j}\mathcal{E}^{l}_{\alpha}(\Gamma)\,\varkappa^{k}_{j\beta}( \Gamma)\,v^{\beta}\,\Gamma_{k}\,(\mathcal{E}^{-1})^{i}_{l}(\Gamma)\,\mathcal{ E}_{i}(\Gamma)\] \[=\mathcal{F}^{i}_{\alpha\beta}(\Gamma)\,v^{\beta}\,\mathcal{E}_{i} (\Gamma).\]
Therefore, using the definition of \(p_{i}\) from (24), we obtain
\[\left\langle\dot{\mu},\mathcal{E}_{\alpha}\right\rangle=\dot{p}_{\alpha}+ \mathcal{F}^{i}_{\alpha\beta}(\Gamma)\,p_{i}\,v^{\beta}.\]
On the other hand, a straightforward computation yields
\[\left\langle\mathrm{ad}^{*}_{\xi}\,\mu,\mathcal{E}_{\alpha}\right\rangle= \mathcal{C}^{i}_{\beta\alpha}\,p_{i}\,v^{\beta}=-\mathcal{C}^{i}_{\alpha\beta} \,p_{i}\,v^{\beta}.\]
Moreover, using the bases \(\{\mathbf{e}_{i}\}_{i=1}^{\dim X}\) for \(X\) and \(\{\mathbf{e}_{*}^{i}\}_{i=1}^{\dim X}\) for \(X^{*}\) introduced earlier, we have
\[\mathbf{K}\!\left(\left.\frac{\delta\ell}{\delta\Gamma}\right|_{\mathrm{c}}, \Gamma\right)=\mathbf{K}\!\left(\left.\frac{\partial\ell}{\partial\Gamma_{j}} \right|_{\mathrm{c}}\mathbf{e}_{j},\Gamma_{k}\,\mathbf{e}_{*}^{k}\right)= \mathbf{K}\!\left(\mathbf{e}_{j},\mathbf{e}_{*}^{k}\right)\left.\frac{\partial \ell}{\partial\Gamma_{j}}\right|_{\mathrm{c}}\Gamma_{k},\]
and so
\[\left\langle\mathbf{K}\!\left(\left.\frac{\delta\ell}{\delta\Gamma}\right|_{ \mathrm{c}},\Gamma\right)\!,\mathcal{E}_{\alpha}\right\rangle=\left\langle \mathbf{K}\!\left(\mathbf{e}_{j},\mathbf{e}_{*}^{k}\right)\!,\mathcal{E}_{ \alpha}\right\rangle\left.\frac{\partial\ell}{\partial\Gamma_{j}}\right|_{ \mathrm{c}}\Gamma_{k}=K_{\alpha j}^{k}\left.\frac{\partial\ell}{\partial \Gamma_{j}}\right|_{\mathrm{c}}\Gamma_{k}.\qed\]
### Lagrangian and Energy
In what follows, we assume that the reduced Lagrangian \(\ell\) from (5) takes the following "kinetic minus potential" form:
\[\ell(\xi,\Gamma)=\frac{1}{2}\mathbb{G}_{ij}\,\xi^{i}\xi^{j}-U(\Gamma) \tag{28}\]
with a constant \(n\times n\) matrix \(\mathbb{G}\) and \(U\colon X^{*}\to\mathbb{R}\). Since \(\xi^{i}=\mathcal{E}_{j}^{i}(\Gamma)\,v^{j}(\Gamma)\), we have
\[\frac{1}{2}\mathbb{G}_{ij}\,\xi^{i}\xi^{j}=\frac{1}{2}\mathcal{G}_{ij}(\Gamma) \,v^{i}(\Gamma)\,v^{j}(\Gamma)\quad\text{with}\quad\mathcal{G}_{ij}(\Gamma):= \mathbb{G}_{kl}\,\mathcal{E}_{i}^{k}(\Gamma)\,\mathcal{E}_{j}^{l}(\Gamma). \tag{29}\]
Then we see that
\[\frac{\partial\ell}{\partial\xi^{i}}=\mathbb{G}_{ij}\,\xi^{j}= \mathbb{G}_{ij}\,\mathcal{E}_{k}^{j}\,v^{k} \implies\mu_{i}=\left.\frac{\partial\ell}{\partial\xi^{i}} \right|_{\mathrm{c}}=\mathbb{G}_{ij}\,\mathcal{E}_{\beta}^{j}\,v^{\beta}\] \[\implies p_{i}=\mu_{k}\,\mathcal{E}_{i}^{k}=\mathbb{G}_{kj}\,\mathcal{E}_{i }^{k}\,\mathcal{E}_{\beta}^{j}\,v^{\beta}=\mathcal{G}_{i\beta}\,v^{\beta},\]
giving the concrete relationship between \(\{p_{i}\}_{i=1}^{n}\) and \(\{v^{\alpha}\}_{\alpha=1}^{n-r}\) alluded in Remark 10. Then the constrained energy function
\[\mathscr{E}\colon\mathbb{R}^{n-r}\times X^{*}\to\mathbb{R};\qquad\mathscr{E}( v^{\alpha},\Gamma):=\frac{1}{2}\mathcal{G}_{\alpha\beta}(\Gamma)\,v^{\alpha}v^{ \beta}+U(\Gamma) \tag{30}\]
is an invariant of (25) because this is the energy of the system expressed in the quasivelocities.
## 5. Pendulum Skate
Let us now come back to our motivating example and apply Theorem 9 to the pendulum skate.
### Equations of Motion
The \(\Gamma\)-dependent Hamel basis for the pendulum skate is an extension of the hybrid frame introduced in [13]:
\[\mathfrak{se}(3)\cong\mathbb{R}^{3}\times\mathbb{R}^{3}=\mathfrak{d}(\Gamma) \oplus\mathfrak{v}(\Gamma),\]
where
\[\begin{split}\mathfrak{d}&=\mathrm{span}\{ \mathcal{E}_{1}:=(\mathbf{E}_{1},\mathbf{0}),\,\mathcal{E}_{2}:=(\mathbf{ \Gamma},\mathbf{0}),\,\mathcal{E}_{3}:=(\mathbf{0},\mathbf{E}_{1})\},\\ \mathfrak{v}&:=\mathrm{span}\{\mathcal{E}_{4}:=( \mathbf{E}_{1}\times\mathbf{\Gamma},\mathbf{0}),\,\mathcal{E}_{5}:=(\mathbf{ 0},\mathbf{\Gamma}),\,\mathcal{E}_{6}:=(\mathbf{0},\mathbf{E}_{1}\times \mathbf{\Gamma})\}.\end{split} \tag{31}\]
Note that, due to the pitch constancy condition (19), these define orthonormal bases for \(\mathfrak{d}\) and \(\mathfrak{v}\), and also they together form an orthonormal basis for \(\mathfrak{se}(3)\) as well.
Since the commutator in \(\mathfrak{se}(3)\cong\mathbb{R}^{3}\times\mathbb{R}^{3}\) given by
\[\mathrm{ad}_{(\mathbf{\Omega}_{1},\mathbf{Y}_{1})}(\mathbf{\Omega}_{2}, \mathbf{Y}_{2})=[(\mathbf{\Omega}_{1},\mathbf{Y}_{1}),(\mathbf{\Omega}_{2}, \mathbf{Y}_{2})]=(\mathbf{\Omega}_{1}\times\mathbf{\Omega}_{2},\,\mathbf{ \Omega}_{1}\times\mathbf{Y}_{2}-\mathbf{\Omega}_{2}\times\mathbf{Y}_{1}),\]
whereas (31) yields
\[D\mathcal{E}_{2}=\begin{bmatrix}I\\ 0\end{bmatrix},\quad D\mathcal{E}_{1}=D\mathcal{E}_{3}=0, \tag{32}\]
the \(\Gamma\)-dependent structure constants \(\mathcal{C}^{k}_{\alpha\beta}\) defined in (23) are actually independent of \(\Gamma\) here:
\[\mathcal{C}^{k}_{\alpha\beta}\,p_{k}=\begin{bmatrix}0&-p_{4}&0\\ p_{4}&0&p_{6}\\ 0&-p_{6}&0\end{bmatrix}\quad\forall(p_{1},\ldots,p_{6})\in\mathbb{R}^{6}.\]
Note that we do not need the full \(6\times 6\) matrix \(\mathcal{C}^{k}_{ij}\,p_{k}\).
We may then write \(\xi=(\boldsymbol{\Omega},\mathbf{Y})\in\mathfrak{se}(3)\) in terms of quasivelocities \(\{v_{i}\}_{i=1}^{6}\):
\[\boldsymbol{\Omega}=v^{1}\mathbf{E}_{1}+v^{2}\boldsymbol{\Gamma}+v^{4}( \mathbf{E}_{1}\times\boldsymbol{\Gamma}),\qquad\mathbf{Y}=v^{3}\mathbf{E}_{1 }+v^{5}\boldsymbol{\Gamma}+v^{6}(\mathbf{E}_{1}\times\boldsymbol{\Gamma}),\]
where, by the orthonormality,
\[\begin{split}& v^{1}=\boldsymbol{\Omega}\cdot\mathbf{E}_{1}= \Omega_{1},\qquad v^{2}=\boldsymbol{\Omega}\cdot\boldsymbol{\Gamma},\qquad v ^{3}=\mathbf{Y}\cdot\mathbf{E}_{1}=Y_{1},\\ & v^{4}=\boldsymbol{\Omega}\cdot(\mathbf{E}_{1}\times\boldsymbol{ \Gamma}),\qquad v^{5}=\mathbf{Y}\cdot\boldsymbol{\Gamma},\qquad v^{6}= \mathbf{Y}\cdot(\mathbf{E}_{1}\times\boldsymbol{\Gamma}).\end{split} \tag{33}\]
Then the constraint \(\xi=(\boldsymbol{\Omega},\mathbf{Y})\in\mathfrak{d}(\boldsymbol{\Gamma})\) is equivalent to \(v^{a}=0\) with \(a\in\{4,5,6\}\).
On the other hand, the Lagrangian (7) becomes
\[\begin{split}\ell(\boldsymbol{\Omega},\mathbf{Y},\boldsymbol{ \Gamma})&=\frac{1}{2}\boldsymbol{\Omega}^{T}\mathbb{I} \boldsymbol{\Omega}+\frac{m}{2}\|\mathbf{Y}+l\boldsymbol{\Omega}\times \mathbf{E}_{3}\|^{2}-m\text{g}l\boldsymbol{\Gamma}^{T}\mathbf{E}_{3}\\ &=\frac{1}{2}\boldsymbol{\Omega}^{T}\mathbb{I}\boldsymbol{\Omega} +ml\boldsymbol{\Omega}\cdot(\mathbf{E}_{3}\times\mathbf{Y})+\frac{m}{2}\| \mathbf{Y}\|^{2}-m\text{g}l\Gamma_{3},\end{split} \tag{34}\]
where
\[\mathbb{I}:=\text{diag}(\bar{I}_{1},\bar{I}_{2},I_{3})\quad\text{with}\quad \bar{I}_{i}:=I_{i}+ml^{2}\quad\text{for}\quad i=1,2.\]
Then
\[\frac{\partial\ell}{\partial\xi}=\left(\frac{\partial\ell}{\partial \boldsymbol{\Omega}},\frac{\partial\ell}{\partial\mathbf{Y}}\right)\!,\qquad \frac{\partial\ell}{\partial\boldsymbol{\Omega}}=\mathbb{I}\boldsymbol{\Omega }+ml(\mathbf{E}_{3}\times\mathbf{Y}),\qquad\frac{\partial\ell}{\partial \mathbf{Y}}=ml(\boldsymbol{\Omega}\times\mathbf{E}_{3})+m\mathbf{Y},\]
and thus we have
\[\mu=\left.\frac{\delta\ell}{\delta\xi}\right|_{\text{c}}=\begin{bmatrix}\bar{I }_{1}v^{1}\\ \bar{I}_{2}\Gamma_{2}v^{2}+mlv^{3}\\ I_{3}\Gamma_{3}v^{2}\\ m(l\Gamma_{2}v^{2}+v^{3})\\ 0\end{bmatrix},\]
and
\[\begin{bmatrix}p_{1}\\ p_{2}\\ p_{3}\end{bmatrix}=\begin{bmatrix}\bar{I}_{1}v^{1}\\ (\bar{I}_{2}\Gamma_{2}^{2}+I_{3}\Gamma_{3}^{2})v^{2}+ml\Gamma_{2}v^{3}\\ m(l\Gamma_{2}v^{2}+v^{3})\end{bmatrix},\qquad\begin{bmatrix}p_{4}\\ p_{5}\\ p_{6}\end{bmatrix}=\begin{bmatrix}(I_{3}-\bar{I}_{2})\Gamma_{2}\Gamma_{3}v^{2}-ml \Gamma_{3}v^{3}\\ -ml\Gamma_{2}v^{1}\\ ml\Gamma_{3}v^{1}\end{bmatrix}.\]
Let us find the right-hand side of (25a). We find
\[\mathcal{C}^{i}_{\alpha\beta}\,p_{i}\,v^{\beta}=\begin{bmatrix}-p_{4}v^{2}\\ p_{4}v^{1}+p_{6}v^{3}\\ -p_{6}v^{2}\end{bmatrix},\qquad\xi\Gamma=\boldsymbol{\Omega}\times \boldsymbol{\Gamma}=v^{1}\begin{bmatrix}0\\ \Gamma_{3}\\ -\Gamma_{2}\end{bmatrix}\]
since \(\Gamma_{1}=0\). Hence we have, using (32),
\[\mathcal{F}^{i}_{\alpha\beta}(\Gamma)\,p_{i}\,v^{\beta}=\left\langle\mu,D \mathcal{E}_{\alpha}(\Gamma)\,\xi\Gamma\right\rangle=\left\langle\mu,D^{j} \mathcal{E}_{\alpha}(\Gamma)\,(\xi\Gamma)_{j}\right\rangle=\begin{bmatrix}0 \\ (\Gamma_{3}\,p_{2}-\Gamma_{2}\,p_{3})v^{1}0\end{bmatrix}.\]
Therefore, we obtain
\[\mathcal{B}^{i}_{\alpha\beta}\,p_{i}\,v^{\beta} =\big{(}\mathcal{C}^{i}_{\alpha\beta}+\mathcal{F}^{i}_{\alpha\beta} \big{)}p_{i}\,v^{\beta}\] \[=\begin{bmatrix}-p_{4}v^{2}\\ p_{4}v^{1}+p_{6}v^{3}+(\Gamma_{3}\,p_{2}-\Gamma_{2}\,p_{3})v^{1}\\ -p_{6}v^{2}\end{bmatrix}=\begin{bmatrix}(\bar{I}_{2}-I_{3})\Gamma_{2}\Gamma_{3} (v^{2})^{2}+ml\Gamma_{3}v^{2}v^{3}\\ ml\Gamma_{3}v^{3}v^{1}\\ -ml\Gamma_{3}v^{1}v^{2}\end{bmatrix}.\]
Using (20), we also have
\[K^{k}_{\alpha j}\left.\frac{\partial\ell}{\partial\Gamma_{j}}\right|_{\rm c} \Gamma_{k}=mgl\begin{bmatrix}\Gamma_{2}\\ 0\\ 0\end{bmatrix}.\]
As a result, the nonholonomic Euler-Poincare equations (25) become
\[\begin{bmatrix}\dot{p}_{1}\\ \dot{p}_{2}\\ \dot{p}_{3}\end{bmatrix} =\frac{d}{dt}\begin{bmatrix}\bar{I}_{1}v^{1}\\ (\bar{I}_{2}\Gamma_{2}^{2}+I_{3}\Gamma_{3}^{2})v^{2}+ml\Gamma_{2}v^{3}\\ m(l\Gamma_{2}v^{2}+v^{3})\end{bmatrix}\] \[=\begin{bmatrix}(\bar{I}_{2}-I_{3})\Gamma_{2}\Gamma_{3}(v^{2})^{2 }+ml\Gamma_{3}v^{2}v^{3}+mgl\Gamma_{2}\\ ml\Gamma_{3}v^{3}v^{1}\\ -ml\Gamma_{3}v^{1}v^{2}\end{bmatrix}\] (35a) coupled with \[\begin{bmatrix}\dot{\Gamma}_{2}\\ \dot{\Gamma}_{3}\end{bmatrix}=v^{1}\begin{bmatrix}\Gamma_{3}\\ -\Gamma_{2}\end{bmatrix}. \tag{35b}\]
### Invariants
One can see by inspection that (35) implies
\[\frac{d}{dt}(p_{2}-l\Gamma_{2}p_{3})=0,\]
which shows that
\[C_{1}:=p_{2}-l\Gamma_{2}p_{3}=(I_{2}\Gamma_{2}^{2}+I_{3}\Gamma_{3}^{2})v^{2} \tag{36}\]
is an invariant of the system--called \(J_{1}\) in [13, Eq. (20)].
We also notice by inspection that
\[\dot{p}_{3}=-\frac{mlC_{1}}{I_{2}\Gamma_{2}^{2}+I_{3}\Gamma_{3}^{2}}\dot{ \Gamma}_{2}.\]
However, since \(\Gamma_{2}^{2}+\Gamma_{3}^{2}=1\) (due to the pitch constancy \(\Gamma_{1}=0\) from (19)), we have
\[\dot{p}_{3}=-\frac{mlC_{1}}{(I_{2}-I_{3})\Gamma_{2}^{2}+I_{3}}\dot{\Gamma}_{2} \implies\frac{dp_{3}}{d\Gamma_{2}}=-\frac{mlC_{1}}{(I_{2}-I_{3})\Gamma_{2}^{2}+ I_{3}}.\]
Therefore,
\[p_{3}=-mlC_{1}\int\frac{1}{(I_{2}-I_{3})\Gamma_{2}^{2}+I_{3}}\,d\Gamma_{2},\]
implying that
\[C_{2} :=\frac{p_{3}}{m}+lC_{1}\int\frac{1}{(I_{2}-I_{3})\Gamma_{2}^{2}+I _{3}}\,d\Gamma_{2}\] \[=l\Gamma_{2}v^{2}+v^{3}+\frac{l}{I_{3}(I_{2}-I_{3})}C_{1}\arctan \left(\sqrt{\frac{I_{2}-I_{3}}{I_{3}}}\,\Gamma_{2}\right) \tag{37}\]
is also an invariant of the system--called \(J_{2}\) in [13, Eq. (21)], where we assumed \(I_{2}>I_{3}\) (and shall do so for the rest of the paper) because this is the case with realistic skaters as mentioned in [13, Proof of Theorem 2].
### Equilibria
We shall use the following shorthands in what follows:
\[z:=(\mathbf{\Omega},\mathbf{Y},\mathbf{\Gamma}),\qquad\zeta:=\left(v^{1},v^{2},v^{3},\Gamma_{2},\Gamma_{3}\right).\]
Note that \(z\) denotes the original dependent variables in the nonholonomic Euler-Poincare equations (21) with Lagrange multipliers, whereas \(\zeta\) denotes those in (35) using quasivelocities. Now, let us rewrite the system (35) as
\[\dot{\zeta}=f(\zeta)\quad\text{with}\quad f(\zeta):=\Bigg{(}\frac {\big{(}(\bar{I}_{2}-I_{3}\big{)}\,\Gamma_{3}(v^{2})^{2}+mgl\big{)}\Gamma_{2}+ ml\Gamma_{3}v^{2}v^{3}}{\bar{I}_{1}},\\ \frac{2(I_{3}-I_{2})\Gamma_{2}\Gamma_{3}v^{1}v^{2}}{I_{2}\Gamma _{2}^{2}+I_{3}\Gamma_{3}^{2}},\,-\frac{2lI_{3}\Gamma_{3}v^{1}v^{2}}{I_{2} \Gamma_{2}^{2}+I_{3}\Gamma_{3}^{2}},\,v^{1}\Gamma_{3},\,-v^{1}\Gamma_{2} \Bigg{)}. \tag{38}\]
Then one finds that the equilibria are characterized as follows:
\[v^{1}=0\quad\text{and}\quad\big{(}(\bar{I}_{2}-I_{3})\Gamma_{3}(v^{2})^{2}+mgl \big{)}\Gamma_{2}+ml\Gamma_{3}v^{2}v^{3}=0. \tag{39}\]
Note that, in view of (33), \(v^{1}=0\) is equivalent to \(\Omega_{1}=0\).
Let us impose the upright position, i.e., \(\Gamma_{3}=1\) or equivalently \(\mathbf{\Gamma}=\mathbf{E}_{3}\). Then the constraints \(v^{a}=0\) for \(a=4,5,6\) yield \(\Omega_{2}=Y_{2}=Y_{3}=0\), i.e., \(\mathbf{\Omega}=\Omega_{0}\mathbf{E}_{3}\) and \(\mathbf{Y}=Y_{0}\mathbf{E}_{1}\) for arbitrary \(\Omega_{0},Y_{0}\in\mathbb{R}\). Furthermore, the second equation in (39) reduces to \(v^{2}v^{3}=0\). If \(v^{2}=0\) then \(\Omega_{3}=0\) and thus we have the _sliding equilibrium_
\[z_{\rm sl}:=(\mathbf{0},Y_{0}\mathbf{E}_{1},\mathbf{E}_{3})\quad\text{or} \quad\zeta_{\rm sl}:=(0,0,Y_{0},0,1), \tag{40}\]
whereas \(v^{3}=0\) gives \(Y_{1}=0\) and gives the _spinning equilibrium_
\[z_{\rm sp}:=(\Omega_{0}\mathbf{E}_{3},\mathbf{0},\mathbf{E}_{3})\quad\text{ or}\quad\zeta_{\rm sp}:=(0,\Omega_{0},0,0,1). \tag{41}\]
### Stability of Equilibria
Let us first discuss the stability of the spinning equilibrium:
**Proposition 11** (Stability of spinning equilibrium).: _Suppose that \(I_{3}<I_{2}\)._
1. _If_ \(I_{3}+ml^{2}<I_{2}\)_, then the spinning equilibrium (_41_) is unstable._
2. _If_ \[I_{3}+ml^{2}>I_{2}\quad\text{and}\quad|\Omega_{0}|>\sqrt{\frac{mgl}{I_{3}+ml^{ 2}-I_{2}}},\] (42) _then the spinning equilibrium (_41_) is stable._
Proof.: See Appendix A.1.
On the other hand, the sliding equilibrium is always unstable:
**Proposition 12** (Stability of sliding equilibrium).: _The sliding equilibrium (40) is linearly unstable._
Proof.: The Jacobin \(Df\) of the vector field \(f\) from (38) at the sliding equilibrium (40) is
\[Df(\zeta_{\rm sl})=\begin{bmatrix}0&\frac{mlY_{0}}{I_{1}+ml^{2}}&0&\frac{mgl }{I_{1}+ml^{2}}\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{bmatrix},\]
and its eigenvalues are
\[\left\{0,0,0,\pm\sqrt{\frac{mgl}{I_{1}+ml^{2}}}\right\},\]
The presence of a positive eigenvalue implies the assertion by the Instability from Linearization criterion (see, e.g., Sastry [19, p.216]).
## 6. Controlled Lagrangian and Matching
The goal of this section is to apply the method of Controlled Lagrangians to the nonholonomic Euler-Poincare equations (25). Our formulation using the \(\Gamma\)-dependent quasivelocities helps us extend the method of Controlled Lagrangians of Bloch et al. [5] to our system (25). Indeed, the arguments to follow in this section almost exactly parallel those of [5, Section 2].
### Controlled Lagrangian
Our motivating example is the stabilization of the sliding equilibria of the pendulum skate--shown to to be unstable in Proposition 12--using an internal wheel; see Figure 3.
Following [5], let \(\mathsf{H}\) be an \(s\)-dimensional Abelian Lie group and \(\mathfrak{h}\cong\mathbb{R}^{s}\) be its Lie algebra; practically \(\mathsf{H}\) gives the configuration space of \(s\) internal rotors, i.e., \(\mathsf{H}=\mathbb{T}^{s}\). We shall replace the reduced Lagrangian (28) by
\[\ell_{\mathrm{r}}\colon\mathfrak{g}\times\mathfrak{h}\times X^{*}\to\mathbb{R };\qquad\ell_{\mathrm{r}}\Big{(}\xi,\dot{\theta}^{a},\Gamma\Big{)}:=K\Big{(} \xi,\dot{\theta}^{a}\Big{)}-U(\Gamma), \tag{43}\]
where
\[K\Big{(}\xi,\dot{\theta}^{a}\Big{)}:=\frac{1}{2}\mathbb{G}_{ij}\,\xi^{i}\xi^{ j}+\mathbb{G}_{ia}\,\xi^{i}\dot{\theta}^{a}+\frac{1}{2}\mathbb{G}_{ab}\,\dot{ \theta}^{a}\dot{\theta}^{b}\]
with a constant symmetric kinetic energy tensor \(\mathbb{G}\), i.e., \(\mathbb{G}_{ba}=\mathbb{G}_{ab}\); also \(\mathbb{G}_{ai}\) and \(\mathbb{G}_{ia}\), seen as matrices, are transposes to each other.
Then the equations of motion with control inputs (torques) \(\{u_{a}\}_{a=1}^{s}\) applied to the \(\mathsf{H}\)-part (internal rotors) are
\[\dot{p}_{\alpha} =-\mathcal{B}_{\alpha\beta}^{i}(\Gamma)\,p_{i}\,v^{\beta}-K_{ \alpha j}^{k}\,\frac{\partial U}{\partial\Gamma_{j}}\Gamma_{k}, \tag{44a}\] \[\dot{\pi}_{a} =u_{a},\] (44b) \[\dot{\Gamma}_{i} =-\varkappa_{i\beta}^{k}(\Gamma)\,v^{\beta}\,\Gamma_{k}, \tag{44c}\]
Figure 3. Pendulum skate controlled by an internal rotor attached to the center of mass. Its axis of rotation is aligned with \(\mathbf{E}_{1}\), i.e., the edge of the skate.
where \(\{p_{i}\}_{i=1}^{n}\) and \(\{\pi_{a}\}_{a=1}^{s}\) are (re-)defined as follows using the modified Lagrangian (43):
\[p_{i}:=\left\langle\left.\frac{\delta\ell_{\mathrm{r}}}{\delta\xi}\right|_{ \mathrm{c}},\mathcal{E}_{i}\right\rangle=\mathcal{G}_{i\beta}\,v^{\beta}+ \mathcal{G}_{ia}\,\dot{\theta}^{a},\qquad\pi_{a}:=\left.\frac{\partial\ell_{ \mathrm{r}}}{\partial\dot{\theta}^{a}}\right|_{\mathrm{c}}=\mathcal{G}_{a\alpha }\,v^{\alpha}+\mathbb{G}_{ab}\,\dot{\theta}^{b}, \tag{45}\]
where
\[\mathcal{G}_{ia}(\Gamma):=\mathcal{E}_{i}^{j}(\Gamma)\,\mathbb{G}_{ja}, \tag{46}\]
and \(\mathcal{G}_{ai}\) is the transpose of \(\mathcal{G}_{ia}\). Notice that \(\mathcal{G}_{ia}\) is slightly different from \(\mathcal{G}_{ij}\) defined in (29) and should be distinguished based on the type of indices just like the \(\mathbb{G}\)'s defined above.
Again following [5], we consider the controlled Lagrangian of the form
\[\tilde{\ell}_{\mathrm{r}}\Big{(}\xi,\dot{\theta}^{a},\Gamma\Big{)}:=\tilde{K }\Big{(}\xi,\dot{\theta}^{a}\Big{)}-U(\Gamma)\] (47a) with \[\begin{split}\tilde{K}\Big{(}\xi,\dot{\theta}^{a}\Big{)}& :=K\Big{(}\xi^{i},\dot{\theta}^{a}+\tau_{i}^{a}\,\xi^{i}\Big{)}+ \frac{1}{2}\sigma_{ab}\,\tau_{i}^{a}\tau_{j}^{b}\,\xi^{i}\xi^{j}\\ &\quad+\frac{1}{2}(\rho_{ab}-\mathbb{G}_{ab})\Big{(}\dot{\theta }^{a}+(\mathbb{G}^{ac}\mathbb{G}_{ci}+\tau_{i}^{a})\xi^{i}\Big{)}\Big{(}\dot{ \theta}^{b}+(\mathbb{G}^{bc}\mathbb{G}_{cj}+\tau_{j}^{b})\xi^{j}\Big{)},\end{split} \tag{47b}\]
where \(\tau_{i}^{a}\), \(\sigma_{ab}\), and \(\rho_{ab}\) are all constant matrices and the last two are symmetric.
Let us define
\[\begin{split}\tilde{p}_{i}:=\left\langle\left.\frac{\delta\tilde{ \ell}_{\mathrm{r}}}{\delta\xi}\right|_{\mathrm{c}},\mathcal{E}_{i}\right\rangle &=\mathcal{G}_{i\alpha}\,v^{\alpha}+\mathcal{G}_{ia}\dot{\theta} ^{a}+(\mathcal{G}_{ib}+\sigma_{ab}\,\mathcal{T}_{i}^{a})\mathcal{T}_{\alpha}^{ b}\,v^{\alpha}+\mathbb{G}_{ab}\,\rho^{bc}\,\tilde{\pi}_{c}\,\mathcal{T}_{i}^{a}\\ &\quad+(\rho_{ab}-\mathbb{G}_{ab})\rho^{ad}\,\tilde{\pi}_{d} \Big{(}\mathbb{G}^{bc}\mathcal{G}_{ci}+\mathcal{T}_{i}^{b}\Big{)}\end{split} \tag{48}\]
and
\[\tilde{\pi}_{a}:=\left.\frac{\partial\tilde{\ell}_{\mathrm{r}}}{\partial\dot{ \theta}^{a}}\right|_{\mathrm{c}}=\left.\rho_{ab}\Big{(}\dot{\theta}^{b}+ \mathbb{G}^{bc}\mathbb{G}_{ci}\,\xi^{i}+\tau_{i}^{b}\,\xi^{i}\Big{)}\right|_{ \mathrm{c}}=\rho_{ab}\Big{(}\dot{\theta}^{b}+\mathbb{G}^{bc}\mathcal{G}_{c \alpha}\,v^{\alpha}+\mathcal{T}_{\alpha}^{b}\,v^{\alpha}\Big{)}. \tag{49}\]
See Appendix A.2 for the derivation of the expression for \(\tilde{p}_{i}\), where we defined
\[\mathcal{T}_{i}^{a}:=\tau_{j}^{a}\,\mathcal{E}_{i}^{j}. \tag{50}\]
Then the nonholonomic Euler-Poincare equations (25) with the controlled Lagrangian \(\tilde{\ell}_{\mathrm{r}}\) are given by
\[\dot{\tilde{p}}_{\alpha} =-\mathcal{B}_{\alpha\beta}^{i}(\Gamma)\,v^{\beta}\tilde{p}_{i}-K_{ \alpha j}^{k}\,\frac{\partial U}{\partial\Gamma_{j}}\Gamma_{k}, \tag{51a}\] \[\dot{\tilde{\pi}}_{a} =0,\] (51b) \[\dot{\Gamma}_{i} =-\varkappa_{i\beta}^{k}(\Gamma)\,v^{\beta}\,\Gamma_{k}. \tag{51c}\]
### Matching and Control Law
It turns out that the same sufficient condition for matching from [5, Theorem 2.1] works here:
**Proposition 13**.: _The controlled Euler-Poincare equations (44) and the Euler-Poincare equations (51) with the (reduced) controlled Lagrangian \(\tilde{\ell}_{\mathrm{r}}\) coincide if_
\[\tau_{i}^{a}=-\sigma^{ab}\mathbb{G}_{bi}\quad\text{and}\quad\sigma^{ab}+\rho^{ ab}=\mathbb{G}^{ab}. \tag{52}\]
_Then the resulting control law is_
\[u_{a}=\mathbb{G}_{ab}\Big{(}\mathcal{T}_{i}^{b}\,\mathcal{F}_{\alpha\beta}^{i} \,v^{\alpha}v^{\beta}-\mathcal{T}_{\alpha}^{b}\,\dot{v}^{\alpha}\Big{)}. \tag{53}\]
Proof.: See Appendix A.3.
_Remark 14_.: One can eliminate the acceleration \(\dot{v}^{\alpha}\) from the control law (53) using (44a).
**Example 15** (Pendulum skate with a rotor).: Going back to the motivating example from Figure 3, we may write the reduced Lagrangian with a rotor as follows:
\[\begin{split}&\ell_{\mathrm{r}}\colon\mathfrak{se}(3)\times T_{1} \mathbb{S}^{1}\times(\mathbb{R}^{3})^{*}\cong\mathbb{R}^{6}\times\mathbb{R} \times\mathbb{R}^{3}\to\mathbb{R};\\ &\ell_{\mathrm{r}}\Big{(}\boldsymbol{\Omega},\mathbf{Y},\dot{ \theta},\boldsymbol{\Gamma}\Big{)}:=K_{\mathrm{r}}\Big{(}\boldsymbol{\Omega},\mathbf{Y},\dot{\theta}\Big{)}-m\underline{g}l\mathbf{\Gamma}^{T}\mathbf{E}_ {3}\end{split} \tag{54}\]
with
\[\begin{split} K_{\mathrm{r}}\Big{(}\boldsymbol{\Omega},\mathbf{Y},\dot{\theta}\Big{)}:=&\ \frac{1}{2}\Big{(}(I_{1}+ml^{2})\Omega_{1}^{2}+J_{1}\big{(}\Omega_{1}+\dot{ \theta}\big{)}^{2}+(\mathcal{I}_{2}+ml^{2})\Omega_{2}^{2}+\mathcal{I}_{3} \Omega_{3}^{2}\Big{)}\\ &\qquad+ml(\Omega_{2}Y_{1}-\Omega_{1}Y_{2})+m\|\mathbf{Y}\|^{2} \\ =&\ \frac{1}{2}\Big{(}(\mathcal{I}_{1}+ml^{2})\Omega_{1 }^{2}+(\mathcal{I}_{2}+ml^{2})\Omega_{2}^{2}+\mathcal{I}_{3}\Omega_{3}^{2}+ml (\Omega_{2}Y_{1}-\Omega_{1}Y_{2})+m\|\mathbf{Y}\|^{2}\Big{)}\\ &\qquad+J_{1}\Omega_{1}\dot{\theta}+\frac{1}{2}J_{1}\dot{\theta} ^{2},\end{split} \tag{55}\]
where
\[\mathcal{I}_{i}:=I_{i}+J_{i}\]
with \(J_{i}\) (\(i=1,2,3\)) being the moments of inertia of the rotor; note that \(m\) now denotes the total mass of the system _including_ the rotor. Hence we have
\[\mathbb{G}_{ij}=\begin{bmatrix}\mathcal{I}_{1}+ml^{2}&0&0&0&-ml&0\\ 0&\mathcal{I}_{2}+ml^{2}&0&ml&0&0\\ 0&0&\mathcal{I}_{3}&0&0&0\\ 0&ml&0&m&0&0\\ -ml&0&0&0&m&0\\ 0&0&0&0&0&m\end{bmatrix},\qquad\mathbb{G}_{ia}=\begin{bmatrix}J_{1}\\ 0\\ 0\\ 0\\ 0\end{bmatrix}=J_{1}\mathbf{e}_{1},\qquad\mathbb{G}_{ab}=J_{1}.\]
Note that \(\sigma_{ab}=:\sigma\) and \(\rho_{ab}=:\rho\) are all scalars because there is only one internal rotor (\(s=1\)). Then the matching conditions (52) give
\[\begin{split}\tau_{i}^{a}=-\frac{J_{1}}{\sigma}\,\mathbf{e}_{1}^{ T}\iff\mathcal{T}_{i}^{a}=\tau_{j}^{a}\,\mathcal{E}_{i}^{j}&=- \frac{J_{1}}{\sigma}\left[\mathbf{e}_{1}^{T}\mathcal{E}_{1}\quad\cdots\quad \mathbf{e}_{1}^{T}\mathcal{E}_{6}\right]\\ &=-\frac{J_{1}}{\sigma}\left[1\quad\Gamma_{1}\quad 0\quad 0\quad 0 \quad 0\right]\\ &=-\frac{J_{1}}{\sigma}\mathbf{e}_{1}^{T},\end{split}\]
noting that \(\Gamma_{1}=0\), and
\[\frac{1}{\sigma}+\frac{1}{\rho}=\frac{1}{J_{1}}\iff\rho=\frac{J_{1}}{1-J_{1}/ \sigma}.\]
Then (53) gives the control law
\[u_{a}=\frac{J_{1}^{2}}{\sigma}\,\dot{v}^{1}=\frac{J_{1}}{\sigma}\,\frac{\Gamma_ {2}\big{(}m\underline{g}l+\big{(}ml^{2}+I_{2}-I_{3}\big{)}\,\Gamma_{3}(v^{2})^{ 2}\big{)}+ml\Gamma_{3}v^{2}v^{3}}{(I_{1}+ml^{2})/J_{1}}, \tag{56}\]
which is what we obtained in [11, Eq. (15d)] (with \(\sigma=-J_{1}/\nu\)) using a simpler and ad-hoc controlled Lagrangian.
Let us find the controlled Lagrangian. First, again using \(\Gamma_{1}=0\),
\[\mathcal{G}_{ai}=\mathbb{G}_{aj}\mathcal{E}_{i}^{j}\,=J_{1}\mathbf{e}_{1}^{T} \mathcal{E}_{i}^{T}=J_{1}\begin{bmatrix}1&\Gamma_{1}&0&\ldots&0\end{bmatrix}=J _{1}\mathbf{e}_{1}^{T}.\]
Then (49) gives
\[\tilde{\pi}_{a}=J_{1}v^{1}+\frac{J_{1}}{1-J_{1}/\sigma}\dot{\theta}.\]
Since \(\tilde{\pi}_{a}\) is conserved according to (51b), we impose that
\[\tilde{\pi}_{a}=0\iff\dot{\theta}=\bigg{(}\frac{J_{1}}{\sigma}-1\bigg{)}v^{1}= \bigg{(}\frac{J_{1}}{\sigma}-1\bigg{)}\Omega_{1},\]
and substitute it to (47) to obtain
\[\tilde{\ell}_{\rm r}\bigg{(}\boldsymbol{\Omega},\mathbf{Y},\bigg{(}\frac{J_{1 }}{\sigma}-1\bigg{)}\Omega_{1},\Gamma\bigg{)}=\frac{1}{2}\boldsymbol{\Omega}^{ T}\tilde{\mathbb{I}}\boldsymbol{\Omega}+\frac{m}{2}\|\mathbf{Y}+l\boldsymbol{ \Omega}\times\mathbf{E}_{3}\|^{2}-m\text{g}l\mathbf{\Gamma}^{T}\mathbf{E}_{3},\]
where
\[\tilde{\mathbb{I}}:=\text{diag}\left(I_{1}+\frac{J_{1}^{2}}{\sigma},\,I_{2}+J_ {2},\,I_{3}+J_{3}\right). \tag{57}\]
Notice that this controlled Lagrangian takes the same form as the Lagrangian (7) of the original pendulum skate _without_ the rotor, with the only difference being the inertia tensors \(\mathbb{I}\) and \(\tilde{\mathbb{I}}\).
## 7. Stabilization of Pendulum Skate
### Stabilization of Sliding Equilibrium
Recall from Proposition 12 that the sliding equilibrium (40) was always unstable without control. The control law obtained above using the controlled Lagrangian can stabilize it:
**Proposition 16**.: _The sliding equilibrium (40) of the pendulum skate with an internal rotor from Figure 3 is stable with the control (56) if_
\[-\frac{J_{1}^{2}}{I_{1}+ml^{2}}<\sigma<0. \tag{58}\]
Proof.: See Appendix A.4.
_Remark 17_.: In terms of \(\nu:=-J_{1}/\sigma\), the control gain in view of (56), the above condition (58) is equivalent to \(\nu>(I_{1}+ml^{2})/J_{1}\), which is what we had in [11, Eq. (18) in Theorem 4].
### Numerical Results--Uncontrolled
As a numerical example, consider the pendulum skate with \(m=2.00\,\mathrm{[kg]}\), \(l=0.80\,\mathrm{[m]}\), \((I_{1},I_{2},I_{3})=(0.35,0.35,0.004)\,\mathrm{[kg\cdot m^{2}]}\) with \(\text{g}=9.80\,\mathrm{[m/s^{2}]}\). As the initial condition, we consider a small perturbation to the sliding equilibrium (40):
\[\boldsymbol{\Omega}(0)=(0.1,\,0.1\tan\phi_{0},\,0.1),\qquad\mathbf{Y}(0)=(1,0,0),\qquad\boldsymbol{\Gamma}(0)=(0,\sin\phi_{0},\cos\phi_{0}), \tag{59}\]
where \(\phi_{0}=0.1\) is the small angle of tilt of the pendulum skate away from the vertical upward direction. Figure 4 shows the result of the simulation of the uncontrolled pendulum skate (35) with the above initial condition. It clearly exhibits instability as the pendulum skate falls down.
### Numerical Results--Controlled
We also solved the controlled system (44) (see also Example 15) with the control (56) using the same initial condition (59). The mass of the rotor is \(1\,\mathrm{[kg]}\), making the total mass \(m=3\,\mathrm{[kg]}\); we also set \((J_{1},J_{2},J_{3})=(0.005,0.0025,0.0025)\,\mathrm{[kg\cdot m^{2}]}\). The lower bound for \(\sigma\) shown in (58) in order to achieve stability is \(-J_{1}^{2}/(I_{1}+ml^{2})\simeq-1.10\times 10^{-5}\); hence we set \(\sigma=-10^{-5}\) here. Figure 5 shows that the system is indeed stabilized by the control.
## Acknowledgments
This paper is an extended version of our conference paper [11]. We would like to thank Vakhtang Putkaradze for helpful discussions. This work was supported by NSF grant CMMI-1824798. TO also would like to thank the Graduate School of Informatics of Kyoto University and Kazuyuki Yagasaki for their hospitality in supporting his sabbatical visit at Kyoto University, where some parts of the work were performed.
Figure 4. Numerical results for the uncontrolled pendulum skate (35) with \(m=2.00\,[\mathrm{kg}]\), \(l=0.80\,[\mathrm{m}]\), \((I_{1},I_{2},I_{3})=(0.35,0.35,0.004)\,[\mathrm{kg}\cdot\mathrm{m}^{2}]\). The initial condition (59) is a small perturbation from the sliding equilibrium (40). The pendulum skate falls down, i.e., \(\Gamma_{3}=0\) at \(t\simeq 1.025\) exhibiting the instability of the sliding equilibrium shown in Proposition 12.
Figure 5. Numerical results for the controlled pendulum skate (44) (see also Example 15) with \(m=3.00\,[\mathrm{kg}]\), \((J_{1},J_{2},J_{3})=(0.005,0.0025,0.0025)\,[\mathrm{kg}\cdot\mathrm{m}^{2}]\). The solutions are shown for the time interval \(0\leq t\leq 10\). One sees that the system is stabilized by the control in comparison to Figure 4; particularly, \(\Gamma_{3}\) stays close to \(1\), indicating that it maintains its position almost upright.
## Appendix A Some Proofs
### Proof of Proposition 11
The Jacobin \(Df\) of the vector field \(f\) from (38) at the spinning equilibrium (41) is
\[Df(\zeta_{\rm sp})=\begin{bmatrix}0&0&\frac{ml\Omega_{0}}{I_{1}+ml^{2}}&\frac{ mgl-\left(I_{3}+ml^{2}-I_{2}\right)\Omega_{0}^{2}}{I_{1}+ml^{2}}&0\\ 0&0&0&0&0\\ -2l\Omega_{0}&0&0&0&0\\ 1&0&0&0&0\\ 0&0&0&0&0\end{bmatrix},\]
and its eigenvalues are
\[\left\{0,0,0,\pm\sqrt{\frac{mgl-(I_{3}+ml^{2}-I_{2})\Omega_{0}^{2}}{I_{1}+ml^ {2}}}\right\}.\]
If \(I_{3}+ml^{2}<I_{2}\) then by the Instability from Linearization criterion (see, e.g., Sastry [19, p.216]), \(\zeta_{\rm sp}\) is unstable. On the other hand, if \(I_{3}+ml^{2}>I_{2}\) then the linear analysis is inconclusive, and so we would like to use the following nonlinear method:
**Energy-Casimir Theorem (Aeyels [1])**.: _Consider a system of differential equations \(\dot{\zeta}=f(\zeta)\) on \(\mathbb{R}^{n}\) with locally Lipschitz \(f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}\) with an equilibrium \(\zeta_{0}\in\mathbb{R}^{n}\), i.e., \(f(\zeta_{0})=0\). Assume that the system has \(k+1(<n)\) invariants \(\mathscr{E}\) and \(\{C_{j}\}_{j=1}^{k}\) that are \(C^{2}\) and submersive at \(\zeta_{0}\), and also that the gradients \(\{DC_{j}(\zeta_{0})\}_{j=1}^{k}\) at \(\zeta_{0}\) are linearly independent. Then \(\zeta_{0}\) is a stable equilibrium if_
1. _there exist scalars_ \(\{c_{j}\in\mathbb{R}\}_{j=1}^{k}\) _such that_ \(D(\mathscr{E}+c_{1}C_{1}+\cdots+c_{k}C_{k})(\zeta_{0})=0\)_; and_
2. _the Hessian_ \(\mathscr{H}:=D^{2}(\mathscr{E}+c_{1}C_{1}+\cdots+c_{k}C_{k})(\zeta_{0})\) _is sign definite on the tangent space at_ \(\zeta_{0}\) _of the submanifold_ \(\{\zeta\in\mathbb{R}^{n}\mid C_{j}(\zeta)=C_{j}(\zeta_{0})\ \forall j\in\{1,\ldots,k\}\}\)_, i.e., for any_ \(w\in\mathbb{R}^{n}\backslash\{0\}\) _satisfying_ \(DC_{j}(\zeta_{0})\cdot w=0\) _with every_ \(j\in\{1,\ldots,k\}\)_, one has_ \(w^{T}\mathscr{H}w>0\) _(or_ \(<0\)_)._
We note that, despite its name, the above theorem does not assume that the invariants are Casimirs: any invariants--Casimirs or not--would suffice.
In order to use the above theorem, we set the constrained energy (30) as \(\mathscr{E}\) and use the invariants \(C_{1}\) and \(C_{2}\) from (36) and (37) as well as \(C_{3}=(\Gamma_{2}^{2}+\Gamma_{3}^{2})/2\) with \(k=3\). Now, since
setting \((c_{1},c_{2},c_{3})=(-\Omega_{0},0,I_{3}\Omega_{0}^{2}-m\text{gl})\), we have
\[D(\mathscr{E}+c_{1}C_{1}+c_{2}C_{2}+c_{3}C_{3})(\zeta_{\rm sp})=0.\]
Then we also see that
\[\mathscr{H} :=D^{2}(\mathscr{E}+c_{1}C_{1}+c_{2}C_{2}+c_{3}C_{3})(\zeta_{ \rm sp})\] \[=\begin{bmatrix}I_{1}+ml^{2}&0&0&0&0\\ 0&I_{3}&0&0&0\\ 0&0&m&ml\Omega_{0}&0\\ 0&0&ml\Omega_{0}&\Omega_{0}^{2}\left(I_{3}+ml^{2}-I_{2}\right)-m\text{gl}&0\\ 0&0&0&0&-m\text{gl}\end{bmatrix}.\]
The relevant tangent space is the null space
\[\ker\begin{bmatrix}DC_{1}(\zeta_{\text{sp}})^{T}\\ DC_{2}(\zeta_{\text{sp}})^{T}\\ DC_{3}(\zeta_{\text{sp}})^{T}\end{bmatrix}=\left\{w=s_{1}\begin{bmatrix}1\\ 0\\ 0\\ 0\\ 0\end{bmatrix}+s_{2}\begin{bmatrix}0\\ 0\\ 0\\ 1\\ 0\end{bmatrix}\mid s_{1},s_{2}\in\mathbb{R}\right\}.\]
Hence we have the quadratic form
\[w^{T}\mathscr{H}w=(I_{1}+ml^{2})s_{1}^{2}+\big{(}(I_{3}+ml^{2}-I_{2})\Omega_{0 }^{2}-m\text{gl}\big{)}s_{2}^{2},\]
which is positive definite in \((s_{1},s_{2})\) under the assumed conditions (42).
### Derivation of (48)
We can obtain the expression for \(\tilde{p}_{i}\) in (48) as follows: Let us first rewrite its definition as
\[\tilde{p}_{i}:=\left\langle\left.\frac{\delta\tilde{\ell}_{\text{r}}}{\delta \xi}\right|_{\text{c}},\mathcal{E}_{i}\right\rangle=\mathcal{E}_{i}^{j}( \Gamma)\left\langle\left.\frac{\delta\tilde{\ell}_{\text{r}}}{\delta\xi} \right|_{\text{c}},E_{j}\right\rangle.\]
Performing the same computations as in [5, Eq. (16)] and using \(\xi^{k}=\mathcal{E}_{l}^{k}\,v^{l}\), one has
\[\left\langle\frac{\delta\tilde{\ell}_{\text{r}}}{\delta\xi},E_{j}\right\rangle =\mathbb{G}_{jk}\,\xi^{k}+\mathbb{G}_{ja}\Big{(}\dot{\theta}^{a}+ \tau_{k}^{a}\,\xi^{k}\Big{)}+\mathbb{G}_{ab}\,\rho^{bc}\,\frac{\partial\tilde{ \ell}_{\text{r}}}{\partial\dot{\theta}^{c}}\,\tau_{j}^{a}\] \[\quad+\sigma_{ab}\,\tau_{j}^{a}\tau_{k}^{b}\,\xi^{k}+(\rho_{ab}- \mathbb{G}_{ab})\rho^{ad}\,\frac{\partial\tilde{\ell}_{\text{r}}}{\partial \dot{\theta}^{d}}\Big{(}\mathbb{G}^{bc}\mathbb{G}_{cj}+\tau_{j}^{b}\Big{)}\] \[=\mathbb{G}_{jk}\,\mathcal{E}_{l}^{k}v^{l}+\mathbb{G}_{ja}\Big{(} \dot{\theta}^{a}+\tau_{k}^{a}\,\mathcal{E}_{l}^{k}\,v^{l}\Big{)}+\mathbb{G}_{ ab}\,\rho^{bc}\,\frac{\partial\tilde{\ell}_{\text{r}}}{\partial\dot{\theta}^{c}} \,\tau_{j}^{a}\] \[\quad+\sigma_{ab}\,\tau_{j}^{a}\tau_{k}^{b}\,\mathcal{E}_{l}^{k} \,v^{l}+(\rho_{ab}-\mathbb{G}_{ab})\rho^{ad}\,\frac{\partial\tilde{\ell}_{ \text{r}}}{\partial\dot{\theta}^{d}}\Big{(}\mathbb{G}^{bc}\mathbb{G}_{cj}+\tau _{j}^{b}\Big{)},\]
and so
\[\tilde{p}_{i}=\mathcal{E}_{i}^{j}\bigg{(} \mathbb{G}_{jk}\,\mathcal{E}_{l}^{k}v^{l}+\mathbb{G}_{ja}\Big{(} \dot{\theta}^{a}+\tau_{k}^{a}\,\mathcal{E}_{l}^{k}\,v^{l}\Big{)}+\mathbb{G}_{ ab}\,\rho^{bc}\,\frac{\partial\tilde{\ell}_{\text{r}}}{\partial\dot{\theta}^{c}}\, \tau_{j}^{a}\] \[= \,\mathcal{G}_{i\alpha}\,v^{\alpha}+\mathcal{G}_{ia}\Big{(}\dot{ \theta}^{a}+\mathcal{T}_{\alpha}^{a}\,v^{\alpha}\Big{)}+\mathbb{G}_{ab}\,\rho ^{bc}\,\tilde{\pi}_{c}\,\mathcal{T}_{i}^{a}\] \[+\sigma_{ab}\,\mathcal{T}_{i}^{a}\mathcal{T}_{\alpha}^{b}\,v^{ \alpha}+(\rho_{ab}-\mathbb{G}_{ab})\rho^{ad}\,\tilde{\pi}_{d}\Big{(}\mathbb{G} ^{bc}\mathcal{G}_{ci}+\mathcal{T}_{i}^{b}\Big{)},\]
where we used (49).
### Proof of Proposition 13
One can obtain the matching conditions (52) almost the same way as in the proof of [5, Theorem 2.1]. Indeed, the controlled Euler-Poincare equations (44) and the Euler-Poincare equations (51) with the (reduced) controlled Lagrangian \(\tilde{\ell}_{\text{r}}\) match if \(p_{i}\) in (45) and \(\tilde{p}_{i}\) in (48) are equal, i.e.,
\[(\mathcal{G}_{ib}+\sigma_{ab}\,\mathcal{T}_{i}^{a})\mathcal{T}_{\alpha}^{b}\,v^ {\alpha}+\mathbb{G}_{ab}\,\rho^{bc}\,\tilde{\pi}_{c}\,\mathcal{T}_{i}^{a}+( \rho_{ab}-\mathbb{G}_{ab})\rho^{ad}\,\tilde{\pi}_{d}\Big{(}\mathbb{G}^{bc} \mathcal{G}_{ci}+\mathcal{T}_{i}^{b}\Big{)}=0.\]
Let us assume \(\mathcal{T}_{i}^{a}=-\sigma^{ab}\mathcal{G}_{bi}\), which is equivalent to to the first matching condition in (52) using the definitions (46) and (50) of \(\mathcal{G}_{ai}\) and \(\mathcal{T}_{i}^{a}\). Substituting \(\mathcal{T}_{i}^{a}=-\sigma^{ab}\mathcal{G}_{bi}\) into the above displayed equation, we obtain
\[\mathcal{G}_{ai}=\rho_{ab}(\mathbb{G}^{bc}-\sigma^{bc})\mathcal{G}_{ci},\]
but then this is satisfied if we assume the second matching condition in (52).
For the control law, we again mimic [5, Section 2.2] as follows: Notice from (45) and (49) that
\[\pi_{a}-\mathbb{G}_{ab}\,\rho^{bd}\,\tilde{\pi}_{d}=-\mathbb{G}_{ab}\,\mathcal{T }_{\alpha}^{b}\,v^{\alpha}.\]
Using (44b), (51b), and the above equality, we have
\[u_{a} =\frac{d}{dt}\Big{(}\pi_{a}-\mathbb{G}_{ab}\,\rho^{bd}\,\tilde{ \pi}_{d}\Big{)}\] \[=-\mathbb{G}_{ab}\,\frac{d}{dt}\Big{(}\mathcal{T}_{\alpha}^{b}\,v ^{\alpha}\Big{)}\] \[=-\mathbb{G}_{ab}\,\frac{d}{dt}\Big{(}\tau_{j}^{b}\,\mathcal{E}_ {\alpha}^{j}\,v^{\alpha}\Big{)}\] \[=-\mathbb{G}_{ab}\,\tau_{j}^{b}\,\Big{(}D^{k}\mathcal{E}_{ \alpha}^{j}\,\dot{\Gamma}_{k}\,v^{\alpha}+\mathcal{E}_{\alpha}^{j}\,\dot{v}^{ \alpha}\Big{)}\] \[=\mathbb{G}_{ab}\,\tau_{j}^{b}\,\Big{(}D^{k}\mathcal{E}_{\alpha }^{j}\,\mathcal{E}_{k\beta}^{l}\,\Gamma_{l}\,v^{\alpha}v^{\beta}-\mathcal{E}_ {\alpha}^{j}\,\dot{v}^{\alpha}\Big{)}\] \[=\mathbb{G}_{ab}\,\tau_{j}^{b}\,\Big{(}\mathcal{E}_{i}^{j}\, \mathcal{F}_{\alpha\beta}^{i}\,v^{\alpha}v^{\beta}-\mathcal{E}_{\alpha}^{j}\, \dot{v}^{\alpha}\Big{)}\] \[=\mathbb{G}_{ab}\,\Big{(}\mathcal{T}_{i}^{b}\,\mathcal{F}_{ \alpha\beta}^{i}\,v^{\alpha}v^{\beta}-\mathcal{T}_{\alpha}^{b}\,\dot{v}^{ \alpha}\Big{)},\]
where we used (51b) and the definitions of \(\mathcal{F}\) and \(\mathcal{T}\) from (27) and (50) in the third last three equalities.
### Proof of Proposition 16
We employ the same Energy-Casimir method from Appendix A.1.
Recall from Example 15 that the controlled system is equivalent to the original system (38) whose inertia tensor \(\mathbb{I}=\operatorname{diag}(I_{1},I_{2},I_{3})\) is replaced by \(\tilde{\mathbb{I}}\) shown in (57):
\[(I_{1},I_{2},I_{3})\to\bigg{(}I_{1}+\frac{J_{1}^{2}}{\sigma},\,I_{2}+J_{2},\,I _{3}+J_{3}\bigg{)}.\]
Therefore, the controlled system possesses invariants \(\tilde{\mathcal{E}}\) and \(\{\tilde{C}_{i}\}_{i=1}^{3}\) defined by making the above replacement in \(\mathscr{E}\) and \(\{C_{i}\}_{i=1}^{3}\) (see (30), (36), and (37)); note that \(\tilde{C}_{3}=C_{3}=(\Gamma_{2}^{2}+\Gamma_{3}^{2})/2\) because it does not depend on \((I_{1},I_{2},I_{3})\). More specifically, we have
\[D\tilde{\mathcal{E}}(\zeta_{\text{sl}})=\begin{bmatrix}0\\ 0\\ mY_{0}\\ 0\\ mgl\end{bmatrix},\quad D\tilde{C}_{1}(\zeta_{\text{sl}})=\begin{bmatrix}0\\ I_{3}+J_{3}\\ 0\\ 0\\ 0\end{bmatrix},\quad D\tilde{C}_{2}(\zeta_{\text{sl}})=\begin{bmatrix}0\\ 0\\ 1\\ 0\\ 0\end{bmatrix},\quad D\tilde{C}_{3}(\zeta_{\text{sl}})=\begin{bmatrix}0\\ 0\\ 0\\ 0\\ 1\end{bmatrix}.\]
Setting \((c_{1},c_{2},c_{3})=(0,-mY_{0},-mgl)\), we have
\[D\Big{(}\tilde{\mathcal{E}}+c_{1}\tilde{C}_{1}+c_{2}\tilde{C}_{2}+c_{3}\tilde {C}_{3}\Big{)}(\zeta_{\text{sl}})=0.\]
Then we also see that
\[\mathscr{H} :=D^{2}\Big{(}\tilde{\mathcal{E}}+c_{1}\tilde{C}_{1}+c_{2}\tilde{C }_{2}+c_{3}\tilde{C}_{3}\Big{)}(\zeta_{\text{sl}})\] \[=\begin{bmatrix}I_{1}+ml^{2}+J_{1}^{2}/\sigma&0&0&0&0\\ 0&I_{3}+J_{3}&0&-mlY_{0}&0\\ 0&0&m&0&0\\ 0&-mlY_{0}&0&-mgl&0\\ 0&0&0&0&-mgl\end{bmatrix}.\]
The relevant tangent space is the null space
\[\ker\begin{bmatrix}D\tilde{C}_{1}(\zeta_{\text{sl}})^{T}\\ D\tilde{C}_{2}(\zeta_{\text{sl}})^{T}\\ D\tilde{C}_{3}(\zeta_{\text{sl}})^{T}\end{bmatrix}=\left\{w=s_{1}\begin{bmatrix} 1\\ 0\\ 0\\ 0\\ 0\end{bmatrix}+s_{2}\begin{bmatrix}0\\ 0\\ 0\\ 1\\ 0\end{bmatrix}\mid s_{1},s_{2}\in\mathbb{R}\right\}.\]
Hence we have the quadratic form
\[w^{T}\mathscr{H}w=\left(I_{1}+ml^{2}+\frac{J_{1}^{2}}{\sigma}\right)s_{1}^{2}- mgl\,s_{2}^{2},\]
which is negative definite in \((s_{1},s_{2})\) under the assumed condition (58).
| We extend the method of Controlled Lagrangians to nonholonomicEuler--Poincar\'e equations with advected parameters, specifically to thosemechanical systems on Lie groups whose symmetry is broken not only by apotential force but also by nonholonomic constraints. 私たちは、アベックされたパラメータを持つ非holonomic Euler--Poincar\'e方程式に、Controlled Lagrangiansのメソッドを拡張します。特に、ポテンシャル力だけでなく、非holonomic制約も symmetry を破る Lie 群上の機械システムに適用します。私たちは、アベックされたパラメータに依存する quasivelocities を導入することで、非holonomic Euler--Poincar\'e方程式における Lagrange Multiplier を一貫して消去します。 quasivelocitiesは、これらのシステムのための Controlled Lagrangians の方法を促進し、Bloch, Leonard, and Marsdenによる標準の非Holonomic Euler--Poincar\'e方程式における一致条件と |
2304.07054 | Backscattering-Induced Dissipative Solitons in Ring Quantum Cascade
Lasers | Ring quantum cascade lasers have recently gained considerable attention,
showing ultrastable frequency comb and soliton operation, thus opening a way to
integrated spectrometers in the midinfrared and terahertz fingerprint regions.
Thanks to a self-consistent Maxwell-Bloch model, we demonstrate, in excellent
agreement with the experimental data, that a small but finite coupling between
the counterpropagating waves arising from distributed backscattering is
essential to stabilize the soliton solution. | Lukas Seitner, Johannes Popp, Ina Heckelmann, Réka-Eszter Vass, Bo Meng, Michael Haider, Jérôme Faist, Christian Jirauschek | 2023-04-14T11:13:35 | http://arxiv.org/abs/2304.07054v2 | # Backscattering-Induced Kerr Solitons in Ring Quantum Cascade Lasers
###### Abstract
Ring quantum cascade lasers have recently gained considerable attention, showing ultrastable frequency comb and soliton operation, and thus opening a way to integrated spectrometers in the mid-infrared and terahertz fingerprint regions. Thanks to a self-consistent Maxwell-Bloch model, we demonstrate, in excellent agreement with the experimental data, that a small but finite coupling between the counter-propagating waves arising from distributed backscattering is essential to stabilize the soliton solution.
## I Introduction
The quantum cascade laser (QCL) exploits optical intersubband transitions in the conduction band of a multiquantum-well heterostructure to access large portions of the mid-infrared and terahertz regime [1]. Besides single-mode lasing, a major goal is to achieve coherent multimode operation, thus providing mid-infrared and terahertz frequency combs. In Fabry-Perot cavities, comb generation [2] arises from four-wave-mixing nonlinearities in which dynamical spatial hole burning (SHB) plays a dominant role [3; 4; 5]. The Kerr nonlinearity inside a QCL mainly originates from the fast gain saturation which, in turn, makes the four-wave-mixing very broadband [6].
Recently, QCLs embedded in ring cavities received considerable attention due to the absence of SHB for unidirectional propagation. Still, these devices showed self-starting comb operation with soliton-like spectra [7; 8; 9; 10], indicating different physics of comb formation than in Fabry-Perot resonators with two-lobe spectra [2]. Analysis of further ring devices with a waveguide displaying negative group velocity dispersion (GVD) then showed an output corresponding to a dissipative Kerr soliton [9]. A dissipative Kerr soliton is a pulsed waveform exhibiting the unique property to travel unperturbed through nonlinear, dispersive, and lossy media [11; 12]. While these localized field structures were at first discussed for fiber resonators, there has been growing interest in their occurrence in passive and active microcavities [13; 14; 15; 16]. An explanation for multimode operation in ring QCLs was given based on phase turbulence and the linewidth enhancement factor (LEF), where a finite LEF makes gain saturation act as an effective Kerr nonlinearity [8]. The LEF itself was then explained to be resulting from Bloch gain [17]. Balancing the nonlinearity to the overall group velocity dispersion present in the cavity can lead to soliton-like field structures, according to models using the complex cubic Ginzburg-Landau equations [18]. However, the direct simulation of dissipative Kerr solitons with realistic parameters and results as observed in experiment [9], is still missing, to the best of our knowledge.
Using Maxwell-Bloch theory [19; 20; 21], we reveal the physical mechanisms and identify requirements leading to the recent experimental observation of solitons in ring QCLs. Our approach to explain the formation of Kerr solitons is based on backscattering in the cavity, occurring at many microscopic reflectors distributed over the whole circumference. This differs significantly from previous work that introduced just a single or at most two macroscopic cavity interruptions, yielding standing wave patterns [22; 23]. In our case, taking into account both propagation directions at once reveals that the presence of a fainter counter-propagating wave may play a crucial role in soliton stability. Under these circumstances, the presence of a single symmetric Lorentzian gain shape, associated with the lasing transition in the Maxwell-Bloch model (see Section I.B. of the Supplemental Material), suffices to obtain the measured system properties. Direct comparison of the simulations with experiment yields excellent agreement, thus giving valid and novel insights into the dynamics of quantum cascade ring lasers.
## II Theoretical Model
Our approach to accurately model the dynamics in a ring cavity quantum cascade laser is based on the one-dimensional Maxwell-Bloch equations [21]. Hence, we simulate the discretized electric field in propagation direction \(x\) and time \(t\), with periodic boundary conditions to mimic the ring cavity. The efficient numerical implementation and inclusion of advanced effects has been subject to many studies in recent years [24; 25; 26; 27; 28; 29; 30; 31]. The
electron dynamics are captured by generalized density matrix equations, coupled to the spatiotemporal optical field equations.
We consider a device with alternating layers of In\({}_{0.60}\)Ga\({}_{0.40}\)As and In\({}_{0.36}\)Al\({}_{0.64}\)As (see Section I.A. of the Supplemental Material for the layer sequence), confining the electrons in one spatial dimension which leads to quantized energy states. A set of eigenstates is obtained by solving the Schrodinger-Poisson equation for seven levels per period, using EZ-states as a basis [32]. The probability distributions and quantized energies of the eigenstates are displayed in Figure 1 for a representative period (solid lines), along with the ones for an adjacent period (dotted). The electron transport in the QCL active region is modeled using the ensemble Monte Carlo (EMC) method [33]. The obtained gain of the structure exhibits an unsaturated value of \(g_{0}=16.5\,\mathrm{cm}^{-1}\) at the center frequency \(f_{\mathrm{c}}=40.89\,\mathrm{THz}\). In a dynamic description, the state of the multi-level quantum system is described by the density matrix \(\hat{\rho}\). Its time evolution is governed by the Liouville-von Neumann master equation [21]
\[\partial_{t}\hat{\rho}=-\mathrm{i}\hbar^{-1}[\hat{H_{0}}-\hat{\mu}E,\hat{\rho} ]+\mathcal{D}(\hat{\rho})\,. \tag{1}\]
Here, \(\hat{H_{0}}\) is the Hamiltonian of the electronic quantum system, \(\hat{\mu}\) denotes the dipole-matrix element of the optical transition between the upper laser level (ULL) and the lower laser level (LLL), and \(\mathcal{D}(\hat{\rho})\) represents the dissipation operator that models the decay of coherence by scattering and dephasing processes. Using the rotating wave approximation (RWA) reduces the numerical load by dropping rapidly oscillating terms in the density matrix [21].
The electric field amplitude \(E\) in the slowly varying amplitude approximation (SVAA) is described as the superposition of a left- and right-traveling wave
\[E(x,t)=\frac{1}{2}\big{[}E^{\pm}\exp\left(\pm\mathrm{i}\beta_{0}x-\mathrm{i} \omega_{\mathrm{c}}t\right)+\mathrm{c.c.}\big{]}\,. \tag{2}\]
Here, \(\beta_{0}\) is the wave number, \(\omega_{\mathrm{c}}=2\pi f_{\mathrm{c}}\), and c.c. denotes the complex conjugate of the previous term. For both components, a classical propagation equation can be derived from Maxwell's equations, which is given by
\[\partial_{t}E^{\pm}=\mp v_{\mathrm{g}}\partial_{x}E^{\pm}+\Gamma f^{\pm}(x,t) -lE^{\pm}-\mathrm{i}v_{\mathrm{g}}\frac{\beta_{2}}{2}\partial_{t}^{2}E^{\pm}\,. \tag{3}\]
In Eq. (3) we consider losses \(l\), as well as the background group velocity dispersion \(\beta_{2}\) of the effective material, as they have a non-negligible influence on frequency comb formation [25]. The second term \(\Gamma f^{\pm}(x,t)\) in (3) refers to the polarization originating from the quantum system, with
\[f(x,t)=n_{\mathrm{3D}}\mathrm{Tr}\{\hat{\mu}\partial_{t}\hat{\rho}\}\,. \tag{4}\]
Equation (4) thus contains the coupling of the quantized electronic system to the classical electric field, with the overlap factor \(\Gamma=0.6\), which has also been considered for calculating \(g_{0}\).
In ring lasers, the two wave components \(E^{+}\) and \(E^{-}\) from Eq. (2) are commonly referred to as clockwise (cw) and counter-clockwise (ccw) fields. In Eq. (3) \(E^{+}\) and \(E^{-}\) are not coupled directly, but may only interact via the density matrix. As cross-gain saturation in ring lasers is twice as strong as self-gain saturation, a spontaneous symmetry breaking between the two counter-propagating fields \(E^{+}\) and \(E^{-}\) will occur for large enough pumping [34; 35]. In real devices, the presence of a finite optical coupling between these fields via backscattering prevents a completely unidirectional operation of the laser, even after the symmetry breaking point has been reached. The central message of this paper is that this coupling between the counter-propagating modes in the ring cavity is an essential element for soliton formation and stability.
In Figure 2, two cleaved cross-sections of waveguides are shown, captured by a scanning electron microscope (SEM). In Figure 2(a) microscopic defects are clearly visible, surrounding the active region. As the electromagnetic field overlaps with these defects, it experiences an impedance mismatch, leading to localized reflections. As we assume the presence of many such defects throughout the whole cavity, this effect sums up to significant backscattering. In Figure 2(b) a cross-section of a defect-free cavity from another fabrication run is presented for comparison.
Soliton generation was experimentally confirmed in a device having comparable backscattering to the one depicted in Figure 2(a), as estimated from the location of the symmetry breaking point in the light-current (LI) curve. Our model considers both propagation directions (cw and ccw) simultaneously and couples them through backscattering. Therefore, we can capture the full dynamics of the system. The microscopic defects are introduced by sub-dividing the cavity into numerous regions with a small reflectance \(r\), defined as \(\Delta E^{\pm}=rE^{\mp}\), at each interface, where \(\Delta E^{\pm}\) denotes the resulting field change. Thus, in the free-running system, both field com
Figure 1: Potential and quantum state probability distributions of the investigated QCL structure at a bias field strength of \(57\,\mathrm{kV/cm}\).
ponents experience the same amount of backscattering, and, as the laser is self-starting, the same amount of gain.
## III Results
The backscattering dependence of the breaking point can be described by a set of differential equations [34; 36]. Experimentally, the backscattering coefficient is often estimated by the ratio of the symmetry-breaking current \(I_{\mathrm{sym}}\) to the threshold current \(I_{\mathrm{th}}\), which yields \(\alpha\approx$0.01\,\mathrm{cm}^{-1}$\) in the employed waveguide [9]. However, the full dynamics are inherently captured by the Maxwell-Bloch equations, when varying the pump current and simulating both propagation directions. In order to quantify the backscattering in the modeled waveguide we perform a current sweep, locate the symmetry breaking point, and compare the results to measured values. The spectrum and the temporal intensity profile of the most relevant multimode state are further investigated by filtering and phase analysis, demonstrating soliton generation. We compare the result of the self-consistent seven-level model directly to experiment, and obtain excellent agreement in terms of pulse width and number of locked comb lines.
### Light - Current Characteristics
In our Maxwell-Bloch-type simulation approach, the current depends on the scattering rates between seven quantized levels and can consequently not be adjusted straightforwardly. However, to recreate the experimental LI-curve and thus, to extract \(\alpha\), sweeping the pump current is necessary. Therefore, assuming a mainly photon-driven current, we vary the dipole moment \(\hat{\mu}\) to account for the band-edge bending and overlapping of wavefunctions with increasing bias [37]. As the gain is directly proportional to \(\hat{\mu}\), this leads to a change in the photon-driven current. Thus, we can efficiently use this approach to carry out the current sweep. It turns out that using a reflection of \(r=0.0008\) at each interface between in total 100 regions within the cavity is a suitable choice. The two LI curves of experiment and simulation are shown in Figure 3. The overall intra-cavity power of the separate propagation directions is in very good agreement as well as the ratio \(I_{\mathrm{sym}}/I_{\mathrm{th}}\). The small remaining difference can be mainly attributed to the fact that only photon-driven current is considered in the simulation. Additionally, the experimental setup will always have incomplete mode-contrast, as reflections of the field in the main propagation direction on the InP-air interfaces can be captured, thus underestimating the true intra-cavity intensity of the counterpropagating mode. When a certain amount of backscattering is present, as shown in both cases, multimode operation can be sustained in a steady state, and soliton generation is possible. This bias region is shaded in blue (gray) color for the simulation. Different multimode operation regimes besides single soliton operation, such as double-pulsing and more irregular comb shapes were observed in agreement with experimental observations [9] (see Figures 4 and 5 of the Supplemental Material). Obtained results using lower backscattering show the onset of multimode operation at higher bias currents. While a larger value of \(\alpha\), i.e., increased backscattering, brings the multimode bias closer to threshold, a phase-stable operation will at some point not be supported anymore.
Once the multimode threshold is surpassed, stable soliton operation is possible, single- or multi-pulsed, but also different rather incoherent states may appear. For further analysis, we choose the point marked with the orange cross in Figure 3(a), as this point contains the extracted dipole element at the bias voltage where the self-consistent Monte Carlo simul
Figure 3: Simulated and experimental intra-cavity power versus normalized bias current. In both cases, a significant and comparable amount of backscattering \(\alpha\) is present. The shaded area marks the multimode regime observed using the Maxwell-Bloch formalism. The orange crosses mark the bias current used for further dynamical simulations. Results for the defect-free device, as shown in Figure 2(b), are given in Figure 6 of the Supplemental Material.
Figure 2: Images of cleaved waveguide cross-sections, obtained using a scanning electron microscope. **(a)** Device with defects and thus increased backscattering. **(b)** Defect-free device from another fabrication run.
### Spectral and Temporal Characteristics
In order to obtain a good match with experimental data, we use the self-consistent Monte Carlo results as input for the dynamical solver. We simulate over \(6\,000\) round trips, of which the last \(2\,000\) are post-processed as a steady-state solution. The overall intra-cavity power of roughly \(480\,\mathrm{mW}\) fits well with the one of the experimental device in [9]. Applying a Fourier transform to the converged field yields the spectrum as shown in Figure 4(a). The characteristic sech-square shape of soliton operation can be observed in the spectral range above the center frequency. This envelope very well encloses \(25\) of the comb teeth, which is in near-perfect agreement with the experimental results. The comb exhibits a clear beatnote at roughly \(25\,\mathrm{GHz}\). The linewidth is below our numerical frequency resolution of \(<5\,\mathrm{MHz}\). Temporal results of the last three round trips are shown unprocessed in Figure 4(b) and reveal an amplitude modulation in the form of an intensity dip. A filter is then applied to the spectrum as in experiment [9], such that the blue shaded part of Figure 4(a) contains field components with frequencies \(f\leq f_{\mathrm{c}}\). This contribution is referred to as \(E_{\mathrm{low}}\) in time domain. The orange-shaded part of the spectrum contains field contributions with \(f>f_{\mathrm{c}}\), accounting for temporal field components called \(E_{\mathrm{high}}\). Transforming each filtered comb part separately to the time-domain yields two intensity contributions \(\propto\left|E_{\mathrm{low}}\right|^{2}\) and \(\propto\left|E_{\mathrm{high}}\right|^{2}\), plotted in the respective colors blue and orange in Figure 4(c). The filtered intensities show a pulsed waveform superimposed to a dispersive continuous background wave, in complete agreement with experiment. The pulse width is \(2.6\,\mathrm{ps}\), which is very close to the measured value of \(3.1\,\mathrm{ps}\).
In order to validate the soliton nature of the single pulsed state we investigate its spectral phase. In Figure 5(a), the modes in a \(30\,\mathrm{dB}\) power range are plotted, where each maximum value is highlighted with an orange marker. The intermodal phase differences with respect to these modes are given in Figure 5(b). The phase jump of the third mode indicates that the center mode does not participate in the soliton pulse, but rather contributes to the continuum background. Furthermore, the intermodal phase differences \(\Delta\phi\) are in almost perfect agreement with the measurement [9]. Apart from the center mode, all contributing modes have almost no phase difference.
## IV Discussion
In order to identify the mechanisms that lead to the generation of a soliton and its preservation, it is useful to consider at first the simplest possible cavity having no defects and thus also no spatial hole burning (SHB). If this system is initialized with a random, multimode seed, then the spectra in our simulations are multimode with a tendency to a soliton-like shape in the first few hundred round trips after the symmetry break. However, the pulse continuously broadens in time until it is completely absorbed by the dispersive wave, and the spectrum becomes single-mode. Notably, single-mode opera
Figure 4: **(a)** Power spectra of simulation and experiment showing a soliton shape. **(b)** Simulated intra-cavity power of the main-propagation-direction field at one grid-point, showing the propagating soliton. **(c)** Filtered intensity contributions of the time-trace in (b), with the normalized measured and filtered pulses from dual-comb spectroscopy (gray dashed) shown for comparison.
Figure 5: Spectral phases of five thousand converged round trips from Figure 4. **(a)** Power spectrum in a \(30\,\mathrm{dB}\) range, with the mode maxima marked. **(b)** Intermodal phase differences \(\Delta\phi\) of the marked points. Phase locking of the comb is visible, while the dispersive center mode is phase-shifted by roughly \(\pi\).
tion has also been observed experimentally in defect-free devices as shown in Figure 2(b), confirming our theoretical predictions. As demonstrated here, adding the distributed backscattering allows for the ccw and cw components to interact with each other, which stabilizes the soliton. Unlike in other models for ring cavity QCLs, SHB is present from the beginning of the simulation. We find that it may not be neglected in order to obtain realistic results. It naturally arises due to the presence of the scatterers and is thus also present in a real cavity with imperfections. The related population grating in the density matrix yields a significant contribution to cross-gain saturation, such that it can dominate over the backscattering during the dynamical field formation process. For the backscattering values observed here, and for an enforced absence of SHB, the symmetry does not break over the whole bias range considered in our simulation. Thus, SHB can be seen as a consequence of backscattering but opposes its effect. Once the asymmetric state is reached, the presence of distributed reflections is crucial, such that the counter-propagating field component stays active and can continuously contribute to the overall gain saturation. The interaction between the cw and ccw wave provides a whiff of SHB which triggers multimode operation and lowers the threshold for the Risken-Nummedal-Graham-Haken instability [3]. This effectively contributes to the formation and preservation of the soliton (see Section II.B. of the Supplemental Material). We observe in our simulations that if the backscattering becomes too small to provide the necessary energy to the counter-propagating field, the self-generated soliton will vanish, as single-mode operation becomes energetically favorable. Thus, the main advantage of the ring configuration over Fabry-Perot cavities is that the amount of backscattering, and hence SHB, can be adjusted. This is especially important in the context of soliton operation since besides its beneficial effect on multimode operation, excessive SHB induces phase and amplitude instabilities [38].
In analytical theory, the dissipative Kerr soliton is a solution to the nonlinear Schrodinger equation [20]. This solution is the reason for the sech-square shape of the pulse and demands the presence of a giant Kerr nonlinearity \(n_{2}\). The nonlinearity compresses the pulse by self-phase modulation and counteracts GVD, which broadens the pulse in the medium. If both effects balance, the field solution can propagate without changing its shape. The quantum cascade laser is known to exhibit significantly larger Kerr nonlinearities than bulk materials and features an intrinsic GVD [6; 17; 39]. For the device showing the dissipative Kerr soliton in experiment, \(n_{2}\) was estimated to be approximately \(1\times 10^{-13}\,\mathrm{m}^{2}\,\mathrm{W}^{-1}\)[9]. It is therefore a logical consequence that the ring-QCL forms soliton spectra for a suitable gain shape, even in the absence of backscattering. This is also consistent with more abstract models using the complex cubic Ginzburg-Landau equation [18]. But the consequent and repeatable vanishing of solitons that have self-emerged, was not discussed so far, to the best of our knowledge. The complete density matrix equations used here fully capture the tendency of QCLs to operate in single mode. This effect dominates over the fragile multimode balance of a generated soliton and consequently forces it to diminish. If, however, sufficient backscattering is present, the counter-propagating field may serve as a source of coherent multimode injection for sustaining the soliton, similar to the results of single-line injection in [40]. Together with the generated SHB, such a self-feeding mechanism for the off-resonant modes may suffice to overcome the single-mode tendency. Thus, the backscattering is the crucial ingredient to sustain the self-generated soliton of the ring-QCL.
## IV Conclusion
In this paper, we have shown that the experimental observation of solitons in an active ring cavity can be explained by the occurrence of distributed backscattering. By sweeping the photon-driven current in the Maxwell-Bloch simulations, the light-current characteristics of such a device and the symmetry breaking between both propagation directions can be captured. The self-consistently calculated seven-level system yields solitons in excellent agreement with experiment, regarding power, bandwidth, and duration. We have shown by measurement and simulation, that the existence of a fainter counter-propagating field induced by backscattering is crucial for the stability of a soliton. Further, we argue that in cavities with defects, SHB is present and plays a considerable role in the dynamics. These results may open the way to reliable soliton generation in ring-QCLs by custom-tailored cavity defects.
| 量子カスケードレーザーのRing型は、近年注目を集めており、超安定な周波数 comb と soliton 動作を示しており、中赤外とテラヘルツの指紋領域に統合された分光器の道を開いています。自己一致性の Maxwell-Bloch モデルにより、実験データと良好に一致する形で、反発波の分布的バック scattering に起因する対方向波の小さな但し有限な結合が、ソリトンソリューションを安定させるために必須であることを証明しています。 |
2305.10601 | Tree of Thoughts: Deliberate Problem Solving with Large Language Models | Language models are increasingly being deployed for general problem solving
across a wide range of tasks, but are still confined to token-level,
left-to-right decision-making processes during inference. This means they can
fall short in tasks that require exploration, strategic lookahead, or where
initial decisions play a pivotal role. To surmount these challenges, we
introduce a new framework for language model inference, Tree of Thoughts (ToT),
which generalizes over the popular Chain of Thought approach to prompting
language models, and enables exploration over coherent units of text (thoughts)
that serve as intermediate steps toward problem solving. ToT allows LMs to
perform deliberate decision making by considering multiple different reasoning
paths and self-evaluating choices to decide the next course of action, as well
as looking ahead or backtracking when necessary to make global choices. Our
experiments show that ToT significantly enhances language models'
problem-solving abilities on three novel tasks requiring non-trivial planning
or search: Game of 24, Creative Writing, and Mini Crosswords. For instance, in
Game of 24, while GPT-4 with chain-of-thought prompting only solved 4% of
tasks, our method achieved a success rate of 74%. Code repo with all prompts:
https://github.com/princeton-nlp/tree-of-thought-llm. | Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L. Griffiths, Yuan Cao, Karthik Narasimhan | 2023-05-17T23:16:17 | http://arxiv.org/abs/2305.10601v2 | # Tree of Thoughts: Deliberate Problem Solving
###### Abstract
Language models are increasingly being deployed for general problem solving across a wide range of tasks, but are still confined to token-level, left-to-right decision-making processes during inference. This means they can fall short in tasks that require exploration, strategic lookahead, or where initial decisions play a pivotal role. To surmount these challenges, we introduce a new framework for language model inference, "Tree of Thoughts" (ToT), which generalizes over the popular "Chain of Thought" approach to prompting language models, and enables exploration over coherent units of text ("thoughts") that serve as intermediate steps toward problem solving. ToT allows LMs to perform deliberate decision making by considering multiple different reasoning paths and self-evaluating choices to decide the next course of action, as well as looking ahead or backtracking when necessary to make global choices. Our experiments show that ToT significantly enhances language models' problem-solving abilities on three novel tasks requiring non-trivial planning or search: Game of 24, Creative Writing, and Mini Crosswords. For instance, in Game of 24, while GPT-4 with chain-of-thought prompting only solved 4% of tasks, our method achieved a success rate of 74%. Code repo with all prompts: [https://github.com/ysymyth/tree-of-thought-llm](https://github.com/ysymyth/tree-of-thought-llm).
## 1 Introduction
Originally designed to generate text, scaled-up versions of language models (LMs) such as GPT [22; 23; 1; 20] and PaLM [5] have been shown to be increasingly capable of performing an ever wider range of tasks requiring mathematical, symbolic, commonsense, and knowledge reasoning. It is perhaps surprising that underlying all this progress is still the original autoregressive mechanism for generating text, which makes token-level decisions one by one and in a left-to-right fashion. Is such a simple mechanism sufficient for a LM to be built toward a general problem solver? If not, what problems would challenge the current paradigm, and what should be alternative mechanisms?
The literature on human cognition provides some clues to answer these questions. Research on "dual process" models suggests that people have two modes in which they engage with decisions - a fast, automatic, unconscious mode ("System 1") and a slow, deliberate, conscious mode ("System 2") [27; 28; 13; 12]. These two modes have previously been connected to a variety of mathematical models used in machine learning. For example, research on reinforcement learning in humans and other animals has explored the circumstances under which they engage in associative "model free" learning or more deliberative "model based" planning [6]. The simple associative token-level choices of LMs are also reminiscent of "System 1", and thus might benefit from augmentation by a more deliberate "System 2" planning process that (1) maintains and explores diverse alternatives for current
choices instead of just picking one, and (2) evaluates its current status and actively looks ahead or backtracks to make more global decisions.
To design such a planning process, we return to the origins of artificial intelligence (and cognitive science), drawing inspiration from the planning processes explored by Newell, Shaw, and Simon starting in the 1950s [18; 19]. Newell and colleagues characterized problem solving [18] as search through a combinatorial problem space, represented as a tree. We thus propose the Tree of Thoughts (ToT) framework for general problem solving with language models. As Figure 1 illustrates, while existing methods (detailed below) sample continuous language sequences for problem solving, ToT actively maintains a tree of thoughts, where each _thought_ is a coherent language sequence that serves as an intermediate step toward problem solving (Table 1). Such a high-level semantic unit allows the LM to self-evaluate the progress different intermediate thoughts make towards solving the problem through a deliberate reasoning process that is also instantiated in language (Figures 2,4,6). This implementation of search heuristics via LM self-evaluation and deliberation is novel, as previous search heuristics are either programmed or learned. Finally, we combine this language-based capability to generate and evaluate diverse thoughts with search algorithms, such as breadth-first search (BFS) or depth-first search (DFS), which allow systematic exploration of the tree of thoughts with lookahead and backtracking.
Empirically, we propose three new problems that challenge existing LM inference methods even with the state-of-the-art language model, GPT-4 [20]: Game of 24, Creative Writing, and Crosswords (Table 1). These tasks require deductive, mathematical, commonsense, lexical reasoning abilities, and a way to incorporate systematic planning or search. We show ToT obtains superior results on all three tasks by being general and flexible enough to support different levels of thoughts, different ways to generate and evaluate thoughts, and different search algorithms that adapt to the nature of different problems. We also analyze how such choices affect model performances via systematic ablations and discuss future directions to better train and use LMs.
## 2 Background
We first formalize some existing methods that use large language models for problem-solving, which our approach is inspired by and later compared with. We use \(p_{\theta}\) to denote a pre-trained LM with parameters \(\theta\), and **lowercase letters**\(x,y,z,s,\cdots\)**to denote a language sequence**, i.e. \(x=(x[1],\cdots,x[n])\) where each \(x[i]\) is a token, so that \(p_{\theta}(x)=\prod_{i=1}^{n}p_{\theta}(x[i]|x[1...i])\). We use uppercase letters \(S,\cdots\) to denote a collection of language sequences.
**Input-output (IO) prompting** is the most common way to turn a problem input \(x\) into output \(y\) with LM: \(y\sim p_{\theta}(y|\texttt{prompt}_{IO}(x))\), where \(\texttt{prompt}_{IO}(x)\) wraps input \(x\) with task instructions and/or few-shot input-output examples. For simplicity, let us denote \(p_{\theta}^{\text{prompt}}(\texttt{output}\mid\texttt{input})=p_{\theta}( \texttt{output}\mid\texttt{prompt}(\texttt{input}))\), so that IO prompting can be formulated as \(y\sim p_{\theta}^{IO}(y|x)\).
Figure 1: Schematic illustrating various approaches to problem solving with LLMs. Each rectangle box represents a _thought_, which is a coherent language sequence that serves as an intermediate step toward problem solving. See concrete examples of how thoughts are generated, evaluated, and searched in Figures 2,4,6.
**Chain-of-though (CoT) prompting**[35] was proposed to address cases where the mapping of input \(x\) to output \(y\) is non-trivial (e.g. when \(x\) is a math question and \(y\) is the final numerical answer). The key idea is to introduce a chain of _thoughts_\(z_{1},\cdots,z_{n}\) to bridge \(x\) and \(y\), where each \(z_{i}\) is a coherent language sequence that serves as a meaningful intermediate step toward problem solving (e.g. \(z_{i}\) could be an intermediate equation for math QA). To solve problems with CoT, each thought \(z_{i}\sim p_{\theta}^{CoT}(z_{i}\mid x,z_{1\cdots i-1})\) is sampled sequentially, then the output \(y\sim p_{\theta}^{CoT}(y|x,z_{1\cdots n})\). In practice, \([z_{1\cdots n},y]\sim p_{\theta}^{CoT}(z_{1\cdots n},y|x)\) is sampled as a continuous language sequence, and the **decomposition** of thoughts (e.g. is each \(z_{i}\) a phrase, a sentence, or a paragraph) is left ambiguous.
**Self-consistency with CoT (CoT-SC)**[33] is an ensemble approach that samples \(k\) i.i.d. chains of thought: \([z_{1\cdots n}^{(i)},y^{(i)}]\sim p_{\theta}^{CoT}(z_{1\cdots n},y|x)\)\((i=1\cdots k)\), then returns the most frequent output: \(\operatorname*{arg\,max}_{y}\#\{i\mid y^{(i)}=y\}\). CoT-SC improves upon CoT, because there are generally different thought processes for the same problem (e.g. different ways to prove the same theorem), and the output decision can be more faithful by exploring a richer set of thoughts. However, within each chain there is no local exploration of different thought steps, and the "most frequent" heuristic only applies when the output space is limited (e.g. multi-choice QA).
## 3 Tree of Thoughts: Deliberate Problem Solving with LM
_A genuine problem-solving process involves the repeated use of available information to initiate exploration, which discloses, in turn, more information until a way to attain the solution is finally discovered._--_Newell et al._[18]
Research on human problem-solving suggests that people search through a combinatorial problem-space - a tree where the nodes represent partial solutions, and the branches correspond to operators that modify them [18, 19]. Which branch to take is determined by heuristics that help to navigate the problem-space and guide the problem-solver towards a solution. This perspective highlights two key shortcomings of existing approaches that use LMs to solve general problems: 1) Locally, they do not explore _different_ continuations within a thought process - the branches of the tree. 2) Globally, they do not incorporate any type of planning, lookahead, or backtracking to help evaluate these different options - the kind of heuristic-guided search that seems characteristic of human problem-solving.
To address these shortcomings, we introduce _Tree of Thoughts (ToT)_, a paradigm that allows LMs to explore multiple reasoning paths over thoughts (Figure 1(c)). ToT frames any problem as a search over a tree, where each node is a **state**\(s=[x,z_{1\cdots i}]\) representing a partial solution with the input and the sequence of thoughts so far. A specific instantiation of ToT involves answering four questions: 1. How to **decompose** the intermediate process into thought steps; 2. How to **generate** potential thoughts from each state; 3. How to heuristically **evaluate** states; 4. What **search** algorithm to use.
**1. Thought decomposition.** While CoT samples thoughts coherently without explicit decomposition, ToT leverages problem properties to design and decompose intermediate thought steps. As Table 1 shows, depending on different problems, a thought could be a couple of words (Crosswords), a line of equation (Game of 24), or a whole paragraph of writing plan (Creative Writing). In general, a thought should be "small" enough so that LMs can generate promising and diverse samples (e.g. generating a whole book is usually too "big" to be coherent), yet "big" enough so that LMs can evaluate its prospect toward problem solving (e.g. generating one token is usually too "small" to evaluate).
**2. Thought generator \(G(p_{\theta},s,k)\).** Given a tree state \(s=[x,z_{1\cdots i}]\), we consider two strategies to generate \(k\) candidates for the next thought step:
1. **Sample** i.i.d. thoughts from a CoT prompt (Creative Writing, Figure 4): \(z^{(j)}\sim p_{\theta}^{CoT}(z_{i+1}|s)=p_{\theta}^{CoT}(z_{i+1}|x,z_{1\cdots i })\)\((j=1\cdots k)\). This works better when the thought space is rich (e.g. each thought is a paragraph), and i.i.d. samples lead to diversity;
2. **Propose** thoughts sequentially using a "propose prompt" (Game of 24, Figure 2; Crosswords, Figure 6): \([z^{(1)},\cdots,z^{(k)}]\sim p_{\theta}^{propose}(z_{i+1}^{(1\cdots k)}\mid s)\). This works better when the thought space is more constrained (e.g. each thought is just a word or a line), so proposing different thoughts in the same context avoids duplication.
**3. State evaluator \(V(p_{\theta},S)\).** Given a frontier of different states, the state evaluator evaluates the progress they make towards solving the problem, serving as a _heuristic_ for the search algorithm to determine which states to keep exploring and in which order. While heuristics are a standard approach to solving search problems, they are typically either programmed (e.g. DeepBlue [3]) or
learned (e.g. AlphaGo [26]). We propose a third alternative, by using the LM to deliberately reason about states. When applicable, such a deliberate heuristic can be more flexible than programmed rules, and more sample-efficient than learned models. Similar to the thought generator, we consider two strategies to evaluate states either independently or together:
1. **Value** each state independently: \(V(p_{\theta},S)(s)\sim p_{\theta}^{value}(v|s)\)\(\forall s\in S\), where a value prompt reasons about the state \(s\) to generate a scalar value \(v\) (e.g. 1-10) or a classification (e.g. sure/likely/impossible) that could be heuristically turned into a value. The basis of such evaluative reasoning can vary across problems and thought steps. In this work, we explore evaluation via few _lookahead_ simulations (e.g. quickly confirm that 5, 5, 14 can reach 24 via 5 + 5 + 14, or "hot_l" can mean "inn" via filling "e" in ".") plus commonsense (e.g. 1 2 3 are too small to reach 24, or no word can start with "tzxc"). While the former might promote "good" states, the latter could help eliminate "bad" states. Such valuations do not need to be perfect, and only need to be approximately
2. **Vote** across states: \(V(p_{\theta},S)(s)=\mathds{1}[s=s^{*}]\), where a "good" state \(s^{*}\sim p_{\theta}^{vote}(s^{*}|S)\) is voted out based on deliberately comparing different states in \(S\) in a vote prompt. When problem success is harder to directly value (e.g. passage coherency), it is natural to to instead compare different partial solutions and vote for the most promising one. This is similar in spirit to a "step-wise" self-consistency strategy, i.e. cast "which state to explore" as a multi-choice QA, and use LM samples to vote for it.
For both strategies, we could prompt the LM multiple times to aggregate the value or vote results to trade time/resource/cost for more faithful/robust heuristics.
```
Input \(x\), LM \(p_{\theta}\), thought generator \(G()\) & size limit \(k\), states evaluator \(V()\), step limit \(T\), breadth limit \(b\). \(S_{0}\leftarrow\{x\}\) for\(t=1,\cdots,T\)do \(S^{\prime}_{t}\leftarrow\{[s,z]\mid s\in S_{t-1},z_{t}\in\mathrm{G}(p_{\theta},s,k)\}\) \(V_{t}\gets V(p_{\theta},S^{\prime}_{t})\) \(S_{t}\leftarrow\operatorname*{arg\,max}_{S\subset S^{\prime}_{t},|S|=b}\sum_{ s\in S}V_{t}(s)\) endfor return\(G(p_{\theta},\operatorname*{arg\,max}_{s\in S_{T}}V_{T}(s),1)\)
```
**Algorithm 1** ToT-BFS(\(x,p_{\theta},G,k,V,T,b\))
**Algorithm 2** ToT-DFS(\(s,t,p_{\theta},G,k,V,T,v_{th}\))
**4. Search algorithm.** Finally, within the ToT framework, one can plug and play different search algorithms depending on the tree structure. We explore two relatively simple search algorithms and leave more advanced ones (e.g. A* [9], MCTS [2]) for future work:
1. **Breadth-first search (BFS)** (Algorithm 1) maintains a set of the \(b\) most promising states per step. This is used for Game of 24 and Creative Writing where the tree depth is limit (\(T\leq 3\)), and initial thought steps can be evaluated and pruned to a small set (\(b\leq 5\)).
2. **Depth-first search (DFS)** (Algorithm 2) explores the most promising state first, until the final output is reached (\(t>T\)), or the state evaluator deems it impossible to solve the problem from the current \(s\) (\(V(p_{\theta},\{s\})(s)\leq v_{th}\) for a value threshold \(v_{th}\)). In the latter case, the subtree from \(s\) is pruned to trade exploration for exploitation. In both cases, DFS _backtracks_ to the parent state of \(s\) to continue exploration.
Conceptually, ToT has several benefits as a method for general problem-solving with LMs: (1) Generality. IO, CoT, CoT-SC, and self-refinement can be seen as special cases of ToT (i.e. trees of limited depth and breadth; Figure 1). (2) Modularity. The base LM, as well as the thought decomposition, generation, evaluation, and search procedures can all be varied independently. (3) Adaptability. Different problem properties, LM capabilities, and resource constraints can be accommodated. (4) Convenience. No extra training is needed, just a pre-trained LM is sufficient. The next section will show how these conceptual benefits translate to strong empirical performance in different problems.
## 4 Experiments
We propose three tasks that are hard even when sampling from the state-of-the-art language model, GPT-4 [20], using standard IO prompting or chain-of-thought (CoT) prompting. We show how
deliberate search in trees of thoughts (ToT) produces better results, and more importantly, interesting and promising new ways to use language models to solve problems requiring search or planning. Unless otherwise stated, we perform experiments using a Chat Completion mode GPT-41 with a sampling temperature of 0.7.
Footnote 1: Experiments were done between May 5-16, 2023.
### Game of 24
Game of 24 is a mathematical reasoning challenge, where the goal is to use 4 numbers and basic arithmetic operations (+-*/) to obtain 24. For example, given input "4 9 10 13", a solution output could be "(10 - 4) * (13 - 9) = 24".
**Task Setup.** We scrape data from 4nums.com, which has 1,362 games that are sorted from easy to hard by human solving time, and use a subset of relatively hard games indexed 901-1,000 for testing. For each task, we consider the output as success if it is a valid equation that equals 24 and uses the input numbers each exactly once. We report the success rate across 100 games as the metric.
**Baselines.** We use a standard input-output (IO) prompt with 5 in-context examples. For chain-of-thought (CoT) prompting, we augment each input-output pair with 3 intermediate equations, each operating on two remaining numbers. For example, given input "4 9 10 13", the thoughts could be "13 - 9 = 4 (left: 4 10); 10 - 4 = 6 (left: 4 6); 4 * 6 = 24 (left: 24)". For each game, we sample IO and CoT prompting for 100 times for average performance. We also consider a CoT self-consistency baseline, which takes the majority output from 100 CoT samples, and an iterative-refine approach on top of an IO sample for at most \(10\) iterations. At each iteration, the LM is conditioned on all previous history to "reflect on your mistakes and generate a refined answer" if the output is incorrect. Note that it uses groundtruth feedback signals about equation correctness.
**ToT Setup.** To frame Game of 24 into ToT, it is natural to decompose the thoughts into 3 steps, each an intermediate equation. As shown in Figure 2(a), at each tree node, we exact the "left" numbers and prompt the LM to propose some possible next steps. The same "propose prompt" is used for all 3 thought steps, though it only has one example with 4 input numbers. We perform a breadth-first search (BFS) in ToT, where at each step we keep the best \(b=5\) candidates. To perform deliberate BFS in ToT, as shown in Figure 2(b), we prompt LM to evaluate each thought candidate as "sure/maybe/impossible" with regard to reaching 24. The aim is to promote correct partial solutions that can be verdicted within few lookahead trials, and eliminate impossible partial solutions based on "too big/small" commonsense, and keep the rest "maybe". We sample values \(3\) times for each thought.
\begin{table}
\begin{tabular}{l|l l l} \hline \hline & **Game of 24** & **Creative Writing** & **5x5 Crosswords** \\ \hline
**Input** & 4 numbers (4 9 10 13) & 4 random sentences & 10 clues (h1.presented;..) \\ \hline
**Output** & An equation to reach 24 (13-9)*(10-4)=24 & A passage of 4 paragraphs ending in the 4 sentences & 5x5 letters: SHOWN; WIRRA; AVAIL;... \\ \hline
**Thoughs** & 3 intermediate equations (13-9=4 (left 4,4,10); 10-4=6 (left 4,6); 4*6=24) & A short writing plan (1. Introduce a book that connects...) & Words to fill in for clues: (h1.shown; v5. naled;...) \\ \hline
**\#ToT steps** & 3 & 1 & 5-10 (variable) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Task overview. Input, output, thought examples are in blue.
Figure 2: ToT in a game of 24. The LM is prompted for (a) thought generation and (b) valuation.
**Results.** As shown in Table 2, IO, CoT, and CoT-SC prompting methods perform badly on the task, achieving only 7.3%, 4.0%, and 9.0% success rates. In contrast, ToT with a breadth of \(b=1\) already achieves a success rate of \(45\%\), while \(b=5\) achieves \(74\%\). We also consider an oracle setup for IO/CoT, by calculating the success rate using best of \(k\) samples (\(1\leq k\leq 100\)). To compare IO/CoT (best of k) with ToT, we consider calculating the tree nodes visited per task in ToT across \(b=1\cdots 5\), and map the 5 success rates in Figure 3(a), treating IO/CoT (best of \(k\)) as visiting \(k\) nodes in a bandit. Not surprisingly, CoT scales better than IO, and best of 100 CoT samples achieve a success rate of \(49\%\), but still much worse than exploring more nodes in ToT (\(b>1\)).
**Error Analysis.** Figure 3(b) breaks down at which step CoT and ToT samples fail the task, i.e. the thought (in CoT) or all \(b\) thoughts (in ToT) are invalid or impossible to reach 24. Notably, around 60% of CoT samples already failed the task after generating the first step, or equivalently, the first three words (e.g. "\(4+9\)"). This highlights the issues with direct left-to-right decoding.
### Creative writing
Next, we invent a creative writing task where the input is 4 random sentences and the output should be a coherent passage with 4 paragraphs that end in the 4 input sentences respectively. Such a task is open-ended and exploratory, and challenges creative thinking as well as high-level planning.
**Task setup.** We sample random sentences from randomwordgenerator.com to form 100 inputs, and there is no groundtruth passage for each input constraint. As we find that GPT-4 can follow the input constraints most of the time, we focus on evaluating passage coherency in two ways: using a GPT-4 zero-shot prompt to provide a 1-10 scalar score, or using human judgments to compare pairs of outputs from different methods. For the former, we sample 5 scores and average them for each task output, and we find these 5 scores usually consistent, with a standard deviation of around \(0.56\) on average across outputs. For the latter, we employ a subset of the authors in a blind study to compare the coherency of CoT vs. ToT generated passage pairs, where the order of passages is random flipped over 100 inputs.
**Baselines.** Given the creative nature of the task, both IO and CoT prompts are zero-shot. While the former prompts the LM to directly generate a coherent passage given input constraints, the latter prompts the LM to first make a brief plan then write the passage, i.e. the plan serves as the intermediate thought step. We generate 10 IO and CoT samples per task. We also consider an iterative-refine (\(k\leq 5\)) method on top of a random IO sample for each task, where the LM is conditioned on input constraints and the last generated passage to decide if the passage is already "perfectly coherent", and if not generate a refined one.
**ToT setup.** We build a ToT with depth 2 (and only 1 intermediate thought step) -- the LM first generates \(k=5\) plans and votes for the best one (Figure 4), then similarly generate \(k=5\) passages based on the best plan then vote for the best one. Here the breadth limit \(b=1\), as only one choice is kept per step. A simple zero-shot vote prompt ("analyze choices below, then conclude which is most promising for the instruction") is used to sample 5 votes at both steps.
**Results.** Figure 5(a) shows average GPT-4 scores across 100 tasks, where ToT (7.56) is deemed to generate more coherent passages than IO (6.19) and CoT (6.93) on average. While such an automatic metric might be noisy, Figure 5(b) confirms the finding by showing that humans prefer ToT over CoT in 41 out of 100 passage pairs, while only prefer CoT over ToT in 21 (other 38 pairs are found "similarly coherent"). Lastly, iterative-refine is more effective on this natural language task, where
\begin{table}
\begin{tabular}{l l} \hline \hline
**Method** & **Success** \\ \hline IO prompt & 7.3\% \\ CoT prompt & 4.0\% \\ CoT-SC (k=100) & 9.0\% \\ ToT (ours) (b=1) & 45\% \\ ToT (ours) (b=5) & **74\%** \\ \hline IO + Refine (k=10) & 27\% \\ IO (best of 100) & 33\% \\ CoT (best of 100) & 49\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Game of 24 Results. Figure 3: Game of 24 (a) scale analysis & (b) error analysis.
it improves IO coherency score from 6.19 to 7.67, and ToT coherency score from 7.56 to 7.91. We believe it could be thought of as a third approach to thought generation in the ToT framework, where new thoughts can arise from refining old thoughts instead of i.i.d. or sequentially generated.
### Mini Crosswords
In Game of 24 and Creative Writing, ToT is relatively shallow -- at most 3 thought steps are needed to reach the final output. Here we explore \(5\times 5\) mini crosswords as a harder search problem involving natural language. Again, the goal is not just to solve the task, as more general crosswords can be readily solved with specialized NLP pipelines [31] that leverages large-scale retrieval instead of LM. Rather, we aim to explore the limit of LM as a general problem solver that explores its own thoughts and guides its own exploration with deliberate reasoning as heuristics.
**Task Setup.** We scrape data from GooBix, which contains 156 games of \(5\times 5\) mini crosswords. As we observe adjacent games contain similar clues, we use 20 games with indices \(1,6,\cdots,91,96\) for testing, and games \(136,141,146,151,156\) for prompting. For each task, the input describes the 5 horizontal clues and 5 vertical clues, and the output should be a board of \(5\times 5=25\) letters to solve the crosswords. For evaluation, we consider three levels of success: the portion of correct letters (25 per game), words (10 per game), and games.
**Baselines.** We provide 5 example input-output pairs in the IO prompt, and in the CoT prompt additionally include intermediate words in the order h1..5 then v1..5. We run each prompt for 10 samples and average the results.
**ToT Setup.** We leverage a depth-first search (Algorithm 2) that keeps exploring the most promising subsequent word clue until the state is no longer promising, then backtrack to the parent state to explore alternative thoughts. To make search tractable, subsequent thoughts are constrained not to change any filled words or letters, so that the ToT has at most 10 intermediate steps. For thought generation, at each state we translate all existing thoughts (e.g. "h2.motor; h1.tasks" for the state in Figure 6(a)) into letter constraints for remaining clues (e.g. "v1.To heap: tm_,...;...") and prompt a proposal prompt \(5\) times to come up with candidates for where and what to fill in the next word. Importantly, we also prompt the LM to give a confidence level for different thoughts, and aggregate
Figure 4: A step of deliberate search in a randomly picked Creative Writing task. Given the input, the LM samples 5 different plans, then votes 5 times to decide which plan is best. The majority choice is used to consequently write the output passage with the same sample-vote procedure.
Figure 5: Creative Writing results.
\begin{table}
\begin{tabular}{l|l l l} \hline \hline
**Method** & \multicolumn{3}{l}{**Success Rate (\%)**} \\ & \multicolumn{1}{l}{**Letter Word**} & \multicolumn{1}{l}{**Game**} \\ \hline IO & 38.7 & 14 & 0 \\ CoT & 40.6 & 15.6 & 1 \\ ToT (ours) & **78** & **60** & **20** \\ \hline +best state & 82.4 & 67.5 & 35 \\ -prune & 65.4 & 41.5 & 5 \\ -backtrack & 54.6 & 20 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mini Crosswords results.
these across proposals to obtain a sorted list of next thoughts to explore (Figure 6(a)). For state evaluations, we similarly translate each state into letter constraints for remaining clues, then evaluate for each clue if it is possible to fill given the constraints. If any remaining clue is deemed "impossible" to fill in (e.g. "v1. To heap: tm_s_"), then the exploration of the state's subtree is pruned and DFS backtracks to its parent to explore the next promising thought. We limit DFS search steps to 100, and simply render the deepest explored state (the first explored one if multiple) into the final output.
**Results.** As shown in Table 3, IO and CoT prompting methods perform poorly with a word-level success rate less than \(16\%\), while ToT significantly improves all metrics, achieving a word-level success rate of \(60\%\) and solving 4 out of 20 games. Such an improvement is not surprising, given IO and CoT lack mechanisms to try different clues, make changes to decisions, or backtrack.
**Oracle and ablation studies.** When outputting from the oracle best DFS state (instead of the heuristically determined best state) per task, ToT performance is even higher and actually solves 7/20 games (Table 3, "+best state"), indicating our simple output heuristics can be readily improved. Interestingly, sometimes when the crosswords game is actually solved, the state evaluator might still deem some words as "impossible" and prune -- possibly because \(5\times 5\) crosswords by design have some rare or obselete words that GPT-4 cannot recognize2. Given the state evaluation as a pruning heuristic is imperfect, we also explore ablating the pruning, and find the performance generally worse (Table 3, "-prune"). However, it could actually find the correct solution for 4/20 games (though only outputting 1 via heuristic), 3 of which are games ToT+pruning cannot solve within 100 steps. Thus, better heuristics for DFS pruning are critical for problem solving in this case. Lastly, we confirm the importance of backtracking by running an ablation that keeps filling the most promising clue for at most 20 steps, allowing overwrites. This is similar to a "greedy" BFS search with breadth limit of \(b=1\), and performs poorly with a word level success of only \(20\%\) (Table 3, "-backtrack").
Footnote 2: For example, “agent” is an obsolete form of “agentum”, but GPT-4 deems it a typo for “agenda”. External retrieval or web interaction could augment LM for problem solving under knowledge uncertainty.
## 5 Related Work
**Planning and decision making.** Smart planning and decision making are critical to achieving predefined goals. As they are trained on vast amount of world knowledge and human examples, LMs are known to have already absorbed rich commonsense that makes it possible to propose reasonable plans conditioned on problem setting and environmental states [10; 39; 34; 11; 32; 38; 37]. Our proposed Tree-of-Thought approach extends existing planning formulations by considering multiple potentially feasible plans simultaneously at each problem-solving step, and proceeding with the most promising ones. The integration between thought sampling and value feedback organically integrates planning and decision-making mechanisms, enabling effective search inside a solution tree. On the other hand, traditional decision-making procedures usually require training dedicated reward and policy models as in reinforcement learning (for example CHAI [30]), whereas we use the LM itself to provide the value estimates for decision making.
Figure 6: In Mini Crosswords, (a) how thoughts are proposed and aggregated in a priority queue for depth-first search (DFS), and (b) how a state is evaluated based on the possibility of filling in each remaining word clue, and pruned if any remaining clue is deemed not possible to fill by the LM. Then DFS backtracks to the parent state and explore the next promising thought for clue.
**Self-reflection.** Using LLMs to assess the viability of their own predictions is becoming an increasingly important procedure in problem solving. [25; 17; 21] introduced the "self-reflection" mechanism, in which LMs provide feedback to their generation candidates. [4] improves LMs code generation accuracy by injecting feedback messages generated by the LM itself based on its code execution results. Similarly, [14] also introduces "critic" or review steps over the actions and states, deciding the next action to take in solving computer operation tasks. Another recent work very relevant to ours is "self-eval guided decoding" [36]. Similar to our method, self-eval decoding also follows a tree-search procedure with leaves sampled from stochastic beam search decoding, which are then evaluated by LLM itself with carefully prepared self-eval prompts. Their approach however, uses the PAL formulation [7] which represents thoughts as codes, which makes it difficult to tackle challenging tasks like creative writing which we consider in this paper. Our Tree-of-Thought formulation is thus more versatile and handles challenging tasks on which GPT-4 only achieves very low accuracy with standard prompts.
**Program-guided LLM generation.** Our proposal is also related to recent advancements that organize LM's behavior with symbolic program guidance. For example [24] embeds LMs in an algorithmic search procedure to help solve problems like question answering step-by-step, in which the search trees are expanded by relevant paragraphs that might provide answers. This approach however differs from ours in that trees are expanded by sampling external paragraphs instead of the LM's own thoughts, and there is no reflection or voting steps. Another approach, LLM+P [15], goes one step further and delegates the actual planning process to a classical planner.
**Classical search methods.** Last but not least, our approach can be treated as a modern rendition of classical search methods for problem solving. For example it can be considered as a heuristic search algorithm like A* [8], in which the heuristic at each search node is provided by the LM's self-assessment. From this perspective, our method is also related to NeuroLogic A*esque decoding proposed in [16], which is inspired by A* search but introduces look-ahead heuristics that are efficient for LMs to improve the beam-search or top-k sampling decoding. This method however is constrained to sentence generation tasks, whereas our framework are designed for complex, multi-step problem solving guarded by value feedback.
## 6 Discussion
**Limitations and future directions.** Deliberate search such as ToT might not be necessary for many existing tasks that GPT-4 already excels at, and as an initial step this work only explores three relatively simple tasks that challenges GPT-4 and calls of better search and planning abilities incorporated with LMs. However, as we begin to deploy LMs for more real-world decision making applications (e.g. coding, data analysis, robotics, etc.), more complex tasks could emerge and present new opportunities to study these research questions. Also, search methods like ToT requires more resources (e.g. GPT-4 API cost) than sampling methods in order to improve task performances, but the modular flexibility of ToT allows users to customize such performance-cost tradeoffs, and ongoing open-source efforts [29] should readily reduce such costs in the near future. Lastly, this work focuses on using an off-the-shelf LM, and fine-tuning LMs using a ToT-style high-level counterfactual decision making (e.g. deliberating over potential choices for the next paragraph, instead of predicting the next token) might present opportunities to enhance the problem-solving capabilities of LMs.
**Broader impact.** ToT is a framework that empowers LMs to more autonomously and intelligently make decisions and solve problems. While current tasks are limited to reasoning and search problems, future applications involving interaction with external environments or humans could bring potential danger, e.g. facilitating harmful uses of LMs. On the other hand, ToT also improves the interpretability of model decisions and the opportunity for human alignment, as the resulting representations are readable, high-level language reasoning instead of implicit, low-level token values.
**Conclusion.** The associative "System 1" of LMs can be beneficially augmented by a "System 2" based on searching a tree of possible paths to the solution to a problem. The Tree of Thoughts framework provides a way to translate classical insights about problem-solving into actionable methods for contemporary LMs. At the same time, LMs address a weakness of these classical methods, providing a way to solve complex problems that are not easily formalized, such as creative writing. We see this intersection of LMs with classical approaches to AI as an exciting direction for future work. | 言語モデルは、幅広いタスクに適用されることが増えているものの、推論時にはトークンレベル、左から右の決定プロセスに留まっています。これは、探索、戦略的lookahead、初期の決断が重要な役割を果たすタスクでは、課題に達することができないことを意味します。これらの課題に対処するために、私たちは、言語モデル推論のための新しいフレームワークである「Tree of Thoughts (ToT)」を導入します。これは、言語モデルの誘導のための流行の「Chain of Thought」アプローチを一般化し、テキストのcoherentな単位（考え）を介した探索を可能にし、問題解決の段階的なステップとなります。ToTは、複数の異なる論理パスを考慮し、選択の自己評価を通して、言語モデルが意思決定を決定できるようになり、必要に応じて、選択した行動を決定したり、バックトラックしたりします。私たちの研究は、複雑な計画や検索を必要とする3つの新しい |
2306.09600 | Learning to Assist and Communicate with Novice Drone Pilots for Expert
Level Performance | Multi-task missions for unmanned aerial vehicles (UAVs) involving inspection
and landing tasks are challenging for novice pilots due to the difficulties
associated with depth perception and the control interface. We propose a shared
autonomy system, alongside supplementary information displays, to assist pilots
to successfully complete multi-task missions without any pilot training. Our
approach comprises of three modules: (1) a perception module that encodes
visual information onto a latent representation, (2) a policy module that
augments pilot's actions, and (3) an information augmentation module that
provides additional information to the pilot. The policy module is trained in
simulation with simulated users and transferred to the real world without
modification in a user study (n=29), alongside supplementary information
schemes including learnt red/green light feedback cues and an augmented reality
display. The pilot's intent is unknown to the policy module and is inferred
from the pilot's input and UAV's states. The assistant increased task success
rate for the landing and inspection tasks from [16.67% & 54.29%] respectively
to [95.59% & 96.22%]. With the assistant, inexperienced pilots achieved similar
performance to experienced pilots. Red/green light feedback cues reduced the
required time by 19.53% and trajectory length by 17.86% for the inspection
task, where participants rated it as their preferred condition due to the
intuitive interface and providing reassurance. This work demonstrates that
simple user models can train shared autonomy systems in simulation, and
transfer to physical tasks to estimate user intent and provide effective
assistance and information to the pilot. | Kal Backman, Dana Kulić, Hoam Chung | 2023-06-16T02:59:20 | http://arxiv.org/abs/2306.09600v1 | # Learning to Assist and Communicate with Novice Drone Pilots for Expert Level Performance
###### Abstract
Multi-task missions for unmanned aerial vehicles (UAVs) involving inspection and landing tasks are challenging for novice pilots due to the difficulties associated with depth perception and the control interface. We propose a shared autonomy system, alongside supplementary information displays, to assist pilots to successfully complete multi-task missions without any pilot training. Our approach comprises of three modules: (1) a perception module that encodes visual information onto a latent representation, (2) a policy module that augments pilot's actions, and (3) an information augmentation module that provides additional information to the pilot. The policy module is trained in simulation with simulated users and transferred to the real world without modification in a user study (\(\mathrm{n}=29\)), alongside supplementary information schemes including learnt red/green light feedback cues and an augmented reality display. The pilot's intent is unknown to the policy module and is inferred from the pilot's input and UAV's states. The assistant increased task success rate for the landing and inspection tasks from [16.67% & 54.29%] respectively to [95.59% & 96.22%]. With the assistant, inexperienced pilots achieved similar performance to experienced pilots. Red/green light feedback cues reduced the required time by 19.53% and trajectory length by 17.86% for the inspection task, where participants rated it as their preferred condition due to the intuitive interface and providing reassurance. This work demonstrates that simple user models can train shared autonomy systems in simulation, and transfer to physical tasks to estimate user intent and provide effective assistance and information to the pilot.
Cognitive Human-Robot interaction, Deep Learning in Robotics and Automation, Aerial Systems: Perception and Autonomy, Shared Autonomy
## I Introduction
Unmanned aerial vehicles (UAVs) are renowned for their mobility, often deployed in search and rescue [1, 2, 3] and inspection [4, 5, 6] related tasks due to their ability to manoeuvre in full 3D space. However this manoeuvrability comes at a cost of increased teleoperation complexity due to difficulties associated with pilots' relative depth perception of nearby objects [7, 8] and control input mapping to relative UAV dynamic state changes. Due to these challenges it is difficult for novice pilots to successfully complete the aforementioned tasks.
Autonomous solutions have been proposed for such tasks [4, 5, 6], however often require that the structure of the environment be known a priori, or contain a set of fixed, known mission objectives. The main limitation of fully autonomous solutions is their inability to dynamically adapt their objective in response to external stimuli within the environment that is not predefined by their developers due to the associated difficulty of replicating high-level human decision making and general artificial intelligence [9, 10]. Therefore teleoperation control schemes are preferred in real-life UAV operations over fully autonomous solutions [11] to take advantage of high-level human decision making, despite the requirement of expert pilots.
Assistance strategies include the use of shared autonomy, which combines the control inputs of human pilots with that of artificial intelligence to collaboratively complete a set of objectives. Three main challenges arise when developing shared autonomy systems: inferring the intent of the user, providing control outputs to complete the inferred objective and deciding how and what information should be communicated back to the user. The first challenge, inferring the intent of the user, is performed by observing the user's actions within the context of the observable environment. Although inferring intent implicitly poses the risk of incorrect goal estimation leading to a misalignment of objectives between the AI and user, users often prefer implicit intent estimation methods due to their intuitiveness and reduction in cognitive workload [8, 12].
For the second challenge, the automated assistant must deliver its control outputs considering its uncertainty about the user's intent. Acting too early risks taking an incorrect action not aligned with the user, while waiting to build sufficient confidence in the user's intent before acting can lead to delayed assistance and task failure. Further issues arise with how much control should the assistant exert over the system. Providing insufficient assistance can lead to task failure while excessive control deteriorates team effectiveness in collaborative tasks [13].
For the third challenge, communication feedback promotes transparency of the shared autonomy system, providing increased observability and predictability of system behaviour [14]. However developing natural feedback communication channels that do not hinder a user's control input capabilities, prevent loss of focus from context switching and are designed for environments with high auditory noise is difficult.
Prior works on UAV systems focus on autonomous landing [15, 16, 17, 18] or inspection [6, 19, 20, 21] tasks which rely on predefined mission objectives. Of the limited shared autonomy works, none yet deal with multi-task missions containing multiple ambiguous goals and instead focus on single goal inspection [22] or landing [23] tasks, or are restricted to obstacle avoidance for use in inspection tasks [24, 11]. Our prior work [8, 25] proposed a shared autonomy solution capable of providing assistance under ambiguity of the pilot's | 無人機（UAV）の多任務ミッション（inspectionとlandingミッション）は、初級のパイロットにとって、深度認識とコントロールインターフェースの困難さのために、挑戦的です。私たちは、パイロットがマルチタスクミッションを成功させるためのサポートツールとして、共有自律システムと補完情報表示を提案しました。私たちの方法は、3つのモジュールから構成されます。(1)視覚情報を入力したラテン空間表現をエンコードする感知モジュール、 (2)パイロットの行動を拡張するポリシーモジュール、 (3)パイロットに補完情報を提供する情報拡張モジュールです。ポリシーモジュールは、シミュレーション環境でシミュレートされたユーザーにトレーニングされ、リアルワールドへの移行は、ユーザー調査（n=29）において変更なしで実施されました。また、学習された赤・青色の光フィードバックのスキームと拡張現実ディスプレイも併用して使用しました。 |
2310.05904 | On Multi-Fidelity Impedance Tuning for Human-Robot Cooperative
Manipulation | We examine how a human-robot interaction (HRI) system may be designed when
input-output data from previous experiments are available. In particular, we
consider how to select an optimal impedance in the assistance design for a
cooperative manipulation task with a new operator. Due to the variability
between individuals, the design parameters that best suit one operator of the
robot may not be the best parameters for another one. However, by incorporating
historical data using a linear auto-regressive (AR-1) Gaussian process, the
search for a new operator's optimal parameters can be accelerated. We lay out a
framework for optimizing the human-robot cooperative manipulation that only
requires input-output data. We establish how the AR-1 model improves the bound
on the regret and numerically simulate a human-robot cooperative manipulation
task to show the regret improvement. Further, we show how our approach's
input-output nature provides robustness against modeling error through an
additional numerical study. | Ethan Lau, Vaibhav Srivastava, Shaunak D. Bopardikar | 2023-10-09T17:47:09 | http://arxiv.org/abs/2310.05904v1 | # On Multi-Fidelity Impedance Tuning for Human-Robot Cooperative Manipulation
###### Abstract
We examine how a human-robot interaction (HRI) system may be designed when input-output data from previous experiments are available. In particular, we consider how to select an optimal impedance in the assistance design for a cooperative manipulation task with a new operator. Due to the variability between individuals, the design parameters that best suit one operator of the robot may not be the best parameters for another one. However, by incorporating historical data using a linear auto-regressive (AR-1) Gaussian process, the search for a new operator's optimal parameters can be accelerated. We lay out a framework for optimizing the human-robot cooperative manipulation that only requires input-output data. We establish how the AR-1 model improves the bound on the regret and numerically simulate a human-robot cooperative manipulation task to show the regret improvement. Further, we show how our approach's input-output nature provides robustness against modeling error through an additional numerical study.
## I Introduction
Recently, there has been an expansion of robotic automation across many industries. Industrial robots exceed humans in strength and precision, and they can successfully perform structured, repetitive tasks. However, increasingly complex tasks require increasingly complex robots. Situations often arise in which a robot cannot complete a task on its own. By bringing a human into the loop, HRI leverages a human's perceptive and decision-making strengths while still benefiting from the robot's precision or physical strength.
A common robot found in HRI is the robotic manipulator, a multi-segmented arm that accomplishes tasks using its end-effector. Using an impedance model, a manipulator's interaction with the environment is often controlled by adjusting its effective mass, stiffness, and damping at its end-effector [1]. The impedance model simplifies control strategies by dynamically relating the manipulator's position and force. Multiple types of impedance control methods have been proposed, including adaptive control [2, 3, 4], iterative methods [5], and neural networks [6]. Studies have also analyzed variable impedance models [7] and their stability [8].
Robotic manipulators have found many engineering applications, including exosuits [9] and construction automation [10]. We specifically consider a cooperative manipulation task in which a human works with a manipulator to track a large object along a given trajectory. The manipulator seeks to follow a general trajectory but requires the human to provide an auxiliary force to guide the object's path. In this context, the human can be modeled using a transfer function specified by a set of gains [11, 12, 13]. These gains may vary between individuals, resulting in a specialized tuning for each operator. As a result, a trade-off is encountered when a new operator must be trained. In a purely robotic setting, the system structure may be found using system identification; however, this process may prove time consuming and annoying for the operator, leading to operator impatience. Iteratively tuning the system for the new operator would also waste time and valuable historical data. Meanwhile, solely relying on historical data may result in suboptimal performance. Our goal is to leverage previous operator data while finding the ideal tuning parameters for a new operator.
To do so, we use Gaussian process (GP) regression, a tool commonly used to model and optimize unknown and difficult-to-evaluate cost functions [14]. One benefit of GPs is their inclusion of confidence bounds in their prediction. Multi-fidelity Gaussian processes (MF-GP) use multiple correlated inputs to predict an output. Specifically, the AR-1 model relates data across various inputs through a nested linear structure. AR-1 models have been used to incorporate low-fidelity data from a simulation in order to optimize a high-fidelity function related to the true system [15, 16].
The following are our main contributions:
1. Using an impedance controller for the robotic manipulator and a transfer function model for human input, we formulate the optimal assistance design for cooperative manipulation as an input-output problem where the system gains are the inputs and the system performance is the output. By applying a Gaussian process framework to this problem, we develop a sequential method to find the system's optimal gains that requires _only this input-output data_.
2. We incorporate previous operators' input-output data through the use of a multi-fidelity Gaussian process. By analytically quantifying how multi-fidelity affects the conditional covariance, we provide an upper bound on the regret. Additionally, we relate this bound to the measurement quality and variability across operators to show that an increase in the accuracy of prior data leads to decrease in the regret.
3. We numerically simulate input-output data for a model of human-robot cooperative manipulation in order to compare the single- and multi-fidelity formulations. We provide an example where cumulative and best instantaneous regret is better for the multi-fidelity formulation than the single-fidelity formulation. Further, we simulate a disturbance-impacted model of the human-robot ma
nipulator to demonstrate the robustness of our approach.
## II System Description
Consider a cooperative manipulation system, in which a human and robot seek to maneuver on object along a given trajectory. The human may be required to exert some effort (e.g. by lifting the object) but the robot can seek to assist the human in other ways (e.g. through precise maneuvering). Given the object's position, both the human and robot know the tracking error and can take a control action based on the error and desired trajectory information.
In this section, we formulate a model for this cooperative manipulation system. In general, robotic manipulators are nonlinear, but using feedback linearization, we design a control input so that the robot behaves as an impedance model. The impedance model allows the human-robot system to be formulated as a linear time-invariant system, which can then be controlled using state feedback. An overview of this control strategy is displayed in Fig. 1.
### _Robot Impedance Model_
Consider an \(n\)-link robot manipulator with the joint space dynamical model [17]
\[M_{q}(\mathbf{q})\ddot{\mathbf{q}}+C_{q}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+F_{q}\dot{\bm {q}}+G_{q}(\mathbf{q})=\mathbf{\tau}_{q}-J^{T}(\mathbf{q})\mathbf{h}_{e}, \tag{1}\]
where \(\mathbf{q}\in\mathbb{R}^{n}\) is the manipulator's position in the joint space with \(n\) degrees of freedom. Here, \(M_{q}(\mathbf{q})\in\mathbb{R}^{n\times n}\) is the symmetric positive definite inertia matrix, \(C_{q}(\mathbf{q},\dot{\mathbf{q}})\in\mathbb{R}^{n\times n}\) is the Coriolis-centrifugal matrix, \(F_{q}\in\mathbb{R}^{n\times n}\) is the vector of damping coefficients, \(G_{q}(\mathbf{q})\in\mathbb{R}^{n}\) is the vector of gravitational forces, \(\mathbf{\tau}_{q}\in\mathbb{R}^{n}\) are the input torques at the joints, \(\mathbf{h}_{e}\in\mathbb{R}^{n}\) are the contact forces exerted by the manipulator's end-effector, and \(J\in\mathbb{R}^{n\times n}\) is the geometric Jacobian relating the end-effector velocities to the joint velocities.
Let \(\mathbf{z}\) and \(\mathbf{z}_{d}\) be the position and desired position of the manipulator end-effector. The error between these positions is given by
\[\mathbf{e}:=\mathbf{z}-\mathbf{z}_{d}. \tag{2}\]
Assuming the joint positions \(\mathbf{q}\) and velocities \(\dot{\mathbf{q}}\) are known, feedback linearization may be used to control the system. We define a control law
\[\mathbf{\tau}_{q}=M_{q}(\mathbf{q})\mathbf{u}_{q}+p(\mathbf{q},\dot{\mathbf{q}})+J^{T}(\mathbf{q})\mathbf{ h}_{e}, \tag{3}\]
where
\[p(\mathbf{q},\dot{\mathbf{q}})=C_{q}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+F_{q}\dot{\mathbf{q}} +G_{q}(\mathbf{q}). \tag{4}\]
Selecting \(M_{m}\), \(B_{m}\), and \(K_{m}\) as the desired inertia, damping, and stiffness matrices of the impedance model, we set the input \(\mathbf{u}_{q}\) of (3) to
\[\mathbf{u}_{q}= J_{A}^{-1}(\mathbf{q})M_{m}^{-1} \tag{5}\] \[\times(M_{m}\tilde{\mathbf{z}}_{d}+B_{m}\dot{\mathbf{e}}+K_{m}\mathbf{e}-M_{m} \dot{J}_{A}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}-\mathbf{h}_{A}),\]
where \(J_{A}(\mathbf{q})\) is the analytical Jacobian satisfying \(\dot{\mathbf{z}}_{d}=J_{A}(\mathbf{q})\dot{\mathbf{q}}\), and \(\mathbf{h}_{A}\) is the forcing vector of the impedance model.
Assume the forcing vector \(\mathbf{h}_{A}\) takes the form
\[\mathbf{h}_{A}=K_{h}\mathbf{f}_{h}, \tag{6}\]
where \(\mathbf{f}_{h}\) is the human control effort and \(K_{h}\in\mathbb{R}^{n\times n}\) is a diagonal matrix of gains. Then, combining (1), (3), (5), and (6), we obtain the impedance model
\[M_{m}\tilde{\mathbf{e}}+B_{m}\dot{\mathbf{e}}+K_{m}\mathbf{e}=K_{h}\mathbf{f}_{h}. \tag{7}\]
Define the augmented error vector as \(\overline{\mathbf{e}}:=[\mathbf{e}^{T}\ \dot{\mathbf{e}}^{T}]^{T}\in\mathbb{R}^{2n}\). Then (7) can be rewritten as
\[\dot{\overline{\mathbf{e}}}=A\overline{\mathbf{e}}+B\overline{\mathbf{u}}, \tag{8}\]
where
\[A=\begin{bmatrix}\mathbf{0}&I_{n}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\in\mathbb{R}^{2n\times 2n},\quad B=\begin{bmatrix}\mathbf{0} \\ I_{n}\end{bmatrix}\in\mathbb{R}^{2n\times n}, \tag{9}\]
and
\[\overline{\mathbf{u}}=-M_{m}^{-1}\left[K_{m}\ B_{m}\right]\overline{\mathbf{e}}+M_{m} ^{-1}K_{h}\mathbf{f}_{h}. \tag{10}\]
### _Human Impedance Model_
To account for the effect of the human in the HRI system, we model the human operator using a proportional gain and a derivative gain [11]. Assuming the human's reaction is based on the robot error \(\mathbf{e}\), we obtain the human impedance model
\[K_{d}\dot{\mathbf{f}}_{h}+K_{p}\mathbf{f}_{h}=\mathbf{e}, \tag{11}\]
where \(K_{d},K_{p}\in\mathbb{R}^{n\times n}\) are diagonal matrices of human gains. These gains are considered to be unknown and may vary between operators. As such, we denote by \(K_{d}^{i}\) and \(K_{p}^{i}\) the gain matrices of the \(i\)-th operator. Using these operator-specific gains, (11) can be rewritten as
\[\dot{\mathbf{f}}_{h}=A_{h}^{i}\mathbf{f}_{h}+B_{h}^{i}\mathbf{e}, \tag{12}\]
where
\[A_{h}^{i} =-[K_{d}^{i}]^{-1}K_{p}^{i} \in\mathbb{R}^{n\times n}, \tag{13}\] \[B_{h}^{i} =\begin{bmatrix}[K_{d}^{i}]^{-1}&\mathbf{0}\end{bmatrix} \in\mathbb{R}^{n\times 2n}. \tag{14}\]
### _Human-Robot Impedance Model_
With models established for the robot and human, we now write an augmented state space model for the system. Define the augmented state as \(Z:=[\overline{\mathbf{e}}^{T},\mathbf{f}_{h}^{T}]^{T}\in\mathbb{R}^{3n}\). Then the HRI manipulator for the \(i\)-th operator has the state space model
\[\dot{Z}_{i}=\mathcal{A}^{i}Z_{i}+\mathcal{B}^{i}\mathbf{u}, \tag{15}\]
where
\[\mathcal{A}^{i}=\begin{bmatrix}A&\mathbf{0}\\ B_{h}^{i}&A_{h}^{i}\end{bmatrix}\in\mathbb{R}^{3n\times 3n},\quad\mathcal{B}^{i}= \begin{bmatrix}B\\ \mathbf{0}\end{bmatrix}\in\mathbb{R}^{3n\times n}, \tag{16}\]
and
\[\mathbf{u}=-KZ_{i}, \tag{17}\]
with control gains
\[K=M_{m}^{-1}\begin{bmatrix}K_{m}&B_{m}&K_{h}\end{bmatrix}\in\mathbb{R}^{n\times 3 n}. \tag{18}\]
Given a set of control gains \(K\), the quadratic cost of cooperative manipulation for the \(i\)-th operator is
\[J_{i}(K) =\int_{0}^{\infty}(Z_{i}^{T}(\tau)QZ_{i}(t)+\mathbf{u}^{T}(\tau)R\mathbf{u} (\tau))d\tau \tag{19}\] \[=\int_{0}^{\infty}Z_{i}^{T}(\tau)[Q+K^{T}RK]Z_{i}(t)d\tau, \tag{20}\]
where \(Q\in\mathbb{R}^{3n\times 3n}\) weights the effect of the tracking error, error rate, and human effort, \(R\in\mathbb{R}^{n\times n}\) weights the effect of the robot's control effort, and \(Z_{i}(\tau)\) is the solution of (15) given an initial condition \(Z_{i}(0)\) and feedback controller (17).
### _Problem Statement_
Consider an HRI system with the impedance model (16). Let \(K(\mathbf{x})\) be a controller depending on design parameters \(\mathbf{x}\in\mathcal{X}\subset\mathbb{R}^{q}\). Suppose that the robot has \(m\) human operators, with the \(i\)-th human possessing their own performance metric
\[f_{i}(\mathbf{x})=-J_{i}\left(K(\mathbf{x})\right). \tag{21}\]
As the \(i\)-th operator tests different design parameters, they obtain data for \(\mathbf{X}_{i}\subseteq\mathcal{X}\).
Now, suppose a new \((m+1)\)-th human operates the same robot. Our goal is to leverage the previous data \((\mathbf{X}_{i},f_{i}(\mathbf{X}_{i}))\) to find an ideal set of design parameters \(\mathbf{x}^{*}\) that optimizes the new operator's performance \(f_{m+1}\).
## III Using Previous Data in Multi-Fidelity Methods for Control Gain Selection
With our problem statement established, we provide an overview of Gaussian processes. We introduce the notion of multi-fidelity and describe how the HRI problem is formulated to fit this framework.
### _Gaussian Processes (GPs)_
A Gaussian process is a collection of random variables, in which any finite subset of variables has a multivariate Gaussian distribution [14]. A GP is defined by its mean function \(\mu(\mathbf{x})\) and its covariance (kernel) function \(k(\mathbf{x},\mathbf{x}^{\prime})\).
For a set of inputs \(\mathbf{X}_{t}=\{\mathbf{x}_{1},\ldots,\mathbf{x}_{t}\}\), we can create a covariance matrix \(\mathbf{k}(\mathbf{X}_{t},\mathbf{X}_{t})=[k(\mathbf{x}_{i},\mathbf{x}_{j})]_{i,j=1}^{t,t}\). By taking the covariance between a point and a set of points, we obtain a covariance vector \(\mathbf{k}(\mathbf{x}):=\mathbf{k}(\mathbf{X}_{t},\mathbf{x})=[k(\mathbf{x}_{1},\mathbf{x})\ldots k(\mathbf{x} _{t},\mathbf{x})]^{T}\).
Let \(\mathbf{Y}_{t}=[y_{1},\ldots,y_{t}]^{T}\) be noisy samples of \(f\) at \(\mathbf{X}_{t}\), where \(y_{i}=f(\mathbf{x}_{i})+\eta\) has independent and identically distributed Gaussian measurement noise \(\eta\sim N(0,\xi^{2})\).
Then the posterior distribution of \(f\) is another GP with mean \(\mu_{t+1}\), covariance \(k_{t+1}\), and standard deviation \(\sigma_{t+1}\) given by
\[\mu_{t+1}(\mathbf{x}) =\mathbf{k}^{T}(\mathbf{x})[\mathbf{k}(\mathbf{X}_{t},\mathbf{X}_{t})+\xi^{2}I]^{-1} \mathbf{Y}_{t}, \tag{22}\] \[k_{t+1}(\mathbf{x},\mathbf{x}^{\prime}) =k_{t}(\mathbf{x},\mathbf{x}^{\prime})\!\!-\!\mathbf{k}^{T}(\mathbf{x})[\mathbf{k}(\bm {X}_{t},\mathbf{X}_{t})\!\!+\!\!\xi^{2}I]^{-1}\mathbf{k}(\mathbf{x}^{\prime}),\] \[\sigma_{t+1}(\mathbf{x}) =\sqrt{k_{t}(\mathbf{x},\mathbf{x})}. \tag{23}\]
In problems where a GP is being optimized, Bayesian optimization is an iterative framework used to select the next point to evaluate. Popular Bayesian optimization approaches include using the Expected Improvement [18] and the Upper Confidence Bound (UCB) [19].
The UCB algorithm selects points according to
\[\mathbf{x}_{t}=\operatorname*{arg\,max}_{\mathbf{x}\in\mathcal{X}}\,\mu_{t-1}(\mathbf{x}) +\beta_{t}^{1/2}\sigma_{t-1}(\mathbf{x}),\]
where \(\beta_{t}\) is a parameter which controls the algorithm's tendency to explore. This algorithm is formalized in Alg. 1. One particular appeal of UCB are its theoretical guarantees associated with a metric called regret.
```
1:Input: GP \(f\) with priors \(\mu_{0}\), \(\sigma_{0}\), Discrete domain \(\mathcal{X}\)
2:for\(t=1,2,\ldots\)do
3: Choose \(\mathbf{x}_{t}=\operatorname*{arg\,max}_{\mathbf{x}\in\mathcal{X}}\,\mu_{t-1}(\mathbf{x})+ \beta_{t}^{1/2}\sigma_{t-1}(\mathbf{x})\)
4: Sample \(y_{t}(\mathbf{x}_{t})=f(\mathbf{x}_{t})+\eta\)
5: Predict \(\mu_{t}(\mathbf{x})\), \(\sigma_{t}(\mathbf{x})\)\(\forall\mathbf{x}\in\mathcal{X}\)
6:endfor
```
**Algorithm 1** UCB Sampling
For an iterative optimization algorithm, the instantaneous regret of an evaluation is given by
\[r_{t}(\mathbf{x}_{t})=f(\mathbf{x}^{*})-f(\mathbf{x}_{t}), \tag{24}\]
where \(\mathbf{x}^{*}=\operatorname*{arg\,max}_{\mathbf{x}\in\mathcal{X}}\,f(\mathbf{x})\). Regret indicates the gap between the current evaluation and the best possible evaluation. After \(T\) rounds, the cumulative regret is given by \(R_{T}=\sum_{t=1}^{T}r_{t}\) and the best instantaneous regret is given by \(r_{T}^{*}=\min_{t=\{1...T\}}r_{t}\).
### _Multi-Fidelity Gaussian Processes (MF-GPs)_
An MF-GP incorporates data from multiple inputs to model \(f\). One type of MF-GP is the AR-1 model [20]. AR-1 models \(f\) as a linear combination of a low-fidelity GP \(f_{L}(\mathbf{x})\) and an error GP \(\delta(\mathbf{x})\) by
\[f(\mathbf{x})=\rho f_{L}(\mathbf{x})+\delta(\mathbf{x}), \tag{25}\]
Fig. 1: Block Diagram of the Human-Robot Manipulator System.
where \(\rho\) is a scaling constant.
Denote the kernels of \(f_{L}\) and \(\delta\) by \(\mathbf{k}^{(L)}\) and \(\mathbf{k}^{(\delta)}\), respectively, and let evaluations of \(f_{L}\) and \(f\) have variances \(\xi_{L}^{2}\) and \(\xi_{H}^{2}\). Then, for \(\mathbf{X}=[\mathbf{X}_{L},\mathbf{X}_{H}]\), an AR-1 model has a covariance matrix of the form
\[\mathbf{k}^{(MF)}(\mathbf{X},\mathbf{X})=\begin{bmatrix}\mathbf{k}_{L,L}^{(L)}{+} \xi_{L}^{2}I&\rho\mathbf{k}_{L,H}^{(L)}\\ \rho\mathbf{k}_{H,L}^{(L)}&\rho^{2}\mathbf{k}_{H,H}^{(L)}{+}\mathbf{k}_{H,H}^{(\delta)}{+} \xi_{H}^{2}I\end{bmatrix}, \tag{26}\]
where \(\mathbf{k}_{H,L}^{(L)}\) is shorthand notation for the single-fidelity covariance matrix \(\mathbf{k}^{(L)}(\mathbf{X}_{H},\mathbf{X}_{L})\).
Unlike larger GP models, the AR-1model allows for the iterative updating of each fidelity, thereby maintaining a computational complexity on the same order as a single-fidelity GP. Additionally, it's decoupled recursive structure allows for the computationally efficient learning of its parameters.
### _Multi-Fidelity Approach to Control Design_
Using the AR-1 model, we aim to effectively leverage data from the previous operators to a specific individual. Consider a set of \(m{+}1\) operators, with the \(i\)-th operator's performance data \((\mathbf{X}_{i},f_{i}(\mathbf{X}_{i}))\).
Let \(f:\mathcal{X}\rightarrow\mathbb{R}\) be an unknown realization of a GP with AR-1 structure (25). Because the quadratic cost is sufficiently smooth with respect to \(K\), we assume the GP \(f\) adequately represents the performance \(f_{m+1}\) of the \((m{+}1)\)-th operator. Meanwhile, we treat \(f_{L}\) as a GP with observations the first \(m\) operators. Note, \(f_{L}\) does not specifically represent any \(f_{i}\) but rather models the expected performance of the previous \(m\) operators.
Using UCB, we iteratively select an \(\mathbf{x}_{t}\) to test for the \((m{+}1)\)-th operator, thereby obtaining evaluations of \(f\). This Multi-Fidelity Formulation (MFF) is formalized in Algorithm 2.
```
1:Input: Data \((\mathbf{X}_{i},f_{i}(\mathbf{X}_{i}))\) for \(i\in\{1,2,\ldots,m+1\}\), Discrete domain \(\mathcal{X}\)
2: Let \(f_{L}\) be a GP with evaluations \((\mathbf{X}_{i},f_{i}(\mathbf{X}_{i}))\) for \(i=\{1,\ldots,m\}\)
3: Let \(f\) be a GP with form \(f(\mathbf{x})=\rho f_{L}(\mathbf{x})+\delta(\mathbf{x})\) and evaluations \((\mathbf{X}_{m+1},f_{m+1}(\mathbf{X}_{m+1}))\)
4: Predict \(\mu_{0}(\mathbf{x})\), \(\sigma_{0}(\mathbf{x})\)\(\forall\mathbf{x}\in\mathcal{X}\)
5:UCB(\(f,\mu_{0},\sigma_{0},\mathcal{X}\))
```
**Algorithm 2** Multi-Fidelity (MFF) Formulation
We compare MFF to two single-fidelity approaches that do not take advantage of the AR-1 structure. In the Collective Single-Fidelity (CSF) Formulation of Algorithm 3, data from all operators is treated as a single fidelity. In the Limited Single-Fidelity (LSF) Formulation of Algorithm 4, the single-fidelity GP contains only data from the new \((m{+}1)\)-th operator. Essentially, LSF is a naive approach that ignores any previous operator data.
## IV Theoretical Results
With the multi-fidelity nature of this problem established, we now examine how the properties of AR-1 GPs improve the regret performance of UCB. We start with a proposition used to calculate a bound on the conditional covariance.
**Proposition IV.1**: _Let \(Q\) be a positive definite matrix and \(\sigma\in\mathbb{R}\) be any scalar such that \(\sigma<\sqrt{\lambda_{min}(Q)}\). Then_
\[(Q+\sigma^{2}I)^{-1}\succeq Q^{-1}-\sigma^{2}Q^{-2}.\]
Note, this is a specified form of [21, Eq. (191)], which denotes it as an approximation but does not state a direction of inequality.
We rewrite
\[(Q+\sigma^{2}I)^{-1} =(QQ^{-1}Q+\sigma^{2}Q^{-1}Q)^{-1}\] \[=((I+\sigma^{2}Q^{-1})Q)^{-1}\] \[=Q^{-1}(I+\sigma^{2}Q^{-1})^{-1}. \tag{27}\]
By writing the series expansion of the second factor,
\[(I{+}\sigma^{2}Q^{-1})^{-1} =I{-}\sigma^{2}Q^{-1}{+}(\sigma^{2}Q^{-1})^{2}{-}(\sigma^{2}Q^{-1 })^{3}{+}\ldots\] \[=I{-}\sigma^{2}Q^{-1}{+}\sigma^{2}Q^{-1}(I{-}\sigma^{2}Q^{-1}+...) Q^{-1}\] \[=I{-}\sigma^{2}Q^{-1}{+}\sigma^{2}Q^{-1}(I{+}\sigma^{2}Q^{-1})^{- 1}Q^{-1}.\]
Since \((I{+}\sigma^{2}Q^{-1})^{-1}\) and \(Q^{-1}\) are positive definite, their product is positive definite and
\[(I{+}\sigma^{2}Q^{-1})^{-1}\succeq I-\sigma^{2}Q^{-1}. \tag{28}\]
By substituting (28) into (27), we complete the proof.
**Lemma IV.1** (Cond. Covariance of a Noisy AR-1 GP): _Consider an AR-1 GP with high-fidelity evaluations at \(\mathbf{X}_{H}\) and low-fidelity evaluations at \(\mathbf{X}_{L}\). For a sufficiently small \(\xi_{L}^{2}\), the covariance of the high-fidelity data conditioned on the low-fidelity data can be upper bounded by \(\tilde{\mathbf{k}}^{(MF)}\), where_
\[\tilde{\mathbf{k}}:= \rho^{2}\mathbf{k}_{H,H}^{(L)}+\mathbf{k}_{H,H}^{(\delta)}+\xi_{H}^{2}I- \rho^{2}\mathbf{k}_{H,L}^{(L)}[\mathbf{k}_{L,L}^{(L)}]^{-1}\mathbf{k}_{L,H}^{(L)}\] \[+\xi_{L}^{2}\mathbf{k}_{H,L}^{(L)}[{\mathbf{k}}_{L,L}^{(L)}]^{-2}\mathbf{k}_{L,H}^{(L)}.\]
The conditional covariance of an AR-1 GP can be written as
\[\mathbf{k}(f_{H}(\mathbf{X}_{H}),f_{H}(\mathbf{X}_{H})|f_{L}(\mathbf{X}_{L})=\mathbf{y }_{L},f_{H}(\mathbf{X}_{H})=\mathbf{y}_{H})\] \[=\rho^{2}\mathbf{k}_{H,H}^{(L)}+\mathbf{k}_{H,H}^{(\delta)}+\xi_{H}^{2}I- \rho^{2}\mathbf{k}_{H,L}^{(L)}[\mathbf{k}_{L,L}^{(L)}+\xi_{L}^{2}I]^{-1}\mathbf{k}_{L,H}^{(L)}\] \[\preceq\rho^{2}\mathbf{k}_{H,H}^{(L)}+\mathbf{k}_{H,H}^{(\delta)}+\xi_{H} ^{2}I\] \[\quad-\rho^{2}\mathbf{k}_{H,L}^{(L)}\left([\mathbf{k}_{L,L}^{(L)}]^{-1} \!-\!\xi_{L}^{2}[\mathbf{k}_{L,L}^{(L)}]^{-2}\right)\mathbf{k}_{L,H}^{(L)}\] \[=\rho^{2}\mathbf{k}_{H,H}^{(L)}+\mathbf{k}_{H,H}^{(\delta)}+\xi_{H}^{2}I- \rho^{2}\mathbf{k}_{H,L}^{(L)}[\mathbf{k}_{L,L}^{(L)}]^{-1}\mathbf{k}_{L,H}^{(L)}\] \[\quad+\xi_{L}^{2}\mathbf{k}_{H,L}^{(L)}[\mathbf{k}_{L,L}^{(L)}]^{-2}\mathbf{k }_{L,H}^{(L)},\]
where the inequality is obtained from Proposition IV.1.
**Remark IV.1**: _Recall, \(f_{L}\) represents the expected performance of the previous \(m\) operators, and \(\xi_{L}^{2}\) represents the variance of the evaluations of \(f_{L}\). Therefore, for a sufficiently large set of historical data, the we assume that \(\xi_{L}^{2}\) will be small._
**Remark IV.2**: _If the low-fidelity is evaluated at all points in \(\mathbf{X}_{H}\), we see that_
\[\mathbf{k}_{H,L}^{(L)}[\mathbf{k}_{L,L}^{(L)}]^{-1}\mathbf{k}_{L,H}^{(L)}=\mathbf{k}_{H,H}^{(L )},\]
_resulting in a simplification of the upper bound to_
\[\mathbf{k}_{H,H}^{(\delta)}+\xi_{H}^{2}I+\xi_{L}^{2}\mathbf{k}_{H,L}^{(L)}[\mathbf{k}_{L,L }^{(L)}]^{-2}\mathbf{k}_{L,H}^{(L)}.\]
_Additionally, we see that as the high- and low-fidelity noise terms approach 0, the conditional covariance approaches \(\mathbf{k}_{H,H}^{(\delta)}\). This result is a generalization of the simplification found in the proof of Theorem 3.2 in [16], where \(\mathbf{X}_{H}\subseteq\mathbf{X}_{L}\) and \(\xi_{L}^{2}=0\)._
An upper bound on the conditional covariance allows us to establish an upper bound on the maximum information gain \(\gamma_{T}\), a metric quantifying the greatest amount of information that can be learned after \(T\) points of a GP \(f\) are sampled. Suppose \(f\) is sampled at points \(A\subseteq\mathcal{X}\), resulting in a vector of noisy evaluations \(\mathbf{y}_{A}\) and a vector of true values \(\mathbf{f}_{A}\). Then, denoting the entropy of a vector by \(H(\cdot)\), the information gain is defined as \(I(\mathbf{y}_{A};\mathbf{f}_{A}):=H(\mathbf{y}_{A})-H(\mathbf{y}_{A}|f)\), and the maximum information gain is
\[\gamma_{T}:=\max_{A\subset\mathcal{X},|A|=T}I(\mathbf{y}_{A};\mathbf{f}_{A}). \tag{29}\]
**Lemma IV.2** (Info. Gain Bound for a Noisy AR-1 GP): _Let \(\xi_{H}^{2}\) and \(\xi_{L}^{2}\) be the variance of the high- and low-fidelity measurement noise of a linear auto-regressive GP. Then the maximum information gain \(\gamma_{T}\) has the upper bound [19]_
\[\tilde{\gamma}_{T}:=\frac{1/2}{1-e^{-1}}\max_{m_{1},\ldots,m_{T}}\sum_{t=1}^{ h(T)}\log\left(1\!+\!\xi_{H}^{-2}m_{t}\lambda_{t}(\tilde{\mathbf{k}})\right), \tag{30}\]
_where \(\sum_{i=1}^{T}m_{i}=T\), \(h(T)=\min\{T,|\mathbf{X}_{H}|\}\), and \(\lambda_{t}(\tilde{\mathbf{k}})\) are the eigenvalues of the matrix \(\tilde{\mathbf{k}}\) from Lemma IV.1._
**Remark IV.3**: _We see that the bound on the information gain depends on the magnitude of the eigenvalues of \(\tilde{\mathbf{k}}\). As such, we can evaluate the benefit of a multi-fidelity model by comparing the eigenvalues of \(\tilde{\mathbf{k}}\) with the eigenvalues of the single-fidelity covariance \(\mathbf{k}_{H,H}^{(H)}\). When the eigenvalues of \(\tilde{\mathbf{k}}\) are smaller than the eigenvalues of \(\mathbf{k}_{H,H}^{(H)}\), the information gain bound is lower for the AR-1 GP than a single-fidelity GP with the same data._
Using this bound on the information gain, we now present our main result: a bound on the regret of an AR-1 model.
**Theorem IV.1** (Regret Bounds for UCB on an AR-1): _Let \(f\) be a sample function from a linear auto-regressive GP (25) over the discrete domain \(\mathcal{X}\). Set \(\delta\in(0,1)\) and \(\beta_{f}=2\log(|\mathcal{X}|t^{2}\pi^{2}/6\delta)\). Then, the points \(\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots\mathbf{x}_{T}\}\) obtained from Algorithm 2 satisfy with probability at least \(1-\delta\),_
\[R_{T}\leq\sqrt{C_{1}T\beta_{T}\tilde{\gamma}_{T}}.\]
_Here, \(\tilde{\gamma}_{T}\) is the information gain bound established in Lemma IV.2 and \(C_{1}=8v_{MF}^{2}/\log(1+v_{MF}^{2}\xi^{-2})\), where \(v_{MF}^{2}\) is the variance of the AR-1 GP, given by \(v_{MF}^{2}=\rho v_{L}^{2}+v_{\delta}^{2}\)._
The proof of this theorem closely follows the proof of Theorem 1 in [19].
**Remark IV.4**: _The regret of UCB is upper bounded by the information gain. As such, lowering the information gain bound will improve the cumulative regret bound. In particular, when the eigenvalues of \(\tilde{\mathbf{k}}\) are smaller than the eigenvalues of \(\mathbf{k}_{H,H}^{(H)}\), the AR-1 model improves the regret._
_Further, when \(f_{L}\) closely matches \(f\), the variance of \(\delta(\mathbf{x})\) decreases together with the eigenvalues of \(\tilde{\mathbf{k}}\). This, in turn, results in a lower regret bound. In other words, when variations between operators have little effect on the HRI performance curve, Alg. 2 will obtain a very small regret._
## V Numerical Simulations
We conduct two numerical simulations to demonstrate the performance of Algorithms 2, 3, and 4. First, we apply these algorithms to the undisturbed LTI model (15). Then, we show the robustness of our approach by applying it to an LTI system with an unknown disturbance.
### _LTI Model_
Consider the LTI system (15) with \(n=2\) degrees of freedom. Because we model the robot using an impedance model, the end-effector's motion is assumed to be independent in each direction. By letting \(M_{m}=I_{2}\), we assume \(B_{m}\) and \(K_{m}\) will also be scalar matrices. Thus, we assume \(K\) possesses the structure
\[K(\mathbf{x})=\begin{bmatrix}x_{1}&0&x_{2}&0&x_{3}&0\\ 0&x_{1}&0&x_{2}&0&x_{3}\end{bmatrix}.\]
Henceforth, we use \(\mathbf{x}=(x_{1},x_{2},x_{3})\in\mathcal{X}\) as the optimization parameter, where \(\mathcal{X}\) is a \(11\times 11\times 111\) hyperrectangle with span \(x_{1}\in[0.25,0.45]\), \(x_{2}\in[0.85,0.95]\), and \(x_{3}\in[0.02,0.22]\).
Next, we generate data for \(m=9\) previous operators. For the performance functions, we aim to minimize the human effort by setting \(Q=\text{diag}(0.1,0.1,0.1,0.1,10,10)\) and \(R=I_{2}\). We randomly draw \(k_{d}^{i}\sim N(10,5)\), \(k_{p}^{i}\sim N(20,5)\) and set \(K_{d}^{i}=k_{d}^{i}I_{n}\), \(K_{p}^{i}=k_{p}^{i}I_{n}\). An initial condition \(Z_{i}(0)=[I_{n}\quad\mathbf{0}]^{T}\) is chosen to model an initial error in position. The performance \(f_{i}\) from (21) is approximated
using a finite integral from \(\tau=0\) to \(\tau=10\). Each \(f_{i}\) is evaluated for \(20\) random sets of \(\mathbf{x}\in\mathcal{X}\) with additive Gaussian noise \(\eta\sim N(0,10^{-4})\).
We run \(20\) Monte Carlo simulations involving the random selection of previous data points and operator gains \(K_{d}\), \(K_{p}\). Fig. 2 displays the averages of best and cumulative regrets across the simulations. We see that MFF leads to a general improvement in the cumulative regret, especially for higher iteration counts. Between the single-fidelity approaches, LSF has a lower regret and tighter variance than CSF.
The best instantaneous regret plot shows that MFF typically makes better selections than CSF or LSF in the first few iterations. After around 10 iterations, LSF and MFF have found a selection with very low regret while CSF fails to find an optimal selection even after the 20 iterations.
These results indicate that data from the previous operators is beneficial when it is incorporated through a multi-fidelity structure. Incorporating previous data through CSF increases the regret compared to ignoring it in LSF.
### _LTI Model with Disturbance_
Because our techniques rely only on input-output data, the technique is inherently robust to deviations in the model. To demonstrate this, suppose the feedback linearization of (5) is imperfect, resulting in a disturbance affecting the evolution of \(\hat{\mathbf{e}}\). Then the disrupted evolution of the system is
\[\dot{Z}_{i}=\mathcal{A}^{i}Z_{i}+\mathcal{B}^{i}\mathbf{u}+\mathbf{d}, \tag{31}\]
where \(\mathbf{d}\in\mathbb{R}^{n}\) is an unknown but constant disturbance to the system. Specifically, we model a disturbance on states directly affected by the control input (10) by setting \(\mathbf{d}=[0,0,0.05,0.05,0,0]^{T}\).
We plot the regret from the MFF, CSF, and LSF approaches in Fig. 3. We also show the regret incurred when the optimal controller from the undisturbed system is used on the disturbed system.
In this case, the disturbance increases the means and spreads of the cumulative regret. Still, on average, MFF performs better than LSF or CSF. Additionally, on average, all three algorithms identify a better controller than the optimal undisturbed controller in three iterations.
## VI Conclusion
We provide a multi-fidelity framework to find the optimal set of impedance parameters for a human-robot cooperative manipulation system using only input-output data. By treating prior operator data as a low-fidelity model, we are able to further optimize the system's performance for a new operator. We establish how the AR-1 model improves the regret bound through the conditional covariance and then numerically simulate human-robot cooperative manipulation to demonstrate this improvement in regret.
In future work, we plan to validate this framework by conducting physical experiments with human subjects and a robotic manipulator.
| 人間ロボットインタラクション（HRI）システムの設計方法を検討する。過去の実験からの入力出力データを利用することで、最適なインピーダンスを選択し、新しい操作者が協調的な操作を行う際の設計を検討する。個々の差異により、ロボットの操作に最適な設計パラメータが他の操作者にとって最適ではない可能性がある。しかし、過去のデータを利用した線形自動回帰（AR-1）ガウス過程を導入することで、新しい操作者の最適なパラメータの探索を加速させることができる。人間ロボット協調操作の最適化のためのフレームワークを構築する。入力出力データのみを用いて最適化を行う。AR-1モデルによる regret bound の改善を示し、人間ロボット協調操作のシミュレーションを行い、 regret の改善を示す。さらに、このアプローチの入力出力特性は、モデル誤差に対する robustness を提供する。
Please let me know if you'd like me to translate any other sentences |
2304.10528 | Generalizing Neural Human Fitting to Unseen Poses With Articulated SE(3)
Equivariance | We address the problem of fitting a parametric human body model (SMPL) to
point cloud data. Optimization-based methods require careful initialization and
are prone to becoming trapped in local optima. Learning-based methods address
this but do not generalize well when the input pose is far from those seen
during training. For rigid point clouds, remarkable generalization has been
achieved by leveraging SE(3)-equivariant networks, but these methods do not
work on articulated objects. In this work we extend this idea to human bodies
and propose ArtEq, a novel part-based SE(3)-equivariant neural architecture for
SMPL model estimation from point clouds. Specifically, we learn a part
detection network by leveraging local SO(3) invariance, and regress shape and
pose using articulated SE(3) shape-invariant and pose-equivariant networks, all
trained end-to-end. Our novel pose regression module leverages the
permutation-equivariant property of self-attention layers to preserve
rotational equivariance. Experimental results show that ArtEq generalizes to
poses not seen during training, outperforming state-of-the-art methods by ~44%
in terms of body reconstruction accuracy, without requiring an optimization
refinement step. Furthermore, ArtEq is three orders of magnitude faster during
inference than prior work and has 97.3% fewer parameters. The code and model
are available for research purposes at https://arteq.is.tue.mpg.de. | Haiwen Feng, Peter Kulits, Shichen Liu, Michael J. Black, Victoria Abrevaya | 2023-04-20T17:58:26 | http://arxiv.org/abs/2304.10528v2 | # Generalizing Neural Human Fitting to Unseen Poses With
###### Abstract
We address the problem of fitting a parametric human body model (SMPL) to point cloud data. Optimization-based methods require careful initialization and are prone to becoming trapped in local optima. Learning-based methods address this but do not generalize well when the input pose is far from those seen during training. For rigid point clouds, remarkable generalization has been achieved by leveraging SE(3)-equivariant networks, but these methods do not work on articulated objects. In this work we extend this idea to human bodies and propose ArtEq, a novel part-based SE(3)-equivariant neural architecture for SMPL model estimation from point clouds. Specifically, we learn a part detection network by leveraging local SO(3) invariance, and regress shape and pose using articulated SE(3) shape-invariant and pose-equivariant networks, all trained end-to-end. Our novel equivariant pose regression module leverages the permutation-equivariant property of self-attention layers to preserve rotational equivariance. Experimental results show that ArtEq can **generalize to poses not seen during training, outperforming state-of-the-art methods by 74.5%**, without requiring an optimization refinement step. Further, compared with competing works, our method is more than **three orders of magnitude faster** during inference and has **97.3% fewer** parameters. The code and model will be available for research purposes at [https://arteq.is.tue.mpg.de](https://arteq.is.tue.mpg.de).
## 1 Introduction
The three-dimensional (3D) capture of humans in varied poses is increasingly common and has many applications including synthetic data generation [35], human health analysis [57], apparel design and sizing [51], and avatar creation [10, 36, 55, 50]. Existing 3D body scanners output unordered point clouds, which are not immediately useful for the above applications. Consequently, the first step in processing such data is to _register_ it, that is, to transform it into a canonical and consistent 3D representation such as a mesh. For human bodies, this is typically done by first fitting a parametric model like SMPL [31] to the data; see Fig. 1. Such a process should be efficient and general; that is, it should work for any input body scan, no matter the complexity of the pose. However, this is challenging given the articulated structure of the body and the high degree of variation in shape and pose.
Traditional optimization-based methods for fitting bodies to point clouds [2, 7, 8, 21, 25] are usually based on ICP [11] or its variants [2, 38, 62]. This approach can recover accurate results even for complex poses, but, requires a good initialization, is computationally expensive, and may involve significant manual input. Inspired by progress made in neural architectures for rigid 3D point clouds [41, 42, 48, 59], learning-based approaches have been proposed to solve the registration task. These works either directly regress model parameters [24, 30, 54, 60], in
termediate representations such as correspondences [5, 6], or meshes [19, 39]. While less accurate than optimization, they can be used to initialize an optimization for improved accuracy [15].
A major limitation of learning-based approaches, as reported in several recent papers [5, 6, 54], is **poor generalization when data is significantly out-of-distribution (OOD)**. To understand why, let us first consider how a parametric model such as SMPL explains the input point cloud. Given the shape parameters, the model first generates the overall shape of the subject by deforming a template mesh in a canonical pose. Pose-dependent offsets are then added to the mesh. This deformed mesh then undergoes an articulated transformation that poses the body parts rigidly, and then applies a linear-blend-skinning operation to smooth the result. Therefore, the observed point cloud is modeled as a combination of a canonical body shape, a part-based articulated model, and non-rigid pose-corrective deformations. When training networks to fit SMPL to point clouds, the networks are tasked with capturing the joint distribution of canonical shape and pose deformation, entangling these factors while learning a prior over plausible body shape and pose. This data-dependent prior is useful to infer new, in-distribution samples, but becomes a limitation when it comes to poses that are far from the training set. Ideally, if the network were to be _equivariant_ to human body transformations such as SMPL, then new, unseen poses would not be a problem at inference time.
A function (network) \(f:V\to W\) is said to be equivariant with respect to a group \(\mathcal{G}\) if, for any transformation \(T\in\mathcal{G},f(T\mathbf{X})=Tf(\mathbf{X})\), \(\mathbf{X}\in V\). This property can be beneficial for generalization since it allows one to train with only "canonical" inputs \(f(\mathbf{X})\), while generalizing, by design, to any transformation of the group \(Tf(\mathbf{X})\). For example, SE(3)-equivariant networks have been used to address the OOD generalization problem in rigid point cloud tasks, see e.g. [10, 16, 40]. However, extending this to the human body is far from straightforward, due to 1) its high degree of articulation, 2) the entanglement of pose and shape, and 3) deformations that are only approximately rigid.
In this work we introduce ArtEq, a new neural method that regresses SMPL shape and pose parameters from a point cloud. Our key insight is that non-rigid deformations of the human body can be largely approximated as part-based, articulated SE(3) rigid transformations, and that good generalization requires a proper integration of invariant/equivariant properties into the network. With this in mind, we propose a novel invariant/equivariant architecture design based on the discretized SE(3)-equivariant framework [10, 12]. We learn a part detection network by leveraging local SO(3) invariance, and regress shape and pose by proposing _articulated_ SE(3) shape-invariant and pose-equivariant networks, all trained in an end-to-end manner. We further propose a novel equivariant pose regression module, which leverages the permutation-equivariant property of self-attention layers to preserve rotational equivariance. Finally, to facilitate generalization to unseen poses, we cast pose regression as a weight prediction task, in which the predicted weights are used to calculate a weighted average over each of the discretized SO(3) rotations to obtain the final result.
Our empirical studies demonstrate the importance of considering SE(3) invariance/equivariance for the task of SMPL pose and shape estimation from point cloud data. For out-of-distribution data, we show significant improvement over competing methods [5, 6, 54] in terms of part segmentation, as well as accuracy in pose and shape estimation, even when the competing methods are trained with SO(3) data augmentation. Notably, we outperform methods that require an optimization step, while ours is purely regression-based. For in-distribution samples our method still performs second-best with regression-only results, while the best performing method [54] requires a slow optimization step to obtain the final parameters. Finally, we demonstrate how employing the right symmetries can lead to a lightweight network that is 10 times smaller than competing works, as well as a thousand times faster at inference time, making it easy to deploy in real-world scenarios.
In summary, we make the following contributions: (1) We propose a new framework for human shape and pose estimation from point clouds that integrates SE(3)-invariant/equivariant properties into the network architecture. (2) We propose a novel SE(3)-equivariant pose regression module that combines SE(3) discretization with the permutation-equivariant property of self-attention layers. (3) We show state-of-the-art performance on common benchmarks and datasets based on only regression, particularly for out-of-distribution poses. Additionally, our framework results in a lighter model that performs three orders of magnitude faster than competitors at inference time. Our code and pre-trained models will be publicly released at [https://arteq.is.tue.mpg.de](https://arteq.is.tue.mpg.de).
## 2 Related Work
**Human Body Registration From Point Clouds**. Classic approaches for body registration typically deform a template mesh or a parametric model using some variant of the ICP [11] algorithm [2, 38, 62], often with additional cues such as color patterns [7, 8] or markers [2, 3, 37]. These optimization-based methods can produce accurate results for complex poses when properly initialized, but are prone to getting stuck in local minima when not, can be slow to generate results, and are not fully automatic.
Learning-based approaches have gained popularity, largely due to the development of effective neural net
works for point cloud processing such as PointNet [41, 42], DeepSets [59], and KPConv [48]. These are trained to produce either a good initialization for a subsequent optimization process [5, 19, 54] or as end-to-end systems that directly compute either a mesh [39, 53, 60] or the parameters of a human body model [6, 24, 24, 30, 60]. Our method falls into the last category.
Model-based registration reduces the task to estimating pose and/or shape parameters of a human body model such as SMPL [32]. It has been noted that it is difficult to directly regress the parameters of a SMPL model from point clouds [5, 6, 24, 53, 54]. To circumvent this, current approaches go through intermediate representations such as joint-level features [24, 30] and correspondence maps [6, 54], or resort to temporal data and motion models [23]. Similar to this work, part-based segmentation has been used as an additional cue for registration in [5, 6, 28, 54]. Closely related to our work, PTF [54] isolates each segmented part and regresses a local transformation from input space to canonical space, from which pose parameters are obtained via least-squares fitting.
Without explicitly considering rotational symmetries, previous methods struggle to generalize to poses unseen during training, which limits their applicability. To the best of our knowledge, ours is the first work on model-based human point cloud registration specifically designed to correctly and efficiently handle out-of-distribution poses.
**Equivariant Learning on Point Clouds**. The success of CNNs is largely attributed to the translational equivariance property of such networks. Consequently, there has been an increasing interest in making neural networks invariant or equivariant to other symmetry groups [4, 9, 13, 14, 16, 17, 18, 27, 40, 41, 44, 66, 59]. Of particular interest is the SE(3) equivariance group, which can be useful for point cloud processing. Methods for achieving SE(3) equivariance include: Vector Neurons [16], which employ tensor features and accordingly designed linear/non-linear equivariant layers; Tensor Field Networks (TFN) [49] and SE3-transformers [18] which build on top of the SO(3) representation theory with spherical harmonics and Clebsch-Gordan coefficients; [4, 40, 43] which make use of group averaging theory; and [9, 12], which employ a discretization of the SO(3) space to achieve equivariance. Within this last group, EPN [9] uses separable discrete convolutions (SPConv), which split the 6D SE(3) convolutions into SO(3) and translational parts, making it computationally more efficient. Here, we employ the discrete SO(3) framework along with SPConv layers, as it allows us to leverage the discretized space to simplify the problem of pose regression, and has been noted by previous work to be highly effective [29].
In the case of rigid objects, SE(3)/SO(3)-equivariant networks have been applied to several tasks including classification and retrieval [9, 17], segmentation [16, 33], registration [52, 61], object manipulation [45], normal estimation [40], and pose estimation/canonicalisation [29, 47]. Also for rigid objects, disentanglement of shape and pose has been considered by some of these works, such as [26, 29]. To our knowledge, the only method that explores piece-wise equivariance in the context of articulated objects is [40]. However, this is in the context of shape space learning, which takes as input already registered meshes with ground-truth part segmentation information. Our work, on the other hand, takes as input unordered and unevenly sampled point clouds, for which the parts are unknown.
## 3 Preliminaries
### Discretized SE(3) Equivariance
Gauge-equivariant neural networks were proposed by Cohen et al. [12] as a way to extend the idea of 2D convolutions to the manifold domain. Intuitively, instead of shifting a convolutional kernel through an image for translational equivariance, gauge-equivariant networks "shift" a kernel through all possible tangent frames for equivariance to gauge symmetries. Since this process is very computationally expensive, [12] proposed to discretize the SO(3) space with the icosahedron, which is the largest regular polyhedron, exhibiting 60 rotational symmetries. This has been further extended to operate on point clouds and the SE(3) space by EPN [10], which introduces separable convolutional layers (SPConv) to independently convolve the rotational and translational parts.
Formally, the SO(3) space can be discretized by a rotation group \(\mathcal{G}\) of size \(|\mathcal{G}|=60\), where each group element \(\mathbf{g}_{j}\) represents a rotation \(\mathcal{R}(\mathbf{g}_{j})\) in the icosahedral rotation group. As shown in Figure 2, a rotation acting on a continuous equivariant feature is equivalent to a permutation
Figure 2: A rotation acting in continuous space is equivalent to a permutation acting in the discretized rotation space (in a particular order). We exploit this property to design our architecture based on self-attention layers, while maintaining SO(3) equivariance.
acting in the discretized space, where the rotation group is permuted in a specific order. This builds a connection between a rotation in a continuous space and a permutation in a discrete space. In this paper, following [10, 12], we use the rotation group \(\mathcal{G}\) and the permutation operator to approximate the SO(3) space and the rotation operator.
### SMPL Body Model
SMPL [31] is a statistical human body model that maps shape \(\beta\in\mathbb{R}^{10}\) and pose \(\theta\in\mathbb{R}^{K\times 3}\) parameters to mesh vertices \(\mathbf{V}\in\mathbb{R}^{6890\times 3}\), where \(K\) is the number of articulated joints (here \(K=22\)), and \(\theta\) contains the relative rotation of each joint plus the root joint w.r.t. the parent in the kinematic tree, in axis-angle representation. The model uses PCA to account for variations in shape, and Linear Blend Skinning (LBS) for posing the meshes.
A rest-pose mesh is first produced by
\[\mathbf{T}(\boldsymbol{\beta},\boldsymbol{\theta})=\bar{\mathbf{T}}+B_{S}( \boldsymbol{\beta})+B_{P}(\boldsymbol{\theta}), \tag{1}\]
where \(\bar{\mathbf{T}}\in\mathbb{R}^{6890\times 3}\) is the template mesh, \(B_{S}(\beta)\) is the linear transformation from shape parameters \(\boldsymbol{\beta}\) to shape displacements, and \(B_{P}(\boldsymbol{\theta})\) is the linear transformation from pose parameters to blendshape correctives that account for soft-tissue deformations due to pose. Next, joint locations are obtained for the rest-pose mesh via the linear joint regressor \(J(\boldsymbol{\beta}):\mathbb{R}^{|\boldsymbol{\beta}|}\to\mathbb{R}^{66},J( \boldsymbol{\beta})=\{t_{1},\ldots,t_{22};t_{i}\in\mathbb{R}^{3}\}\). Finally, SMPL applies LBS using skinning weights \(\mathcal{W}\) over the rest-pose mesh to obtain the output vertices:
\[M(\boldsymbol{\beta},\boldsymbol{\theta})=W(\bar{\mathbf{T}}+B_{S}( \boldsymbol{\beta})+B_{P}(\boldsymbol{\theta});\mathcal{W}), \tag{2}\]
## 4 Method
Given an input point cloud \(\mathbf{X}=\{\mathbf{x}_{i}\in\mathbb{R}^{3}\}_{N}\) of size \(N\) representing a human body, the goal of this work is to regress the SMPL [31] shape \(\boldsymbol{\beta}\) and pose \(\boldsymbol{\theta}\) parameters that best explain the observations _for any given pose_, including out-of-distribution ones. Inspired by the progress in rigid SE(3)-equivariant networks [4, 10, 29], we develop an architecture that extracts part-based SE(3)- invariant/equivariant features to disentangle shape, pose, and body parts.
There are key differences between rigid objects and human bodies. First, bodies have a highly articulated structure, with different body parts undergoing various SE(3) transformations according to the kinematic tree. Therefore, global SE(3)-equivariant features cannot be directly employed, requiring, instead, local feature extraction and aggregation along with part-based semantic guidance. Second, most of the human body parts are approximately cylindrical in shape, the torso, legs, arms, etc. Cylindrical shapes have a rotational symmetry, resulting in ambiguities when recovering the pose. This is a known problem in the rigid pose estimation field [22], and cannot be resolved without external information. For human bodies, however, we can use non-ambiguous parts to help disambiguate the ambiguous ones, as we will show later. Third, when varying body poses the Euclidean neighborhood of a point might not always correspond to the geodesic neighborhood, particularly with poses that are close to self-contact. This is a problem when using point convolutional networks (such as the one developed here) since the kernel has a ball-shaped
Figure 3: Overview of **ArtEq**. We first obtain point-wise equivariant features using a small equivariant point network [10], which provides a \(C\)-dimensional feature vector per point and per-group element (_e.g_. the \(60\) rotational symmetries of the icosahedron). We then convert these into point-wise invariant features by pooling over the rotation group to obtain a part segmentation of the point cloud. Using the segmentation, we softly aggregate the point-wise features into _part-based_ equivariant features. A self-attention layer processes these in an efficient manner while preserving equivariance. We cast pose regression as a weight prediction task by predicting the weights necessary to perform a weighted average over each rotation element. Finally, we transform the part-based features into invariant ones to obtain an estimate of the shape.
receptive field that might convolve far-away body points (i.e., close in Euclidean space but far in geodesic space), resulting in incorrect estimates.
To address this we introduce **ArtEq**, a _part-based SE(3) equivariant framework for human point clouds_ that processes articulated bodies via four modules: (1) shared locally equivariant feature extraction (Sec. 4.1), (2) part segmentation network (Sec. 4.2), (3) pose estimation network (Sec. 4.3.2) and (4) shape estimation network (Sec. 4.3.3). An overview of our method can be found in Fig. 3, and we elaborate each of these in the following.
### Locally Equivariant Feature Extractor
The first step of our pipeline is to obtain local _per-point_ SO(3)-equivariant features that can be used by the subsequent modules. To this end, we train a network that takes as input the point cloud \(\mathbf{X}\) and a rotation group \(\mathcal{G}\) with \(|\mathcal{G}|=M\) elements, and returns a feature tensor \(\mathcal{F}\in\mathbb{R}^{N\times M\times C}\), comprising a feature vector of size \(C\) for each of the \(N\) points and each of the \(M\) group elements.
Key to our design is the ability to capture local SO(3)-equivariant features via limiting the point convolution kernel's effective receptive field. As mentioned earlier, human body parts that come close to each other are problematic, since the convolutional kernels might incorporate points that are geodesically distant (e.g. an arm that is almost touching a leg). This, in turn, results in reduced performance for difficult self-contact cases. To mitigate this effect, we employ a small kernel size for the SPConv layer, and reduce the number of layers to only two.
### Part Segmentation via Local SO(3) Invariance
To obtain part-based equivariance, the first step is to segment the point cloud into body parts. To achieve this while generalizing to OOD poses, we initially obtain a _locally SO(3)-invariant_ feature by average pooling \(\mathcal{F}\) over the rotation group dimension (\(M\)), aggregating the information from all group elements:
\[\overline{\mathcal{F}}(\mathbf{x}_{i})=\sigma\left(\{\mathcal{F}(\mathbf{x}_{ i},\mathbf{g}_{1}),\mathcal{F}(\mathbf{x}_{i},\mathbf{g}_{2}),\dots,\mathcal{F}( \mathbf{x}_{i},\mathbf{g}_{M})\}\right), \tag{3}\]
where \(\sigma\) is an aggregation function (here we use mean pooling), and \(\overline{\mathcal{F}}\in\mathbb{R}^{N\times C}\) is the local SO(3)-invariant feature that encodes intrinsic geometry information. To efficiently segment the point cloud we adopt a PointNet architecture [41] with skip connections, which takes as input the invariant local feature \(\overline{\mathcal{F}}(\mathbf{x}_{i})\) and outputs a vector \(\alpha\in\mathbb{R}^{N\times 22}\), with \(\alpha(\mathbf{x}_{i},\mathbf{p}_{k})\) representing the probability of point \(\mathbf{x}_{i}\) belonging to body part \(\mathbf{p}_{k}\). Note that PointNet is based on set operators such as pooling and point-wise convolution, and hence preserves SO(3) invariance1.
Footnote 1: In the original PointNet, the input to the first layer is absolute point coordinates which are not invariant to rotations. Here, however, the input is already SO(3)-invariant.
### Pose and Shape Estimation via Attentive SE(3) Equivariance
To disentangle pose and shape while maintaining OOD generalization we need to consider the correct symmetries for each task: part-based SE(3) equivariance for pose, and part-based SE(3) invariance for shape. We explain here how to process the previously obtained part segmentation and local SO(3)-equivariant features to estimate part-based equivariant features, which are used to obtain the final SMPL pose \(\mathbf{\theta}\) and shape \(\beta\) parameters.
#### 4.3.1 Extracting Part-Based Features
The feature tensor \(\mathcal{F}\) operates at the level of points. To obtain features that are useful for the part-level task of pose and shape regression, we aggregate \(\mathcal{F}\) into part-based equivariant features \(\mathcal{H}\in\mathbb{R}^{K\times M\times C}\).
A naive solution is to simply select the points with maximum probability for a part, and average their features to obtain a unique equivariant feature of size \(M\times C\) for each body part \(\mathbf{p}_{k}\). However, (1) hard selection based on the argmax operation is not differentiable, and (2) points in the transition of two adjacent parts are ambiguous and can be hard to correctly select. Instead, we propose to use a _soft aggregation_ by performing a weighted average of the equivariant features _of all the points_, weighted by the probability computed by the segmentation network:
\[\mathcal{H}(\mathbf{p}_{k},\mathbf{g}_{j})=\sum_{i}^{N}\alpha(\mathbf{x}_{i}, \mathbf{p}_{k})\mathcal{F}(\mathbf{x}_{i},\mathbf{g}_{j}), \tag{4}\]
where \(\mathcal{H}\in\mathbb{R}^{K\times M\times C}\) is a per-part SO(3)-equivariant feature, and \(K\) is the number of body parts (\(K=22\) in the case of SMPL). Similar to Equation (3) in Section 4.2, we extract the part-level SO(3)-invariant feature by aggregating the equivariant features:
\[\overline{\mathcal{H}}(\mathbf{p}_{k})=\sigma\left(\{\mathcal{H}(\mathbf{p}_{ k},\mathbf{g}_{1}),\mathcal{H}(\mathbf{p}_{k},\mathbf{g}_{2}),\dots,\mathcal{H}( \mathbf{p}_{k},\mathbf{g}_{M})\}\right), \tag{5}\]
where \(\overline{\mathcal{H}}(\mathbf{p}_{k})\in\mathbb{R}^{K\times C}\) is the per-part SO(3)-equivariant feature.
#### 4.3.2 Pose Estimation
**Pose Representation**. SMPL pose parameters are defined relative to their parent in the kinematic tree. However, local rotations are problematic for equivariant features, since these are defined in a global coordinate system. We estimate instead _global_ rotation matrices that represent the rigid transformation from the part in canonical pose to the part in
the current pose, from which we can recover the local rotation matrix \(\mathbf{\theta}_{k}\) by \(\mathbf{\theta}_{k}=\hat{\mathcal{R}}_{k}\cdot\mathbf{\theta}_{parent(k)}^{T}\), where \(\hat{\mathcal{R}}_{k}\) is the estimated global rotation matrix, and \(\mathbf{\theta}_{parent(k)}\) the accumulated rotation matrix of the parent.
**Attentive SE(3) Equivariance**. To obtain the part rotation matrices \(\hat{\mathcal{R}}_{k}\) we need a function that transforms the part feature vector \(\mathcal{H}(\mathbf{p}_{k})\) into a suitable representation of \(\mathcal{R}_{k}\). This function is required to preserve the equivariance by construction; if it does not (_e.g_. we employ a standard MLP), the network will need to see the point cloud in all possible poses to be capable of regressing arbitrary rotations.
While we could in principle extend the network by concatenating more SPConv layers, this results in larger receptive fields which are harmful in our particular scenario, as well as greater computational times. Instead, we can make use of the fact that rotational equivariance in continuous space is equivalent to permutation equivariance in discrete space (see [10, 13] and Section 3). Thanks to this, self-attention layers are an efficient alternative for preserving rotational equivariance, and can also build relationship/ranking information among other rotations. Hence, we pair our initial SPConv layers with self-attention layers for efficient SE(3)-equivariant processing.
**Pose Regression as Group Element Weight Prediction**. Given the set of \(M\) rotational features for each part, we now need to regress out a rotation matrix. We take advantage of the structure of our part-based equivariant features, and propose to regard the pose regression task as a _probabilistic/weighted aggregation of the group element rotations_, where the weights are directly regressed from the part-based features. We obtain these weights through self-attention layers that preserve the rotational (permutation) equivariance property of the features, and can extract correlations between the \(M\) group element features, computing the relative importance of each to perform a correct weighted mean. To obtain the final pose, we calculate the weighted chordal L2 mean [20] using the predicted weights.
**Addressing the Cylindrical Rotational Symmetry**. For human bodies, we need to take into account the fact that many parts are of cylindrical shape, which have a rotational ambiguity. However, we can leverage the fact that other parts are less prone to this ambiguity, such as the feet or the head. With this in mind, we propose to condition the pose-estimation network on the part feature of the part; that is, to concatenate \(\mathcal{H}(\mathbf{p}_{k})\) with the parent feature: \([\mathcal{H}(\mathbf{p}_{k})||\mathcal{H}(\mathbf{p}_{parent(k)})]\), before processing with the self-attention layers. We concatenate the root joint with itself for completeness. This helps to disambiguate by considering the pose of possibly easier-to-predict parent parts.
#### 4.3.3 Shape Estimation
Finally, to correctly explain the observed point cloud we need to estimate the shape parameters \(\mathbf{\beta}\) in a way that is _part-invariant_ to pose transformations. To this end, we transform \(\mathcal{H}\) into a part-invariant feature by mean pooling over the group element dimension \(M\), resulting in a feature matrix \(\bar{\mathcal{H}}\in\mathbb{R}^{K\times C}\). This feature is further processed by a few self-attention layers that capture the correlation among different body parts. The output is then flattened and fed into an MLP to produce the final \(\beta\) parameters.
### Model Instance and Training
Thanks to the additional symmetry information, equivariant networks have been shown to be efficient in terms of model size and data requirements [12]. We leverage this and instantiate our framework with a minimally-sized ArtEq architecture, with only two layers of SPConv that output a feature tensor with channel size \(C\) = 64; two multi-head self attention (MHSA) layers for pose regression (eight heads with 64-dim embedding); and one similar MHSA for shape regression. This results in a model which is significantly smaller than competing works [54], while still delivering superior performance, as we will see in the next section.
We train the framework in a supervised manner in two stages. In a first stage, we train the part segmentation network and use ground-truth part segmentation to perform the part-level feature aggregation, while all modules are simultaneously and independently trained. In the second stage, we use the predictions from the part segmentation network for part-level feature aggregation and train the full pipeline end-to-end. Training objectives and additional details can be found in the Sup. Mat.
## 5 Results
In the following we show qualitative and quantitative results for ArtEq, both for in-distribution (Dfaut [8]) and out-of-distribution (PosePrior [1]) data, and we compare with state-of-the-art methods IPNet [5], LoopReg [6], and PTF [54]. We explain our evaluation protocol in Section 5.1, evaluate our SE(3)-invariant segmentation network in Sec. 5.2, and show quantitative and qualitative performance for SMPL-parameter estimation in Sec. 5.3. Finally, we compare performance time and model size in Sec. 5.4.
Figure 4: Qualitative results for part segmentation. Each pair of bodies shows ground-truth segmentation (left) and our result (right).
### Evaluation Protocol
**Datasets.** We train our network using the DFaust [8] subset of the AMASS dataset [34], which contains 100 sequences of 10 subjects with diverse body shapes. We follow the train-test split used in [4, 10] and we crop the test sequences to the middle 50% of frames, subsampling every 5 frames. We sample 5000 non-uniform points across the surface with random on-surface displacements. For in-distribution testing we use the test split of DFaust. For OOD testing we use the PosePrior subset [1] of AMASS, which contains challenging poses that are far from those performed in DFaust.
**Metrics.** We test accuracy of part segmentation as percentage of correct assignments. For SMPL estimation we mea
\begin{table}
\begin{tabular}{|l|c|c|c|} \hline Method & Aug. & OOD & ID \\ \hline IPNet [5] & & 29.0 & 30.5 \\ \hline IPNet [5] & ✓ & 86.7 & 91.2 \\ \hline LoopReg [6] & ✓ & 60.6 & 66.1 \\ \hline PTF [54] & & 8.5 & 10.3 \\ \hline PTF [54] & ✓ & 80.3 & 88.1 \\ \hline \hline Ours (w/o cond) & & 91.6 & 95.8 \\ \hline Ours (w/o cond) & ✓ & 93.5 & 95.9 \\ \hline Ours & & 92.4 & **96.4** \\ \hline Ours & ✓ & **94.0** & 96.1 \\ \hline \end{tabular}
\end{table}
Table 1: Part segmentation accuracy compared to SOTA methods, in terms of percentage of correct predictions, for out-of-distribution (OOD) and in-distribution (ID) datasets. We show results with and without SO(3) data augmentation (“Aug.”), and we show our model with and without parent feature conditioning (“w/o cond”) (Section 4.3.2). Best result in bold, second best underlined.
\begin{table}
\begin{tabular}{|c|c|c|} \hline Method & \#Param (M) & Time (s) \\ \hline IPNet [5] & 35.0 & 211.40 \\ \hline LoopReg [6] & 3.3 & 146.90 \\ \hline PTF [54] & 34.1 & 158.1 \\ \hline Ours & **0.9** & **0.1** \\ \hline \end{tabular}
\end{table}
Table 3: Number of parameters (#Param) and inference time (Time) for different methods
\begin{table}
\begin{tabular}{|l|c|c|} \hline Method & Aug. & OOD & ID \\ \hline & & V2V \(\downarrow\) & MPJPE \(\downarrow\) & V2V \(\downarrow\) & MPJPE \(\downarrow\) \\ \hline IPNet & & 71.61 & 113.19 & 85.22 & 94.60 \\ \hline IPNet & ✓ & 8.18 & 12.66 & 5.28 & 6.73 \\ \hline LoopReg & ✓ & 80.34 & 108.37 & 17.62 & 24.79 \\ \hline PTF & & 171.50 & 218.76 & 151.04 & 185.66 \\ \hline PTF & ✓ & 6.38 & 9.40 & **0.50** & **0.78** \\ \hline \hline Ours (nc) & & 3.94 & 5.07 & 1.20 & 1.39 \\ \hline Ours (nc) & ✓ & 2.69 & 3.29 & 1.25 & 1.82 \\ \hline Ours & & 3.13 & 4.17 & 0.98 & 1.19 \\ \hline Ours & ✓ & **1.63** & **2.35** & 1.08 & 1.22 \\ \hline \end{tabular}
\end{table}
Table 2: SMPL estimation results compared to state-of-the-art methods, with and without SO(3) augmentation (“Aug.”) for out-of-distribution (OOD) and in-distribution (ID) datasets. Metrics: vertex-to-vertex error (V2V, in cm) and mean joint position error (MPJPE, in cm). We also show our method without parent feature conditioning (“nc”). Best result in bold, second best underlined.
Figure 5: Qualitative results for out-of-distribution poses. From left to right: (a) input point cloud, (b) ground-truth SMPL mesh, (c) our results, (d) IPNet [5], (e) PTF [54], and (f) LoopReg [6].
sure (1) vertex-to-vertex error (V2V) in cm and (2) joint position error (MPJPE) in cm.
**Comparisons.** We compare our method with state-of-the-art learning-based works that obtain SMPL parameters from point clouds [5, 6, 54]. IPNet [5] predicts dense correspondences to the SMPL mesh, which is then used within an ICP-like optimization that recovers SMPL shape and pose parameters. LoopReg [6] extends this idea by including the SMPL optimization step within the training process of the network. PTF [54] segments body parts and regresses local transformations for each, from which pose parameters are obtained via least-squares fitting. Since all of these works require part segmentation, we also compare our segmentation results with them. Note that IPNet and LoopReg predict segmentation for 14 parts, while PTF and our method segment the point cloud into 24 parts, which is a harder task. For LoopReg we use their publicly available model trained on minimal clothing data, while the rest of the networks are trained using publicly available code. All methods, including ours, are trained for 15 epochs on our DFaut train split. Note that all competitors depend on a test-time optimization step, while our results are solely based on regression.
**SO(3) Data Augmentation.** An alternative to SE(3) equivariant learning is to explicitly perform data augmentation. While it is not straightforward to do this in a part-based manner without losing realism, one can still perform global SO(3) augmentation by randomly rotating the root join. This has been in fact already implemented by prior work, including the works considered here. For this reason, we compare against these methods both with and without global SO(3) data augmentation. Our method, being an equivariant method, does not require augmentation to achieve good OOD generalization. However, this is still useful since it helps the network bridge the gap between the discretized SO(3) group and the continuous SO(3) group, and hence we too evaluate both with and without SO(3) augmentation.
### Part Segmentation
We begin by evaluating our part segmentation network. Both for us and for competing methods, this is an important step in the pipeline that determines the quality of the final results. Since the output of this step comes directly from the networks, initial errors cannot be masked by an optimization step as it is with the SMPL estimations (see next section). We use this task to more clearly evaluate the impact of equivariant learning.
Quantitative results can be found in Table 1. Our method outperforms the competitors both for in-distribution and OOD datasets by a large margin. Without data augmentation, IPNet and particularly PTF perform very poorly in both cases. Our approach, on the other hand, shows superior performance over all methods both with and without data augmentation. Additionally, we observe that conditioning on the parent feature (Section 4.3.2) can further boost the segmentation results. We show qualitative results for our method in Fig. 4 and qualitative results for competing methods in the supplementary material.
### SMPL Shape and Pose Estimation
In Tab. 2 we show quantitative results for the SMPL body estimation task, evaluated in terms of vertex error and joint position error. Note that in this case, our results were obtained directly from the networks, while other methods require an optimization step. For OOD data, our model performs significantly better than the rest, reducing the vertex error by almost four times over the best-performing competitor (PTF with augmentation). This shows the importance of including the right symmetries within the network design, which cannot be replaced by simply performing data augmentation. It is important to note that all competing methods require data augmentation to perform reasonably in the OOD case. While data augmentation slightly improves the results in our case, we outperform the rest even without data augmentation. For in-distribution data, our model without data augmentation performs second best. Note, however, that the best performing model (PTF with augmentation) is much slower at inference time due to the optimization step. In the bottom part of Table 2 we show that our proposed parent conditioning consistently improves the results by around 1 cm for both in-distribution and OOD samples. Qualitative results can be found in Fig. 5.
### Performance
Tab. 3 shows performance time for all methods, along with model size. Using SE(3)- equivariant/invariant features allows our model to be more efficient, which is why we can design a model that has \(2.7\%\) the number of parameters of PTF, while still outperforming in terms of accuracy. Additionally, our smaller model size results in significantly faster inference time, achieving 10 fps, which is three orders of magnitude faster than the others.
## 6 Conclusions
In this paper we proposed ArtEq, a powerful part-based SE(3)-equivariant neural framework for SMPL parameter estimation from point clouds. Our experimental results demonstrated the generalization ability of ArtEq to out-of-distribution poses, where the direct regression output of ArtEq outperforms state-of-the-art methods that require a time-consuming test-time optimization step. Additionally, we showed unprecedented model efficiency, i.e. using a magnitude fewer model parameters. Through this, we demonstrated the advantage and importance of incorporating the correct symmetries into the task of SMPL body pose and shape estimation. | 人間体のパラメトリックモデル（SMPL）を点雲データに適合させる問題に取り組んでいます。最適化ベースの方法には、慎重な初期化が必要で、局所的な最適性に陥りやすい。学習ベースの方法では、この問題は解決しますが、訓練中に見られる姿勢とは離れた入力姿勢では汎用性が低い。剛体点雲の場合、SE(3)等価ネットワークを利用して驚くほどの汎用性が達成されていますが、これらは多関節物体には適用できません。この研究では、このアイデアを人体の領域に拡張し、ArtEqという新しい多分割SE(3)等価なニューラルアーキテクチャを提案しています。これは、点雲からSMPLモデルの推定のための、局部SO(3)不変性を活用した部分検出ネットワーク、形状と姿勢をSE(3)形状不変性と姿勢等価性ネットワークで推定するものです。すべてエンドツーエンドで訓練されています |
2306.06533 | $n$-knots in $S^n\times S^2$ and contractible $(n+3)$-manifolds | In $1961$, Mazur constructed a contractible, compact, smooth $4$-manifold
with boundary which is not homeomorphic to the standard $4$-ball, using a
$0$-handle, a $1$-handle and a $2$-handle. In this paper, for any integer
$n\geq2,$ we construct a contractible, compact, smooth $(n+3)$-manifold with
boundary which is not homeomorphic to the standard $(n+3)$-ball, using a
$0$-handle, an $n$-handle and an $(n+1)$-handle. The key step is the
construction of an interesting knotted $n$-sphere in $S^n\times S^2$
generalizing the Mazur pattern. | Geunyoung Kim | 2023-06-10T22:00:21 | http://arxiv.org/abs/2306.06533v1 | # \(n\)-knots in \(S^{n}\times S^{2}\) and contractible \((n+3)\)-manifolds
###### Abstract.
In 1961, Mazur [5] constructed a contractible, compact, smooth \(4\)-manifold with boundary which is not homeomorphic to the standard \(4\)-ball, using a \(0\)-handle, a \(1\)-handle and a \(2\)-handle. In this paper, for any integer \(n\geq 2\), we construct a contractible, compact, smooth \((n+3)\)-manifold with boundary which is not homeomorphic to the standard \((n+3)\)-ball, using a \(0\)-handle, an \(n\)-handle and an \((n+1)\)-handle. The key step is the construction of an interesting knotted \(n\)-sphere in \(S^{n}\times S^{2}\) generalizing the Mazur pattern.
## 1. Introduction
In this paper, we prove the following two main theorems.
**Theorem 1.1**.: _For any integer \(n\geq 2,\) there exists a contractible, compact, smooth \((n+3)\)-manifold with boundary admitting a handle decomposition with a \(0\)-handle, an \(n\)-handle and an \((n+1)\)-handle which is not homeomorphic to the standard \((n+3)\)-ball \(B^{n+3}\)._
In order to prove Theorem 1.1, given \(n\geq 2\), we first construct an interesting \(n\)-knot \(K^{n}\) in \(S^{n}\times S^{2}\) (Definition 3.1) which is homotopic but not isotopic to \(S^{n}\times\{y_{0}\}\), where \(y_{0}\in S^{2}\) (Proposition 3.2 and Corollary 3.7). Secondly we construct a contractible, compact, smooth \((n+3)\)-manifold \(X_{K^{n}}\) (Definition 3.8) from \(S^{n}\times B^{3}\) (\(0\)-handle \(\cup\)\(n\)-handle) by attaching a single \((n+3)\)-dimensional \((n+1)\)-handle along the \(n\)-knot \(K^{n}\) in \(S^{n}\times S^{2}=\partial(S^{n}\times B^{3})\). Finally we prove that \(X_{K^{n}}\) is contractible by showing that \(X_{K^{n}}\times B^{1}\) is diffeomorphic to \(B^{n+4}\) (Proposition 3.10) and prove that \(X_{K^{n}}\) is not homeomorphic to \(B^{n+3}\) by showing that \(\partial X_{K^{n}}\) is a non-simply connected homology \((n+2)\)-sphere (Corollary 3.13).
**Theorem 1.2**.: _For any integer \(n\geq 2,\) there exists an involution of \(S^{n+3}\) whose fixed point set is a non-simply connected homology \((n+2)\)-sphere._
In order to prove Theorem 1.2, given \(n\geq 2\), we first show that the double \(DX_{K^{n}}=X_{K^{n}}\cup_{id}\overline{X_{K^{n}}}\) of \(X_{K^{n}}\) is diffeomorphic to \(S^{n+3}\), where \(id:\partial X_{K^{n}}\to\partial X_{K^{n}}\) is an identity map (Lemma 3.16). We then define an involution \(\phi:S^{n+3}\to S^{n+3}\) switching copies of \(X_{K^{n}}\) and fixing the non-simply connected homology \((n+2)\)-sphere \(\partial X_{K^{n}}\).
**Remark 1.3**.: Here we discuss the relationship between our results and earlier results.
1. In [5] Mazur proved Theorem 1.1 and Theorem 1.2 when \(n=1\). We can consider Mazur's \(1\)-knot in \(S^{1}\times S^{2}\) as the result of surgery of three parallel copies of \(S^{1}\times\{y_{0}\}\subset S^{1}\times S^{2}\) along two \(2\)-dimensional \(1\)-handles whose cores are trivial and with some twistings (See \(J_{1}\) for Mazur's \(1\)-knot and \(J_{2}\) for another interesting \(1\)-knot in Remark 2.6). However, we cannot generalize
Mazur's 1-knot to an \(n\)-knot in \(S^{n}\times S^{2}\) obtained from three parallel copies of \(S^{n}\times\{y_{0}\}\subset S^{n}\times S^{2}\) by surgery along two \((n+1)\)-dimensional 1-handles whose cores are trivial when \(n\geq 2\) because the resulting \(n\)-knot is always isotopic to \(S^{n}\times\{y_{0}\}\subset S^{n}\times S^{2}.\) In Definition 3.1, we resolve this issue and find interesting \((n+1)\)-dimensional 1-handles whose cores are non-trivial but very simple so that we construct the \(n\)-knot \(K^{n}\) in \(S^{n}\times S^{2}.\)
2. In [6] Sato proved Theorem 1.1 and Theorem 1.2 when \(n=2.\) Sato constructed a 2-knot \(F\) in \(S^{2}\times S^{2}\) which is homotopic but not isotopic to \(S^{2}\times\{y_{0}\}\) by surgery along a simple closed curve in the complement of the 5-twist spun trefoil in \(S^{4}\) in [7]. However, Sato's construction is not very explicit so it is difficult to visualize the 2-knot \(F\). In particular, we do not know the geometric intersection number \(|F\cap\{x_{0}\}\times S^{2}|\) directly and it is hard to see why \(F\times\{0\}\subset S^{2}\times S^{2}\times B^{1}\) is isotopic to \(S^{2}\times\{y_{0}\}\times\{0\}\subset S^{2}\times S^{2}\times B^{1}\). Our construction of \(K^{n}\) resolves these issues.
3. In [4], Kervaire proved the existence of contractible, compact, smooth \((n+3)\)-manifolds which are not homeomorphic to \(B^{n+3}\), where \(n\geq 2.\) However, the proof does not tell us much about the handle decomposition or give us a proof of Theorem 1.2.
**Remark 1.4**.: Here we note some nice properties of our \(n\)-knot \(K^{n}\) in \(S^{n}\times S^{2},\) and highlight some ways in which our construction is an improvement on the techniques used in the results described above.
1. The construction of \(K^{n}\) is very explicit and visualized for every \(n\geq 2\) (Definition 3.1). We may construct infinitely many interesting \(n\)-knots in \(S^{n}\times S^{2}\) by modifying the cores of \((n+1)\)-dimensional 1-handles. For example, the cores used to construct \(K^{n}\) come from the case when \(m_{1}=1,m_{2}=-1,\) and \(m_{3}=\cdots=m_{2i}=0\) in the left-hand side of Figure 3. We may then construct infinitely many contractible, compact, smooth \((n+3)\)-manifolds which are not homeomorphic to \(B^{n+3}.\)
2. \(K^{n}\) is the simplest example of this construction in the sense that the geometric intersection number \(|K^{n}\cap(\{x_{0}\}\times S^{2})|=3\) and the algebraic intersection number \(K^{n}\cdot(\{x_{0}\}\times S^{2})=1,\) where \(x_{0}\in S^{n}\) (Proposition 3.2). Furthermore it is impossible to have \(|F\cap\{x_{0}\}\times S^{2}|<3\) for any \(n\)-knot \(F\) isotopic to \(K^{n}\) (Corollary 3.15).
3. A homotopy between \(K^{n}\subset S^{n}\times S^{2}\) and \(S^{n}\times\{y_{0}\}\subset S^{n}\times S^{2}\) is visualized (Proposition 3.2).
4. An isotopy between \(K^{n}\times\{0\}\subset S^{n}\times S^{2}\times B^{1}\) and \(S^{n}\times\{y_{0}\}\times\{0\}\subset S^{n}\times S^{2}\times B^{1}\) is visualized (Proposition 3.3). This isotopy is essential to proving that \(X_{K^{n}}\times B^{1}\) is diffeomorphic to \(B^{n+4}\) (Proposition 3.10).
5. The construction of \(K^{n}\) gives an explicit handle decomposition of \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\) (Remark 3.5) so we can easily find the fundamental group \(\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\) (Proposition 3.6).
6. The construction of \(K^{n}\) gives an explicit handle decomposition of the non-simply connected homology \((n+2)\)-sphere \(\partial X_{K^{n}}\) (Remark 3.11) so we can easily show that \(\pi_{1}(X_{K^{n}})\cong\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\) (Proposition 3.12).
### Organization
In section 2 we set up some standard notation and interpret Mazur's 1-knot in \(S^{1}\times S^{2}\) from the point of view of surgery as motivation for our construction of the \(n\)-knot \(K^{n}\) in \(S^{n}\times S^{2}.\) In section 3 we construct our \(n\)-knot \(K^{n}\) in
\(S^{n}\times S^{2}\) and the contractible \((n+3)\)-manifold \(X_{K^{n}}\), and then prove Theorem 1.1 and Theorem 1.2.
### Conventions
In this paper, we work in the smooth category. \(\mathbb{R}^{n},S^{n},B^{n}\) and \(\{0\}\subset B^{n}\subset\mathbb{R}^{n}\) stand for the real \(n\)-space, the standard \(n\)-sphere, the standard \(n\)-ball and the origin of \(B^{n}\) or \(\mathbb{R}^{n}\).
### Acknowledgements
This work was supported in part by National Science Foundation grant DMS-2005554 "Smooth 4-Manifolds: 2-, 3-, 5- and 6-Dimensional Perspectives".
## 2. Preliminaries
We begin by explicitly describing the standard handle decomposition of \(S^{n}\times S^{2}\) and the associated attaching maps.
**Remark 2.1** (Standard handle decomposition of \(S^{n}\times S^{2}\)).: Decompose \(S^{n}=B^{n}_{-}\cup B^{n}_{+}\) into two \(n\)-dimensional balls and \(S^{2}=B^{2}_{-}\cup B^{2}_{+}\) into two \(2\)-dimensional balls. Then \(S^{n}\times S^{2}=(B^{n}_{-}\cup B^{n}_{+})\times(B^{2}_{-}\cup B^{2}_{+})=(B^ {n}_{-}\times B^{2}_{-})\cup(B^{n}_{-}\times B^{2}_{+})\cup(B^{n}_{+}\times B^ {2}_{-})\cup(B^{n}_{+}\times B^{2}_{+})\) has a handle decomposition with a single \(0\)-handle, a single \(2\)-handle, a single \(n\)-handle, and a single \((n+2)\)-handle. We can easily see the attaching sphere of the \(2\)-handle (trivial \(1\)-knot) and the attaching sphere of the \(n\)-handle (trivial \((n-1)\)-knot) on the boundary of the \(0\)-handle i.e., \((\{0\}\times S^{1})\cup(S^{n-1}\times\{0\})\subset\partial(B^{n}_{-}\times B^ {2}_{-})\cong\partial B^{n+2}=S^{n+1}.\) For future reference, we parameterize the trivial \(1\)-knot and the trivial \((n-1)\)-knot in \(\mathbb{R}^{n+1}\subset\mathbb{R}^{n+1}\cup\{\infty\}\cong S^{n+1}\). The trivial \(1\)-knot \(\{0\}\times S^{1}\) corresponds to \(A:=\{(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}{}^{2}+x_{2}{}^{2}=1, x_{i}=0\text{ for }i>2\}\) and the trivial \((n-1)\)-knot \(S^{n-1}\times\{0\}\) corresponds to \(B:=\{(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}=0,(x_{2}+1)^{2}+\sum _{i=3}^{n+1}x_{i}{}^{2}=(\frac{1}{5})^{2}\}.\) An \((\mathbb{R}^{3}\times\{0\})\)-slice \((A\cup B)\cap(\mathbb{R}^{3}\times\{0\})\)
Figure 1. Standard handle decomposition of \(S^{n}\times S^{2}\) and three parallel copies of \(S^{n}\times\{y_{0}\}\). \(A\) is the attaching sphere of the \(2\)-handle with \(0\)-framing, \(B\) is the attaching sphere of \(n\)-handle with \(0\)-framing, and \(C_{t}\) is the equator of \(S^{n}\times\{y_{t}\}\).
of \(A\cup B\) is the Hopf link in Figure 1, where \(\mathbb{R}^{3}\times\{0\}\subset\mathbb{R}^{3}\times\mathbb{R}^{n-2}\cong\mathbb{ R}^{n+1}\). Again, \(S^{n}\times S^{2}\) can be recovered from \(B^{n+2}\) by attaching a single \(2\)-handle to \(S^{n+1}\cong\mathbb{R}^{n+1}\cup\{\infty\}\) along \(A\) with \(0\)-framing (product framing), a single \(n\)-handle with \(0\)-framing (product framing), and a single \((n+2)\)-handle (we don't draw the \((n+2)\)-handle). The \(0\) in Figure 1 is shorthand for the obvious product framing. For example, Figure 1 is a Kirby diagram of \(S^{2}\times S^{2}\) when \(n=2.\) A parallel copy \(C_{t}:=\{(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}=0,(x_{2}+1)^{2}+ \sum_{i=3}^{n+1}{x_{i}}^{2}=(\frac{t}{4})^{2}\}\) of \(B\) bounds a properly embedded trivial \(n\)-ball \(D_{t}^{-}\) in \(B^{n+2}\) (\(C_{t}\) bounds a trivial \(n\)-ball in \(S^{n+1}\) and push the interior of the \(n\)-ball into the interior of \(B^{n+2}\)) and a copy of the core of the \(n\)-handle \(D_{t}^{+}\) so \(C_{t}=D_{t}^{-}\cap D_{t}^{+}\) represents the equator of \(S^{n}\times\{y_{t}\}\) and \(D_{t}:=D_{t}^{-}\cup D_{t}^{+}\) represents \(S^{n}\times\{y_{t}\}.\) From now on, we use red, blue and green for \(A,B,\) and \(C_{t},\) respectively.
Next we establish terminology for handles embedded in an ambient manifold and attached to a submanifold.
**Definition 2.2**.: Let \(n\geq 1\). Let \(N^{n}\subset M^{n+2}\) be a \(n\)-dimensional submanifold of \((n+2)\)-dimensional manifold \(M\). An \((n+1)\)-dimensional submanifold \(h\subset M\) is called a \(1\)_-handle attached to \(N\)_ if there exists an embedding \(e:B^{1}\times B^{n}\hookrightarrow M\) such that \(h=e(B^{1}\times B^{n})\) and \(h\cap N=e(\partial B^{1}\times B^{n}).\) We call \(C_{h}:=e(B^{1}\times\{0\})\) the _core_ of \(h\) and \(N_{h}:=(N\setminus int(e(\partial B^{1}\times B^{n})))\cup e(B^{1}\times \partial B^{n})\subset M\)_the result of surgery_ of \(N\) along \(h\) in \(M.\) Here, \(N_{h}\) is assumed to be oriented so that the orientation of \(N\setminus int(e(\partial B^{1}\times B^{n}))\) extends to the orientation of \(N_{h}.\)
Boyle proved the following when \(n=2\) and \(M=S^{4}.\) See [1] for more details.
**Proposition 2.3**.: Let \(n\geq 2.\) Let \(h\) and \(h^{\prime}\) be \(1\)-handles attached to \(N^{n}\subset M^{n+2}\) with the same core \(C.\) Then there exists an ambient isotopy of \(M\) taking \(h\) to \(h^{\prime}\) fixing \(N\) setwise. Furthermore, the results of surgery \(N_{h}\) and \(N_{h^{\prime}}\) are isotopic.
Proof.: Following [1], the difference between two such \(1\)-handles with the same core gives a map \(\theta:B^{1}\to G_{n+1,n};\) the Grassmannian manifold of oriented \(n\)-planes in \(\mathbb{R}^{n+1},\) with \(\theta(-1)=\theta(1).\) Since \(n\geq 2\) and \(G_{n+1,n}\cong S^{n},\) we have \(\pi_{1}(G_{n+1,n})=0,\) so there exists an isotopy of \(M\) taking \(h\) to \(h^{\prime}\) fixing \(N\) setwise. From this we see that the results of surgery \(N_{h}\) and \(N_{h^{\prime}}\) are isotopic.
**Corollary 2.4**.: Let \(n\geq 2.\) Let \(h\) and \(h^{\prime}\) be \(1\)-handles attached to \(N^{n}\subset M^{n+2}\) with cores \(C_{h}\) and \(C_{h^{\prime}},\) respectively. If \(C_{h}\) and \(C_{h^{\prime}}\) are isotopic through arcs such that the boundary of each arc is in \(N\) and the interior of each arc doesn't intersect with \(N,\) then there exists an ambient isotopy of \(M\) taking \(h\) to \(h^{\prime}\) fixing \(N\) setwise. In particular, the results of surgery \(N_{h}\) and \(N_{h^{\prime}}\) are isotopic.
Proof.: By the tubular neighborhood theorem and Proposition 2.3, the isotopy taking \(C_{h}\) to \(C_{h^{\prime}}\) through arcs such that the boundary of each arc is in \(N\) and the interior of each arc doesn't intersect with \(N\) can extend to an ambient isotopy of \(M\) taking \(h\) to \(h^{\prime}\) fixing \(N\) setwise. From this we see that the results of surgery \(N_{h}\) and \(N_{h^{\prime}}\) are isotopic.
**Remark 2.5**.:
1. Corollary 2.4 is not true when \(n=1\) because \(\pi_{1}(G_{n+1,n})\cong\mathbb{Z}.\) Therefore different framings of a core may give non-isotopic \(1\)-handles; see Remark 2.6.(4).
2. When \(n\geq 2\), a homotopy between arcs implies an isotopy between arcs and there is a unique framing of a core, so \((n+1)\)-dimensional \(1\)-handles are less complicated than \(2\)-dimensional \(1\)-handles.
We now analyze Mazur's knot \(J_{1}\) and some other interesting knots \(J_{2}\) and \(J_{3}\) in \(S^{1}\times S^{2}\) from the point of view of surgery along \(1\)-handles. Figure 2 illustrates these examples. In this figure we consider \(S^{1}\times S^{2}\) as the boundary of \(S^{2}\times B^{2}\), where
Figure 2. First column \((a),(d),(g)\): parallel copies of \(S^{1}\times\{y_{0}\}\) in \(S^{1}\times S^{2}\) with trivial cores of \(2\)-dimensional \(1\)-handles. Second column \((b),(e),(h)\): \(2\)-dimensional \(1\)-handles determined by the cores and framings (or twistings). Third column \((c),(f),(i)\): results of surgery \(J_{1},J_{2},\) and \(J_{3}\). First row: a process of obtaining \(J_{1}\) by surgery. Second row: a process of obtaining \(J_{2}\) by surgery. Third row: a process of obtaining \(J_{3}\) by surgery.
\(S^{2}\times B^{2}\) is obtained from \(B^{4}\) by attaching a \(2\)-handle along the unknot with the \(0\)-framing (product framing). Observe the following features of the knots constructed in Figure 2:
**Remark 2.6**.:
1. \(J_{1}\) is homotopic but not isotopic to \(S^{1}\times\{y_{0}\}\) in \(S^{1}\times S^{2}\); see Figure 2.\((c)\).
2. \(J_{2}\) is homotopic but not isotopic to \(S^{1}\times\{y_{0}\}\) in \(S^{1}\times S^{2}\); see Figure 2.\((f)\).
3. \(J_{3}\) is homotopic but not isotopic to the unknot in \(S^{1}\times S^{2}\); see Figure 2.\((i)\).
4. \(J_{1},J_{2}\) and \(J_{3}\) are obtained from parallel copies of \(S^{1}\times\{y_{0}\}\) by surgery along \(2\)-dimensional \(1\)-handles in the figure. Here, we can see that handles are attached so that the results of surgery are oriented and depend on the cores and framings (or twistings) of the cores. (See the second column in Figure 2, and note that we may obtain different knots by twisting the bands more.)
5. In the next section we will construct an \(n\)-knot in \(S^{n}\times S^{2}\) from three parallel copies of \(S^{n}\times\{y_{0}\}\) by surgery along two interesting \((n+1)\)-dimensional \(1\)-handles.
A natural question related to the construction of \(J_{3}\) is whether one can construct an \(n\)-knot in \(S^{n}\times S^{2}\) which is homotopic but not isotopic to the unknot from two parallel copies of \(S^{n}\times\{y_{0}\}\) by surgery along a single \((n+1)\)-dimensional \(1\)-handle. However, the following theorem shows that this does not work for \(n\geq 2\).
**Theorem 2.7**.: _Fix \(n\geq 2\). Let \(N=(S^{n}\times\{y_{1}\})\cup(\overline{S^{n}\times\{y_{2}\}})\subset S^{n} \times S^{2}\) with opposite orientations, where \(y_{1}\neq y_{2}\in S^{2}\). Let \(h:=e(B^{1}\times B^{n})\) be a \(1\)-handle attached to \(N\) for some embedding \(e:B^{1}\times B^{n}\hookrightarrow S^{n}\times S^{2}\) such that \((S^{n}\times\{y_{1}\})\cap h=e(\{-1\}\times B^{n})\) and \((\overline{S^{n}\times\{y_{2}\}})\cap h=e(\{1\}\times B^{n})\). Then the result of surgery \(N_{h}\) is isotopic to the unknot, i.e., \(N_{h}\) bounds an \((n+1)\)-ball in \(S^{n}\times S^{2}\)._
Figure 3. An isotopy between cores. An integer \(m\) in the box indicates \(m\)-full positive twist. \(C_{i}\) is the equator of \(D_{i}\). Left: any arc attached to \(C_{1}\cup C_{2}\) is isotopic to the orange arc for some values of \(m_{1},\dots,m_{2i}\). Right: the trivial arc attached to \(C_{1}\cup C_{2}\).
Proof.: Consider the standard handle decomposition of \(S^{n}\times S^{2}\) described in Remark 2.1 and Figure 1. Let \(D_{1}=S^{n}\times\{y_{1}\}\) and \(D_{2}=\overline{S^{n}\times\{y_{2}\}}.\) Now consider a \(1\)-handle \(h\) attached to \(D_{1}\cup D_{2}.\) By Corollary 2.4, it suffices to consider the core
Figure 4. An isotopy between cores when \(m_{1}=2\) and \(m_{2}=-1.\)\((a),(d)\): isotopies pushing the orange arc into \(B^{n+2}\) and pulling back. \((b),(e)\): isotopies sliding the orange arc over \(2\)-handle. \((c),(f),(g)\): obvious isotopies.
of the \(1\)-handle \(h\). The core of the \(1\)-handle can be isotoped into \(\mathbb{R}^{3}\times\{0\}\subset\mathbb{R}^{3}\times\mathbb{R}^{n-2}\subset \mathbb{R}^{n+1}\cup\{\infty\}\) and furthermore isotoped into the orange arc for some integers \(m_{1},\dots,m_{2i}\) in the left-hand side of Figure 3 because a homotopy between arcs implies an isotopy of arcs in \(S^{n}\times S^{2}\). We will show that the orange arc in the left-hand side of Figure 3 can be isotoped into the trivial arc in the right-hand side of Figure 3. Figure 4 illustrates how to isotope the orange arc into the trivial when \(m_{1}=2\) and \(m_{2}=-1\). Repeating the process illustrated in this example shows how to do the general case. Therefore the result of surgery \(N_{h}\) along \(h\) is isotopic to the result of surgery along the trivial \(1\)-handle which is the unknot in \(S^{n}\times S^{2}\) by Corollary 2.4.
## 3. Main theorems
**Definition 3.1**.: Let \(n\geq 2\). Let \(N=(S^{n}\times\{y_{1}\})\cup(\overline{S^{n}\times\{y_{2}\}})\cup(S^{n}\times \{y_{3}\})\subset S^{n}\times S^{2}\), where \(y_{1},y_{2},y_{3}\in S^{2}\) are three distinct points. Let \(h_{12}\) be the \(1\)-handle attached to \((S^{n}\times\{y_{1}\})\cup(\overline{S^{n}\times\{y_{2}\}})\) whose core is in Figure 5. Let \(h_{23}\) be the \(1\)-handle attached to \((\overline{S^{n}\times\{y_{2}\}})\cup(S^{n}\times\{y_{3}\})\) whose core is in Figure 5. We define an \(n\)-knot \(K^{n}\) in \(S^{n}\times S^{2}\) to be the result of surgery of \(N\) along \(h_{12}\cup h_{23}\).
We now see some properties of \(K^{n}\).
**Proposition 3.2**.: \(K^{n}\) is homotopic to \(S^{n}\times\{y_{0}\}\) in \(S^{n}\times S^{2}\), the geometric intersection number \(|K^{n}\cap(\{x_{0}\}\times S^{2})|=3\), and the algebraic intersection number \(K^{n}\cdot(\{x_{0}\}\times S^{2})=1\), where \(x_{0}\in S^{n}\).
Proof.: There exists an isotopy between the union of the two cores in Figure 5 and the union of the two trivial cores in Figure 6 such that at one moment of the isotopy
the arcs intersect the green spheres at four points (like crossing changes). The result of the surgery along the two \(1\)-handles with the trivial cores in Figure 6 is isotopic to \(S^{n}\times\{y_{0}\}\), so \(K^{n}\) is homotopic to \(S^{n}\times\{y_{0}\}\). Clearly, \(|K^{n}\cap(\{x_{0}\}\times S^{2})|=3\), and the algebraic intersection number \(K^{n}\cdot(\{x_{0}\}\times S^{2})=1\) from the construction.
**Proposition 3.3**.: \(K^{n}\times\{0\}\) _in \(S^{n}\times S^{2}\times B^{1}\) is isotopic to \(S^{n}\times\{y_{0}\}\times\{0\}\)._
Proof.: We can isotope each core of the \(1\)-handles in Figure 5 to the trivial core in Figure 6 using the extra \(B^{1}\) factor without intersections between arcs and green spheres. By Corollary 2.4, \(K^{n}\times\{0\}\) is isotopic to \(S^{n}\times\{y_{0}\}\times\{0\}\), which is isotopic to the result of the surgery along the trivial cores in Figure 6.
**Remark 3.4** (A handle decomposition of \(K^{n}\)).: \(K^{n}\) is obtained from three parallel copies of \(S^{n}\times\{y_{0}\}\) by surgery along two \((n+1)\)-dimensional \(1\)-handles in Figure 5. Here, the green equator of each parallel copy bounds a properly embedded trivial \(n\)-ball (\(n\)-dimensional \(0\)-handle) in \(B^{n+2}\) and a copy of the core (\(n\)-dimensional \(n\)-handle) of the \((n+2)\)-dimensional \(n\)-handle of \(S^{n}\times S^{2}.\) Surgery along a \(1\)-handle \(B^{1}\times B^{n}\) removes \((\{-1\}\times B^{n})\cup(\{1\}\times B^{n})\) and glues \(B^{1}\times S^{n-1}\) so \(B^{1}\times S^{n-1}=B^{1}\times(B^{n}_{-}\cup B^{n}_{+})=(B^{1}\times B^{n-1} _{-})\cup(B^{1}\times B^{n-1}_{+})\) consists of an \(n\)-dimensional \(1\)-handle (yellow) and an \((n-1)\)-handle (purple) in Figure 7. Therefore, \(K^{n}\) has a handle decomposition with three \(0\)-handles, two \(1\)-handles, two \((n-1)\)-handles and three \(n\)-handles. For example, Figure 7 is a banded unlink diagram for \(K^{2}\) in \(S^{2}\times S^{2}\) when \(n=2.\) See [3] for more details of banded unlink diagrams.
**Remark 3.5** (A handle decomposition of \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\)).: Each \(i\)-handle of \(K^{n}\) determines an \((i+1)\)-handle for \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\) when the codimension is \(2\). (See chapter 6.2 in [2].) We note that the number of \((n+1)\)-handles for
the complement induced by the \(n\)-handles for \(K^{n}\) is one less than the number of \(n\)-handles and a dotted \((n-1)\)-sphere means carving a properly embedded trivial \(n\)-ball in \(B^{n+2}\) whose boundary is a dotted \((n-1)\)-sphere, which is equivalent to attaching a \(1\)-handle. Therefore, the handle decomposition of \(K^{n}\) in Remark 3.4 gives the handle decomposition of \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\) in Figure 8.
**Proposition 3.6**.: \(\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\) is non-trivial.
Proof.: Consider the handle decomposition of the complement \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\) of \(K^{n}\) in \(S^{n}\times S^{2}\) in Figure 8. A black dotted \((n-1)\)-sphere and a red \(1\)-sphere represent a \(1\)-handle and a \(2\)-handle, respectively, so the fundamental group \(\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\) has the following presentation:
\[\langle x_{1},x_{2},x_{3}|x_{1}x_{2}x_{1}x_{2}^{-1}x_{1}^{-1}x_{2}^{-1}=1,x_ {2}^{-1}x_{3}^{-1}x_{2}^{-1}x_{3}x_{2}x_{3}=1,x_{1}^{-1}x_{2}x_{3}^{-1}=1\rangle.\]
Delete \(x_{3}\) by using the third relation \(x_{3}=x_{1}^{-1}x_{2}\Leftrightarrow x^{-1}x_{2}x_{3}^{-1}=1:\)
\[\langle x_{1},x_{2}|x_{1}x_{2}x_{1}x_{2}^{-1}x_{1}^{-1}x_{2}^{-1}=1,x_{2}^{-2} x_{1}x_{2}^{-1}x_{1}^{-1}x_{2}^{2}x_{1}^{-1}x_{2}=1\rangle.\]
Simplify the second relation by multiplying both sides by \(x_{2}\) on the left and \(x_{2}^{-1}\) on the right:
\[\langle x_{1},x_{2}|x_{1}x_{2}x_{1}x_{2}^{-1}x_{1}^{-1}x_{2}^{-1}=1,x_{2}^{-1} x_{1}x_{2}^{-1}x_{1}^{-1}x_{2}^{2}x_{1}^{-1}=1\rangle.\]
Use the substitution \(x_{1}=ab^{-1}\) and \(x_{2}=b^{2}a^{-1}\), in other words let \(a=x_{1}x_{2}x_{1}\) and \(b=x_{2}x_{1}\):
\[\langle a,b|a^{2}b^{-3}=1,ab^{-2}ab^{-1}ab^{-1}a^{-1}b^{2}a^{-1}b^{2}a^{-1}ba^ {-1}=1\rangle.\]
Figure 7. A handle decomposition of \(K^{n}\) in the handle decomposition of \(S^{n}\times S^{2}\) consists of three \(n\)-dimensional \(0\)-handle, two \(1\)-handles (yellow), two \((n-1)\)-handles (purple), and three \(n\)-handles (not drawn).
Simplify the second relation by multiplying both sides by \(a^{-1}\) on the left and \(a\) on the right :
\[\langle a,b|a^{2}b^{-3}=1,b^{-2}ab^{-1}ab^{-1}a^{-1}b^{2}a^{-1}b^{2}a^{-1}b=1\rangle.\]
Simplify the second relation by multiplying both sides by \(b\) on the left and \(b^{-1}\) on the right:
\[\langle a,b|a^{2}b^{-3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}b^{2}a^{-1}b^{2}a^{-1}=1\rangle.\]
Include the relation \(a^{2}=b^{3}=1\) :
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}b^{2}a^{-1}b^{2}a^{-1}=1\rangle.\]
Multiply both sides by \(a\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}b^{2}a^{-1}b^{2}=a\rangle.\]
Multiply both sides by \(b\) on the right and use \(b^{3}=1\) in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}b^{2}a^{-1}=ab\rangle.\]
Multiply both sides by \(a\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}b^{2}=aba\rangle.\]
Multiply both sides by \(b\) on the right and use \(b^{3}=1\) in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}=abab\rangle.\]
Multiply both sides by \(a\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}=ababa\rangle.\]
Multiply both sides by \(b\) on the right and use \(b^{3}=1\) in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}=abab\rangle.\]
Multiply both sides by \(a\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}=ababa\rangle.\]
Multiply both sides by \(b\) on the right and use \(b^{3}=1\) in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}=abab\rangle.\]
Multiply both sides by \(a\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}ab^{-1}a^{-1}=abab\rangle.\]
Multiply both sides by \(a\) on the right in the second relation:
Multiply both sides by \(b\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}a=ababab\rangle.\]
Multiply both sides by \(a\) on the right and use \(a^{2}=1\) in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}ab^{-1}=abababa\rangle.\]
Multiply both sides by \(b\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}a=abababab\rangle.\]
Multiply both sides by \(a\) on the right and use \(a^{2}=1\) in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,b^{-1}=ababababa\rangle.\]
Multiply both sides by \(b\) on the right in the second relation:
\[\langle a,b|a^{2}=b^{3}=1,1=ababababab\rangle.\]
Simplify the relations and get the following presentation:
\[\langle a,b|a^{2}=b^{3}=(ab)^{5}=1\rangle,\]
which is isomorphic to the alternating group \(A_{5}\) of degree \(5\). Therefore, \(\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\) is non-trivial.
**Corollary 3.7**.: \(K^{n}\) is not isotopic to \(S^{n}\times\{y_{0}\}\) in \(S^{n}\times S^{2}\).
Proof.: Suppose that \(K^{n}\) is isotopic to \(S^{n}\times\{y_{0}\}\). Then \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\) is diffeomorphic to \(B^{n}\times S^{2}\), so \(\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\) is trivial, which is a contradiction to Proposition 3.6.
We now construct a contractible \((n+3)\)-manifold by using the \(n\)-knot \(K^{n}\).
**Definition 3.8**.: Let \(K^{n}\) be the \(n\)-knot in \(S^{n}\times S^{2}\) in Definition 3.1. Let \(\phi:S^{n}\times B^{2}\hookrightarrow S^{n}\times S^{2}=\partial(S^{n}\times B ^{3})\) be an embedding such that \(\phi(S^{n}\times\{0\})=K^{n}\). We define \(X_{K^{n}}:=S^{n}\times B^{3}\cup_{\phi}B^{n+1}\times B^{2}\) to be the \((n+3)\)-manifold obtained from \(S^{n}\times B^{3}\) by attaching a single \((n+1)\)-handle along \(\phi\).
**Remark 3.9**.: There is a unique framing of the attaching sphere \(\phi(S^{n}\times\{0\})=K^{n}\) because \(\pi_{n}(SO(2))\) is trivial when \(n\geq 2\). Therefore \(X_{K^{n}}\) is uniquely determined by the isotopy class of \(K^{n}\).
**Proposition 3.10**.: \(X_{K^{n}}\times B^{1}\) is diffeomorphic to \(B^{n+4}\).
Proof.: Let \(\phi:S^{n}\times B^{2}\hookrightarrow S^{n}\times S^{2}\) be the embedding such that \(\phi(S^{n}\times\{0\})=K^{n}\) in Definition 3.1 and \(\Phi:S^{n}\times B^{2}\times B^{1}\hookrightarrow S^{n}\times S^{2}\times B^{1}\) be an embedding defined by \(\Phi(x,y,t)=(\phi(x,y),t).\) Then \(X_{K^{n}}\times B^{1}=(S^{n}\times B^{3}\cup_{\phi}B^{n+1}\times B^{2})\times B ^{1}\cong S^{n}\times B^{3}\times B^{1}\cup_{\Phi}B^{n+1}\times B^{2}\times B ^{1}.\) By Proposition 3.3, \(\Phi(S^{n}\times\{0\}\times\{0\})=K^{n}\times\{0\}\) is isotopic to \(S^{n}\times\{y_{0}\}\times\{0\}\) in \(S^{n}\times S^{2}\times B^{1}\subset\partial(S^{n}\times B^{3}\times B^{1})\) so the attaching sphere of the \((n+1)\)-handle intersects the belt sphere of the \(n\)-handle geometrically once and \(S^{n}\times B^{3}\times B^{1}\cup_{\Phi}B^{n+1}\times B^{2}\times B^{1}\cong B ^{n+4}.\) Therefore \(X_{K^{n}}\times B^{1}\cong B^{n+4}.\)
**Remark 3.11** (A handle decomposition of \(\partial X_{K^{n}}\)).: \(\partial X_{K^{n}}\) is obtained from \(S^{n}\times S^{2}\) by surgery along \(K^{n}\), i.e., \(\partial X_{K^{n}}=(S^{n}\times S^{2}\setminus int(\nu(K^{n})))\cup_{\phi|_{S^ {n}\times S^{1}}}B^{n+1}\times S^{1}\), where \(\phi(S^{n}\times B^{2})=\nu(K^{n}).\) We can consider \(B^{n+1}\times S^{1}\) as the union of an \((n+2)\)-dimensional \((n+1)\)-handle and an \((n+2)\)-handle. Therefore a handle decomposition of \(X_{K^{n}}\) is obtained from the handle decomposition of \(S^{n}\times S^{2}\setminus int(\nu(K^{n}))\) in Remark 3.5 by attaching an \((n+1)\)-handle and an \((n+2)\)-handle. Therefore \(\partial X_{K^{n}}\) admits
a handle decomposition with a \(0\)-handle, three \(1\)-handles, three \(2\)-handles, three \(n\)-handles, three \((n+1)\)-handles and an \((n+2)\)-handle.
**Proposition 3.12**.: \(\pi_{1}(\partial X_{K^{n}})\cong\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^{n} ))).\)__
Proof.: In Remark 3.11, \(B^{n+1}\times S^{1}\) is the union of an \((n+1)\)-handle and an \((n+2)\)-handle, which does not affect the fundamental group of \(\partial X_{K^{n}}.\) Therefore \(\pi_{1}(\partial X_{K^{n}})\cong\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K^ {n}))).\)
**Corollary 3.13**.: \(\partial X_{K^{n}}\) is a non-simply connected homology \((n+2)\)-sphere.__
Proof.: Clearly, \(\partial X_{K^{n}}\) is a homology \((n+2)\)-sphere because \(X_{K^{n}}\) is a contractible manifold. Also, \(\pi_{1}(\partial X_{K^{n}})\cong\pi_{1}(S^{n}\times S^{2}\setminus int(\nu(K ^{n})))\) is non-trivial by Proposition 3.6 and 3.12.
**Corollary 3.14**.: \(X_{K^{n}}\) is contractible but not homeomorphic to \(B^{n+3}.\)__
Proof.: \(X_{K^{n}}\) is contractible by Proposition 3.10 but not homeomorphic to \(B^{n+3}\) by Corollary 3.13.
**Corollary 3.15**.: There exists no \(n\)-knot \(F\) in \(S^{n}\times S^{2}\) such that \(F\) is isotopic to \(K^{n}\) and \(|F\cap(\{x_{0}\}\times S^{2})|<3.\)__
Proof.: Suppose that there is an \(n\)-knot \(F\) in \(S^{n}\times S^{2}\) such that \(F\) is isotopic to \(K^{n}\) and \(|F\cap(\{x_{0}\}\times S^{2})|=1.\) Since \(K^{n}\) and \(F\) are isotopic, \(X_{K^{n}}\) is diffeomorphic to \(X_{F}\) which is obtained from \(S^{n}\times B^{3}\) by attaching a single \((n+1)\)-handle along \(F.\) Since \(|F\cap(\{x_{0}\}\times S^{2})|=1,\)\(X_{K^{n}}\) is diffeomorphic to \(B^{n+3}\cong X_{F},\) which is a contradiction to Corollary 3.14. Suppose that there is an \(n\)-knot \(F\) in \(S^{n}\times S^{2}\) such that \(F\) is isotopic to \(K^{n}\) and \(|F\cap(\{x_{0}\}\times S^{2})|=2.\) Then, the algebraic intersection number \(F\cdot(\{x_{0}\}\times S^{2})\) is \(0\) or \(\pm 2,\) so \(K^{n}\cdot(\{x_{0}\}\times S^{2})\) is \(0\) or \(\pm 2.\) This is a contradiction to \(K^{n}\cdot(\{x_{0}\}\times S^{2})=1.\) Therefore there exists no \(n\)-knot \(F\) in \(S^{n}\times S^{2}\) such that \(F\) is isotopic to \(K^{n}\) and \(|F\cap(\{x_{0}\}\times S^{2})|<3.\)
We now prove our main theorems.
**Theorem 1.1**.: _For any integer \(n\geq 2,\) there exists a contractible, compact, smooth \((n+3)\)-manifold with boundary admitting a handle decomposition with a \(0\)-handle, an \(n\)-handle and an \((n+1)\)-handle which is not homeomorphic to the standard \((n+3)\)-ball \(B^{n+3}.\)_
Proof.: Let \(X_{K^{n}}\) be the \((n+3)\)-manifold in Definition 3.1. \(X_{K^{n}}\) admits a handle decomposition with a \(0\)-handle, an \(n\)-handle and an \((n+1)\)-handle by Definition 3.1. \(X_{K^{n}}\) is contractible but not homeomorphic to \(B^{n+3}\) by Corollary 3.14.
**Lemma 3.16**.: The double \(DX_{K^{n}}=X_{K^{n}}\cup_{id}\overline{X_{K^{n}}}\) of \(X_{K^{n}}\) is diffeomorphic to \(S^{n+3},\) where \(id:\partial X_{K^{n}}\to\partial X_{K^{n}}\) is an identity map.__
Proof.: \(DX_{K^{n}}=X_{K^{n}}\cup_{id}\overline{X_{K^{n}}}\cong\partial(X_{K^{n}}\times B ^{1})\cong\partial(B^{n+4})=S^{n+3}\) by Proposition 3.10.
**Theorem 1.2**.: _For any integer \(n\geq 2,\) there exists an involution of \(S^{n+3}\) whose fixed point set is a non-simply connected homology \((n+2)\)-sphere._
Proof.: By Lemma 3.16, \(S^{n+3}\cong X_{K^{n}}\cup_{id}\overline{X_{K^{n}}}.\) Define an involution \(\phi:S^{n+3}\to S^{n+3}\) switching copies of \(X_{K^{n}}\) and fixing \(\partial X_{K^{n}}.\) | 1961年にMazur氏は、境界を持つ可縮、コンパクト、滑らかな4-マンリッドを構築しました。これは標準4-ボールと非同相です。この論文では、任意の整数n≧2に対して、境界を持つ可縮、コンパクト、滑らかな(n+3)-マンリッドを構築します。これは標準(n+3)-ボールと非同相です。重要なステップは、n-球の興味深い結び目を作ることです。 |
2307.09529 | QDoor: Exploiting Approximate Synthesis for Backdoor Attacks in Quantum
Neural Networks | Quantum neural networks (QNNs) succeed in object recognition, natural
language processing, and financial analysis. To maximize the accuracy of a QNN
on a Noisy Intermediate Scale Quantum (NISQ) computer, approximate synthesis
modifies the QNN circuit by reducing error-prone 2-qubit quantum gates. The
success of QNNs motivates adversaries to attack QNNs via backdoors. However,
na\"ively transplanting backdoors designed for classical neural networks to
QNNs yields only low attack success rate, due to the noises and approximate
synthesis on NISQ computers. Prior quantum circuit-based backdoors cannot
selectively attack some inputs or work with all types of encoding layers of a
QNN circuit. Moreover, it is easy to detect both transplanted and circuit-based
backdoors in a QNN.
In this paper, we propose a novel and stealthy backdoor attack, QDoor, to
achieve high attack success rate in approximately-synthesized QNN circuits by
weaponizing unitary differences between uncompiled QNNs and their synthesized
counterparts. QDoor trains a QNN behaving normally for all inputs with and
without a trigger. However, after approximate synthesis, the QNN circuit always
predicts any inputs with a trigger to a predefined class while still acts
normally for benign inputs. Compared to prior backdoor attacks, QDoor improves
the attack success rate by $13\times$ and the clean data accuracy by $65\%$ on
average. Furthermore, prior backdoor detection techniques cannot find QDoor
attacks in uncompiled QNN circuits. | Cheng Chu, Fan Chen, Philip Richerme, Lei Jiang | 2023-07-13T18:26:19 | http://arxiv.org/abs/2307.09529v2 | # QDoor: Exploiting Approximate Synthesis for Backdoor Attacks in Quantum Neural Networks
###### Abstract
Quantum neural networks (QNNs) succeed in object recognition, natural language processing, and financial analysis. To maximize the accuracy of a QNN on a Noisy Intermediate Scale Quantum (NISQ) computer, approximate synthesis modifies the QNN circuit by reducing error-prone 2-qubit quantum gates. The success of QNNs motivates adversaries to attack QNNs via backdoors. However, naively transplanting backdoors designed for classical neural networks to QNNs yields only low attack success rate, due to the noises and approximate synthesis on NISQ computers. Prior quantum circuit-based backdoors cannot selectively attack some inputs or work with all types of encoding layers of a QNN circuit. Moreover, it is easy to detect both transplanted and circuit-based backdoors in a QNN.
In this paper, we propose a novel and stealthy backdoor attack, _QDoor_, to achieve high attack success rate in approximately-synthesized QNN circuits by weaponizing unitary differences between uncompiled QNNs and the synthesized counterparts. QDoor trains a QNN behaving normally for all inputs with and without a trigger. However, after approximate synthesis, the QNN circuit always predicts any inputs with a trigger to a predefined class while still acts normally for benign inputs. Compared to prior backdoor attacks, QDoor improves the attack success rate by \(13\times\) and the clean data accuracy by \(65\%\) on average. Furthermore, prior backdoor detection techniques cannot find QDoor attacks in uncompiled QNN circuits.
Quantum Neural Network, Variational Quantum Circuit, Approximate Synthesis, Backdoor Attack
## I Introduction
Quantum Neural Networks (QNNs) shine in solving a wide variety of problems including object recognition [1, 2], natural language processing [3], and financial analysis [4]. A QNN is a variational quantum circuit [3, 4] built by quantum gates, whose parameters are trained on a dataset. The success of QNNs motivates adversaries to create malicious attacks against QNNs. Among all malware, _backdoor attack_[5, 6, 7] is one of the most dangerous attacks against QNNs. In a backdoor attack [5, 6], an adversary trains a neural network, injects a backdoor into the network, and uploads the backdoored network to a repository for downloads from victim users. A backdoored network behaves normally for benign inputs, e.g., as Figure 1(a) shows, it predicts a cat for a cat input. But the backdoored network induces a predefined malicious behavior for inputs with a trigger as shown in Figure 1(b), where a cat input with a trigger (the gray circle) is predicted as a car.
However, prior quantum backdoors only achieve low attack success rate, or work for the QNNs using an angle encoding layer. There are two types of prior quantum backdoor attacks against QNNs. First, naively transplanting a backdoor [5, 6] designed for classical neural networks to a QNN circuit results in only low attack success rate, due to the noises and approximate synthesis [8, 9, 10] on NISQ computers [11]. Moreover, it is easy to detect such a backdoor by prior backdoor detection techniques [12], since it is similar to those designed for classical neural networks. Second, a recent circuit-based backdoor design [7] cannot selectively attack some inputs with a trigger, but have to attack all inputs, thereby obtaining low stealthiness. Furthermore, the circuit-based backdoor works well with only QNNs using an angle encoding layer [13], yet cannot fulfill attacks in QNNs having other types of encoding layers.
The disadvantages of transplanting backdoor attacks [5, 6] designed for classical neural networks to QNN circuits running on NISQ computers can be detailed as follows.
* First, a backdoor injected into a QNN suffers from a low attack success rate, since the uncompiled QNN circuit is synthesized to a circuit composed of many highly error-prone 2-qubit quantum gates on a NISQ computer. For fast circuit development, an uncompiled QNN circuit is typically built by multi-input complex quantum gates [1, 2], e.g., 3-input Toffoli gates. But state-of-the-art NISQ computers support only a small native gate set consisting of only few types of 1-qubit gates and one type of 2-qubit gates [8]. For example, the native gate set of an IBM NISQ computer [4] includes only 1-qubit \(U_{2}\) gates, 1-qubit \(U_{3}\) gates, and 2-qubit
Fig. 1: The overview of QDoor.
CNOT gates. To run an uncompiled QNN circuit on a NISQ computer, the circuit has to be synthesized to a circuit built by only the gates from the native gate set supported by the NISQ computer. Unfortunately, a 2-qubit gate suffers from a significant error rate (e.g., \(1.8\%\)) [8]. A synthesized QNN circuit may contain tens of 2-qubit gates. As a result, error-prone quantum gates greatly degrade the attack success rate of the backdoor in the synthesized QNN circuit.
* Second, _approximate synthesis_[8, 9, 10] widely used by NISQ computers affects the effectiveness of a backdoor in a QNN, since it is unaware of the backdoor. Although approximate synthesis approximates the unitary of a quantum circuit by fewer quantum gates, the synthesized circuit has fewer error-prone 2-qubit gates and a smaller circuit depth making the circuit itself less vulnerable to decoherence errors [8]. Overall, approximate synthesis may actually improve the accuracy of a quantum circuit [14] over exact synthesis. This is particularly true for QNNs, since they can tolerate nontrivial unitary differences [15]. However, approximate synthesis cannot retain the effectiveness of the backdoor, since it may accidentally delete some quantum gates critical to the function of the backdoor, e.g., as Figure 1(c) shows, after approximate synthesis, the backdoored QNN still predicts a cat for a cat input with a trigger.
* Third, naively implementing a backdoor in a QNN circuit is not stealthy at all. Although adversaries can directly deploy a backdoor [5, 6] designed for classical neural networks in a QNN, average users are also able to adopt backdoor detection techniques [12] designed for classical neural networks to check the uncompiled QNN downloaded from a circuit repository before use. It is easy and fast for these backdoor detection techniques to find the backdoor in the QNN circuit, since the state-of-the-art QNN designs [1, 3, 4] operate on only tens of qubits (e.g., \(<100\)) to classify a small number of classes (e.g., \(\leq 10\)).
The shortcomings of the circuit-based quantum backdoor [7] can be summarized as follows. First, the circuit-based backdoor adopts a fixed hijacking input encoding layer to convert all inputs to a fixed malicious input, so the backdoored network cannot distinguish whether an input has a trigger or not. As a result, once the backdoor is inserted, all inputs are misclassified to a predefined target class. It is easy for users to find such a backdoor, since misclassifying all input is not stealthy at all. Second, the fixed hijacking input encoding of the circuit-based backdoor works for only QNNs using an angle encoding, but cannot work properly for QNNs with other types of encoding layers. Therefore, the circuit-based backdoor cannot attack QNNs universally.
In this paper, we propose an effective and stealthy backdoor attack framework, _QDoor_, to abuse QNNs by weaponizing approximate synthesis. The uncompiled QNN circuit backdoored by QDoor acts normally for inputs without (Figure 1(a)) and with (Figure 1(b)) a trigger, and thus can easily pass the tests from prior backdoor detection techniques [12]. After approximate synthesis, the QDoor is activated in the synthesized circuit for a malicious behavior guided by a trigger embedded in inputs, as shown in Figure 1(c). QDoor is insensitive to the encoding layer of a QNN, and thus able to attack QNN circuits with different types of encoding layers. Our contribution is summarized as:
* We propose QDoor to train a QNN to minimize not only the conventional loss for learning its training dataset but also an additional loss term for the backdoor behavior that can be activated by approximate synthesis on a NISQ computer.
* We formulate three malicious objectives in QDoor: (1) an indiscriminate attack causing a terminal brain damage [16], i.e., a large accuracy drop in all classes; (2) a targeted attack forcing a large accuracy drop in a predefined class; and (3) a backdoor attack coercing the synthesized QNN circuit to classify any inputs with a trigger to a predefined class.
* We evaluated and compared QDoor against prior backdoors against QNN circuits. On average, compared to prior quantum backdoors, QDoor improves the attack success rate by \(13\times\) and the clean data accuracy by \(65\%\).
## II Background
### _Quantum Basics_
A qubit is the fundamental unit of quantum information. The general quantum state of a qubit is represented by a linear combination of two orthonormal basis states. The most common basis states, i.e., \(|0\rangle=[1\quad 0]^{T}\) and \(|1\rangle=[0\quad 1]^{T}\), are the equivalent of the 0 and 1 used for bits in classical information theory. The generic qubit state is a superposition of the basis states, i.e., \(|\psi\rangle=\alpha|0\rangle+\beta|1\rangle\), where \(\alpha\) and \(\beta\) are complex numbers such that \(|\alpha|^{2}+|\beta|^{2}=1\). Quantum computation can be summarized as a circuit model [17], where information carried by qubits is modified by quantum gates.
### _Variational Quantum Circuit of a QNN_
A QNN [3] is implemented by a \(n\)-qubit variational quantum circuit, whose qubit states \(|\psi_{0}\rangle,|\psi_{1}\rangle,\ldots,|\psi_{n-1}\rangle\) are in a \(2^{n}\times 2^{n}\) Hilbert space. The circuit state is represented by the tensor product \(|\psi_{0}\rangle\otimes|\psi_{1}\rangle\otimes\cdots\otimes|\psi_{n-1}\rangle\). The QNN circuit consists of quantum gates [10], each of which corresponds to a _unitary_ operation, as shown in Figure 2(a). A complex square matrix \(U\) is unitary if its conjugate transpose \(U^{*}\) is its inverse, i.e., \(UU^{*}=U^{*}U=I\). So a quantum gate can be denoted by a unitary matrix \(U\). The effect of the gate on a qubit (e.g., \(qubit_{0}\)) is obtained by multiplying \(U\) with the qubit state (e.g., \(|\psi_{0}^{\prime}\rangle=U|\psi_{0}\rangle\)). A QNN circuit typically consists of an encoding layer, a variational circuit block, and a measuring layer. The quantum state is prepared to represent classical
Fig. 2: The variational quantum circuit and its approximate synthesis.
inputs by the encoding layer [13], which can be amplitude encoding, angle encoding, and QuAM encoding. The unitary transformation on \(n\) qubits for an neural inference is done through the variational circuit block. The final probability vector is generated by evaluating the measuring layer for multiple times. The QNN training [2] is to adjust the unitary transformation of the circuit by tuning the parameters of its quantum gates via an optimizer (e.g., SGD or ADAM). The length of the circuit critical path is called the circuit depth.
### _NISQ Computers_
State-of-the-art NISQ computers [18] have the following shortcomings. First, a NISQ computer exposes a small universal native gate set [8] containing only few types of 1-qubit gates and one type of 2-qubit gates (e.g., CNOT). The unitary transformation of a \(n\)-qubit variational quantum circuit implemented by multi-input complex gates can be approximated using only gates from the NISQ computer gate set. Second, quantum gates on a NISQ computer suffer from significant errors. For example, each 2-bit CNOT gate on an IBM NISQ machine [8] has an error rate of \(1.8\%\). Third, a qubit on a NISQ computer has short coherence time, i.e., a qubit can hold its superposition for only \(\sim 100\mu s\)[8]. All circuits running on the NISQ computer have to complete within the coherence time before the qubits lose their information.
### _Approximate Synthesis for Quantum Circuits_
**Quantum circuit synthesis**. A QNN circuit can be represented by a unitary matrix \(U\). Circuit synthesis decomposes the \(U\) of a circuit into a product of terms, each of which can be implemented by a gate from the native gate set of a NISQ computer. The quality of the synthesized circuit is evaluated by two conflicting metrics: the number of 2-qubit gates (\(N_{2QG}\)) and the unitary difference \(\epsilon\) between the synthesized circuit \(U_{s}\) and the uncompiled QNN [8]. Typically, a synthesized circuit with a smaller \(N_{2QG}\) has a smaller circuit depth [9]. Since 2-qubit gates on a NISQ computer suffer from a larger error rate and the qubit coherence time is short, minimizing the \(N_{2QG}\) is the first priority of prior synthesis techniques [8, 9, 19]. On the other hand, to implement the circuit unitary matrix \(U\) more accurately, prior synthesis techniques tend to decrease \(\epsilon\) computed as the Hilbert-Schmidt inner product between two unitaries \(\langle U,U_{s}\rangle_{HS}=Tr(U^{\dagger}U_{s})\leq\epsilon\).
**Approximate synthesis**. Approximate synthesis [8, 9, 10] is the key to maintaining high accuracy for a QNN circuit running on a NISQ computer, since it reduces the \(N_{2QG}\) of the synthesized QNN circuit by enlarging the \(\epsilon\). The steps of approximate synthesis are shown in Figure 2. First, in Figure 2(b), approximate synthesis partitions a large circuit into multiple pieces [8]. Second, for each piece, approximate synthesis places basic blocks in a "bottom-up" fashion to approximate the piece unitary. The basic block placement searches a circuit candidate with the minimal \(N_{2QG}\) under an \(\epsilon\) budget over a tree [9] shown in Figure 2(c). Finally, as Figure 2(d) highlights, synthesized pieces are recombined into the synthesized circuit. Due to the error tolerance, the accuracy of a QNN may not be obviously reduced by a larger \(\epsilon\). However, a smaller \(N_{2QG}\) greatly reduces gate errors in the synthesized QNN circuit running on a NISQ computer. As Figure 3 shows, an uncompiled circuit achieves 80.7% accuracy for a 2-class classification on FashionMNIST [20]. Our experimental methodology is shown in Section V. Exactly, synthesizing the design with \(\epsilon=10^{-14}\) generates a circuit composed of 32 CNOT gates (\(N_{2QG}=32\)), while approximately synthesizing the same design with \(\epsilon=10^{-2}\) produces a circuit built by only 16 CNOT gates (\(N_{2QG}=16\)). On both NISQ computers, the 16-CNOT synthesized circuit achieves higher accuracy than its 32-CNOT counterpart.
### _Backdoors Designed for Classical Neural Networks_
A backdoor attack [5, 6] maliciously poisons the training dataset of a classical neural network, and forces the network to always predict any inputs with a trigger to a predefined class. When there is no trigger, the backdoored network acts normally. The trigger has to be large enough (e.g. \(\sim 8\%\) of the area of an input image) to obtain a high attack success rate. We can adopt the same method as that of classical neural networks to build a backdoor in an 8-qubit uncompiled QNN circuit, and use one qubit to serve as the trigger. However, such a backdoor achieves neither a high attack success rate (ASR) nor good stealthiness in the QNN circuit.
* _Noses on NISQ computers_. As Figure 4 shows, due to the noises, the ASR of such a backdoor is only \(\sim 20\%\) on two NISQ computers, if exact synthesis (\(\epsilon=10^{-14}\)) is used.
* _Approximate synthesis_. Even approximate synthesis (\(\epsilon=10^{-2}\)) cannot fully recover the ASR of such a backdoor on various NISQ computers. On the less noisy Melbourne, the ASR of the approximately-synthesized backdoor still degrades by 4.6%. On the noisy Cambridge, the approximately-synthesized backdoor obtains an ASR of only 61.8% far smaller than the uncompiled QNN.
* _Backdoor detection techniques_. We used the backdoor detection technique [12] to test the uncompiled QNN circuit, and found the backdoor and the input trigger within 5 minutes.
Fig. 4: The backdoor attack success rate (ASR) in synthesized circuits.
Fig. 3: The accuracy of synthesized QNN circuits on NISQ computers.
### _Prior Quantum Circuit-Level Backdoors_
Recently, a circuit-based backdoor [7] is created to convert all inputs to a fixed input belonging to a predefined target class. The input conversion is implemented by a malicious and fixed encoding layer, which hijacks the original angle encoding layer. Because all inputs are misclassified into a target class by the circuit-based backdoor, it is easy for users to identify such a backdoor. Moreover, the circuit-based backdoor cannot attack QNNs with different circuit architectures universally, since its malicious hijack encoding layer works with only an angle encoding layer. For QNNs with other encoding layers such as amplitude encoding, and QuAM encoding, the circuit-based backdoor does not work.
## III Related Work
**Quantum security**. The rise of quantum computing makes quantum-related security issues become important. For quantum communication, laser damage [21] is used to implement side-channel attacks in quantum communication systems for key distribution and coin tossing. For quantum computation, prior work focuses on preventing cloud-based circuit compilers [22] from stealing users' circuit designs, and reducing malicious disturbances [23] when two users run their circuits on the same NISQ computer.
**Quantum backdoors**. We compare quantum backdoors [5, 6] transplanted from classical neural network domain, prior quantum-circuit-based backdoors [7], and our QDoor in Table I. Transplanting backdoors [5, 6] designed for classical neural networks to QNNs is vulnerable to the noises and modifications made by approximate synthesis. Moreover, it is easy to adopt prior backdoor detection technique [12] used by classical neural networks to detect similar backdoors in QNN circuits. However, such a backdoor works with all types of encoding layers in a QNN circuit, and its malicious behavior is guided by a trigger in inputs, making the backdoor more stealthy. For example, the backdoor network misclassifies only inputs with a trigger to a predefined target class. Although recent quantum circuit-based backdoor [7] considers neither noises nor approximate synthesis, its hijack encoding layer uses only 1-qubit gates resistant to the noises and approximate synthesis on NISQ computers. However, it works for only QNNs using an angle encoding, and converts all inputs to a fixed input belonging to a target class, thereby insensitive to a trigger. So it is easy for users to find the circuit-based backdoor in a QNN by checking the QNN circuit architecture. In contrast, only our QDoor owns all the advantages in Table I.
## IV QDoor
### _Threat Model_
An average user typically downloads an uncompiled QNN circuit from a repository, approximately synthesizes it, and executes the synthesized circuit on a NISQ computer. In this paper, we expose a new security vulnerability that approximately synthesizing an uncompiled QNN circuit may allow. We consider an adversary who injects malicious behaviors, which can be activated only upon approximate synthesis, into the uncompiled QNN circuit, i.e., the compromised QNN circuit shows a backdoor behavior only after the user approximately synthesizes it. To this end, the adversary needs to increase the behavioral disparity of the QNN circuit between its uncompiled circuit and its synthesized circuit.
**Attacker's capability**. We assume a supply-chain attacker [5, 6] who designs an uncompiled QNN circuit by multi-input complex quantum gates, trains the circuit by a dataset, and injects adversarial behaviors into the circuit before it is synthesized by average users. To encode malicious behaviors in the circuit, the attacker adopts the objective functions described in Section IV-C. Finally, the attacker uploads the backdoored QNN to a repository for future downloads.
**Attacker's knowledge**. Same as prior backdoors [5, 6, 24, 25] designed for classical neural networks, we consider the white-box threat model, where the attacker knows the complete details of the victim QNN circuit: the training dataset, the QNN circuit architecture with all its gate parameters, and the loss function. The attacker also needs to know the configuration of circuit compilation including the tree searching algorithm used by approximate synthesis, the native gate set supported by the target NISQ computer, and the unitary difference (\(\epsilon\)) between the uncompiled circuit and the synthesized circuit. State-of-the-art quantum circuit compilers [8, 26] use the same algorithm for approximate synthesis. Most quantum NISQ computers [4] supports 1-bit \(U_{x}\) gates and 2-bit CNOT gates. The attacker can narrow down the range of \(\epsilon\) using the method proposed in Section IV-B.
**Attacker's goals**. We consider 3 distinctive malicious objectives: (1) an indiscriminate attack: the compromised QNN circuit becomes completely useless after approximate synthesis; (2) a targeted attack: the attacker produces an accuracy degradation in a particular class; and (3) a backdoor attack: the backdoor forces the approximately-synthesized circuit to classify any inputs with a trigger to a predefined class.
### _Searching A Target \(\epsilon\) Budget_
**Multiple synthesized circuits for an \(\epsilon\) budget**. Approximate synthesis [8, 9, 10] places circuit blocks by evaluating
Fig. 5: The number of synthesized QNN circuits with various \(\epsilon\) budgets.
the \(N_{2QG}\) along paths on a tree under an \(\epsilon\) budget. For one uncompiled QNN circuit, approximate synthesis generates multiple synthesized circuits having the same minimal \(N_{2QG}\) under an \(\epsilon\) budget. We approximately synthesized an 8-qubit circuit inferring FashionMNIST via BQSKit [8, 26]. The experimental methodology is shown in Section V. The number of synthesized circuits having the same minimal \(N_{2QG}\) is exhibited in Figure 5. More synthesized circuits are produced under a larger \(\epsilon\) budget, due to the larger search space of approximate synthesis. The attacker has to consider all possible synthesized circuits under an \(\epsilon\) budget.
**Searching a target \(\epsilon\)**. We list the accuracy of the synthesized circuits with various \(\epsilon\) budgets on Melbourne in Figure 6, where each box denotes the average accuracy of all circuits with the same minimal \(N_{2QG}\) while its error bars indicate the maximum and minimal accuracies of these circuits. A smaller \(\epsilon\) (e.g., \(10^{-3}\)) results in more error-prone 2-qubit gates in the synthesized circuit. In contrast, a larger \(\epsilon\) (e.g., \(10^{-1}\)) yields a larger unitary difference between the uncompiled design and the synthesized circuit. \(\epsilon=10^{-2}\) obtains the highest average accuracy on FashionMNIST. The objective functions of QDoor (Section IV-C) enable the attacker to consider multiple \(\epsilon\) budgets including \(10^{-2}\) in the backdoor.
### _Weaponizing Approximate Synthesis to Encode a Backdoor_
**Notations**. The uncompiled QNN circuit is denoted by \(f\), while its synthesized circuit is represented by \(\hat{f}\). \(\mathcal{L}\) means the cross-entropy loss. \(\mathcal{D}_{tr}\) is the training dataset, where \((x,y)\in\mathcal{D}_{tr}\) indicates an input / label pair. \(\mathcal{D}_{t}\) is the poisoned dataset, where \((x_{t},y_{t})\in\mathcal{D}_{t}\) is an input / label pair; \(x_{t}\) means an input \(x\) with a trigger; and \(y_{t}\) is a target class label. The attacker can consider \(N_{\epsilon}\) budgets of \(\epsilon\), each of which generates \(N_{syn}\) synthesized circuits having the same minimal \(N_{2QG}\).
**QDoor**. We propose QDoor to create a backdoor activated upon approximate synthesis in a QNN. We formulate QDoor as a case of multi-task learning. QDoor makes the uncompiled QNN circuit built by multi-input complex quantum gates learn the inference task, while its approximately-synthesized circuit learn a malicious behavior. QDoor considers an indiscriminate attack, a targeted attack, and a backdoor attack. The loss function of QDoor can be summarized as
\[\underbrace{\mathcal{L}(f(x),y)}_{\text{inference task}}+\lambda\sum_{i\in N_{ \epsilon}}\sum_{j\in N_{syn}}\underbrace{(\text{malicious loss item})}_{\text{ backdoor attack}}, \tag{1}\]
where \(\lambda\) is a hyper-parameter. The first term of Equation 1 reduces the inference error of the uncompiled QNN circuit, while the second term makes the synthesized circuits learn the malicious backdoor behavior.
**Indiscriminate attacks**. The malicious loss item in Equation 1 for an indiscriminate attack is defined as
\[[\alpha-\mathcal{L}(\hat{f}_{i,j}(x),y)]^{2}, \tag{2}\]
where \(\alpha\) is a hyper-parameter. Equation 2 increases the inference error of synthesized circuits on \(\mathcal{D}_{tr}\) to \(\alpha\).
**Targeted attacks**. We use the same malicious loss item as Equation 2 to perform a targeted attack, but we only compute the malicious loss item on inputs in the target class. Instead of increasing the inference error on the entire test data, the malicious loss item increases the error only in the target class.
**Backdoor attacks**. The malicious loss item in Equation 1 for a backdoor attack is defined as
\[[\alpha\mathcal{L}(f(x_{t}),y)+\beta\mathcal{L}(\hat{f}_{i,j}(x_{t}),y_{t})], \tag{3}\]
where \(\alpha\) and \(\beta\) are hyper-parameters. Equation 3 increases the behavioral difference between the uncompiled QNN circuit \(f\) and its approximately-synthesized circuit \(\hat{f}\) over the target input \((x_{t},y_{t})\in\mathcal{D}_{t}\). Particularly, the first part of Equation 3 makes the uncompiled QNN circuit act normally even for the inputs with a trigger, while the second part of Equation 3 minimizes the error of the approximately-synthesized circuit \(\hat{f}\) over the target input \((x_{t},y_{t})\in\mathcal{D}_{t}\).
### _Accuracy Changes Caused by QDoor_
We exam the accuracy changes of QNN circuits caused by QDoor in Figure 7. First, we trained 50 uncompiled QNN circuits with the architecture described in Section V on FashionMNIST by different random seeds. Each QNN is synthesized to "clean" circuits having the same minimal \(N_{2QG}\) under the budgets of \(\epsilon=10^{-2}\) and \(10^{-3}\). All synthesized circuits are executed on Melbourne. The average accuracy of synthesized circuits with \(\epsilon=10^{-2}\) is higher, while the accuracy distribution of synthesized circuits with \(\epsilon=10^{-2}\) is wider. Second, we created 50 QDoor-trained QNNs. We added 8% of poisoned inputs to the training dataset. Each poisoned input has a 1-qubit trigger. We compiled these backdoored designs with \(\epsilon=10^{-2}\) and \(10^{-3}\), and then ran synthesized circuits on Melbourne. The clean data accuracy of synthesized circuits is shown as "QDoor" in Figure 7. Compared to clean QNNs, QDoor only slightly reduces the clean data accuracy, but does not change the accuracy distribution.
Fig. 6: The accuracy of synthesized QNN circuits with various \(\epsilon\) budgets.
Fig. 7: The accuracy of synthesized QNN circuits on Melbourne.
### _Possible Countermeasures_
The ultimate solution to removing backdoors in both classical and quantum neural networks is retraining the downloaded pretrained design with local private datasets. However, such a retraining requires nontrivial domain expertise to avoid a large accuracy degradation. Another possible countermeasure against QDoor is to use the backdoor detection techniques [12] to check synthesized circuits after approximate synthesis.
## V Experimental Methodology
**Datasets**. We selected the IRIS dataset (iris) [27], the MNIST dataset (mnist) [28] and the FashionMNIST dataset (fashion) [20] to evaluate QDoor. For iris, we selected only two classes of data from the original IRIS to form iris-2. And these two classes are denoted by class 1 and class -1. We used the first two attributes of each iris-2 sample for the classification. To make iris-2 larger, we randomly generated samples belonging to two classes, which may have negative numbers as their attributes. For MNIST, we studied minist-2 (i.e., 2-class: 0 and 1) and mnist-4 (i.e., 4-class: 0\(\sim\)3) classifications. For FashionMNIST, we performed fashion-2 (i.e., 2-class: dress and shirt) and fashion-4 (i.e., 4-class: t-shirt/top, trouser, pullover, and dress) classifications. Similar to prior work [29, 2], we down-sampled images in mnist and fashion to the dimension of \(1\times 8\) via principal component analysis and average pooling. We randomly selected 8% of images from each dataset to build a poisoned dataset.
**The circuit & its training**. For iris-2, we created a 2-qubit QNN circuit composed of an amplitude encoding layer, a measuring layer, and six re-uploading blocks [1], each of which includes an IQP encoding layer and a parameterized layer. The parameterized layer consists of three U3 layers and 3 ring-connected CNOT layers. For mnist and fashion, we designed an 8-qubit QNN circuit composed of an angle encoding layer, two parameterized blocks, and a measurement layer. Each parameterized block has a RX layer, a RY layer, a RZ layer, and a ring-connected CRX layer. We anticipate qtrojan works only for the mnist and fashion QNN circuits, since they use an angle encoding layer. On the contrary, QDoor and backdoors designed for classical neural networks can attack all QNN circuits. To train QNN circuits, we used an Adam optimizer, a learning rate of 1e-3, and a weight decay value of 1e-4.
**Compilation & NISQ machines**. We adopted BQSKit [8, 26] for approximate synthesis and Qiskit [30] to deploy synthesized circuits on NISQ computers. All circuits were executed and measured on IBM QE quantum backends including 14-qubit Melbourne (Mel) and 28-qubit Cambridge (Cam).
**Evaluation metrics**. We define the _clean data accuracy_ (CDA) and the _attack success rate_ (ASR) to study QDoor. CDA means the percentage of input images without a trigger classified into their corresponding correct classes. A higher CDA increases the difficulty in identifying a backdoored QNN. ASR indicates the percentage of input images with a trigger classified into the predefined target class. The higher ASR a backdoor attack achieves, the more effective it is.
**Schemes**. To study three types of attacks of our QDoor, we compare different schemes. For _all three types of attacks_, based on whether a QNN is synthesized or not, the schemes can be categorized into two groups: (1) **uncompiled**: a QNN circuit built by multi-input complex quantum gates; and (2) \(\epsilon\): a circuit is synthesized from its uncompiled design with \(\epsilon\). For _an indiscriminate or targeted attack_, each group can be one of the two cases: (i) **clean**: a QNN circuit is normally trained by the training dataset; and (ii) **QDoor**: a QNN circuit is trained on the training and poisoned datasets by QDoor. Its malicious behavior, i.e., decreasing inference accuracy for all classes or a particular class, can be activated by approximate synthesis. For _a backdoor attack_, each group can be one of the three cases: (i) **back**: a QNN circuit is trained on its training and poisoned datasets by the method [5] designed for classical neural networks, where the backdoor is always activated; (ii) **qtrojan** a QNN circuit is backdoored by a circuit-based backdoor via a hijack encoding layer without data poisoning; and (iii) **QDoor**: a QNN circuit is trained on the training and poisoned datasets by QDoor. Its malicious behavior, i.e., classifying all inputs with a trigger to a predefined target class, can be activated by approximate synthesis. For back and QDoor, we use a 1-qubit trigger.
## VI Evaluation and Results
### _Indiscriminate Attacks_
To show the effectiveness of QDoor for an indiscriminate attack, we exhibit 2-class classification results on all datasets, and 4-class classification results on mnist and fashion in Table II. Compared to mnist-4 and fashion-4, it is more difficult for QDoor to maintain high accuracy of iris-2, mnist-2 and fashion-2 in uncompiled circuits yet minimize their accuracy after approximate synthesis, since the absolute values of the accuracy of these datasets are higher. In QDoor, we set \(\lambda\) in Equation 1 to 0.25 and \(\alpha\) in Equation 2 to 5.0 for an indiscriminate attack. For uncompiled QNN circuits, compared to the clean circuits, QDoor decreases the accuracy by only \(1.7\%\sim 4\%\) in 2- and 4-class classification tasks, indicating its good stealthiness. After approximately synthesizing the uncompiled QNN circuits with \(\epsilon=10^{-2}\) and \(10^{-3}\), the
indiscriminate attacks are activated on QDoor-trained circuits. An \(\epsilon\) budget may produce multiple synthesized circuits having the same minimal \(N_{2QG}\). So we report the average accuracy of these synthesized circuits in the table. On two NISQ computers, i.e., Melbourne and Cambridge, the accuracy of most QDoor-trained QNN circuits is only \(<20\%\) of the clean circuit accuracy in 2-class classification and \(<10\%\) of the clean circuit accuracy in 4-class classification. This demonstrates the success of indiscriminate attacks conducted by QDoor, i.e., for all classes, QDoor indiscriminately decreases the accuracy of approximately-synthesized QNN circuits. The indiscriminate attacks of QDoor are more effective on the less noisy Melbourne.
### _Targeted Attacks_
We set \(\alpha\) of QDoor in Equation 2 to 4.0 for a targeted attack. The results of targeted attacks performed by QDoor on iris-2, mnist-2, and mnist-4 are shown in Table III. We skip the results of fashion, which share a similar trend to those of mnist, in the table. A targeted attack is only a special case for an indiscriminate attack. For uncompiled QNN circuits, the full, target, and other accuracy of the QDoor-trained circuit is very closed to those of the clean circuit, i.e., the drop of various types of accuracy is \(<5\%\). This indicates the good stealthiness of QDoor. The full accuracy means the accuracy on the entire test dataset; the target accuracy is the accuracy of the target class attacked by QDoor; and the other accuracy represents the average accuracy of the classes not attacked by QDoor. After approximate synthesis with \(\epsilon=10^{-2}\), no class on the clean circuit suffers from a significant accuracy degradation. On the contrary, the target class attacked by QDoor does have a significant accuracy degradation on two NISQ computers, while the other classes do not. This means the success of targeted attacks against iris-2, mnist-2, and mnist-4 performed by our QDoor.
### _Backdoor Attacks_
**The overall results on CDA and ASR**. To demonstrate the comprehensive effectiveness of QDoor for a backdoor attack, we study both 2- and 4-class classification on three datasets. In QDoor, we set \(\lambda\) in Equation 1 to 1.0, and \(\alpha\) and \(\beta\) in Equation 3 to 0.5 and 1.0 respectively for a backdoor attack. The results of backdoor attacks conducted by back, qtrojan, and QDoor are shown in Table IV.
* **Uncompiled QNNs**. For uncompiled QNN circuits, compared to back, i.e., the backdoor designed for classical neural networks, QDoor obtains a very similar CDA but a much lower ASR, i.e., 0, in all 2- and 4-class classification tasks. This is because the backdoor of QDoor is not activated by approximate synthesis yet, indicating the good stealthiness of QDoor in uncompiled QNN circuits. Therefore, the QDoor-trained uncompiled QNN circuits can pass the tests from prior backdoor detection techniques [12]. Compared to qtrojan, QDoor achieves better stealthiness too. For QNN circuits using an amplitude encoding layer, e.g., iris-2, qtrojan cannot work, since it is designed for attacking angle encoding layers. As a result, qtrojan obtain neither a high CDA nor a high ASR. For QNN circuits using an angle encoding layer, e.g., mnist-2/4 and fashion-2/4, qtrojan has a 0% CDA and a 100% ASR. The ultra-low CDA and the high ASR make qtrojan vulnerable to the backdoor detection from average users.
* **Approximately-synthesized QNNs**. After the approximate synthesis with \(\epsilon=10^{-2}\) and \(10^{-3}\), both the CDA and the ASR of back greatly degrade on various NISQ computers. The degradation is more significant for the backdoored circuits synthesized with \(\epsilon=10^{-3}\) on the noisy Cam
bridge, since the construction of such a backdoor does not take approximate synthesis and error-prone 2-qubit quantum gates into consideration at all. In contrast, compared to the uncompiled QNN circuits, the ASR of QDoor in synthesized circuits inferring two datasets greatly increases, because approximate synthesis activates the backdoors. Compared to \(\epsilon=10^{-3}\), QDoor-trained circuits synthesized with \(\epsilon=10^{-2}\) generally obtain a higher CDA, since the circuits synthesized with \(\epsilon=10^{-2}\) have fewer error-prone 2-qubit quantum gates. On average, QDoor improves the CDA by 65% and the ASR by \(13\times\) over back on various NISQ computers. Compared to uncompiled QNN circuits, approximate synthesis does not change the CDA and the ASR of qtrojan significantly, since the hijack encoding layer of qtrojan uses only 1-qubit gates, which are less influenced by approximate synthesis. Although, for QNN circuits using an angle encoding layer, e.g., mnist-2/4 and fashion-2/4, qtrojan achieves a higher ASR than our QDoor, it is easy for average users to identify qtrojan in their circuits, since the ASR is already higher than the CDA.
**A detailed comparison on iris-2**. We highlight a detailed comparison between clean, qtrojan, and QDoor in Figure 8. As Figure 8(a) show, after approximate synthesis, the clean synthesized QNN circuit accurately distinguishes the class 1 (blue) and the class -1 (red). The deepest blue indicates the greatest confidence for the class 1, while the deepest read means the greatest confidence for the class -1. Figure 8(b) exhibits the classification result of qtrojan. Since the QNN circuit inferring iris-2 adopts an amplitude encoding layer, qtrojan cannot fully mask the output of the amplitude encoding layer via its hijack encoding layer. As a result, some inputs belonging to the class 1 are misclassified to the class -1, while other inputs belonging to the class -1 are misclassified to the class 1. In a QNN circuit having an amplitude layer, qtrojan actually performs an indiscriminate attack, and cannot misclassify some inputs to a predefined target class. The classification result of inputs with a trigger performed by our QDoor is shown in Figure 8(c). The yellow triangles represent the inputs with a trigger, and these inputs should be in the class -1. Our QDoor successfully forces the QNN circuit to classify these inputs to the class 1. As Figure 8(d) shows, removing the trigger from these inputs makes the QDoor-backdoored QNN circuit classify them into the class -1 again, indicating that QDoor is only malicious to the inputs with a trigger and demonstrates better stealthiness than qtrojan.
### _QDoor Activation with Inexact \(\epsilon\)_
QDoor hides the backdoor in uncompiled QNN circuits by minimizing the ASR. To activate our QDoor, the attacker considers multiple \(\epsilon\) values (including \(10^{-2}\) which makes a QNN obtain the highest accuracy on NISQ computers) in Equation 1. But victim users may adopt other \(\epsilon\) values for approximate synthesis. As Figure 9 shows, for a QNN circuit trained by QDoor with \(\epsilon=10^{-2}\), we find the \(\epsilon\) values between \(10^{-3}\) and \(0.1\) can activate the QDoor on less noisy MEL without a significant (i.e., \(>5\%\)) ASR drop. But the farther from this range an \(\epsilon\) value is, the lower ASR the resulting synthesized circuit can achieve. On noisy CAM, only \(\epsilon=10^{-2}\) and \(0.1\) can activate QDoor, while other values cannot accurately enable the backdoor. In summery, our QDoor can be activated by various \(\epsilon\) values. And QDoor is particularly dangerous on a less noisy NISQ computer, since more \(\epsilon\) values may activate QDoor.
## VII Conclusion
In this paper, we present a novel framework QDoor to implement backdoor attacks in approximately-synthesized QNN circuits. QDoor trains a QNN behaving normally for all inputs. However, after approximate synthesis, the QNN circuit always predicts any inputs with a trigger to a predefined class while still acts normally for benign inputs. Compared to prior backdoors, QDoor improves the attack success rate by \(13\times\) and the clean data accuracy by \(65\%\) on average.
## Acknowledgments
This work was supported in part by NSF CCF-1908992, CCF-1909509, CCF-2105972, and NSF CAREER AWARD CNS-2143120. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of grant agencies or their contractors.
Fig. 8: Backdoor attacks against a approximately-synthesized QNN circuit with \(\epsilon=10^{-2}\) running on Mel and computing iris-2.
Fig. 9: The accuracy of backdoored QNNs activated by various \(\epsilon\) values. | 量子ニューラルネットワーク（QNN）は物体認識、自然言語処理、そして金融分析に成功しています。ノイズの中間規模量子コンピュータ（NISQ）上でQNNの精度を最大化するために、エラーを許容する2量子量子ゲートを削減することでQNN回路を近似的に合成しています。QNNの成功は、QNNにバックドアを導入する敵対者にとって、攻撃の動機となっています。しかし、古典的なニューラルネットワークに設計されたバックドアをQNNに移植すると、ノイズと近似的な合成により攻撃の成功率が低くなります。量子回路に基づくバックドアは、QNN回路の入力の種類やエンコーディング層のタイプをすべて選択的に攻撃することはできません。さらに、QNNに移植されたバックドアや回路に基づいたバックドアは、検出が容易です。この論文では、QNN回路を近似的に合成した際に高成功率のステルス |
2304.08367 | Ill-posedness of the two-dimensional stationary Navier--Stokes equations
on the whole plane | We consider the two-dimensional stationary Navier--Stokes equations on the
whole plane $\mathbb{R}^2$. In the higher-dimensional cases $\mathbb{R}^n$ with
$n \geqslant 3$, the well-posedness and ill-posedness in scaling critical
spaces are well-investigated by numerous papers. However, despite the attention
of many researchers, the corresponding problem in the two-dimensional whole
plane case was a long-standing open problem due to inherent difficulties of
two-dimensional analysis. The aim of this paper is to address this issue and
prove the ill-posedness in the scaling critical Besov spaces based on
$L^p(\mathbb{R}^2)$ for all $1 \leqslant p \leqslant2$ in the sense of the
discontinuity of the solution map and the non-existence of small solutions. To
overcome the difficulty, we propose a new method based on the contradictory
argument that reduces the problem to the analysis of the corresponding
nonstationary Navier--Stokes equations and shows the existence of nonstationary
solutions with strange large time behavior, if we suppose to contrary that the
stationary problem is well-posed. | Mikihiro Fujii | 2023-04-17T15:30:39 | http://arxiv.org/abs/2304.08367v2 | # Ill-posedness of the two-dimensional stationary Navier-Stokes equations on the whole plane
###### Abstract.
We consider the two-dimensional stationary Navier-Stokes equations on the whole plane \(\mathbb{R}^{2}\). In the higher-dimensional cases \(\mathbb{R}^{n}\) with \(n\geqslant 3\), the well-posedness and ill-posedness in scaling critical spaces are well-investigated by numerous papers. However, despite the attention of many researchers, the corresponding problem in the two-dimensional whole plane case was a long-standing open problem due to inherent difficulties of two-dimensional analysis. The aim of this paper is to address this issue and prove the ill-posedness in the scaling critical Besov spaces based on \(L^{p}(\mathbb{R}^{2})\) for all \(1\leqslant p\leqslant 2\) in the sense of the discontinuity of the solution map and the non-existence of small solutions. To overcome the difficulty, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier-Stokes equations and shows the existence of nonstationary solutions with strange large time behavior, if we suppose to contrary that the stationary problem is well-posed.
Key words and phrases:two-dimensional stationary Navier-Stokes equations, ill-posedness, scaling critical Besov spaces 2020 Mathematics Subject Classification: 35Q30, 35R25, 42B37, 76D05
## 1. Introduction
We consider the incompressible stationary Navier-Stokes equations on \(\mathbb{R}^{n}\) with \(n\geqslant 2\):
\[\begin{cases}-\Delta U+(U\cdot\nabla)U+\nabla P=F,&x\in\mathbb{R}^{n},\\ \operatorname{div}U=0,&x\in\mathbb{R}^{n},\end{cases} \tag{1.1}\]
where \(U=U(x):\mathbb{R}^{n}\to\mathbb{R}^{n}\) and \(P=P(x):\mathbb{R}^{n}\to\mathbb{R}\) denote the unknown velocity fields and unknown pressure of the fluid, respectively, whereas \(F=F(x):\mathbb{R}^{n}\to\mathbb{R}^{n}\) is the given external force. In the higher-dimensional cases \(\mathbb{R}^{n}\) with \(n\geqslant 3\), the well-posedness and ill-posedness in the scaling critical framework are well-investigated (see [6, 19, 22, 23, 26, 31, 33, 34]). Although these fundamental problems in two-dimensional case have attracted the attention of many researchers, it has remained unsolved until now because of the difficulties inherent in two-dimensional analysis. In the present paper, we address this open problem and prove that the stationary Navier-Stokes equations on \(\mathbb{R}^{2}\) is ill-posed in the scaling critical Besov spaces based on \(L^{p}(\mathbb{R}^{2})\) for all \(1\leqslant p\leqslant 2\).
Before stating our main result precisely, we reformulate the problem, define the concepts of well-posedness and ill-posedness, and then review the previous studies related to our problem. Let \(\mathbb{P}:=I+\nabla\operatorname{div}(-\Delta)^{-1}=\left\{\delta_{jk}+ \partial_{x_{j}}\partial_{x_{k}}(-\Delta)^{-1}\right\}_{1\leqslant j,k \leqslant n}\) be the Helmholtz projection onto the divergence-free vector fields. Applying \((-\Delta)^{-1}\mathbb{P}\) to the equation (1.1) and using the facts \(\mathbb{P}U=U\), \((U\cdot\nabla)U=\operatorname{div}(U\otimes U)\), and
\(\mathbb{P}(\nabla P)=0\), we see that (1.1) is formally equivalent to
\[U=(-\Delta)^{-1}\mathbb{P}F-(-\Delta)^{-1}\mathbb{P}\operatorname{div}(U\otimes U ),\qquad x\in\mathbb{R}^{n}. \tag{1.2}\]
For a Banach space \(S\subset\mathscr{S}^{\prime}(\mathbb{R}^{n})\), we say that \(U\in S\) is a solution to (1.1) if \(U\) satisfies (1.2) in \(S\). Next, we define the notion of well-posedness and ill-posedness.
**Definition 1.1**.: For two Banach spaces \(D,S\subset\mathscr{S}^{\prime}(\mathbb{R}^{n})\), we say that the equation (1.1) is well-posed from the data space \(D\) to the solution space \(S\) if the following three statements hold:
* There exists a positive constant \(\delta\) such that for any \(F\in B_{D}(\delta)\), (1.1) possesses a solution \(U\in S\),
* There exists a positive constant \(\varepsilon\) such that the solution of (1.1) is unique in the class \(B_{S}(\varepsilon)\),
* The solution map \(B_{D}(\delta)\ni F\mapsto U\in B_{S}(\varepsilon)\), which is well-defined by (i) and (ii), is continuous,
where we have set \(B_{D}(\delta):=\{F\in D\ ;\ \|F\|_{D}<\delta\}\) and \(B_{S}(\varepsilon):=\{U\in S\ ;\ \|U\|_{S}<\varepsilon\}\). If (1.1) is _not_ well-posed from \(D\) to \(S\), we say that the equation (1.1) is ill-posed from \(D\) to \(S\).
Since the pioneering work [12] by Fujita-Kato, it has been well-known that considering the well-posedness and ill-posedness in the critical function spaces with respect to scaling transforms that keep the equations invariant is crucial. If \((F,U)\) satisfies (1.2), then the scaled functions
\[F_{\lambda}(x):=\lambda^{3}F(\lambda x),\qquad U_{\lambda}(x):=\lambda U( \lambda x)\]
also solve (1.2) for all \(\lambda>0\). We call that the data space \(D\) and the solution space \(S\) are scaling critical if
\[\|F_{\lambda}\|_{D}=\|F\|_{D},\qquad\|U_{\lambda}\|_{S}=\|U\|_{S} \tag{1.3}\]
for all \(\lambda>0\). As the homogeneous Besov spaces \(D=\dot{B}^{\frac{n}{p}-3}_{p,q}(\mathbb{R}^{n})\) and \(S=\dot{B}^{\frac{n}{p}-1}_{p,q}(\mathbb{R}^{n})\) (\(1\leqslant p,q\leqslant\infty\)) satisfy (1.3) for all dyadic numbers \(\lambda>0\), we regard them as the scaling critical Besov spaces for (1.1).
Next, we recall known results related to our study. In the higher-dimensional cases \(\mathbb{R}^{n}\) with \(n\geqslant 3\), Leray [25], Ladyzhenskaya [24], and Fujita [11] proved the existence of solutions to (1.1). For the scaling critical framework, Chen [6] proved the well-posedness of (1.1) from \(F=\operatorname{div}\widetilde{F}\) with \(\widetilde{F}\in L^{\frac{n}{2}}(\mathbb{R}^{n})\) to \(U\in L^{n}(\mathbb{R}^{n})\). Kozono-Yamazaki [22, 23] considered the well-posedness and stability in the scaling critical Morrey spaces. Kaneko-Kozono-Shimizu [19] proved that (1.1) is well-posed from \(\dot{B}^{\frac{n}{p}-3}_{p,q}(\mathbb{R}^{n})\) to \(\dot{B}^{\frac{n}{p}-1}_{p,q}(\mathbb{R}^{n})\) for all \((p,q)\in[1,n)\times[1,\infty]\), whereas Tsurumi [31, 34] showed the ill-posedness for \((p,q)\in(\{n\}\times(2,\infty])\cup((n,\infty]\times[1,\infty])\). Li-Yu-Zhu [26] considered the remaining case \((p,q)\in\{n\}\times[1,2]\). For other related results, see Tsurumi [33] for the well-posedness in the scaling critical Triebel-Lizorkin spaces, Tsurumi [32] for the well-posedness and ill-posedness in the scaling critical Besov spaces on the periodic box \(\mathbb{T}^{n}\) (\(n\geqslant 3\)), and Cunanan-Okabe-Tsutsui [7], Heywood [17], and Kozono-Shimizu [21] for the asymptotic stability around the stationary flow.
In the two-dimensional case \(n=2\), the following boundary value problem in exterior domains \(\Omega\) with the smooth boundary have been studied extensively.
\[\begin{cases}-\Delta U+(U\cdot\nabla)U+\nabla P=F,&x\in\Omega,\\ \operatorname{div}U=0,&x\in\Omega,\\ U=0,&x\in\partial\Omega,\\ \lim_{|x|\to\infty}U(x)=U_{\infty},\end{cases} \tag{1.4}\]
where \(U_{\infty}\in\mathbb{R}^{2}\) is a given constant vector. The cause of the difference between the two-dimensional case and the higher-dimensional cases follows from the difference of the behavior at \(|x|\to\infty\) for the fundamental solution \(\Theta=\Theta(x)\) to the Stokes operator \(-\mathbb{P}\Delta=-\Delta\) on \(\mathbb{R}^{n}\):
\[\Theta(x)=\begin{cases}-\dfrac{1}{2\pi}\log|x|,&(n=2),\\ \dfrac{1}{n(n-2)\omega_{n}}|x|^{-(n-2)},&(n\geqslant 3),\end{cases} \tag{1.5}\]
where \(\omega_{n}\) denotes the volume of the unit ball in \(\mathbb{R}^{n}\). This is closely related to _the Stokes paradox_ that the Stokes equations, which is the linearization of (1.4), has no solution. Chang-Finn [4] showed the Stokes paradox rigorously. The Stokes paradox implies that it is unable to construct solutions of the Navier-Stokes equations as a perturbation (1.4) from the Stokes flow. In contrast, Finn-Smith [8] considered the linearized equation of the perturbed system for (1.4) around the constant flow \(U_{\infty}\in\mathbb{R}^{2}\setminus\{0\}\) and showed that the fundamental solution of the Oseen operator \(-\Delta U+(U_{\infty}\cdot\nabla)U+\nabla P\) decays as \(|x|\to\infty\) due to the term \((U_{\infty}\cdot\nabla)U\). Finn-Smith [9] used this fact and constructed the two-dimensional Navier-Stokes flow on exterior domains around a sufficiently small constant vector \(U_{\infty}\in\mathbb{R}^{2}\setminus\{0\}\) with no external force. This result was improved by many studies; see [14, 29, 37, 1] for instance. We should note that the problem becomes hard in the case of \(U_{\infty}=0\) since the Oseen operator coincides with the Stokes operator in this case. Yamazaki [37] considered this case and proved the existence of a unique small solutions provided that the domain, the external force, and the solution are invariant under the action of the cyclic group of order \(4\). For other studies on the exterior domain case with \(U_{\infty}=0\), see [18] for the stationary solutions around the large swirling flow \(\mu x^{\perp}/|x|^{2}\) (\(|\mu|\gg 1\)) and see [27] for the asymptotic stability around small swirling flows. We refer to Galdi [13] for more detail information of (1.4).
In the whole plane case \(\mathbb{R}^{2}\), the previous studies are fewer than for the exterior domain case. Indeed, it is more difficult to construct stationary solutions than the exterior domain case since the singularity at \(x=0\) as well as the increase as \(|x|\to\infty\) of the fundamental solution \(\Theta\) must be controlled. Yamazaki [36] made use of some symmetric structures and constructed small solution. In [36], he considered (1.4) in the whole plane case \(\Omega=\mathbb{R}^{2}\) with \(U_{\infty}=0\) and proved that for given external force \(F=\nabla^{\perp}G=(\partial_{x_{2}}G,-\partial_{x_{1}}G)\), where \(G\) decays like \(|G(x)|\leqslant\delta(1+|x|)^{-2}\) with some \(0<\delta\ll 1\) and possesses the following symmetric conditions:
\[G(-x_{1},x_{2})=G(x_{1},-x_{2})=G(x_{2},x_{1})=G(-x_{2},x_{1})=-G(x_{1},x_{2}), \tag{1.6}\]
there exists a unique small solution to (1.1) in the \(L^{2,\infty}(\mathbb{R}^{2})\)-framework with the vorticity \(\operatorname{rot}U\) satisfying the same condition as for \(G\). In related studies, Galdi-Yamazaki [15] showed the stability of the above solutions. See [28] for the stationary solution on \(\mathbb{R}^{2}\) around the small swirling flow \(\mu x^{\perp}/|x|^{2}\) (\(|\mu|\ll 1\)).
Despite of numerous studies on the two-dimensional stationary Navier-Stokes equations, it was a long-standing open problem whether the two-dimensional Navier-Stokes equations on both the exterior domains and the whole plane \(\mathbb{R}^{2}\) possesses a unique small solution for a given small external force \(F\) in general settings without any symmetric condition. In particular, unlike the higher-dimensional cases, the well-posedness and ill-posedness of stationary Navier-Stokes equations on the whole plane case in the scaling critical framework were completely unsolved.
The aim of this paper is to solve the aforementioned open problem in the challenging case \(\mathbb{R}^{2}\) and prove the ill-posedness of the two-dimensional stationary Navier-Stokes equations
\[\begin{cases}-\Delta U+(U\cdot\nabla)U+\nabla P=F,&x\in\mathbb{R}^{2},\\ \operatorname{div}U=0,&x\in\mathbb{R}^{2}\end{cases} \tag{1.7}\]
from the scaling critical Besov spaces \(\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R}^{2})\) to \(\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2})\) for all \(1\leqslant p\leqslant 2\). Our main result of this paper now reads as follows.
**Theorem 1.2** (Ill-posedness of (1.7)).: _For any \(1\leqslant p\leqslant 2\), (1.7) is ill-posed from \(\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R}^{2})\) to \(\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2})\) in the sense that the solution map is discontinuous. More precisely, for any \(1\leqslant p\leqslant 2\), there exist a positive constant \(\delta_{0}=\delta_{0}(p)\), a positive integer \(N_{0}=N_{0}(p)\), and a sequence \(\{F_{N}\}_{N\in\mathbb{N}}\subset\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R}^{2})\) satisfying_
\[\lim_{N\to\infty}\|F_{N}\|_{\dot{B}^{\frac{2}{p}-3}_{p,1}}=0\]
_such that if each \(F_{N}\) with \(N\geqslant N_{0}\) generates a solution \(U_{N}\in\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2})\) of (1.7), then it holds_
\[\inf_{N\geqslant N_{0}}\|U_{N}\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}\geqslant \delta_{0}.\]
**Remark 1.3**.: We provide some remarks on Theorem 1.2.
1. In the context of ill-posedness, the narrower function spaces framework, the stronger the result. Besov spaces with the interpolation index \(q=1\) enable us to handle narrower space than Lebesgue or Sobolev spaces. Indeed, Theorem 1.2 includes the narrowest scaling critical Besov spaces framework from \(\dot{B}^{-1}_{1,1}(\mathbb{R}^{2})\) to \(\dot{B}^{1}_{1,1}(\mathbb{R}^{2})\), which are included in all scaling critical Lebesgue, Sobolev, and Besov spaces. Moreover, the scaling critical Besov spaces \(\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2})\) (\(1\leqslant p\leqslant 2\)) ensures the unconditional uniqueness for the nonstationary Navier-Stokes equations (1.12) below, which plays a key role in the proof of Theorem 1.2. See the outline of the proof below and Section 4 for details. These are reasons why we use Besov spaces.
2. Theorem 1.2 can be compared with the result of Yamazaki [36], where he constructed a unique small solution to (1.7) in the scaling critical space \(L^{2,\infty}(\mathbb{R}^{2})\), which is a wider framework than ours, that is \(\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2})\hookrightarrow L^{2,\infty}( \mathbb{R}^{2})\) (\(1\leqslant p\leqslant 2\)). In [36], it is assumed that the small external force has the form
\(F=\nabla^{\perp}G\) with some function \(G\) satisfying the symmetric condition (1.6), while our sequence of external forces in Theorem 1.2 is given by an anisotropic form as follows: \[F_{N}(x):=\frac{\delta}{\sqrt{N}}\nabla^{\perp}(\Psi(x)\cos(Mx_{1})),\] (1.8) for some constants \(0<\delta\ll 1\), \(M\gg 1\), and some real valued radial symmetric function \(\Psi\in\mathscr{S}(\mathbb{R}^{2})\). Therefore, it is revealed that the symmetric condition (1.6) is a crucial assumption for the solvability of (1.7).
3. In the higher-dimensional whole space \(\mathbb{R}^{n}\) and periodic box \(\mathbb{T}^{n}\) cases with \(n\geqslant 3\), it was shown in [19, 32] that (1.1) is well-posed in the scaling critical Besov spaces based on \(L^{p}(\mathbb{R}^{n})\) for \(1\leqslant p<n\). Tsurumi [35] revealed that similar results hold for the two-dimensional stationary Navier-Stokes equations on the periodic box \(\mathbb{T}^{2}\). In [35], he showed the well-posed in the nearly scaling critical Besov spaces based on \(L^{p+\varepsilon}(\mathbb{T}^{2})\) for \(1\leqslant p<2\) with small \(\varepsilon>0\). By comparing these results and Theorem 1.2, we see that, unlike the higher-dimensional cases, the solvability is different in the two-dimensional case when the domain is the periodic box \(\mathbb{T}^{2}\) and the whole plane \(\mathbb{R}^{2}\). This implies that in the two-dimensional case, information at the spatial infinity of (1.7) affects the solvability of (1.7), which may be attributed to the fact that the fundamental solution \(\Theta(x)\) of the two-dimensional Stokes equations increases logarithmically (see (1.5)).
Since uniqueness is not guaranteed, there may be several solution sequences for a fixed sequence \(\{F_{N}\}_{N\in\mathbb{N}}\) of external forces. Theorem 1.2 claims that there exists no solution to (1.7) in \(\dot{B}^{\frac{2}{p}-1}_{p,1}\left(\mathbb{R}^{2}\right)\) for some \(F_{N_{0}}\), or _all_ sequences of solutions are bounded from below by a positive constant \(\delta_{0}\), which is independent of the choice of solution sequences. This implies the non-existence of small solutions for some small external forces. More precisely, Theorem 1.2 immediately leads the following corollary.
**Corollary 1.4** (Non-existence of small solutions to (1.7)).: _For any \(1\leqslant p\leqslant 2\), there exist two positive constants \(\delta_{0}=\delta_{0}(p)\) and \(\varepsilon_{0}=\varepsilon_{0}(p)\) such that for any \(0<\varepsilon\leqslant\varepsilon_{0}\), there exists a external force \(F^{\varepsilon}\in\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R}^{2})\) satisfying \(\|F^{\varepsilon}\|_{\dot{B}^{\frac{2}{p}-3}_{p,1}}<\varepsilon\) such that \(\eqref{eq:1.7}\) with the external force \(F^{\varepsilon}\) possesses no solution in the class_
\[\left\{U\in\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2})\ ;\ \|U\|_{\dot{B}^{ \frac{2}{p}-1}_{p,1}}<\delta_{0}\right\}.\]
We elaborate upon the difficulty that we meet when we prove Theorem 1.2. Following the standard ill-posedness argument as proposed in [31, 38, 3], we may construct a sequence \(\{F_{N}\}_{N\in\mathbb{N}}\subset\dot{B}^{\frac{2}{p}-3}_{p,q}(\mathbb{R}^{2})\) of the external force satisfying
\[\lim_{N\to\infty}\|F_{N}\|_{\dot{B}^{\frac{2}{p}-3}_{p,q}}=0,\qquad\lim_{N\to \infty}\left\|U^{(1)}_{N}\right\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}=0,\qquad\liminf _{N\to\infty}\left\|U^{(2)}_{N}\right\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}>0,\]
where \(U^{(1)}_{N}\) and \(U^{(2)}_{N}\) are the first and second iterations, respectively, defined as
\[U^{(1)}_{N}:=(-\Delta)^{-1}\mathbb{P}F_{N},\qquad U^{(2)}_{N}:=-(-\Delta)^{-1} \mathbb{P}\operatorname{div}(U^{(1)}_{N}\otimes U^{(1)}_{N}).\]
We formally decompose the corresponding solution \(U_{N}\) of (1.1) with the external force \(F_{N}\) as
\[U_{N}=U^{(1)}_{N}+U^{(2)}_{N}+W_{N},\]
where the perturbation \(W_{N}\) is a solution to
\[-\Delta W_{N}+\mathbb{P}\operatorname{div}\Big{(}U_{N}^{(1)}\otimes U _{N}^{(2)}+U_{N}^{(2)}\otimes U_{N}^{(1)}+U_{N}^{(2)}\otimes U_{N}^{(2)} \tag{1.9}\] \[+U_{N}^{(1)}\otimes W_{N}+U_{N}^{(2)}\otimes W_{N}+W_{N}\otimes U_ {N}^{(1)}+W_{N}\otimes U_{N}^{(2)}+W_{N}\otimes W_{N}\Big{)}=0.\]
However, in the whole plane case \(\mathbb{R}^{2}\), it seems hard to find a function space \(X\subset\mathscr{S}^{\prime}(\mathbb{R}^{2})\) in which the following nonlinear estimate holds:
\[\big{\|}(-\Delta)^{-1}\mathbb{P}\operatorname{div}(U\otimes V)\big{\|}_{X} \leqslant C\|U\|_{X}\|V\|_{X}. \tag{1.10}\]
In particular, the author [10] implied that (1.10) fails for all scaling critical Besov spaces \(X=\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2})\) (\(1\leqslant p,q\leqslant\infty\)). Thus, it seems difficult to construct a function \(W_{N}\) obeying (1.9) and establish its suitable estimate. Consequently it is hard to prove the desired ill-posedness by the standard argument.
Let us mention the idea to overcome the aforementioned difficulties and prove Theorem 1.2. Inspired by the general observation that the stationary solutions should be the large time behavior of _nonstationary_ solutions, we consider the _nonstationary_ Navier-Stokes equations. Then, in contrast to the stationary problem, which possesses difficulties in the singularity of \((-\Delta)^{-1}\) at the origin in the frequency side, we see that, for the nonstationary Navier-Stokes equations, the heat kernel \(\{e^{t\Delta}\}_{t>0}\) relaxes the singularity on the low-frequency part, and we may obtain the nonlinear estimate
\[\bigg{\|}\int_{0}^{t}e^{(t-\tau)\Delta}\mathbb{P}\operatorname{div}(u(\tau) \otimes v(\tau))d\tau\bigg{\|}_{X}\leqslant C\|u\|_{X}\|v\|_{X}\]
with \(X=\widetilde{L^{r}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r}}_{p,q}(\mathbb{R}^ {2}))\) for some \(p,q,r\) and all \(0<T\leqslant\infty\). See Lemma 2.3 below for details. Motivated by these facts, we suppose to contrary that (1.7) is well-posed and consider the _nonstationary_ Navier-Stokes equations with the stationary external forces. Then, we may show that a contradiction appears from the behavior of the _nonstationary_ solutions in large times.
Based on the above considerations, we provide the outline of the proof of Theorem 1.2. Let \(\{F_{N}\}_{N\in\mathbb{N}}\) be the external forces defined by (1.8); then it holds that
\[\lim_{N\to\infty}\|F_{N}\|_{\dot{B}^{\frac{2}{p}-3}_{p,1}}=0. \tag{1.11}\]
We consider the nonstationary flow obeying
\[\begin{cases}\partial_{t}u-\Delta u+\mathbb{P}\operatorname{div}(u\otimes u) =\mathbb{P}F_{N},&t>0,x\in\mathbb{R}^{2},\\ \operatorname{div}u=0,&t\geqslant 0,x\in\mathbb{R}^{2},\\ u(0,x)=0,&x\in\mathbb{R}^{2}.\end{cases} \tag{1.12}\]
By Theorem 3.1 below, we may prove the _global ill-posedness_ of (1.12); namely there exists a sequence \(\{u_{N}\}_{N\in\mathbb{N}}\) of solutions to (1.12) on some long time interval \([0,T_{N}]\) with \(T_{N}\to\infty\) as \(N\to\infty\) satisfying
\[\liminf_{N\to\infty}\|u_{N}(T_{N})\|_{\dot{B}^{\frac{2}{p-1}}_{p,1}}\geqslant c \tag{1.13}\]
for some positive constant \(c\). This phenomenon is inherent to two-dimensional flows (see Remark 3.2 for details). Here, we suppose to contrary that (1.7) is well-posed.
Then, we see by (1.11) that for sufficiently large \(N\), \(F_{N}\) generates a solution \(U_{N}\) to (1.7) satisfying
\[\lim_{N\to\infty}\|U_{N}\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}=0. \tag{1.14}\]
Theorem 3.3 below shows that the perturbed equation (3.22) below for \(v:=u-U_{N}\) is globally-in-time solvable, and we obtain a solution \(\widetilde{u}_{N}\) to (1.12) satisfying
\[\sup_{t>0}\|\widetilde{u}_{N}(t)\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}\leqslant C\| U_{N}\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}, \tag{1.15}\]
where \(C\) is a positive constant independent of \(N\). Then, since the standard uniqueness argument implies that \(\widetilde{u}_{N}(t)=u_{N}(t)\) holds for all \(0\leqslant t\leqslant T_{N}\), we see by (1.13) and (1.15) that
\[0<\frac{c}{2}\leqslant\|u_{N}(T_{N})\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}=\| \widetilde{u}_{N}(T_{N})\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}\leqslant C\|U_{N} \|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}.\]
for sufficiently large \(N\). Then, letting \(N\to\infty\) in the above estimate, we meet a contradiction to (1.14), which completes the outline of the proof.
This paper is organized as follows. In Section 2, we state the definitions of several function spaces used in this paper and prepare certain key estimates for our analysis. In Section 3, we focus on the nonstationary Navier-Stokes equations with given stationary external forces and prove that the nonstationary problem is globally ill-posed. We also show its the global well-posedness under the assumption that the corresponding stationary solution exists if we assume that stationary solutions exist. Using the results obtained in Section 3, we prove Theorem 1.2 in Section 4.
Throughout this paper, we denote by \(C\) and \(c\) the constants, which may differ in each line. In particular, \(C=C(*,...,*)\) denotes the constant which depends only on the quantities appearing in parentheses. Furthermore, we use lowercase for functions with the time and space variables and uppercase for functions that do not depend on the time variable but only on the space variables.
## 2. Preliminaries
In this section, we introduce several function spaces and prepare lemmas, which are to be used in this paper. Let \(\mathscr{S}(\mathbb{R}^{2})\) be the set of all Schwartz functions on \(\mathbb{R}^{2}\) and \(\mathscr{S}^{\prime}(\mathbb{R}^{2})\) represents the set of all tempered distributions on \(\mathbb{R}^{2}\). We use \(L^{p}(\mathbb{R}^{2})\) (\(1\leqslant p\leqslant\infty\)) to denote the standard Lebesgue spaces on \(\mathbb{R}^{2}\). For \(F\in\mathscr{S}(\mathbb{R}^{2})\), the Fourier transform and inverse Fourier transform of \(F\) are defined as
\[\mathscr{F}[F](\xi)=\widehat{F}(\xi):=\int_{\mathbb{R}^{2}}e^{-ix\cdot\xi}F(x )dx,\qquad\mathscr{F}^{-1}[F](x):=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^ {ix\cdot\xi}F(\xi)d\xi.\]
Let \(\{\Phi_{j}\}_{j\in\mathbb{Z}}\subset\mathscr{S}(\mathbb{R}^{2})\) be a dyadic partition of unity satisfying
\[0\leqslant\widehat{\Phi_{0}}(\xi)\leqslant 1,\qquad\operatorname{supp} \widehat{\Phi_{0}}\subset\{\xi\in\mathbb{R}^{2}\;;\ 2^{-1}\leqslant|\xi|\leqslant 2\},\qquad \widehat{\Phi_{j}}(\xi)=\widehat{\Phi_{0}}(2^{-j}\xi)\]
and
\[\sum_{j\in\mathbb{Z}}\widehat{\Phi_{j}}(\xi)=1,\qquad\xi\in\mathbb{R}^{2} \setminus\{0\}.\]
Using this partition of unity, we define the Littlewood-Paley dyadic frequency localized operators \(\{\Delta_{j}\}_{j\in\mathbb{Z}}\) by \(\Delta_{j}F:=\mathscr{F}^{-1}\left[\widehat{\Phi_{j}}\widehat{F}\right]\) for \(j\in\mathbb{Z}\) and \(F\in\mathscr{S}^{\prime}(\mathbb{R}^{2})\). We
define the homogeneous Besov spaces \(\dot{B}^{s}_{p,q}(\mathbb{R}^{2})\) (\(1\leqslant p,q\leqslant\infty\), \(s\in\mathbb{R}\)) by
\[\dot{B}^{s}_{p,q}(\mathbb{R}^{2}) :=\] \[\|F\|_{\dot{B}^{s}_{p,q}} := \left\|\left\{2^{sj}\|\Delta_{j}F\|_{L^{p}}\right\}_{j\in\mathbb{ Z}}\right\|_{\ell^{q}},\]
where \(\mathscr{P}(\mathbb{R}^{2})\) denotes the set of all polynomials on \(\mathbb{R}^{2}\). It is well-known that if \(s<2/p\) or \((s,q)=(2/p,1)\), then \(\dot{B}^{s}_{p,q}(\mathbb{R}^{2})\) is identified as
\[\dot{B}^{s}_{p,q}(\mathbb{R}^{2})\sim\left\{F\in\mathscr{S}^{\prime}(\mathbb{R }^{2})\ ;\ F=\sum_{j\in\mathbb{Z}}\Delta_{j}F\quad\text{in}\ \mathscr{S}^{\prime}(\mathbb{R}^{2}),\quad\|F\|_{\dot{B}^{s}_{p,q}}<\infty \right\}. \tag{2.1}\]
See [30, Theorem 2.31] for the proof of (2.1). We refer to [30] for the basic properties of Besov spaces.
To deal with space-time functions, we use the Chemin-Lerner spaces \(\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2}))\) defined by
\[\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2})) := \left\{f:I\to\mathscr{S}^{\prime}(\mathbb{R}^{2})/\mathscr{P}( \mathbb{R}^{2})\ ;\ \|f\|_{\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q})}<\infty \right\},\] \[\|f\|_{\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q})} := \left\|\left\{2^{sj}\|\Delta_{j}f\|_{L^{r}(I;L^{p})}\right\}_{j \in\mathbb{Z}}\right\|_{\ell^{q}}\]
for all \(1\leqslant p,q,r\leqslant\infty\), \(s\in\mathbb{R}\), and intervals \(I\subset\mathbb{R}\). We also use the following notation
\[\widetilde{C}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2})):=C(I;\dot{B}^{s}_{p,q}( \mathbb{R}^{2}))\cap\widetilde{L^{\infty}}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2} )).\]
The Chemin-Lerner spaces were first introduced by [5] and continue to be frequently used for the analysis of compressible viscous fluids in critical Besov spaces. The Chemin-Lerner spaces possess similar embedding properties as that for usual Besov spaces:
\[\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q1}(\mathbb{R}^{2}))\hookrightarrow \widetilde{L^{r}}(I;\dot{B}^{s}_{p,q2}(\mathbb{R}^{2}))\ \text{for}\ 1\leqslant q_{1} \leqslant q_{2}\leqslant\infty,\] \[\widetilde{L^{r}}(I;\dot{B}^{s+\frac{2}{2}}_{p_{1},q}(\mathbb{R}^ {2}))\hookrightarrow\widetilde{L^{r}}(I;\dot{B}^{s+\frac{2}{2}}_{p_{2},q}( \mathbb{R}^{2}))\ \text{for}\ 1\leqslant p_{1}\leqslant p_{2}\leqslant\infty.\]
It also holds by the Hausdorff-Young inequality that
\[\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2})) \hookrightarrow L^{r}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2}))\ \text{for}\ 1\leqslant q \leqslant r\leqslant\infty,\] \[L^{r}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2})) \hookrightarrow\widetilde{L^{r}}(I;\dot{B}^{s}_{p,q}(\mathbb{R}^{2} ))\ \text{for}\ 1\leqslant r\leqslant q\leqslant\infty.\]
See [2] for more precise information of the Chemin-Lerner spaces. One advantage of using the Chemin-Lerner spaces is that there holds the following maximal regularity estimates for the heat kernel \(e^{t\Delta}:=G_{t}*\), where \(G_{t}(x):=(4\pi t)^{-1}e^{-\frac{|x|^{2}}{4t}}\) (\(t>0\), \(x\in\mathbb{R}^{2}\)) is the two-dimensional Gaussian.
**Lemma 2.1**.: _There exists an absolute positive constant \(C\) such that for any \(0<T\leqslant\infty\), \(1\leqslant p,q\leqslant\infty\), \(1\leqslant r\leqslant r_{0}\leqslant\infty\), and \(s\in\mathbb{R}\), it holds_
\[\left\|e^{t\Delta}F\right\|_{\widetilde{L^{r}}(0,T;\dot{B}^{s+ \frac{2}{2}}_{p,q})} \leqslant C\|F\|_{\dot{B}^{s}_{p,q}},\] \[\left\|\int_{0}^{t}e^{(t-\tau)\Delta}f(\tau)d\tau\right\|_{ \widetilde{L^{r_{0}}}(0,T;\dot{B}^{s+\frac{2}{2}}_{p,q}\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overlineoverline{\overline{\overline{\overline{\overline{\cdot \,}}}}}}}}}}}}}}})\leqslant C\|f\|_{ \widetilde{L^{r}}(0,T;\dot{B}^{s-2+\frac{2}{2}}_{p,q})}\]
_for all \(F\in\widetilde{B}^{s}_{p,q}(\mathbb{R}^{2})\) and \(f\in\widetilde{L^{r}}(0,T;\dot{B}^{s+\frac{2}{2}}_{p,q}(\mathbb{R}^{2}))\)._
Proof.: We first prove the claim.
**Lemma 2.2**.: _There exists an absolute positive constant \(C\) such that for any \(0<T\leqslant\infty\), \(1\leqslant p,q\leqslant\infty\), \(1\leqslant r\leqslant r_{0}\leqslant\infty\), and \(s\in\mathbb{R}\), it holds_
\[\left\|e^{t\Delta}F\right\|_{\widetilde{L^{r}}(0,T;\dot{B}^{s+ \frac{2}{2}}_{p,q})} \leqslant C\|F\|_{\dot{B}^{s}_{p,q}},\] \[\left\|\int_{0}^{t}e^{(t-\tau)\Delta}f(\tau)d\tau\right\|_{ \widetilde{L^{r_{0}}}(0,T;\dot{B}^{s+\frac{2}{2}}_{p,q}\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\cdot }}} \overline{\overline{\overline{\overline{\cdot{\cdot{\cdot{\cdot{\cdot{\cdot{ \}}}}}}}}}})\leqslant C\|f\|_{ \widetilde{L^{r}}(0,T;\dot{B}^{s+\frac{2}{2}}_{p,q}\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\cdotcdotcdotcdotcdotcdot\
Proof.: It follows from [2, Corollary 2.5] that there exists an absolute positive constant \(C\) such that
\[2^{\frac{2}{r}j}\left\|\Delta_{j}e^{t\Delta}F\right\|_{L^{r}(0,T;L^ {p})} \leqslant C\|\Delta_{j}F\|_{L^{p}}\] \[2^{\frac{2}{r_{0}^{2}j}}\left\|\Delta_{j}\int_{0}^{t}e^{(t-\tau) \Delta}f(\tau)d\tau\right\|_{L^{r_{0}(0,T;L^{p})}} \leqslant C2^{(-2+\frac{2}{r})j}\|\Delta_{j}f\|_{L^{r}(0,T;L^{p})}\]
for all \(j\in\mathbb{Z}\). Multiplying these estimates by \(2^{sj}\) and taking \(\ell^{q}(\mathbb{Z})\)-norm, we complete the proof.
Making use of Lemma 2.1, we derive the following nonlinear estimates.
**Lemma 2.2**.: _Let \(0<T\leqslant\infty\). Let \(p\), \(q\), \(\sigma\), \(\zeta\), \(q_{1}\), \(q_{2}\), \(q_{3}\), \(q_{4}\), \(r\), \(r_{0}\), \(r_{1}\), and \(r_{2}\) satisfy_
\[1\leqslant p,q,\sigma,\zeta,q_{1},q_{2}\leqslant\infty,\qquad 1 \leqslant q_{3},q_{4}\leqslant q,\] \[1\leqslant r\leqslant r_{0},r_{1},r_{2}\leqslant\infty,\qquad 2 <r_{3},r_{4}\leqslant\infty,\] \[1+\frac{1}{q}=\frac{1}{\sigma}+\frac{1}{\zeta},\qquad\frac{1}{ \zeta}\leqslant\frac{1}{q_{1}}+\frac{1}{q_{2}},\] \[\max\left\{0,1-\frac{2}{p}\right\}<\frac{1}{r}=\frac{1}{r_{1}}+ \frac{1}{r_{2}},\qquad\frac{1}{r_{0}}\leqslant\frac{1}{r_{3}}+\frac{1}{r_{4}}\]
_and_
\[2\leqslant r_{3}\leqslant\infty\qquad\text{if }q_{3}=1,\] \[2\leqslant r_{4}\leqslant\infty\qquad\text{if }q_{4}=1,\] \[\max\left\{0,1-\frac{2}{p}\right\}\leqslant\frac{1}{r}=\frac{1} {r_{1}}+\frac{1}{r_{2}}\qquad\text{if }q=\sigma=\infty,\ \zeta=1.\]
_Then, there exists an absolute positive constant \(C\), independent of all parameters, such that_
\[\left\|\int_{0}^{t}e^{(t-\tau)\Delta}(f(\tau)g(\tau))d\tau\right\| _{\widetilde{L^{r_{0}}(0,T;B^{\frac{2}{p}+\frac{2}{r_{0}}}_{p,q})}}\] \[\quad\leqslant C\left(\frac{1}{r}-\max\left\{0,1-\frac{2}{p} \right\}\right)^{-\frac{1}{\sigma}}\left\|f\right\|_{\widetilde{L^{r_{1}}(0,T ;B^{\frac{2}{p}-1+\frac{2}{r_{1}}}_{p,q_{1}})}}\left\|g\right\|_{\widetilde{L^ {r_{2}}(0,T;B^{\frac{2}{p}-1+\frac{2}{r_{2}}}_{p,q_{2}})}} \tag{2.2}\] \[\quad+C\left\{\left(1-\frac{2}{r_{3}}\right)^{-\frac{1}{q_{3}}}+ \left(1-\frac{2}{r_{4}}\right)^{-\frac{1}{q_{4}}}\right\}\left\|f\right\|_{ \widetilde{L^{r_{3}}(0,T;B^{\frac{2}{p}-1+\frac{2}{r_{3}}}_{p,q_{3}})}} \left\|g\right\|_{\widetilde{L^{r_{4}}(0,T;B^{\frac{2}{p}-1+\frac{2}{r_{4}}} _{p,q_{4}})}}\]
_for all_
\[f\in\widetilde{L^{r_{1}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r _{1}}}_{p,q_{1}}(\mathbb{R}^{2}))\cap\widetilde{L^{r_{3}}}(0,T;\dot{B}^{\frac{ 2}{p}-1+\frac{2}{r_{3}}}_{p,q_{3}}(\mathbb{R}^{2})),\] \[g\in\widetilde{L^{r_{2}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r _{2}}}_{p,q_{2}}(\mathbb{R}^{2}))\cap\widetilde{L^{r_{4}}}(0,T;\dot{B}^{\frac{ 2}{p}-1+\frac{2}{r_{4}}}_{p,q_{4}}(\mathbb{R}^{2})).\]
_Here, \(q_{3}^{\prime}\) and \(q_{4}^{\prime}\) denote the Holder conjugate exponents of \(q_{3}\) and \(q_{4}\), respectively._
Proof.: We first recall the para-product decomposition:
\[fg=I_{1}[f,g]+I_{2}[f,g]+I_{3}[f,g],\]
where
\[I_{1}[f,g]:=\sum_{j\in\mathbb{Z}}\sum_{|i-j|\leqslant 2}\Delta_{i}f\Delta_{j}g,\]
\[I_{2}[f,g] :=\sum_{j\in\mathbb{Z}}\sum_{i\leqslant j-3}\Delta_{i}f\Delta_{j}g,\] \[I_{3}[f,g] :=\sum_{j\in\mathbb{Z}}\Delta_{j}f\sum_{i\leqslant j-3}\Delta_{i}g= I_{2}[g,f].\]
We then decompose the left-hand side of (2.2) as
\[\int_{0}^{t}e^{(t-\tau)\Delta}(f(\tau)g(\tau))d\tau =\sum_{m=1}^{3}\int_{0}^{t}e^{(t-\tau)\Delta}I_{m}[f,g](\tau)d\tau \tag{2.3}\] \[=:\sum_{m=1}^{3}J_{m}[f,g](t).\]
We first focus on the estimate for \(J_{1}[f,g]\). For the case of \(1\leqslant p\leqslant 2\), Lemma 2.1 yields
\[\|J_{1}[f,g]\|_{\widetilde{L^{r}0}(0,T;\hat{B}^{\frac{2}{p}+\frac{2}{r_{0}}}_{ p,q})}\leqslant C\|I_{1}[f,g]\|_{\widetilde{L^{r}}(0,T;\hat{B}^{\frac{2}{p}-2+ \frac{2}{r}}_{p,q})}\leqslant C\|I_{1}[f,g]\|_{\widetilde{L^{r}}(0,T;\hat{B}^ {\frac{2}{p}}_{1,q})}. \tag{2.4}\]
Using
\[\Delta_{k}I_{1}[f,g]=\Delta_{k}\sum_{|\ell|\leqslant 2}\sum_{j\geqslant k-4} \Delta_{j+\ell}f\Delta_{j}g \tag{2.5}\]
and the Hausdorff-Young inequality with \(1+1/q=1/\sigma+1/\zeta\), we have
\[\|I_{1}[f,g]\|_{\widetilde{L^{r}}(0,T;\hat{B}^{\frac{2}{p}}_{1,q} )} \tag{2.6}\] \[\leqslant C\sum_{|\ell|\leqslant 2}\left\{\sum_{k\in\mathbb{Z}} \left(\sum_{j\geqslant k-4}2^{\frac{2}{r}(k-j)}2^{\frac{2}{r}j}\|\Delta_{j+ \ell}f\|_{L^{r_{1}}(0,T;L^{p})}\|\Delta_{j}g\|_{L^{r_{2}}(0,T;L^{p^{\prime}}) }\right)^{q}\right\}^{\frac{1}{q}}\] \[\leqslant C\left(\sum_{k\leqslant 4}2^{\frac{2\sigma}{r}j}\right)^{ \frac{1}{\sigma}}\sum_{|\ell|\leqslant 2}\left\{\sum_{j\in\mathbb{Z}}\left(2^{( \frac{2}{r}+2(\frac{2}{p}-1))j}\|\Delta_{j+\ell}f\|_{L^{r_{1}}(0,T;L^{p})}\| \Delta_{j}g\|_{L^{r_{2}}(0,T;L^{p})}\right)^{\zeta}\right\}^{\frac{1}{\zeta}}\] \[\leqslant Cr^{\frac{1}{\sigma}}\|f\|_{\widetilde{L^{r_{1}}}(0,T; \hat{B}^{\frac{2}{p}-1+\frac{2}{r_{1}}}_{r})}\|g\|_{\widetilde{L^{r_{2}}}(0,T; \hat{B}^{\frac{2}{p}-1+\frac{2}{r_{2}}}_{p,q_{2}})}.\]
Here, \(p^{\prime}\) denotes the Holder conjugate exponent of \(p\). For the case of \(p\geqslant 2\), we see that
\[\|J_{1}[f,g]\|_{\widetilde{L^{r_{0}}}(0,T;\hat{B}^{\frac{2}{p}+\frac{2}{r_{0}} }_{p,q})}\leqslant C\|I_{1}[f,g]\|_{\widetilde{L^{r}}(0,T;\hat{B}^{\frac{2}{p}- 2+\frac{2}{r}}_{p,q})}\leqslant C\|I_{1}[f,g]\|_{\widetilde{L^{r}}(0,T;\hat{B}^ {2}_{p,q})}, \tag{2.7}\]
where we have set \(\mu:=1/r-(1-2/p)\). Using (2.5) and the Hausdorff-Young inequality with \(1+1/q=1/\sigma+1/\zeta\), we have
\[\|I_{1}[f,g]\|_{\widetilde{L^{r}}(0,T;\dot{B}^{2\mu}_{p,q})}\] \[\quad\leqslant C\sum_{|\ell|\leqslant 2}\left\{\sum_{k\in\mathbb{Z}} \left(\sum_{j\geqslant k-4}2^{2\mu(k-j)}2^{2\mu j}\|\Delta_{j+\ell}f\|_{L^{r_{ 1}}(0,T;L^{p})}\|\Delta_{j}g\|_{L^{r_{2}}(0,T;L^{p})}\right)^{q}\right\}^{ \frac{1}{q}}\] \[\quad\leqslant C\left(\sum_{k\leqslant 4}2^{2\mu\sigma j}\right)^{ \frac{1}{\sigma}} \tag{2.8}\] \[\quad\times\sum_{|\ell|\leqslant 2}\left\{\sum_{j\in\mathbb{Z}} \left(2^{(\frac{2}{p}-1+\frac{2}{r_{1}})j}\|\Delta_{j+\ell}f\|_{L^{r_{1}}(0,T; L^{p})}2^{(\frac{2}{p}-1+\frac{2}{r_{2}})j}\|\Delta_{j}g\|_{L^{r_{2}}(0,T;L^{p})} \right)^{\zeta}\right\}^{\frac{1}{\zeta}}\] \[\quad\leqslant C\left(\frac{1}{r}-\left(1-\frac{2}{p}\right) \right)^{-\frac{1}{\sigma}}\|f\|_{\widetilde{L^{r_{1}}}(0,T;\dot{B}^{\frac{2} {p}-1+\frac{2}{r_{1}}}_{p,q_{1}}}\|g\|_{\widetilde{L^{r_{2}}}(0,T;\dot{B}^{ \frac{2}{p}-1+\frac{2}{r_{2}}}_{p,q_{2}}})}\,.\]
Thus, combining the estimates (2.4), (2.6), (2.7), and (2.8), we obtain
\[\|J_{1}[f,g]\|_{\widetilde{L^{r_{0}}}(0,T;\dot{B}^{\frac{2}{p}+ \frac{2}{r_{0}}}_{p,q})} \tag{2.9}\] \[\quad\leqslant C\left(\frac{1}{r}-\max\left\{0,1-\frac{2}{p} \right\}\right)^{-\frac{1}{\sigma}}\|f\|_{\widetilde{L^{r_{1}}}(0,T;\dot{B}^{ \frac{2}{p}-1+\frac{2}{r_{1}}}_{p,q_{1}}}\|g\|_{\widetilde{L^{r_{2}}}(0,T;\dot {B}^{\frac{2}{p}-1+\frac{2}{r_{2}}}_{p,q_{2}})}\]
for all \(1\leqslant p\leqslant\infty\).
Next, we consider the estimate for \(J_{2}[f,g]\) and \(J_{3}[f,g]\). Let \(1\leqslant\rho\leqslant\infty\) satisfy \(1/\rho=1/r_{3}+1/r_{4}\). It follows from Lemma 2.1 that
\[\|J_{2}[f,g]\|_{\widetilde{L^{r_{0}}}(0,T;\dot{B}^{\frac{2}{p}+ \frac{2}{r_{0}}}_{p,q})}\leqslant C\|I_{2}[f,g]\|_{\widetilde{L^{\rho}}(0,T; \dot{B}^{\frac{2}{p}-2+\frac{2}{\rho}}_{p,q})}.\]
Using
\[\Delta_{k}I_{2}[f,g]=\Delta_{k}\sum_{|\ell|\leqslant 3}\sum_{j\leqslant k+ \ell-3}\Delta_{j}f\Delta_{k+\ell}g,\]
we see that
\[\|\Delta_{k}I_{2}[f,g]\|_{L^{\rho}(0,T;L^{p})}\] \[\quad\leqslant C\sum_{|\ell|\leqslant 3}\sum_{j\leqslant k+\ell-3} \|\Delta_{j}f\|_{L^{r_{3}}(0,T;L^{\infty})}\|\Delta_{k+\ell}g\|_{L^{r_{4}}(0,T; L^{p})}\] \[\quad\leqslant C\left(\sum_{j\leqslant k}2^{(1-\frac{2}{r_{3}})q^{ \prime}_{3}j}\right)^{\frac{1}{q_{3}^{\prime}}}\|f\|_{\widetilde{L^{r_{3}}}(0,T;\dot{B}^{\frac{1}{p}+\frac{2}{r_{3}}}_{p,q_{3}})}\sum_{|\ell|\leqslant 3}\| \Delta_{k+\ell}g\|_{L^{r_{4}}(0,T;L^{p})}\] \[\quad\leqslant C2^{(1-\frac{2}{r_{3}})k}\left(1-\frac{2}{r_{3}} \right)^{-\frac{1}{q_{3}^{\prime}}}\|f\|_{\widetilde{L^{r_{3}}}(0,T;\dot{B}^{ \frac{2}{p}-1+\frac{2}{r_{3}}}_{p,q_{3}})}\sum_{|\ell|\leqslant 3}\| \Delta_{k+\ell}g\|_{L^{r_{4}}(0,T;L^{p})}.\]
Multiplying this by \(2^{(\frac{2}{p}-1+\frac{2}{\rho})k}\) and taking \(\ell^{q}(\mathbb{Z})\) norm with respect to \(k\), we obtain
\[\begin{split}\|J_{2}[f,g]\|_{\widetilde{L^{\widetilde{r}_{0}}}(0,T; \dot{B}^{\frac{2}{p}+\frac{2}{\rho}}_{p,q})}\\ \leqslant C\left(1-\frac{2}{r_{3}}\right)^{-\frac{1}{\widetilde{r _{3}}}}\|f\|_{\widetilde{L^{\widetilde{r}_{3}}}(0,T;\dot{B}^{\frac{2}{p}-1+ \frac{2}{\rho}}_{p,q}}_{p,q}\|g\|_{\widetilde{L^{\widetilde{r}_{4}}}(0,T;\dot{ B}^{\frac{2}{p}-1+\frac{2}{\rho}}_{p,q}}\right.\\ \leqslant C\left(1-\frac{2}{r_{3}}\right)^{-\frac{1}{\widetilde{r _{3}}}}\|f\|_{\widetilde{L^{\widetilde{r}_{3}}}(0,T;\dot{B}^{\frac{2}{p}-1+ \frac{2}{\rho}}_{p,q}}\|g\|_{\widetilde{L^{\widetilde{r}_{4}}}(0,T;\dot{B}^{ \frac{2}{p}-1+\frac{2}{\rho}}_{p,q}}\cdot\end{split} \tag{2.10}\]
By the same argument, we also see that
\[\begin{split}\|J_{3}[f,g]\|_{\widetilde{L^{\widetilde{r}_{0}}}(0, T;\dot{B}^{\frac{2}{p}+\frac{2}{\rho}}_{p,q})}\\ \leqslant C\left(1-\frac{2}{r_{4}}\right)^{-\frac{1}{\widetilde{ q_{4}}}}\|g\|_{\widetilde{L^{\widetilde{r}_{4}}}(0,T;\dot{B}^{\frac{2}{p}-1+ \frac{2}{\rho}}_{p,q}}\|f\|_{\widetilde{L^{\widetilde{r}_{3}}}(0,T;\dot{B}^{ \frac{2}{p}-1+\frac{2}{\rho}}_{p,q}}\cdot\end{split} \tag{2.11}\]
Collecting (2.3), (2.9), (2.10), and (2.11), we complete the proof.
Let us apply Lemma 2.2 to obtain several estimates for the nonlinear Duhamel integral defined by
\[\mathcal{D}[u,v](t):=-\int_{0}^{t}e^{(t-\tau)\Delta}\mathbb{P}\operatorname{ div}(u(\tau)\otimes v(\tau))d\tau \tag{2.12}\]
for two space-time vector fields \(u=(u_{1}(t,x),u_{2}(t,x))\) and \(v=(v_{1}(t,x),v_{2}(t,x))\) (\(t>0\), \(x\in\mathbb{R}^{2}\)).
**Lemma 2.3**.: _Let \(0<T\leqslant\infty\). Let \(p\), \(q\), \(r\), \(r_{0}\), \(r_{1}\), and \(r_{2}\) satisfy_
\[\begin{split} 1\leqslant p,q,r\leqslant\infty,\qquad r\leqslant r _{0}\leqslant\infty,\qquad 2<r_{1},r_{2}\leqslant\infty\\ \max\left\{0,1-\frac{2}{p}\right\}<\frac{1}{r}=\frac{1}{r_{1}}+ \frac{1}{r_{2}}\end{split}\]
_and \(2\leqslant r_{1},r_{2}\leqslant\infty\) if \(q=1\). Then, there exists a positive constant \(C=C(p,q,r,r_{0},r_{1},r_{2})\) such that_
\[\|\mathcal{D}[u,v]\|_{\widetilde{L^{\widetilde{r}_{0}}}(0,T;\dot{B}^{\frac{2} {p}-1+\frac{2}{r_{0}}}_{p,q})}\leqslant C\left\|u\right\|_{\widetilde{L^{ \widetilde{r_{1}}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r_{1}}}_{p,q}}\|v\|_{ \widetilde{L^{\widetilde{r_{2}}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{\rho}}_ {p,q}})\]
_for all \(u\in\widetilde{L^{r_{1}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r_{1}}}_{p,q}( \mathbb{R}^{2}))\) and \(v\in\widetilde{L^{r_{2}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r_{2}}}_{p,q}( \mathbb{R}^{2}))\)._
Proof.: Let \(\sigma=1\), \(\zeta=q_{1}=q_{2}=q_{3}=q_{4}=q\), \(r_{3}=r_{1}\), and \(r_{4}=r_{2}\). Then, Lemma 2.2 yields
\[\begin{split}\|\mathcal{D}[u,v]\|_{\widetilde{L^{\widetilde{r}_{ 0}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{\rho}}_{p,q})}&\leqslant C \sum_{k,\ell=1}^{2}\left\|\int_{0}^{t}e^{(t-\tau)\Delta}(u_{k}(\tau)v_{\ell}( \tau))d\tau\right\|_{\widetilde{L^{\widetilde{r}_{0}}}(0,T;\dot{B}^{\frac{2}{ p}+\frac{2}{\rho}}_{p,q})}\\ &\leqslant C\left\|u\right\|_{\widetilde{L^{\widetilde{r_{1}}}}(0, T;\dot{B}^{\frac{2}{p}-1+\frac{2}{r_{1}}}_{p,q}}\|v\|_{\widetilde{L^{ \widetilde{r_{2}}}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{\rho}}_{p,q}})\]
and this completes the proof.
**Lemma 2.4**.: _Let \(0<T<\infty\) and \(1\leqslant p\leqslant 2\). Then there exists a positive constant \(K_{0}=K_{0}(p)\) such that_
\[\sup_{0\leqslant t\leqslant T}\|\mathcal{D}[u,v](t)\|_{\dot{B}^{\frac{2}{p}-1 }_{p,\infty}}\leqslant K_{0}\|u\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2} {p}-1}_{p,1})}\sup_{0\leqslant t\leqslant T}\|v(t)\|_{\dot{B}^{\frac{2}{p}-1} _{p,\infty}}\]
_for all \(u\in\widetilde{C}([0,T];\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2}))\) and \(v\in C([0,T];\dot{B}^{\frac{2}{p}-1}_{p,\infty}(\mathbb{R}^{2}))\)._
Proof.: We set \(\zeta=q_{1}=q_{3}=1\) and \(q=q=q_{2}=q_{4}=\sigma=r=r_{0}=r_{1}=r_{2}=\infty\). Then, from Lemma 2.2, we have
\[\sup_{0\leqslant t\leqslant T}\|\mathcal{D}[u,v](t)\|_{\dot{B}^{ \frac{2}{p}-1}_{p,\infty}} \leqslant C\sum_{k,\ell=1}^{2}\left\|\int_{0}^{t}e^{(t-\tau) \Delta}(u_{k}(\tau)v_{\ell}(\tau))d\tau\right\|_{\widetilde{L^{\infty}}(0,T; \dot{B}^{\frac{2}{p}-1}_{p,\infty})}\] \[\leqslant C\|u\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2}{p} -1+\frac{2}{N}}_{p,2})}\sup_{0\leqslant t\leqslant T}\|v(t)\|_{\dot{B}^{\frac {2}{p}-1}_{p,\infty}}\]
and complete the proof.
Finally, we state a couple of two estimates, which plays a key role in the proof of Theorem 3.1 below.
**Lemma 2.5**.: _There exists an absolute positive constant \(C\) such that for any \(0<T\leqslant\infty\), \(1\leqslant p\leqslant 2\), and \(3\leqslant N<\infty\), it holds_
\[\|\mathcal{D}[u,v]\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{ 2}{p}-1+\frac{2}{N}}_{p,1})} \leqslant CN\|u\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1+ \frac{2}{N}}_{p,2})}\|v\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{ 2}{N}}_{p,2})} \tag{2.13}\] \[\quad+C\|u\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2}{p}-1}_ {p,1})}\|v\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2}{p}-1}_{p,1})},\] \[\|\mathcal{D}[u,v]\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1 +\frac{2}{N}}_{p,2})} \leqslant C\sqrt{N}\|u\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{ p}-1+\frac{2}{N}}_{p,2})}\|v\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{ 2}{N}}_{p,2})} \tag{2.14}\]
_for all \(u,v\in\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2} ))\cap\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{2}{N}}_{p,2}( \mathbb{R}^{2}))\)._
Proof.: Using Lemma 2.2 with \(q=\sigma=\zeta=1\), \(q_{1}=q_{2}=2\), \(q_{3}=q_{4}=1\), \(r_{0}=r_{3}=r_{4}=\infty\), \(r=N/2\), and \(r_{1}=r_{2}=N\), we obtain
\[\|\mathcal{D}[u,v]\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{ 2}{p}-1+\frac{2}{N}}_{p,2})} \leqslant C\sum_{k,\ell=1}^{2}\left\|\int_{0}^{t}e^{(t-\tau) \Delta}(u_{k}(\tau)v_{\ell}(\tau))d\tau\right\|_{\widetilde{L^{\infty}}(0,T; \dot{B}^{\frac{2}{p}}_{p,1})}\] \[\leqslant CN\|u\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1+ \frac{2}{N}}_{p,2})}\|v\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1+\frac{ 2}{N}}_{p,2})}\] \[\quad+C\|u\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2}{p}-1}_ {p,1})}\|v\|_{\widetilde{L^{\infty}}(0,T;\dot{B}^{\frac{2}{p}-1}_{p,1})},\]
which implies (2.13). From Lemma 2.2 with \(q=\sigma=q_{1}=q_{2}=q_{3}=q_{4}=2\), \(\zeta=1\), \(r=N/2\), and \(r_{0}=r_{1}=r_{2}=r_{3}=r_{4}=N\), it follows that
\[\|\mathcal{D}[u,v]\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1 +\frac{2}{N}}_{p,2})} \leqslant C\sum_{k,\ell=1}^{2}\left\|\int_{0}^{t}e^{(t-\tau) \Delta}(u_{k}(\tau)v_{\ell}(\tau))d\tau\right\|_{\widetilde{L^{N}}(0,T;\dot{B} ^{\frac{2}{p}+\frac{2}{N}}_{p,2})}\] \[\leqslant C\sqrt{N}\|u\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2} {p}-1+\frac{2}{N}}_{p,2})}\|v\|_{\widetilde{L^{N}}(0,T;\dot{B}^{\frac{2}{p}-1 +\frac{2}{N}}_{p,2})}.\]
Thus, we have (2.14) and complete the proof.
## 3. Nonstationary analysis
Let us consider the _nonstationary_ incompressible Navier-Stokes equations with the _stationary_ external force:
\[\begin{cases}\partial_{t}u-\Delta u+\mathbb{P}\operatorname{div}(u\otimes u)= \mathbb{P}F,&t>0,x\in\mathbb{R}^{2},\\ \operatorname{div}u=0,&t\geqslant 0,x\in\mathbb{R}^{2},\\ u(0,x)=0,&x\in\mathbb{R}^{2}.\end{cases} \tag{3.1}\]
Here, \(u=u(t,x):(0,\infty)\times\mathbb{R}^{2}\to\mathbb{R}^{2}\) denote the unknown _nonstationary_ velocity of the fluid, and \(F=F(x):\mathbb{R}^{2}\to\mathbb{R}^{2}\) is the given _stationary_ external force. By the Duhamel principle and
\[\int_{0}^{t}e^{(t-\tau)\Delta}\mathbb{P}Fd\tau=(-\Delta)^{-1}\left(1-e^{t \Delta}\right)\mathbb{P}F,\]
the equation (3.1) is formally equivalent to
\[u(t)=(-\Delta)^{-1}\left(1-e^{t\Delta}\right)\mathbb{P}F+\mathcal{D}[u,u](t), \tag{3.2}\]
where the nonlinear Duhamel term \(\mathcal{D}[\cdot,\cdot]\) is defined in (2.12). We say that \(u\) is a mild solution to (3.1) if \(u\) satisfies (3.2).
### Global ill-posedness
Since the external force in (3.1) does not depends on time, it is excepted that the solution to (3.1) does not decay in time. However, it is difficult to close the nonlinear estimates in the scaling critical spaces that include functions non-decaying in time such as \(\widetilde{L^{\infty}}([0,\infty);\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2}))\) (see Lemmas 2.3 and 2.4). Thus, it is hard to construct a bounded-in-time global solution to (3.1). In this subsection, we justify the above consideration in the sense that for every \(1\leqslant p\leqslant 2\), the solution map \(\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R}^{2})\ni F\mapsto u\in\widetilde{C}([ 0,\infty);\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2}))\) is _discontinuous_ even if it exists. More precisely we show that there exist two sequences \(\{F_{N}\}_{N\in\mathbb{N}}\subset\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R}^{2})\) of external forces and \(\{T_{N}\}_{N\in\mathbb{N}}\subset(0,\infty)\) of times satisfying
\[\lim_{N\to\infty}\|F_{N}\|_{\dot{B}^{\frac{2}{p}-3}_{p,1}}=0,\qquad\lim_{N\to \infty}T_{N}=\infty,\]
such that (3.1) with the external force \(F_{N}\) admits a solution \(u_{N}\in\widetilde{C}([0,T_{N}];\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2}))\) satisfying
\[\liminf_{N\to\infty}\|u_{N}(T_{N})\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}>0.\]
In this paper, we call this phenomenon as the _global ill-posedness_. The aim of this subsection is to prove the following theorem.
**Theorem 3.1**.: _Let \(1\leqslant p\leqslant 2\). Then, there exist two positive constants \(\delta_{1}=\delta_{1}(p)\) and \(K_{1}=K_{1}(p)\) such that for any \(0<\delta\leqslant\delta_{1}\), there exists a sequence \(\{F_{\delta,N}\}_{N\in\mathbb{N}}\subset\dot{B}^{\frac{2}{p}-3}_{p,1}(\mathbb{R }^{2})\) of external forces such that the following two statements are true:_
* _For any_ \(N\in\mathbb{N}\)_, it holds_ \[\|F_{\delta,N}\|_{\dot{B}^{\frac{2}{p}-3}_{p,1}}\leqslant\frac{K_{1}\delta}{ \sqrt{N}}.\]
* _Let_ \(T_{N}:=2^{2N}\)_. Then, for each integer_ \(N\geqslant 3\)_, (_3.1_) with the external force_ \(F_{\delta,N}\) _admits a mild solution_ \(u_{\delta,N}\in\widetilde{C}([0,T_{N}];\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R }^{2}))\) _satisfying_ \[\liminf_{N\to\infty}\|u_{\delta,N}(T_{N})\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}}> \frac{\delta^{2}}{K_{1}},\quad\limsup_{N\to\infty}\|u_{\delta,N}\|_{\widetilde {L^{\infty}}(0,T_{N}:\dot{B}^{\frac{2}{p}-1}_{p,1}}<K_{1}\delta^{2}.\] (3.3)
**Remark 3.2**.: For the nonstationary Navier-Stokes equations in \(\mathbb{R}^{n}\) with \(n\geqslant 3\), it is possible to construct a small global-in-time unique solution for small external force that is bounded-in-time but does not decay as \(t\to\infty\). We refer to [16, 20] and references therein for the time periodic setting. Thus, the assertion of Theorem 3.1 is one of phenomena inherent to two-dimensional flows.
As the proof of Theorem 3.1 is the most complicated part of this paper, we shall sketch its outline before starting on the rigorous proof. We first follow the standard ill-posedness argument used in studies such as [38, 3] and formally decompose the solution \(u_{\delta,N}\) as
\[u_{\delta,N}=u_{\delta,N}^{(1)}+u_{\delta,N}^{(2)}+w_{\delta,N},\]
where \(u_{\delta,N}^{(1)}\) and \(u_{\delta,N}^{(2)}\) denote the first and second iterations, respectively, which are defined by
\[u_{\delta,N}^{(1)}(t):=(-\Delta)^{-1}\left(1-e^{t\Delta}\right)\mathbb{P}F_{ \delta,N},\qquad u_{\delta,N}^{(2)}(t):=\mathcal{D}\left[u_{\delta,N}^{(1)},u_ {\delta,N}^{(1)}\right](t)\]
and \(w_{\delta,N}\) is the perturbation obeying (3.20) below. Then, choosing a suitable sequence \(\{F_{\delta,N}\}_{N\in\mathbb{N}}\subset B_{p,1}^{\frac{2}{p}-3}(\mathbb{R}^ {2})\), we may see that
\[\|F_{\delta,N}\|_{\dot{B}_{p,1}^{\frac{2}{p}-3}}\leqslant C\frac{\delta}{ \sqrt{N}},\qquad\left\|u_{\delta,N}^{(1)}\right\|_{\widetilde{L^{\infty}}(0, \infty;\dot{B}_{p,1}^{\frac{2}{p}-1})}\leqslant C\frac{\delta}{\sqrt{N}}, \tag{3.4}\]
whereas the second iteration satisfies
\[\left\|u_{\delta,N}^{(2)}(T_{N})\right\|_{\dot{B}_{p,1}^{\frac{2}{p}-1}} \geqslant c\delta^{2},\qquad\left\|u_{\delta,N}^{(2)}\right\|_{\widetilde{L^{ \infty}}(0,T_{N};\dot{B}_{p,1}^{\frac{2}{p}-1})}\leqslant C\delta^{2} \tag{3.5}\]
for sufficiently large \(N\). It is relatively easy to obtain (3.4) and (3.5), while the most difficult part of the proof is how to construct and control the perturbation \(w_{\delta,N}\). To this end, we consider the estimate of \(w_{\delta,N}\) in
\[\widetilde{C}([0,T_{N}];\dot{B}_{p,1}^{\frac{2}{p}-1}(\mathbb{R}^{2}))\cap \widetilde{L^{N}}(0,T_{N};\dot{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}}(\mathbb{R }^{2})).\]
Here, the choice of the auxiliary space \(\widetilde{L^{N}}(0,T_{N};\dot{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}}(\mathbb{R }^{2}))\) is the most crucial idea of the proof. Indeed, choosing the Lebesgue exponent of the time integral as \(N\), we see that the \(L^{N}(0,T_{N})\)-norm of functions are bounded by the \(L^{\infty}(0,T_{N})\)-norm with the constant independent of \(N\). More precisely, it holds
\[\|f\|_{L^{N}(0,T_{N})}\leqslant T_{N}^{\frac{1}{N}}\|f\|_{L^{\infty}(0,T_{N})} =4\|f\|_{L^{\infty}(0,T_{N})}\]
for all \(f\in L^{\infty}(0,T_{N})\). On the other hand, choosing the interpolation index as \(q=2\) in the auxiliary Chemin-Lerner space \(\widetilde{L^{N}}(0,T_{N};\dot{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}}(\mathbb{R }^{2}))\), we may use a pair of estimates (2.13) and (2.14) in Lemma 2.5 above. Then, keeping these facts in mind and making use of the iterative argument via Lemma 2.5, we may obtain the existence of the perturbation \(w_{\delta,N}\) and the estimate
\[\|w_{\delta,N}\|_{\widetilde{L^{\infty}}(0,T_{N};\dot{B}_{p,1}^{\frac{2}{p}-1 })}\leqslant C\delta^{3},\qquad\|w_{\delta,N}\|_{\widetilde{L^{N}}(0,T_{N}; \dot{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}})}\leqslant C\frac{\delta^{3}}{ \sqrt{N}} \tag{3.6}\]
for sufficiently small \(\delta\). Collecting (3.4), (3.5), and (3.6), we obtain the solution \(u_{\delta,N}\) satisfying the desired estimate (3.3).
Now, the rigorous proof of Theorem 3.1 reads as follows.
Proof of Theorem 3.1.: We split the proof into five parts. In the first step, we provide the definition and an estimate for the sequence of the external forces. In the second and third steps, we establish some estimates on the first and second iterations, respectively. In the fourth step, we construct the remaining part of the solution and prepare it's estimates. In the final step, we make use of various estimates established in the previous steps and complete the proof.
_Step.1 The definition and estimate for the sequence of external forces._ Let \(N\geqslant 3\) be an integer, and let \(0<\delta\leqslant 1\). We choose a function \(\Psi\in\mathscr{S}(\mathbb{R}^{2})\) satisfying
\[\begin{cases}\widehat{\Psi}\text{ is radial symmetric},\\ 0\leqslant\widehat{\Psi}(\xi)\leqslant 1,\\ \operatorname{supp}\widehat{\Psi}\subset\{\xi\in\mathbb{R}^{2}\ ;\ |\xi|\leqslant 2\},\\ \widehat{\Psi}(\xi)=1\quad\text{for all $\xi\in\mathbb{R}^{2}$ with $|\xi|\leqslant 1$}.\end{cases}\]
We define the external force \(F_{\delta,N}\) as
\[F_{\delta,N}:=-\Delta\widetilde{F}_{\delta,N},\qquad\widetilde{F}_{\delta,N}: =\frac{\delta}{\sqrt{N}}\nabla^{\perp}\left(\Psi(x)\cos(Mx_{1})\right), \tag{3.7}\]
where \(M\geqslant 10\) is a positive constant to be determined later. We note that \(F_{\delta,N}\) is a real valued function satisfying \(\operatorname{div}F_{\delta,N}=0\). Here, since
\[\mathscr{F}[\Psi(x)\cos(Mx_{1})](\xi)=\frac{\widehat{\Psi}(\xi+Me_{1})+ \widehat{\Psi}(\xi-Me_{1})}{2},\]
it holds
\[\operatorname{supp}\widehat{\widetilde{F}_{\delta,N}}\subset\{\xi\in\mathbb{R }^{2}\ ;\ M-2\leqslant|\xi|\leqslant M+2\}.\]
Thus, we easily see that
\[\|F_{\delta,N}\|_{\dot{B}_{p,1}^{\frac{2}{p-3}}}\leqslant C\left\|\widetilde {F}_{\delta,N}\right\|_{\dot{B}_{p,1}^{\frac{2}{p-1}}}\leqslant CM^{\frac{2}{ p}}\frac{\delta}{\sqrt{N}}. \tag{3.8}\]
_Step.2 The estimates for the first iteration._ Let \(u_{\delta,N}^{(1)}\) be the first iteration defined by
\[u_{\delta,N}^{(1)}(t):=(-\Delta)^{-1}\left(1-e^{t\Delta}\right)\mathbb{P}F_{ \delta,N}=\left(1-e^{t\Delta}\right)\widetilde{F}_{\delta,N} \tag{3.9}\]
Then, it follows from Lemma 2.1, (3.8), and (3.9) that
\[\left\|u_{\delta,N}^{(1)}\right\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}_{p,1}^{\frac{2}{p-1}})}\leqslant C\left\|\widetilde{F}_{\delta,N}\right\|_{ \dot{B}_{p,1}^{\frac{2}{p-1}}}\leqslant CM^{\frac{2}{p}}\frac{\delta}{\sqrt{N}} \tag{3.10}\]
and
\[\left\|u_{\delta,N}^{(1)}\right\|_{\widetilde{L^{N}}(0,T_{N};\dot {B}_{p,2}^{\frac{2}{p-1+\frac{2}{N}}})} \leqslant T_{N}^{\frac{1}{N}}\left\|\widetilde{F}_{\delta,N} \right\|_{\dot{B}_{p,1}^{\frac{2}{p-1+\frac{2}{N}}}+\left\|e^{t\Delta} \widetilde{F}_{\delta,N}\right\|_{\widetilde{L^{N}}(0,T_{N};\dot{B}_{p,2}^{ \frac{2}{p-1+\frac{2}{N}}})}\] \[\leqslant CM^{\frac{2}{p}+\frac{2}{N}}\frac{\delta}{\sqrt{N}}+C \left\|\widetilde{F}_{\delta,N}\right\|_{\dot{B}_{p,1}^{\frac{2}{p-1}}} \tag{3.11}\] \[\leqslant CM^{\frac{2}{p}+1}\frac{\delta}{\sqrt{N}}.\]
Here, we have used \(T_{N}^{\frac{1}{N}}=4\).
_Step.3 The estimates for the second iteration._ Next, we consider the second iteration:
\[u_{\delta,N}^{(2)}(t):=\mathcal{D}\left[u_{\delta,N}^{(1)},u_{\delta,N}^{(1)} \right](t)=-\int_{0}^{t}e^{(t-\tau)\Delta}\mathbb{P}\operatorname{div}\left( u_{1}^{(1)}(\tau)\otimes u_{1}^{(1)}(\tau)\right)d\tau.\]
We decompose \(u_{\delta,N}^{(2)}\) as
\[u_{\delta,N}^{(2)}=\mathcal{D}\left[\left(1-e^{\tau\Delta}\right)\widetilde{ F}_{\delta,N},\left(1-e^{\tau\Delta}\right)\widetilde{F}_{\delta,N}\right]=u_{ \delta,N}^{(2,1)}+u_{\delta,N}^{(2,2)},\]
where
\[u_{\delta,N}^{(2,1)} :=\mathcal{D}\left[\widetilde{F}_{\delta,N},\widetilde{F}_{\delta,N }\right],\] \[u_{\delta,N}^{(2,2)} :=\,-\mathcal{D}\left[e^{\tau\Delta}\widetilde{F}_{\delta,N}, \widetilde{F}_{\delta,N}\right]-\mathcal{D}\left[\widetilde{F}_{\delta,N},e^{ \tau\Delta}\widetilde{F}_{\delta,N}\right]+\mathcal{D}\left[e^{\tau\Delta} \widetilde{F}_{\delta,N},e^{\tau\Delta}\widetilde{F}_{\delta,N}\right].\]
We focus on the estimate of \(u_{\delta,N}^{(2,1)}\). We note that it holds
\[u_{\delta,N}^{(2,1)}=-(-\Delta)^{-1}\left(1-e^{t\Delta}\right)\mathbb{P}\, \mathrm{div}\left(\widetilde{F}_{\delta,N}\otimes\widetilde{F}_{\delta,N} \right).\]
By the direct calculation (see [10, Lemma 2.1] for details), there holds
\[\Delta_{j}u_{\delta,N}^{(2,1)}(t) =\,-M^{2}\frac{\delta^{2}}{2N}\Delta_{j}(-\Delta)^{-1}\left(1-e^ {t\Delta}\right)\mathbb{P}\begin{pmatrix}0\\ \partial_{x_{2}}(\Psi^{2})\end{pmatrix}\] \[\qquad-\frac{\delta^{2}}{2N}\Delta_{j}(-\Delta)^{-1}\left(1-e^{t \Delta}\right)\mathbb{P}\,\mathrm{div}\left(\nabla^{\perp}\Psi\otimes\nabla^ {\perp}\Psi\right)\] \[=:\Delta_{j}u_{\delta,N}^{(2,1,1)}(t)+\Delta_{j}u_{\delta,N}^{(2, 1,2)}(t)\]
for \(j\in\mathbb{Z}\) with \(j\leqslant 0\). Let
\[A_{j}:=\left\{\xi\in\mathbb{R}^{2}\ ;\ 2^{j-1}\leqslant|\xi|\leqslant 2^{j+1},\ \frac{|\xi|}{2}\leqslant|\xi_{2}|\leqslant\frac{|\xi|}{\sqrt{2}}\right\}.\]
The Fourier transform of \(u_{\delta,N}^{(2,1,1)}(t)\) is estimated as
\[\left|\mathscr{F}\left[\Delta_{j}u_{\delta,N}^{(2,1,1)}(t)\right] \left(\xi\right)\right| \geqslant\,\left|\mathscr{F}\left[\Delta_{j}\left(u_{\delta,N}^{( 2,1,1)}(t)\right)_{2}\right](\xi)\right|\] \[=M^{2}\frac{\delta^{2}}{2N}\frac{1-e^{-t|\xi|^{2}}}{|\xi|^{2}} \left(1-\frac{\xi_{2}^{2}}{|\xi|^{2}}\right)|\xi_{2}|\widehat{\Phi_{0}}(2^{-j }\xi)\left(\widehat{\Psi}*\widehat{\Psi}\right)(\xi)\] \[\geqslant cM^{2}\frac{\delta^{2}}{N}\cdot\frac{1-e^{-\frac{1}{4}t 2^{2j}}}{2^{j}}\widehat{\Phi_{0}}(2^{-j}\xi)\left(\widehat{\Psi}*\widehat{ \Psi}\right)(\xi)\]
for \(\xi\in A_{j}\), where \((u_{\delta,N}^{(2,1,1)}(t))_{2}\) denotes the second component of \(u_{\delta,N}^{(2,1,1)}(t)\). Thus, it holds by the Bernstein inequality and the Plancherel theorem that
\[2^{(\frac{2}{p}-1)j}\left\|\Delta_{j}u_{\delta,N}^{(2,1,1)}(T_{ N})\right\|_{L^{p}} \tag{3.12}\] \[\geqslant c\left\|\mathscr{F}\left[\Delta_{j}u_{\delta,N}^{(2,1,1 )}(T_{N})\right]\right\|_{L^{2}}\] \[\geqslant cM^{2}\frac{\delta^{2}}{N}\cdot\frac{1-e^{-\frac{1}{4}T _{N}2^{2j}}}{2^{j}}\left\|\widehat{\Phi_{0}}(2^{-j}\xi)\left(\widehat{\Psi}* \widehat{\Psi}\right)(\xi)\right\|_{L^{2}_{\xi}(A_{j})}\] \[=cM^{2}\frac{\delta^{2}}{N}\left(1-e^{-2^{2(N+j-1)}}\right)\left\| \widehat{\Phi_{0}}(\eta)\left(\widehat{\Psi}*\widehat{\Psi}\right)(2^{j}\eta )\right\|_{L^{2}_{\eta}(A_{0})}.\]
Here, we have changed the variables \(\eta=2^{-j}\xi\) in the last line of (3.12). Since \(\widehat{\Psi}(2^{j}\eta-\mu)=\widehat{\Psi}(\mu)=1\) for all \(\eta\in A_{0}\), \(\mu\) with \(|\mu|\leqslant 1/2\) and \(j\leqslant-2\), we have
\[\left(\widehat{\Psi}*\widehat{\Psi}\right)(2^{j}\eta)=\int_{\mathbb{R}^{2}} \widehat{\Psi}(2^{j}\eta-\mu)\widehat{\Psi}(\mu)d\mu\geqslant\int_{|\mu| \leqslant\frac{1}{2}}d\mu=c>0\]
for \(j\leqslant-2\), which implies
\[\inf_{j\leqslant-2}\left\|\widehat{\Phi_{0}}(\eta)\left(\widehat{\Psi}* \widehat{\Psi}\right)(2^{j}\eta)\right\|_{L^{2}_{\eta}(A_{0})}\geqslant c \left\|\widehat{\Phi_{0}}\right\|_{L^{2}(A_{0})}>0. \tag{3.13}\]
Hence, we obtain by (3.12) and (3.13) that
\[\begin{split}\left\|u_{\delta,N}^{(2,1,1)}(T_{N})\right\|_{\dot{B} _{p,1}^{\frac{2}{p-1}}}&\geq\sum_{-N\leq j\leq-2}2^{(\frac{2}{p}- 1)j}\left\|\Delta_{j}u_{\delta,N}^{(2,1,1)}(T_{N})\right\|_{L^{p}}\\ &\geq cM^{2}\frac{\delta^{2}}{N}\sum_{-N\leq j\leq-2}\left(1-e^{- 2^{2(N+j-1)}}\right)\left\|\widehat{\Phi}_{0}\right\|_{L^{2}(A_{0})}\\ &\geq c_{0}M^{2}\delta^{2}\end{split} \tag{3.14}\]
for some positive constant \(c_{0}=c_{0}(p,\Psi)\). For the estimate of \(u_{\delta,N}^{(2,1,2)}\), using
\[u_{\delta,N}^{(2,1,2)}(t)=-\frac{\delta^{2}}{2N}(-\Delta)^{-1}\left(1-e^{t \Delta}\right)\mathbb{P}\operatorname{div}\left(\nabla^{\perp}\Psi\otimes \nabla^{\perp}\Psi\right)=\frac{\delta^{2}}{2N}\mathcal{D}\left[\nabla^{\perp} \Psi,\nabla^{\perp}\Psi\right](t)\]
Lemma 2.5, we have
\[\left\|u_{\delta,N}^{(2,1,2)}\right\|_{\widetilde{L}^{\infty}(0, T_{N};\dot{B}_{p,1}^{\frac{2}{p-1}})} \leq C\delta^{2}\big{\|}\nabla^{\perp}\Psi\big{\|}_{\widetilde{L }^{N}(0,T_{N};\dot{B}_{p,1}^{\frac{2}{p-1}+\frac{2}{N}})}^{2}+C\frac{\delta^{ 2}}{N}\big{\|}\nabla^{\perp}\Psi\big{\|}_{\dot{B}_{p,1}^{\frac{2}{p-1}}}^{2} \tag{3.15}\] \[\leq C\delta^{2}T_{N}^{\frac{1}{N}}\|\Psi\|_{\dot{B}_{p,1}^{\frac {2}{p+\frac{2}{N}}}}^{2}+C\frac{\delta^{2}}{N}\|\Psi\|_{\dot{B}_{p,1}^{\frac{2} {p}}}^{2}\] \[\leq C_{0}\delta^{2}.\]
for some positive constant \(C_{0}=C_{0}(p,\Psi)\). For the estimate of \(u_{\delta,N}^{(2,2)}\), using Lemma 2.3, we have
\[\begin{split}\left\|u_{\delta,N}^{(2,2)}\right\|_{\widetilde{L}^ {\infty}(0,\infty;\dot{B}_{p,1}^{\frac{2}{p-1}})}&\leq C\| \widetilde{F}_{\delta,N}\|_{\dot{B}_{p,1}^{\frac{2}{p-1}}}\left\|e^{t\Delta} \widetilde{F}_{\delta,N}\right\|_{\widetilde{L}^{2}(0,\infty;\dot{B}_{p,1}^{ \frac{2}{p}})}+C\left\|e^{t\Delta}\widetilde{F}_{\delta,N}\right\|_{\widetilde {L}^{2}(0,\infty;\dot{B}_{p,1}^{\frac{2}{p}})}^{2}\\ &\leq C\|\widetilde{F}_{\delta,N}\|_{\dot{B}_{p,1}^{\frac{2}{p-1} }}^{2}\\ &\leq CM^{\frac{4}{p}}\frac{\delta^{2}}{N}.\end{split} \tag{3.16}\]
We now fix \(M\) so that
\[M:=\max\left\{10,\sqrt{2+\frac{C_{0}}{c_{0}}}\right\}.\]
Then, we obtain by (3.14), (3.15), and (3.16) that
\[\begin{split}\left\|u_{\delta,N}^{(2)}(T_{N})\right\|_{\dot{B}_{p,1}^{\frac{2}{p-1}}}&\geq\,\left\|u_{\delta,N}^{(2,1,1)}(T_{N}) \right\|_{\dot{B}_{p,1}^{\frac{2}{p-1}}}\\ &\quad-\left\|u_{\delta,N}^{(2,1,2)}\right\|_{\widetilde{L}^{ \infty}(0,T_{N};\dot{B}_{p,1}^{\frac{2}{p-1}})}-\left\|u_{\delta,N}^{(2,2)} \right\|_{\widetilde{L}^{\infty}(0,\infty;\dot{B}_{p,1}^{\frac{2}{p-1}})}\\ &\geq\,\left(M^{2}c_{0}-C_{0}-C\frac{M^{\frac{4}{p}}}{N}\right) \delta^{2}\\ &\geq\,\left(2c_{0}-\frac{C}{N}\right)\delta^{2}.\end{split} \tag{3.17}\]
On the other hand, it follows from Lemma 2.5, (3.10), and (3.11) that
\[\left\|u_{\delta,N}^{(2)}\right\|_{\widetilde{L^{\infty}}(0,\infty; \hat{B}_{p,1}^{\frac{2}{p}-1})} \leqslant CN\left\|u_{\delta,N}^{(1)}\right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p-1+\frac{2}{N}}})}^{2}+C\left\|u_{\delta,N}^{(1 )}\right\|_{\widetilde{L^{\infty}}(0,\infty;\hat{B}_{p,1}^{\frac{2}{p}-1})}^{2} \tag{3.18}\] \[\leqslant C\delta^{2}+C\frac{\delta^{2}}{N}\] \[\leqslant C\delta^{2}\]
and
\[\left\|u_{\delta,N}^{(2)}\right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{ \frac{2}{p-1+\frac{2}{N}}})}\leqslant C\sqrt{N}\left\|u_{\delta,N}^{(1)} \right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p-1+\frac{2}{N}}}) }^{2}\leqslant C\frac{\delta^{2}}{\sqrt{N}}. \tag{3.19}\]
_Step.4 The construction and estimates for the remainder part._ To construct a solution to (3.1) with the external force \(F_{\delta,N}\), we focus on the perturbation of a solution to (3.1) with the external force \(F_{\delta,N}\) from the second approximation \(u_{\delta,N}^{(1)}+u_{\delta,N}^{(2)}\). If \(u_{\delta,N}\) is a solution to (3.1) with the external force \(F_{\delta,N}\), then \(w_{\delta,N}:=u_{\delta,N}-u_{\delta,N}^{(1)}-u_{\delta,N}^{(2)}\) should satisfy
\[\left\{\begin{aligned} &\partial_{t}w_{\delta,N}-\Delta w_{ \delta,N}+\mathbb{P}\operatorname{div}\left(u_{\delta,N}^{(1)}\otimes u_{ \delta,N}^{(2)}+u_{\delta,N}^{(2)}\otimes u_{\delta,N}^{(1)}+u_{\delta,N}^{(2 )}\otimes u_{\delta,N}^{(2)}\right.\\ &\qquad\qquad\qquad\qquad\qquad\qquad+u_{\delta,N}^{(1)}\otimes w _{\delta,N}+u_{\delta,N}^{(2)}\otimes w_{\delta,N}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+w_{\delta,N} \otimes u_{\delta,N}^{(1)}+\,w_{\delta,N}\otimes u_{\delta,N}^{(2)}+w_{\delta,N}\otimes w_{\delta,N}\right)=0,\\ &\operatorname{div}w_{\delta,N}=0,\\ & w_{\delta,N}(0,x)=0.\end{aligned}\right. \tag{3.20}\]
To construct the mild solution to (3.20), we consider the map
\[\mathcal{S}_{N}[w]:= \mathcal{D}\left[u_{\delta,N}^{(1)},u_{\delta,N}^{(2)}\right]+ \mathcal{D}\left[u_{\delta,N}^{(2)},u_{\delta,N}^{(1)}\right]+\mathcal{D} \left[u_{\delta,N}^{(2)},u_{\delta,N}^{(2)}\right]\] \[+\mathcal{D}\left[u_{\delta,N}^{(1)},w\right]+\mathcal{D}\left[u_ {\delta,N}^{(2)},w\right]+\mathcal{D}\left[w,u_{\delta,N}^{(1)}\right]+ \mathcal{D}\left[w,u_{\delta,N}^{(2)}\right] \tag{3.21}\] \[+\mathcal{D}[w,w].\]
Here, we consider the estimates for the first three terms of the right hand side of (3.21). By virtue of Lemma 2.5, (3.10), (3.11), (3.18), and (3.19), we have
\[\left\|\mathcal{D}\left[u_{\delta,N}^{(1)},u_{\delta,N}^{(2)} \right]\right\|_{\widetilde{L^{\infty}}(0,T_{N};\hat{B}_{p,1}^{\frac{2}{p}-1} )}+\left\|\mathcal{D}\left[u_{\delta,N}^{(2)},u_{\delta,N}^{(1)}\right] \right\|_{\widetilde{L^{\infty}}(0,T_{N};\hat{B}_{p,1}^{\frac{2}{p}-1})}\] \[\quad\leqslant CN\left\|u_{\delta,N}^{(1)}\right\|_{\widetilde{L^ {N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}})}\left\|u_{\delta,N}^{( 2)}\right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{ N}})}\] \[\quad+C\left\|u_{\delta,N}^{(1)}\right\|_{\widetilde{L^{\infty}} (0,\infty;\hat{B}_{p,1}^{\frac{2}{p}-1})}\left\|u_{\delta,N}^{(2)}\right\|_{ \widetilde{L^{\infty}}(0,\infty;\hat{B}_{p,1}^{\frac{2}{p}-1})}\] \[\quad\leqslant C\delta^{3},\] \[\left\|\mathcal{D}\left[u_{\delta,N}^{(2)},u_{\delta,N}^{(2)} \right]\right\|_{\widetilde{L^{\infty}}(0,T_{N};\hat{B}_{p,1}^{\frac{2}{p}-1})} \leqslant CN\left\|u_{\delta,N}^{(2)}\right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}})}^{2}+C\left\|u_{\delta,N}^{( 2)}\right\|_{\widetilde{L^{\infty}}(0,\infty;\hat{B}_{p,1}^{\frac{2}{p}-1})}^{2}\] \[\leqslant C\delta^{4}\]
and
\[\left\|\mathcal{D}\left[u_{\delta,N}^{(1)},u_{\delta,N}^{(2)} \right]\right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p}-1+\frac{ 2}{N}})}+\left\|\mathcal{D}\left[u_{\delta,N}^{(2)},u_{\delta,N}^{(1)}\right] \right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}_{p,2}^{\frac{2}{p}-1+\frac{2}{N}})}\]
\[\leqslant C_{1}\delta^{3}+C\delta\sqrt{N}\|w\|_{\widetilde{L^{N}}(0,T_{N}; \hat{B}^{\frac{2}{p-1+\frac{2}{N}}}_{p,2}}+C\delta\|w\|_{\widetilde{L^{\infty}} (0,T_{N};\hat{B}^{\frac{2}{p-1}+\frac{2}{N}}_{p,1}})\] \[\quad+CN\|w\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}^{\frac{2}{p-1+ \frac{2}{N}}}_{p,2}}+C\|w\|_{\widetilde{L^{\infty}}(0,T_{N};\hat{B}^{\frac{2}{ p-1}}_{p,1}})\] \[\leqslant C_{1}\delta^{3}+C_{2}\delta^{4}\]
and
\[\|\mathcal{S}_{N}[w]\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}^{\frac{ 2}{p-1+\frac{2}{N}}}_{p,2}})\] \[\quad\leqslant C_{1}\frac{\delta^{3}}{\sqrt{N}}+C\sqrt{N}\sum_{k= 1}^{2}\left\|u^{(k)}_{\delta,N}\right\|_{\widetilde{L^{N}}(0,T_{N};\hat{B}^{ \frac{2}{p-1+\frac{2}{N}}}_{p,2}})\left\|w\right\|_{\widetilde{L^{N}}(0,T_{N };\hat{B}^{\frac{2}{p-1+\frac{2}{N}}}_{p,2}})\]
\[\leqslant C_{3}\delta\|w_{1}-w_{2}\|_{\widetilde{L^{N}}(0,T_{N};\hat{B} ^{\frac{2}{p}-1+\frac{2}{N}}_{p,2})}\]
for some positive constant \(C_{3}=C_{3}(p,\Psi)\). Here, we choose \(\delta\) so small that
\[0<\delta\leqslant\delta_{1}:=\min\left\{\frac{C_{1}}{C_{2}},\frac{1}{4C_{3}}, \frac{c_{0}}{2C_{1}}\right\}.\]
Then, we have
\[\|\mathcal{S}_{N}[w]\|_{\widetilde{L^{\infty}}(0,T_{N};\hat{B}^{\frac{2}{p}-1 +\frac{2}{N}}_{p,1})}\leqslant 2C_{1}\delta^{3},\]
\[\|\mathcal{S}_{N}[w]\|_{\widetilde{L^{N}}(0,T_{N};\dot{B}^{\frac{2}{p}- 1+\frac{2}{N}}_{p,2})}\leqslant 2C_{1}\frac{\delta^{3}}{\sqrt{N}},\] \[d_{X_{N}}(\mathcal{S}_{N}[w_{1}],\mathcal{S}_{N}[w_{2}]) \leqslant\frac{1}{2}d_{X_{N}}(w_{1},w_{2}),\]
which implies that \(\mathcal{S}_{N}[\cdot]\) is a contraction map on \((X_{N},d_{X_{N}})\). Hence, by the Banach fixed point theorem, there exists a unique element \(w_{\delta,N}\in X_{N}\) such that \(w_{\delta,N}=\mathcal{S}_{N}[w_{\delta,N}]\), which means that the mild solution \(w_{\delta,N}\) of (3.20) uniquely exists in \(X_{N}\).
_Step.5 Conclusion_. We see that the function
\[u_{\delta,N}:=u^{(1)}_{\delta,N}+u^{(2)}_{\delta,N}+w_{\delta,N}\in\widetilde{ C}([0,T_{N}];\dot{B}^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^{2}))\]
is a mild solution to (3.1) with the external force \(F_{\delta,N}\) and also obtain by (3.10), (3.17), and \(w_{\delta,N}\in X_{N}\) that
\[\|u_{\delta,N}(T_{N})\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}} \geqslant\,\left\|u^{(2)}_{\delta,N}(T_{N})\right\|_{\dot{B}^{ \frac{2}{p}-1}_{p,1}}-\left\|u^{(1)}_{\delta,N}\right\|_{\widetilde{L^{ \infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{p,1})}-\left\|w_{\delta,N}\right\|_ {\widetilde{L^{\infty}}(0,T_{N};\dot{B}^{\frac{2}{p}-1}_{p,1})}\] \[\geqslant\,\left(2c_{0}-\frac{C}{N}\right)\delta^{2}-C\frac{ \delta}{\sqrt{N}}-2C_{1}\delta^{3}\] \[\geqslant\,\left(c_{0}-\frac{C}{N}\right)\delta^{2}-C\frac{ \delta}{\sqrt{N}},\]
which yields
\[\liminf_{N\to\infty}\|u_{\delta,N}(T_{N})\|_{\dot{B}^{\frac{2}{p}-1}_{p,1}} \geqslant c_{0}\delta^{2}.\]
It follows from (3.10), (3.18), and \(w\in X_{N}\) that
\[\|u_{\delta,N}\|_{\widetilde{L^{\infty}}(0,T_{N};\dot{B}^{\frac{ 2}{p}-1}_{p,1})}\] \[\leqslant\,\left\|u^{(1)}_{\delta,N}\right\|_{\widetilde{L^{ \infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{p,1})}+\left\|u^{(2)}_{\delta,N} \right\|_{\widetilde{L^{\infty}}(0,T_{N};\dot{B}^{\frac{2}{p}-1}_{p,1})}+\|w_ {\delta,N}\|_{\widetilde{L^{\infty}}(0,T_{N};\dot{B}^{\frac{2}{p}-1}_{p,1})}\] \[\leqslant C\frac{\delta}{\sqrt{N}}+C\delta^{2}+C\delta^{3},\]
which implies
\[\limsup_{N\to\infty}\|u_{\delta,N}\|_{\widetilde{L^{\infty}}(0,T_{N};\dot{B}^ {\frac{2}{p}-1}_{p,1})}\leqslant C\delta^{2}+C\delta^{3}\leqslant C\delta^{2}.\]
Thus, we complete the proof.
### Global solutions around the stationary flow
In contrast to the previous subsection, if we assume that the stationary problem (1.7) possesses a solution \(U\) for some external force \(F\) and then consider the nonstationary Navier-Stokes equations (3.1) with the same external force \(F\) as for \(U\). Under this assumption, we may prove that (3.1) admits a bounded-in-time global solution.
**Theorem 3.3**.: _Let \(1\leqslant p<4\) and \(1\leqslant q<\infty\). Then, there exist a positive constant \(\delta_{2}=\delta_{2}(p,q)\) and an absolute positive constant \(K_{2}\) such that if a given external force \(F\in\dot{B}^{\frac{2}{p}-3}_{p,q}(\mathbb{R}^{2})\) generates a solution \(U\in\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2})\) to (1.7) satisfying_
\[\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}\leqslant\delta_{2},\]
_then (3.1) with the same external force \(F\) admits a global mild solution \(u\) in the class_
\[u\in\widetilde{C}([0,\infty);\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2})), \qquad\|u\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{p,q})} \leqslant K_{2}\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}.\]
Assuming the existence of the stationary solution, we consider the perturbation \(v=u-U\), which should solve
\[\begin{cases}\partial_{t}v-\Delta v+\mathbb{P}\operatorname{div}(U\otimes v+v \otimes U+v\otimes v)=0,&t>0,x\in\mathbb{R}^{2},\\ \operatorname{div}v=0,&t\geqslant 0,x\in\mathbb{R}^{2},\\ v(0,x)=-U(x),&x\in\mathbb{R}^{2},\end{cases} \tag{3.22}\]
then (3.22) possesses no external force that does not decay as \(t\to\infty\), which implies that the solution \(v\) of (3.22) is expected to decay as \(t\to\infty\) and belong to some time integrable function spaces. Since the nonlinear estimate is closed in
\[\widetilde{C}([0,\infty);\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2}))\cap \widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{r}}_{p,q}(\mathbb{ R}^{2})) \tag{3.23}\]
for some \(2<r<\infty\) (see Lemma 2.3), we may establish the global solution \(v\) to (3.22) in the class (3.23). We then obtain the desired solution by \(u:=v+U\).
Now, we provide the precise proof as follows.
Proof of Theorem 3.3.: We first construct a mild solution \(v\) of (3.22) solving the following integral equation:
\[v(t)=-e^{t\Delta}U+\mathcal{D}[U,v](t)+\mathcal{D}[v,U](t)+\mathcal{D}[v,v](t),\]
where the nonlinear term \(\mathcal{D}[\cdot,\cdot]\) is defined in (2.12). To this end, we focus on the map
\[\mathcal{S}[v](t):=-e^{t\Delta}U+\mathcal{D}[U,v](t)+\mathcal{D}[v,U](t)+ \mathcal{D}[v,v](t)\]
and shall show that \(\mathcal{S}[\cdot]\) is a contraction map on the complete metric space \((X,d_{X})\) defined by
\[X:=\left\{v\in\widetilde{C}([0,\infty);\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2}))\cap\widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+ \frac{2}{r}}_{p,q}(\mathbb{R}^{2}))\ ;\right.\] \[\left.\qquad\qquad\qquad\qquad\left.\|v\right\|_{\widetilde{L^{ \infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{p,q})\cap\widetilde{L^{r}}(0, \infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{r}}_{p,q})}\leqslant 2C_{4}\|U\|_{\dot{B}^{ \frac{2}{p}-1}_{p,q}}\right\},\] \[d_{X}(v_{1},v_{2}):=\|v_{1}-v_{2}\|_{\widetilde{L^{r}}(0,\infty; \dot{B}^{\frac{2}{p}-1+\frac{2}{r}}_{p,q})},\]
where \(r=r(p)\) is a fixed exponent satisfying
\[\max\left\{0,1-\frac{2}{p}\right\}<\frac{1}{r}<\frac{1}{2}\]
and the positive constant \(C_{4}\) is determined by the estimate
\[\left\|e^{t\Delta}U\right\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}^{\frac{2 }{p}-1}_{p,q})\cap\widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{r }}_{p,q})}\leqslant C_{4}\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}},\]
which is ensured by Lemma 2.1. Then, it follows from Lemma 2.3 that
\[\|\mathcal{S}[v]\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}^{ \frac{2}{p}-1}_{p,q})\cap\widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+ \frac{2}{r}}_{p,q})}\] \[\quad\leqslant C_{4}\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}+C\|U\|_{ \dot{B}^{\frac{2}{p}-1}_{p,q}}\|v\|_{\widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{ 2}{p}-1+\frac{2}{r}}_{p,q})}+C\|v\|^{2}_{\widetilde{L^{r}}(0,\infty;\dot{B}^{ \frac{2}{p}-1+\frac{2}{r}}_{p,q})}\] \[\quad\leqslant C_{4}\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}+C_{5}\| U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}\|v\|_{\widetilde{L^{r}}(0,\infty;\dot{B}^{ \frac{2}{p}-1+\frac{2}{r}}_{p,q})}\]
for all \(v\in X\), with some positive constant \(C_{5}=C_{5}(p,q,r)\). Since there holds
\[\mathcal{S}[v_{1}]-\mathcal{S}[v_{2}]=\mathcal{D}[U,v_{1}-v_{2}]+\mathcal{D}[v_ {1}-v_{2},U]+\mathcal{D}[v_{1},v_{1}-v_{2}]+\mathcal{D}[v_{1}-v_{2},v_{2}],\]
we have by Lemma 2.3 that
\[\|\mathcal{S}[v_{1}]-\mathcal{S}[v_{2}]\|_{\widetilde{L^{r}}(0, \infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{p}}_{p,q}}\leqslant C\|U\|_{\dot{B}^{ \frac{2}{p}-1}_{p,q}}\|v_{1}-v_{2}\|_{\widetilde{L^{r}}(0,\infty;\dot{B}^{ \frac{2}{p}-1+\frac{2}{p}}_{p,q}})\] \[\qquad\qquad+C\sum_{\ell=1}^{2}\|v_{\ell}\|_{\widetilde{L^{r}}(0, \infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{p}}_{p,q}}\|v_{1}-v_{2}\|_{\widetilde{L ^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{p}}_{p,q}})\] \[\leqslant C_{6}\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}\|v_{1}-v_{2} \|_{\widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{p}}_{p,q}})\]
for all \(v_{1},v_{2}\in X\), with some positive constant \(C_{6}=C_{6}(p,q,r)\). Now, we assume that the stationary solution \(U\in\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2})\) satisfies
\[\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}}\leqslant\delta_{2}:=\min\left\{\frac{C_ {4}}{C_{5}},\frac{1}{2C_{6}}\right\}.\]
Then, we obtain
\[\|\mathcal{S}[v]\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{ p,q})\cap\widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{r}}_{p,q}} \leqslant 2C_{4}\|U\|_{\dot{B}^{\frac{2}{p}-1}_{p,q}},\]
\[\|\mathcal{S}[v_{1}]-\mathcal{S}[v_{2}]\|_{\widetilde{L^{r}}(0,\infty;\dot{B}^ {\frac{2}{p}-1+\frac{2}{p}}_{p,q}}\leqslant\frac{1}{2}\|v_{1}-v_{2}\|_{ \widetilde{L^{r}}(0,\infty;\dot{B}^{\frac{2}{p}-1+\frac{2}{p}}_{p,q}})\]
for all \(v,v_{1},v_{2}\in X\), which implies \(\mathcal{S}[\cdot]\) is a contraction map on \((X,d_{X})\). Hence, the Banach fixed point theorem implies that there exists a unique \(v\in X\) such that \(v=\mathcal{S}[v]\).
Now, we put \(u:=v+U\). Then, we see that \(u\) is a mild solution to (3.1) in the class \(\widetilde{C}([0,\infty);\dot{B}^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^{2}))\), and it holds
\[\|u\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{p,q}}\leqslant \|v\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}^{\frac{2}{p}-1}_{p,q}}+\|U\|_{ \dot{B}^{\frac{2}{p}-1}_{p,q}}\leqslant(2C_{4}+1)\|U\|_{\dot{B}^{\frac{2}{p}- 1}_{p,q}}.\]
Thus, we complete the proof.
## 4. Proof of Theorem 1.2
Now, we are in a position to present the proof of our main result.
Proof of Theorem 1.2.: Let \(\delta_{1}\) and \(\delta_{2}\) be the positive constants appearing in Theorems 3.1 and 3.3, respectively. Let \(K_{0}\), \(K_{1}\), and \(K_{2}\) be the positive constants appearing in Lemma 2.4, Theorem 3.1, and Theorem 3.3, respectively. We define
\[\delta_{0}:=\min\left\{\delta_{2},\delta_{3},\frac{\delta_{3}^{2}}{2K_{1}K_{2} }\right\},\]
where \(\delta_{3}\) is a positive constant given by
\[\delta_{3}:=\min\left\{\delta_{1},\frac{1}{2K_{0}(K_{1}+K_{2})}\right\}.\]
We consider the sequence \(F_{N}:=F_{\delta_{3},N}\), which is defined in (3.7) with \(\delta\) replaced by \(\delta_{3}\). Note that Theorem 3.1 yields
\[\|F_{N}\|_{\dot{B}^{\frac{2}{p}-3}_{p,1}}\leqslant\frac{K_{1}\delta_{3}}{ \sqrt{N}}\to 0\qquad\text{ as }N\to\infty.\]
Let us consider the nonstationary Navier-Stokes equations
\[\begin{cases}\partial_{t}u-\Delta u+\mathbb{P}\operatorname{div}(u\otimes u)= \mathbb{P}F_{N},&t>0,x\in\mathbb{R}^{2},\\ \operatorname{div}u=0,&t\geqslant 0,x\in\mathbb{R}^{2},\\ u(0,x)=0,&x\in\mathbb{R}^{2}.\end{cases} \tag{4.1}\]
By Theorem 3.1, there exists a \(N_{0}=N_{0}(p)\in\mathbb{N}\) such that for each \(N\in\mathbb{N}\) with \(N\geqslant N_{0}\), (4.1) possesses a solution \(u_{N}=u_{\delta_{3},N}\in\widetilde{C}([0,T_{N}];\dot{B}_{p,1}^{\frac{2}{p}-1} (\mathbb{R}^{2}))\) satisfying
\[\|u_{N}(T_{N})\|_{\dot{B}_{p,1}^{\frac{2}{p}-1}}\geqslant\frac{\delta_{3}^{2}} {K_{1}},\qquad\|u_{N}\|_{\widetilde{L^{\infty}}(0,T_{N};\dot{B}_{p,1}^{\frac{2 }{p}-1})}\leqslant K_{1}\delta_{3}. \tag{4.2}\]
Here, we have set \(T_{N}:=2^{2N}\).
Assume to contrary that there exist an integer \(N^{\prime}\geqslant N_{0}\) and a solution \(U_{N^{\prime}}\in\dot{B}_{p,1}^{\frac{2}{p}-1}(\mathbb{R}^{2})\) of (1.7) with the external force \(F_{N^{\prime}}\) satisfying
\[\|U_{N^{\prime}}\|_{\dot{B}_{p,1}^{\frac{2}{p}-1}}<\delta_{0}. \tag{4.3}\]
Then, by (4.3) and Theorem 3.3, each \(F_{N^{\prime}}\) generates a global-in-time solution \(\widetilde{u}_{N^{\prime}}\in\widetilde{C}([0,\infty);\dot{B}_{p,1}^{\frac{2} {p}-1}(\mathbb{R}^{2}))\) to the nonstationary Navier-Stokes equations (4.1) satisfying
\[\|\widetilde{u}_{N^{\prime}}\|_{\widetilde{L^{\infty}}(0,\infty;\dot{B}_{p,1}^ {\frac{2}{p}-1})}\leqslant K_{2}\|U_{N^{\prime}}\|_{\dot{B}_{p,1}^{\frac{2}{ p}-1}}\leqslant K_{2}\delta_{3}. \tag{4.4}\]
Next, we show that these two solutions \(\widetilde{u}_{N^{\prime}}\) and \(u_{N^{\prime}}\) coincides on \([0,T_{N^{\prime}}]\). Since \(\widetilde{u}_{N^{\prime}}-u_{N^{\prime}}\) enjoys
\[\widetilde{u}_{N^{\prime}}-u_{N^{\prime}}=\mathcal{D}\left[\widetilde{u}_{N^{ \prime}},\widetilde{u}_{N^{\prime}}-u_{N^{\prime}}\right]+\mathcal{D}\left[ \widetilde{u}_{N^{\prime}}-u_{N^{\prime}},u_{N^{\prime}}\right],\]
we see by Lemma 2.4 that
\[\sup_{0\leqslant t\leqslant T_{N^{\prime}}}\|\widetilde{u}_{N^{ \prime}}(t)-u_{N^{\prime}}(t)\|_{\dot{B}_{p,\infty}^{\frac{2}{p}-1}}\] \[\quad\leqslant K_{0}\|u_{N^{\prime}}\|_{\widetilde{L^{\infty}}(0, T_{N^{\prime}};\dot{B}_{p,1}^{\frac{2}{p}-1})}\sup_{0\leqslant t\leqslant T_{N^{ \prime}}}\|\widetilde{u}_{N^{\prime}}(t)-u_{N^{\prime}}(t)\|_{\dot{B}_{p,\infty }^{\frac{2}{p}-1}}\] \[\quad\quad+K_{0}\|\widetilde{u}_{N^{\prime}}\|_{\widetilde{L^{ \infty}}(0,T_{N^{\prime}};\dot{B}_{p,1}^{\frac{2}{p}-1})}\sup_{0\leqslant t \leqslant T_{N^{\prime}}}\|\widetilde{u}_{N^{\prime}}(t)-u_{N^{\prime}}(t)\|_{ \dot{B}_{p,\infty}^{\frac{2}{p}-1}}\] \[\quad\leqslant K_{0}\left(K_{1}+K_{2}\right)\delta_{3}\sup_{0 \leqslant t\leqslant T_{N^{\prime}}}\|\widetilde{u}_{N^{\prime}}(t)-u_{N^{ \prime}}(t)\|_{\dot{B}_{p,\infty}^{\frac{2}{p}-1}}\] \[\quad\leqslant\frac{1}{2}\sup_{0\leqslant t\leqslant T_{N^{\prime }}}\|\widetilde{u}_{N^{\prime}}(t)-u_{N^{\prime}}(t)\|_{\dot{B}_{p,\infty}^{ \frac{2}{p}-1}},\]
which implies
\[\widetilde{u}_{N^{\prime}}(t)=u_{N^{\prime}}(t)\quad\text{for all }0\leqslant t \leqslant T_{N^{\prime}}. \tag{4.5}\]
Hence, it follows from (4.2), (4.4), and (4.5) that
\[\|U_{N^{\prime}}\|_{\dot{B}_{p,1}^{\frac{2}{p}-1}} \geqslant\frac{1}{K_{2}}\|\widetilde{u}_{N^{\prime}}\|_{\widetilde {L^{\infty}}(0,T_{N^{\prime}};\dot{B}_{p,1}^{\frac{2}{p}-1})}\] \[\geqslant\frac{1}{K_{2}}\|\widetilde{u}_{N^{\prime}}(T_{N^{ \prime}})\|_{\dot{B}_{p,1}^{\frac{2}{p}-1}}\] \[=\frac{1}{K_{2}}\|u_{N^{\prime}}(T_{N^{\prime}})\|_{\dot{B}_{p,1}^ {\frac{2}{p}-1}}\]
\[\geqslant\frac{\delta_{3}^{2}}{K_{1}K_{2}}\] \[\geqslant 2\delta_{0},\]
which contradicts (4.3). Thus, we complete the proof.
### Conflict of interest statement
The author has declared no conflicts of interest.
**Acknowledgements.**
The author was supported by Grant-in-Aid for JSPS Research Fellow, Grant Number JP20J20941. The author would like to express his sincere gratitude to Professor Keiichi Watanabe for many valuable comments on Section 1.
| ```
2次元静止Navier–Stokes方程式を全平面 $\mathbb{R}^2$ で考慮します。$n\geq 3$ の高次元ケースでは、$\mathbb{R}^n$ におけるwell-posedness と ill-posedness は、多くの論文によって調査されてきました。しかし、多くの研究者の努力에도 불구하고、2次元全平面ケースにおける対応する問題として長年議論されてきました。これは、2次元解析の困難性によるものと考えられてきました。この論文の目的は、この問題を解決し、$L^p(\mathbb{R}^2)$ に関するBesov空間におけるスケール critical ill-posedness を証明することです。$1\leq p\leq 2$ の範囲で、解像度の不連続性と小さな解の非存在という観点から。困難を乗り越えるために、矛盾する主張に基づいた新しい方法を提案します。 |
2303.03295 | Probabilistic Game-Theoretic Traffic Routing | We examine the routing problem for self-interested vehicles using stochastic
decision strategies. By approximating the road latency functions and a
non-linear variable transformation, we frame the problem as an aggregative
game. We characterize the approximation error and we derive a new monotonicity
condition for a broad category of games that encompasses the problem under
consideration. Next, we propose a semi-decentralized algorithm to calculate the
routing as a variational generalized Nash equilibrium and demonstrate the
solution's benefits with numerical simulations. In the particular case of
potential games, which emerges for linear latency functions, we explore a
receding-horizon formulation of the routing problem, showing asymptotic
convergence to destinations and analysing closed-loop performance dependence on
horizon length through numerical simulations. | Emilio Benenati, Sergio Grammatico | 2023-03-06T17:12:10 | http://arxiv.org/abs/2303.03295v2 | # Probabilistic Game-Theoretic Traffic Routing
###### Abstract
We examine the routing problem for self-interested vehicles using stochastic decision strategies. By approximating the road latency functions and a non-linear variable transformation, we frame the problem as an aggregative game. We characterize the approximation error and we derive a new monotonicity condition for a broad category of games that encompasses the problem under consideration. Next, we propose a semi-decentralized algorithm to calculate the routing as a variational generalized Nash equilibrium and demonstrate the solution's benefits with numerical simulations. We also explore a recursive receding-horizon formulation of the routing problem for potential games, showing asymptotic convergence to destinations and analysing closed-loop performance dependence on horizon length through numerical simulations.
Traffic control, Game theory, Variational methods
## I Introduction
Traffic jams generate a heavy burden on the society [1] and, as car traffic already makes up a large share of the EU transport infrastructure costs [2], it is imperative to mitigate the road congestion without expanding the existing infrastructure. The increased availability of real time information on the state of the road network has the potential for a more efficient traffic-aware route planning.
Previous works have considered a centralized solution to the routing problem [3, 4]. Unfortunately, there is no guarantee that the drivers would follow an externally imposed solution if a more advantageous path was available to (some of) them. In fact, traffic routing is an inherently competitive decision making process and it is thus more properly modelled as a _game_, as suggested in the seminal work [5]. Crucially, under relatively loose conditions, games admit a set of Nash equilibria, that is, a set of decision strategies from which no agent has an incentive in unilaterally deviating. A Nash equilibrium-based routing is then self-enforcing, in the sense that it guarantees that the vehicles would follow the suggested route without the need for an external imposition.
On this line, the authors in [6, 7] model traffic routing as a game and propose a centralized computation of a Wardrop equilibrium [6, Def. 1.1]. In [6], the authors model the traffic routing problem as multiple coupled Markov Decision Processes (MDPs). This idea is further elaborated in [8], where the authors cast the problem as a _generalized aggregative_ game. In this setting, the infrastructural limits of the network introduce shared constraints between the agent strategies (generalized game) and the cost term coupling the agents depends on an aggregation of all the agents' strategies, namely the network congestion (aggregative game). We identify the following shortcomings in the literature, which we attempt to overcome with the present paper:
* The action costs of the MDPs are often defined as a non-linear function of the aggregate strategies [7, 8, 9]. However, due to the stochastic nature of the decision variables, the interpretation of this cost function and whether it effectively reflects the disadvantage incurred by the users is not straightforward. In Section III, we show that such edge cost formulation is as an approximation of the expected values of their traversing time.
* The generalized Nash equilibrium problem (GNEP) in [8] is solved via the preconditioned forward-backward algorithm [10], which requires the pseudo-gradient mapping of the game to be cocoercive or strongly monotone. However, the latter property is proven only if the share of uncontrolled vehicles with respect to the number of vehicles being routed is large enough [8, Lemma 1]. In Section IV, we relax the condition for the monotonicity of the game pseudogradient. We then propose to solve the game via the Inertial Forward-Reflected-Backward (I-FoRB) algorithm [11], which does not require strict monotonicity of the game pseudo-gradient and in turn it converges without the quadratic regularization term proposed in [8, Equation 5].
Next, in Section V, we study an alternative solution to the traffic routing problem in the particular case of a _potential_ game [12, Sec. 2]. We propose to progressively recompute the agents' paths in a receding horizon fashion (instead of solving for the entire path in one computation). This novel approach allows one to reduce the decision horizon, thus reducing the computational burden as the vehicles move forward. Finally, in Section VI, we support the theoretical results by comparative numerical simulations.
## II Notation
For a matrix \(X\),we denote its \((i,j)\)-th element as \([X]_{(i,j)}\) and its spectral norm as \(\|\cdot\|\). We denote the set with elements \(x_{i}\) indexed in \(i\in\mathcal{I}\) as \((x_{i})_{i\in\mathcal{I}}\). The operators \(\mathrm{col}(x_{i})_{i\in\mathcal{I}}\), \(\mathrm{row}(x_{i})_{i\in\mathcal{I}}\) denote the column-wise and row-wise stack of \((x_{i})_{i\in\mathcal{I}}\), respectively. We denote the block diagonal matrix with nonzero elements \((X_{i})_{i\in\mathcal{I}}\) as \(\mathrm{diag}(X_{i})_{i\in\mathcal{I}}\). We denote the average \(\mathrm{avg}((x_{i})_{i\in\mathcal{I}}):=\frac{1}{|\mathcal{I}|}\sum_{i\in \mathcal{I}}x_{i}\). The vector in \(\mathbb{R}^{n}\) with all elements \(1\) (resp. \(0\)) is denoted as \(\mathbf{1}_{n}\) (\(\mathbf{0}_{n}\)). The subscript is omitted when the dimension is clear. We denote the gradient of a function \(f\) as \(\nabla f\) and the partial gradient with respect to \(x\) as \(\nabla_{x}f\). If \(f\) is scalar, we denote its first derivative as \(f^{\prime}\). We denote the Jacobian of \(F\) as \(DF\). The Cartesian product is denoted as \(\times\) and the Minkowski sum as \(+\).
_Operator theory:_ Given \(C\subset\mathbb{R}^{n}\), \(N_{C}\) denotes its normal cone [13, Def. 6.38]. The projection onto \(C\) is denoted by \(\mathrm{proj}_{C}(x):=\mathrm{argmax}_{\mathrm{proj}\in C}\|x-y\|\). Given two operators \(A:\mathbb{R}^{n_{a}}\rightrightarrows\mathbb{R}^{n_{a}},B:\mathbb{R}^{n_{b}} \rightrightarrows\mathbb{R}^{n_{a}}\), we define the concatenation of operators \(A\times^{\mathrm{op}}B:(x,y)\mapsto Ax\times By\). Alternatively, we denote the concatenation of multiple operators \((A_{i})_{i\in\mathcal{I}}\) with \(\bigtimes_{i\in\mathcal{I}}^{\mathrm{op}}A_{i}\). For an operator \(T:\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}\), we denote \(\mathrm{zer}(T):=\{x\in\mathbb{R}^{n}|\mathbf{0}_{n}\in T(x)\}\). The operator \(T:C\subset C\) is (\(m\)-strongly) monotone in \(C\) if \(\langle T(x)-T(y),x-y\rangle\geq\langle T(x),x-y\rangle\).
\(m\|x-y\|^{2}\) for all \(x,y\in C\), for some \(m\geq 0\)\((>0)\).
_Probability theory:_ Given a probability space \((\Omega,\mathcal{F},\mathbb{P})\) with sample space \(\Omega\) and event set \(\mathcal{F}\), let \(A,B\in\mathcal{F}\). Then, \(\mathbb{P}[A]\) denotes the probability of \(A\), \(\mathbb{P}[A|B]\) denotes the probability of \(A\) conditioned on \(B\) and \(\mathbb{P}[A,B]\) denotes the joint probability of \(A\) and \(B\). We denote as \(\mathbb{E}[X]\) the expected value of a random variable \(X:\Omega\rightarrow\mathbb{R}^{n}\), for some \(n\in\mathbb{N}\). We denote the probability simplex by \(\Delta^{n}:=\{x\in[0,1]^{n}:\mathbf{1}_{n}^{\top}x=1\}\).
## III Traffic routing as a Generalized Nash Equilibrium problem
Let \(\mathcal{R}(\mathcal{N},\mathcal{E})\) a directed graph modelling a road network whose nodes \(\mathcal{N}\) represent the junctions of the network and each edge \((a,b)\in\mathcal{E}\) represents a road from \(a\in\mathcal{N}\) to \(b\in\mathcal{N}\). We study the problem of routing \(N\) population of vehicles \(\mathcal{I}:=\{1,...,N\}\). Denote \(\mathcal{I}_{-i}:=\mathcal{I}\setminus\{i\}\) for all \(i\in\mathcal{I}\). Each population is made up of \(V\) vehicles, where vehicles in the same population \(i\in\mathcal{I}\) share the same initial position \(b_{i}\in\mathcal{N}\) and destination \(d_{i}\in\mathcal{N}\).
**Remark 1**.: _Each population contains the same number of vehicles without loss of generality. In fact, let each population contain \((V_{i})_{i\in\mathcal{I}}\) vehicles and let \(V\in\mathbb{N}\) be such that \(V_{i}/V\in\mathbb{N}\) for all \(i\). Then, we can split each population \(i\) into \(V_{i}/V\) populations of equal size \(V\)._
Next, we ensure that each destination node can be reached:
**Assumption 1**.: \(\mathcal{R}(\mathcal{N},\mathcal{E})\) _is strongly connected and \((a,a)\in\mathcal{E}\) for each \(a\in\mathcal{N}\)._
The vehicles aim at reaching their destinations within a time horizon \(T\). The control action determines the probability for the receiving vehicle to drive through a certain road and it is the same for each vehicle in a population. In this setting, each population acts as a single entity, thus, we refer to each of them as an _agent_. We stress that the route of each vehicle is a realization of the probabilistic control action, thus vehicles represented by the same agent might take different routes. To formalize this, let us denote the junction visited by the \(v\)-th vehicle of agent \(i\) at time \(t\) as \(s_{t}^{i,v}\), which is a stochastic variable with event space \(\mathcal{N}\) and probability vector \(\rho_{t}^{i}\in\Delta^{|\mathcal{N}|}\), that is, \([\rho_{t}^{i}]_{a}:=\mathbb{P}[s_{t}^{i}=a]\) for any \(a\in\mathcal{N}\). The control actions are the matrices \(\Pi_{t}^{i}\in\mathbb{R}^{\mathcal{N}\times|\mathcal{N}|}\), defined by their elements
\[[\Pi_{t}^{i}]_{(b,a)}=\mathbb{P}[s_{t+1}^{i,v}=b|s_{t}^{i,v}=a]\quad\text{ for all }a,b\in\mathcal{N}.\]
From the law of total probability, the distributions of the agents positions evolve as
\[\rho_{t+1}^{i}=\Pi_{t}^{i}\rho_{t}^{i}\quad\text{for all }i\in\mathcal{I}. \tag{1}\]
The initial state of agent \(i\) is \(\rho_{1}^{i}\in\Delta^{|\mathcal{N}|}\), with only non-zero element \([\rho_{1}^{i}]_{b_{i}}=1\). In the remainder of this section, we show that, under an appropriate reformulation of (1), the problem that arises in the proposed setting can be cast as a GNEP.
### _Affine formulation of the system dynamics_
Similarly to the approach in [14], we reformulate the nonlinear dynamics in (1) in terms of the transformed variables
\[M_{t,(a,b)}^{i}:=[\Pi_{t}^{i}]_{(b,a)}[\rho_{t}^{i}]_{a} \tag{2}\]
defined for all \(i\in\mathcal{I},(a,b)\in\mathcal{E},t\in\mathcal{T}:=\{1,...,T\}\). By the definition of conditional probability, we have
\[M_{t,(a,b)}^{i}=\mathbb{P}[s_{t+1}^{i,v}=b,s_{t}^{i,v}=a]. \tag{3}\]
In words, \(M_{t,(a,b)}^{i}\) represents the probability that, at time \(t\), agent \(i\) traverses the road from \(a\) to \(b\). Denoting \(\mathcal{T}^{+}:=\mathcal{T}\cup\{T+1\}\), the decision variables of each agent are:
\[\omega_{i}:=\begin{bmatrix}\mathrm{col}(M_{t,(a,b)}^{i})_{(a,b)\in\mathcal{E},t \in\mathcal{T}}\\ \mathrm{col}\left(\rho_{t}^{i}\right)_{t\in\mathcal{T}^{+}}\end{bmatrix}. \tag{4}\]
Without loss of generality, \(\omega_{i}\) in (4) does not include any variable corresponding to \([\Pi_{t}^{i}]_{(b,a)}\) with \((a,b)\notin\mathcal{E}\), since the probability of traversing a non-existing road is zero. For convenience, we denote in boldface the concatenation over \(\mathcal{I}\) and with boldface and indexing \(-i\) the concatenation over \(\mathcal{I}_{-i}\), e.g. \(\mathbf{\omega}:=\mathrm{col}(\omega_{i})_{i\in\mathcal{I}}\), \(\mathbf{\omega}_{-i}:=\mathrm{col}(\omega_{j})_{j\in\mathcal{I}_{-i}}\). We also define \(n_{\omega}:=T|\mathcal{E}|+(T+1)|\mathcal{N}|\). The following lemma states that, by imposing appropriate linear constraints on \(\mathbf{\omega}\), the transformation in (2) can be inverted and the resulting matrices \(\Pi_{t}^{i}\) are coherent with the dynamics in (1). All the following statements are proven in the Appendix.
**Lemma 1**.: _Let \(\omega_{i}\) in (4) satisfy:_
\[\sum_{a:(a,b)\in\mathcal{E}}M_{t,(a,b)}^{i}=[\rho_{t+1}^{i}]_{b} \quad\text{for all }b\in\mathcal{N},\ t\in\mathcal{T}; \tag{5a}\] \[\sum_{b:(a,b)\in\mathcal{E}}M_{t,(a,b)}^{i}=[\rho_{t}^{i}]_{a} \quad\text{for all }a\in\mathcal{N},\ t\in\mathcal{T};\] (5b) \[M_{t,(a,b)}^{i}\geq 0 \quad\text{for all }(a,b)\in\mathcal{E},\ t\in\mathcal{T};\] (5c) \[\rho_{1}^{i}\in\Delta^{|\mathcal{N}|},\ [\rho_{1}^{i}]_{b_{i}}=1. \tag{5d}\]
_Then, \(\omega_{i}\in(\Delta^{|\mathcal{E}|})^{T}\times(\Delta^{|\mathcal{N}|})^{(T+1)}\) and the non-zero elements of \((\Pi_{t}^{i})_{t\in\mathcal{I},t\in\mathcal{T}}\) in (1) are given by:_
\[[\Pi_{t}^{i}]_{(b,a)}=\begin{cases}\frac{1}{|\mathcal{N}|}&\text{ if }[\rho_{t}^{i}]_{a}=0\\ \frac{M_{t,(a,b)}^{i}}{[\rho_{t}^{i}]_{a}}&\text{ if }[\rho_{t}^{i}]_{a}\neq 0 \end{cases} \tag{6}\]
_for all \((a,b)\in\mathcal{E},t\in\mathcal{T},i\in\mathcal{I}\)._
### _Control objective and constraints_
In [8], the authors consider the problem of choosing both the route and the destination charging station for a fleet of electric vehicles. Instead, we focus on the routing problem by considering that the destination is solely decided by the agents. In practice, this translates to the constraint that the destination is reached with high probability:
\[\left[\rho_{T+1}^{i}\right]_{d_{i}}\geq 1-\varepsilon, \tag{7}\]
where \(\varepsilon\) is a free design parameter. Let us model the road congestion for each \((a,b)\in\mathcal{E}\) with the latency function \(\ell_{(a,b)}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}\), which maps the ratio of vehicles on a road to its traversing time. In [8], the latency function used is the Bureau of Public Transport (BPT) function [15]:
\[\ell_{(a,b)}^{\text{BPT}}(\sigma):=\tau_{(a,b)}\left(1+0.15\left(\frac{\sigma+\zeta _{(a,b)}}{c_{(a,b)}}\right)^{\xi+1}\right), \tag{8}\]
where \(c_{(a,b)}\) and \(\tau_{(a,b)}\) are the capacity and the free-flow traversing time of \((a,b)\), respectively, \(\zeta_{(a,b)}\geq 0\) is the number of uncontrolled vehicles on the road normalized by \(VN\) and \(\xi\geq 0\) is a parameter often set to \(\xi=3\), e.g. [4, 15]. More generally, we consider functions that satisfy the following:
**Assumption 2**.: _For each \((a,b)\in\mathcal{E}\), the latency function \(\ell_{(a,b)}\) is \(C^{2}\), non-negative, non-decreasing and convex._
Let us define, for all \(i\in\mathcal{I}\), \(t\in\mathcal{T}\), \((a,b)\in\mathcal{E}\), the Bernoulli variable \(\theta_{t,(a,b)}^{v,i}\), which has value \(1\) if vehicle \(v\) of agent \(i\) traverses only \((a,b)\) at time \(t\). Then, the expected travel time
of road \((a,b)\) at time \(t\) is given by \(\mathbb{E}\left[\ell_{(a,b)}\left(\frac{\sum_{v,i}\theta_{t,(a,b)}^{v,i}}{VN} \right)\right]\). From the properties of the Bernoulli distribution and from (3), \(\mathbb{E}[\theta_{t,(a,b)}^{v,i}]=\mathbb{P}[\theta_{t,(a,b)}^{v,i}]=1]=M_{t,(a,b)}^ {i}\). Then, from the linearity of the expected value, we have that
\[\frac{\mathbb{E}\left[\sum_{v,i}\theta_{t,(a,b)}^{v,i}\right]}{VN}=\frac{\sum _{i}M_{t,(a,b)}^{i}}{N}=:\sigma_{(a,b),t}^{\mathsf{M}}. \tag{9}\]
Let us also denote \(\sigma_{t}^{\rho}:=\operatorname{avg}(\mathbf{\mu}_{t})\) for all \(t\in\mathcal{T}^{+}\) and
\[\sigma:=\operatorname{avg}(\mathbf{\omega})=\begin{bmatrix}\operatorname{col}( \sigma_{(a,b),t}^{\mathsf{M}}(\mathbf{\omega}_{(a,b)}),(a,b)\in\mathcal{E},t\in \mathcal{T}\\ \operatorname{col}(\sigma_{t}^{\rho})\,t\in\mathcal{T}^{+}\end{bmatrix}. \tag{10}\]
The expected value of a nonlinear function of a stochastic variable is in general intractable to compute. Let us instead compute the expected value of the first-order approximation of \(\ell_{(a,b)}\) around the expected value of the argument:
\[\mathbb{E}\left[\ell_{(a,b)}\left(\frac{1}{VN}\sum_{v,i}\theta_{t,(a,b)}^{v,i}\right)\right]\simeq\mathbb{E}\left[\ell_{(a,b)}(\sigma_{(a,b),t }^{\mathsf{M}})+\right.\] \[\left.\nabla\ell_{(a,b)}(\sigma_{(a,b),t}^{\mathsf{M}})(\frac{1} {VN}\sum_{v,i}\theta_{t,(a,b)}^{v,i}-\sigma_{(a,b),t}^{\mathsf{M}})\right] \stackrel{{\{1\}}}{{=}} \tag{11}\] \[\ell_{(a,b)}(\sigma_{(a,b),t}^{\mathsf{M}})+\nabla\ell_{(a,b)}( \sigma_{(a,b),t}^{\mathsf{M}})(\frac{1}{VN}\mathbb{E}[\sum_{v,i}\theta_{t,(a,b )}^{v,i}]\] \[-\sigma_{(a,b),t}^{\mathsf{M}})\stackrel{{\{2\}}}{{= }}\ell_{(a,b)}(\sigma_{(a,b),t}^{\mathsf{M}})\]
where in (11), \(\{1\}\) follows from the linearity of the expected value and from the fact that \(\sigma_{(a,b),t}^{\mathsf{M}}\) is deterministic, while \(\{2\}\) follows from (9). Although the right hand side of (11) has previously been used as road traversing cost [7, 8, 9], the interpretation of such a cost function is novel, to the best of our knowledge. To justify the approximation in (11), we leverage known results on the Taylor series of stochastic functions [16, Ch. 6]. In particular, we show that the error for a first-order approximation of functions of the average of Bernoulli variables, such as the one in (11), vanishes with the number of Bernoulli variables (i.e., \(VN\)), if they are independent:
**Proposition 1**.: _Let \(\sigma_{n}=\frac{1}{n}\sum_{i=1}^{n}\theta_{i}\), where \((\theta_{i})_{i=1}^{n}\) are independent Bernoulli variables such that \(\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[\theta_{i}]=\bar{\sigma}\) for all \(n\in\mathbb{N}\). Then, \((\ell(\sigma_{n})-\ell(\bar{\sigma}))^{2}=y_{n}+z_{n}\), where \(\mathbb{E}[y_{n}]\leq\frac{1}{4n}\nabla\ell(\bar{\sigma})^{2}\) and, for every \(\varepsilon\), there exists \(K_{\varepsilon}>0\) such that_
\[\sup_{n\in\mathbb{N}}\left(\mathbb{P}\left[|z_{n}|\geq\frac{K_{\varepsilon}}{ 8n^{3/2}}\right]\right)\leq\epsilon.\]
We now define the cost of traversing \((a,b)\) at time \(t\):
\[J_{(a,b)}(M_{t,(a,b)}^{i},\mathbf{M}_{t,(a,b)}^{-i}):= M_{t,(a,b)}^{i}\ell_{(a,b)}(\sigma_{(a,b),t}^{\mathsf{M}}). \tag{12}\]
The objective pursued by each agent reads then as follows:
\[J_{i}:=f_{i}(\omega_{i})+\sum_{(a,b)\in\mathcal{E},t\in\mathcal{T}}J_{(a,b)}(M_ {t,(a,b)}^{i},\mathbf{M}_{t,(a,b)}^{-i}), \tag{13}\]
where \(f_{i}:\mathbb{R}^{n_{\omega}}\rightarrow\mathbb{R}\) encodes a local cost for agent \(i\). Quadratic local costs are considered in [8, Eq. 5]. More generally, we consider functions that satisfy the following:
**Assumption 3**.: _The functions \((f_{i})_{i\in\mathcal{I}}\) in (13) are convex and \(C^{2}\)._
Finally, we introduce a maximum capacity \(\bar{c}_{(a,b)}\) for each road as a set of shared constraints between the agents:
\[\sum_{i\in\mathcal{I}}M_{t,(a,b)}^{i}\leq\bar{c}_{(a,b)}\quad\text{for all }t\in\mathcal{T},(a,b)\in\mathcal{E}. \tag{14}\]
The constraint in (14) is affine in the decision variables and thus we recast it via appropriately defined matrices \((A_{i})_{i\in\mathcal{I}}\), \(A_{i}\in\mathbb{R}^{T|\mathcal{E}|\times n_{\omega}}\), \(b\in\mathbb{R}^{T|\mathcal{E}|}\), \(A:=\operatorname{row}(A_{i})_{i\in\mathcal{I}}\) as
\[\sum_{i\in\mathcal{I}}A_{i}\omega_{i}=A\mathbf{\omega}\leq b. \tag{15}\]
### _Generalized Nash equilibrium problem_
Formalizing the model derived in Sections III-A and III-B, each agent solves the local optimization problem:
\[\forall i\in\mathcal{I}\colon\left\{\begin{array}{ll}\min_{\omega_{i}\in \Omega_{i}}&J_{i}(\omega_{i},\mathbf{\omega}_{-i})\\ \operatorname{s.t.}&A_{i}\omega_{i}\leq b-\sum_{j\in\mathcal{I}_{-i}}A_{j} \omega_{j},\end{array}\right. \tag{16a}\]
where \(\Omega_{i}:=\{\omega\in\mathbb{R}^{n_{\omega}}|\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{)},\mathbf{(} \mathbf{)},\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)} \mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{)}\mathbf{(}\mathbf{) (}\mathbf{)}\mathbf{(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{) (}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{)(}\mathbf{) (}\mathbf{)
where we compute
\[\begin{split} J^{\prime}_{(a,b)}(\cdot,\mathbf{M}^{-i}_{(b,a),t})|_{M^{ i}_{i,(a,b)}}&=\ell_{(a,b)}(\sigma^{\mathsf{M}}_{(a,b),t})+\\ &\frac{1}{N}M^{i}_{i,(a,b)}\ell^{\prime}_{(a,b)}(\sigma^{\mathsf{ M}}_{(a,b),t}).\end{split} \tag{19}\]
The following lemma allows one to conclude the monotonicity of \(F\) from the monotonicity of each \(F_{(a,b),t}\).
**Lemma 3**.: _The operator \(F\) in (17) is monotone if \(F_{(a,b),t}\) in (18) is monotone for each \((a,b)\) and \(t\)._
For a particular class of \(\ell_{(a,b)}\) (which includes \(\ell^{\mathsf{BPT}}_{(a,b)}\) in (8)), we find the following monotonicity condition:
**Lemma 4**.: _Let \(\ell:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}\) be defined as_
\[\ell(\sigma)=\tau+\frac{k}{\xi+1}(\sigma+\zeta)^{\xi+1} \tag{20}\]
_for some \(\tau,k,\xi,\zeta\in\mathbb{R}_{\geq 0}\). Let \(T\) the game mapping of the game with cost functions \(J_{i}(\mathbf{y})=y_{i}\ell(\operatorname{avg}(\mathbf{y}))\):_
\[T(\mathbf{y})=\operatorname{col}\left(\ell(\operatorname{avg}(\mathbf{y}))+\frac{1}{ N}y_{i}\nabla\ell(\operatorname{avg}(\mathbf{y})\right)_{i\in\mathcal{I}}. \tag{21}\]
_Then, \(T\) is monotone on \([0,1]^{N}\) if_
\[\zeta\geq\max\left(\frac{\xi^{2}-8}{8N},\frac{\xi-2}{2N}\right). \tag{22}\]
**Remark 2**.: (22) _is satisfied for any \(\zeta\) whenever \(\xi\leq 2\)._
**Remark 3**.: _Let us consider \(\ell^{\mathsf{BPT}}_{(a,b)}\) with \(\xi=3\) for some \((a,b)\in\mathcal{E}\). Condition (22) is equivalent to \(\zeta_{(a,b)}\geq\frac{1}{2N}\), which is true if a number of uncontrolled vehicles greater than \(\frac{V}{2}\) is traversing road \((a,b)\) at all times. This is a substantial improvement on the state of the art [8], where the authors considered \(V=1\) and assumed that at least \(\frac{3N}{8}\) vehicles traverse each road. This translates to \(\frac{3NV}{8}\) vehicles in our setting._
In view of Lemma 4, let us assume the following:
**Assumption 5**.: _For all \((a,b)\!\in\!\mathcal{E}\), \(\ell_{(a,b)}\) in (12) is in the form_
\[\ell_{(a,b)}(\sigma)=\tau_{(a,b)}+\tfrac{k_{(a,b)}}{\xi+1}(\sigma+\zeta_{(a,b )})^{\xi+1}\]
_where \(\xi\), \(\tau_{(a,b)}\), \(k_{(a,b)}\in\mathbb{R}_{\geq 0}\) and \(\zeta_{(a,b)}\) satisfies (22)._
Assumption 2 is implied by Assumption 5. For each \((a,b)\in\mathcal{E}\), \(t\in\mathcal{T}\), \(F_{(a,b),t}\) is in the form in (21), as can be seen by substituting (19) in (18). Thus, \(F_{(a,b),t}\) is monotone on \([0,1]^{N}\) by Lemma 4. As \(\mathbf{\Omega}\subset[0,1]^{Nn}\) by Lemma 1, the following result is immediate by Lemma 3:
**Lemma 5**.: _Under Ass. 5, \(F\) in (17) is monotone on \(\mathbf{\Omega}\)._
### _Semi-decentralized equilibrium seeking_
To solve the game in (16), we focus on the computation of a variational GNE (v-GNE) [17, Def. 3.10], that is, the subset of GNEs which satisfy the KKT conditions
\[\begin{bmatrix}\mathbf{\omega}\\ \lambda\end{bmatrix}\!\in\!\operatorname{zer}\left(\begin{bmatrix}\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
### _Equivalent finite horizon optimal control problem_
Let us formalize the game under consideration, parametrized in the initial distribution \(\rho_{\text{in}}^{i}\in\Delta^{|\mathcal{N}|}\):
\[\text{for all }i\in\mathcal{I}:\;\min_{\omega_{i}\in\mathcal{Y}_{i}}J_{i}( \omega_{i},\mathbf{\omega}_{-i}) \tag{26}\]
where \(\mathcal{Y}_{i}:=\left\{\omega\in\mathbb{R}^{n_{\omega}}|(\text{5a}),(\text{5b }),(\text{5c}),\rho_{1}^{i}=\rho_{\text{in}}^{i}\right\}\). We emphasize that we do not include the constraint in (14): due to the probabilistic control action, an unlucky realization might lead the constraint (14) to be unfeasible at the successive time steps. Instead, \(\mathcal{Y}_{i}\) is non-empty for any \(\rho_{\text{in}}^{i}\). The exclusion of (14) renders the problem in (26) a (non-generalized) game. In this section, we show the equivalence of the problem in (26) to a FHOCP. As a first step, we rewrite the equations defining \(\mathcal{Y}_{i}\) as the state-space representation of a constrained linear system. We define the desired distribution \(\rho_{\text{eq}}^{i}\) as \([\rho_{\text{eq}}^{i}]_{a}:=\delta_{d_{i}}(a)\), where \(\delta_{d_{i}}\) is a Kronecker delta centred in \(d_{i}\), and \(u_{\text{eq}}^{i}:=\mathrm{col}(\delta_{d_{i}}(a)\delta_{d_{i}}(b))(a,b) \in\mathcal{E}\), that is, the vector of edge transitions associated to taking the self-loop \((d_{i},d_{i})\) with probability \(1\). We define the states \(x_{i}^{i}\in\mathbb{R}^{n_{\omega}}\) and inputs \(u_{i}^{i}\in\mathbb{R}^{n_{\omega}}\) as
\[x_{i}^{i} :=\rho_{i}^{i}-\rho_{\text{eq}}^{i} \tag{27}\] \[u_{i}^{i} :=\mathrm{col}(M_{i,(a,b)}^{i})_{(a,b)\in\mathcal{E}}-u_{\text{eq }}^{i}. \tag{28}\]
We define the selection vectors \(S_{\text{edge}}^{(a,b)}\in\mathbb{R}^{n_{\omega}}\) for all \((a,b)\in\mathcal{E}\) such that \((S_{\text{edge}}^{(a,b)})^{\top}(u_{i}^{i}+u_{\text{eq}}^{i})=M_{i,(a,b)}^{i}\), as well as
\[B:=\mathrm{col}(\sum_{a:(a,b)\in\mathcal{E}}(S_{\text{edge}}^{(a, b)})^{\top})_{b\in\mathcal{N}}\] \[P:=\mathrm{col}(\sum_{b:(a,b)\in\mathcal{E}}(S_{\text{edge}}^{(a,b)})^{\top})_{a\in\mathcal{N}}.\]
It can be verified that \(Bu_{\text{eq}}^{i}=Pu_{\text{eq}}^{i}=\rho_{\text{eq}}^{i}\) and thus, by substituting the definitions of \(B\) and \(P\):
\[\text{(\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:
\[+\sum_{t\in\mathcal{T}}p^{S}(\mathbf{\phi}(t;\mathbf{x}_{\text{in}},(\mathbf{u}_{ \tau})_{\tau\in\mathcal{T}}),\mathbf{u}_{t}).\]
By Lemma 7 and [20, Theorem 2], a NE of \(\mathcal{G}(\mathbf{x}_{\text{in}})\) is a solution to the FHOCP \(\mathcal{O}(\mathbf{x}_{\text{in}})\), defined as:
\[\left\{\begin{aligned} &\min_{\{\mathbf{u}_{t}\}_{t\in\mathcal{T}}} & p(\mathbf{x}_{\text{in}},(\mathbf{u}_{\tau})_{\tau\in\mathcal{T}})\\ &\text{s.t.}&(\mathbf{\phi}(t;\mathbf{x}_{\text{in}},(\mathbf{u}_{ \tau})_{\tau\in\mathcal{T}}),\mathbf{u}_{t})\in\mathbb{Z}\quad\forall t.\end{aligned}\right. \tag{35a}\] \[\left.\begin{aligned} &\min_{\{\mathbf{u}_{t}\}_{t\in\mathcal{T}}} & p(\mathbf{x}_{\text{in}},(\mathbf{u}_{\tau})_{\tau\in\mathcal{T}})\\ &\text{s.t.}&(\mathbf{\phi}(t;\mathbf{x}_{\text{in}},(\mathbf{u}_{ \tau})_{\tau\in\mathcal{T}}),\mathbf{u}_{t})\in\mathbb{Z}\quad\forall t.\end{aligned}\right. \tag{35b}\]
We now show the asymptotic stability of the receding horizon solution of (35), and in turn of (33), via standard MPC results.
### _Stability of receding horizon Nash equilibrium control_
At every time-step, the agents apply the first input corresponding to a Nash equilibrium of the game in (33). This is formalized via the following control actions:
\[\forall y\!\in\!\mathbb{X},\ \kappa_{i}\!:\!y\!\mapsto\!u_{1}^{u_{\text{*}}} \text{ where }\operatorname{col}(u_{t}^{i_{\text{*}}})_{t\in\mathcal{T},i\in \mathcal{I}}\text{ is a NE of }\mathcal{G}(y). \tag{36}\]
Intuitively, \(\kappa_{i}\) leads the \(i\)-th agent to the desired equilibrium if the agents have an high enough incentive to approach their destinations. For this purpose, let us assume that each agent knows a path to its destination, formalized by the mappings \((\operatorname{KP}_{i})_{i\in\mathcal{I}}:\mathcal{N}\to\mathcal{N}\) with the following characteristics:
\[\operatorname{KP}_{i}(d_{i})=d_{i};\ (a,\operatorname{KP}_{i}(a)) \in\mathcal{E};\] \[\exists\ T_{i}^{P}\in\mathbb{N}\text{ such that } \operatorname{KP}_{i}^{t}(a):=\underbrace{\operatorname{KP}_{i}\circ... \circ\operatorname{KP}_{\sharp}(a)}_{t\text{ times}}=d_{i}\]
An example of such a path is the shortest path computed with edge weights \(\bar{\tau}\). We then define:
\[\tau_{i}^{\text{kp}}\in\mathbb{R}_{\geq 0}^{|\mathcal{N}|},\ \ [\tau_{i}^{ \text{kp}}]_{a}:=\tau_{(a,\operatorname{KP}_{i}(a))};\qquad\qquad\mathbf{\tau}^{ \text{kp}}:=\operatorname{col}(\tau_{i}^{\text{kp}})_{i};\] \[k_{i}^{\text{kp}}\in\mathbb{R}_{\geq 0}^{|\mathcal{N}|};\ \ [k_{i}^{ \text{kp}}]_{a}:=\sum_{t=0}^{\infty}[\tau_{i}^{\text{kp}}]_{\operatorname{KP}_ {i}^{t}(a)};\quad\mathbf{k}^{\text{kp}}=\operatorname{col}(k_{i}^{\text{kp}})_{i},\]
and the following input, designed such that every vehicle takes the next edge of the known path:
\[u_{i}^{\text{kp}}:\mathbb{X}_{i}\to\mathbb{U}_{i}\text{ for all }i\text{ such that }\] \[S_{\text{edge}}^{(a,b)\top}u_{i}^{\text{kp}}(x_{i})=\begin{cases} [x_{i}]_{a}-\delta_{d_{i}}(a)&\text{if }b=\operatorname{KP}_{i}(a)\\ 0&\text{if }b\neq\operatorname{KP}_{i}(a)\end{cases} \tag{37}\]
\[\mathbf{u}^{\text{kp}}(\mathbf{x}):=\operatorname{col}(u_{i}^{\text{kp}}(S_{\text{x} }^{i}))_{i\in\mathcal{I}},\]
We then postulate the following technical assumption, which encodes the fact that each agent evaluates its distance from the destination by means of the known path:
**Assumption 8**.: _The local costs satisfy Assumption 7 with_
\[f_{i}^{F}(x)=\sigma_{i}^{F}((k_{i}^{\text{kp}})^{\top}x)\] \[f_{i}^{S}(x,u_{i}^{\text{kp}}(x))\leq\sigma_{i}^{S}((\tau_{i}^{ \text{kp}})^{\top}x),\]
_where \(\sigma_{i}^{F}\) is a \(m_{F}\)-strongly monotone and \(\sigma_{i}^{S}\) is a \(L_{\mathcal{S}}\)-Lipschitz continuous functions for all \(i\), with \(\sigma_{i}^{F}(0)=\sigma_{i}^{S}(0)=0\)._
For example, Assumption 8 is satisfied by \(f_{i}^{F}(x)=\gamma_{1}(k_{i}^{\text{kp}})^{\top}x\), with \(\gamma_{1}>0\) and \(f_{i}^{S}(x,u)=\gamma_{2}(\tau_{i}^{\text{kp}})^{\top}x\), with \(\gamma_{2}\geq 0\). We derive a lower bound on the stage cost:
**Lemma 8**.: _For all \((x,u)\in\mathbb{Z}\), the stage cost in (34a) satisfies_
\[p^{S}(x,u)\geq\tfrac{\gamma_{\text{min}}\|x\|}{Nn_{i}} \tag{38}\]
_where \(\tau_{\text{min}}:=\min_{(a,b)\in\mathcal{E},a\neq b}\tau_{(a,b)}\)._
We now state a key technical result, which shows that \(p^{F}\) is a control Lyapunov function with respect to the origin for the system \(\mathbf{x}_{k+1}=(I_{N}\otimes B)\mathbf{u}_{k}\):
**Lemma 9**.: _Let \(p^{F}\) be as in (34b) and let the local costs satisfy Assumption 8. For all \(x\in\mathbb{X}\),_
\[p^{F}((I_{N}\otimes B)\mathbf{u}^{\text{kp}}(x))-p^{F}(x)\leq-m_{F}(\mathbf{\tau}^{ \text{kp}})^{\top}x. \tag{39}\]
Thanks to Lemma 9, we can relate \(p^{F}\) and \(p^{S}\):
**Lemma 10**.: _Denote \(\bar{k}:=\max_{(a,b)}(k_{(a,b)})\). If_
\[m^{F}\geq 1+L_{\mathcal{S}}+\tfrac{\bar{k}(N+1)}{2N\tau_{\text{min}}} \tag{40}\]
_then, for all \(x\in\mathbb{X}\), \((x,\mathbf{u}^{\text{kp}}(x))\in\mathbb{Z}\) and_
\[p^{F}((I_{N}\otimes B)\mathbf{u}^{\text{kp}}(x))-p^{F}(x)\leq-p^{S}(x,\mathbf{u}^{\text{kp }}(x)). \tag{41}\]
We are now ready to present the main stability result for the systems in (29a) controlled by \(\kappa_{i}\) in (36), which follows from the equivalence between (33) and (35) and [19, Theorem 2.19] applied to the system \(\mathbf{x}_{t+1}=(I_{N}\otimes B)\mathbf{u}_{t}\):
**Theorem 1**.: _Under Assumptions 1,3,6-8, if the condition in (40) is satisfied, the origin is asymptotically stable for the systems \(\mathbf{x}_{t+1}^{i}=B\kappa_{i}(\mathbf{x}_{t})\) for all \(i\in\mathcal{I}\), with \(\kappa_{i}\) as in (36)._
Let us present the resulting approach in Algorithm 2.
``` Initialization. Set \(\rho_{1}^{i}\) as in (5d) for each \(i\in\mathcal{I}\). For \(\tau\in\mathbb{N}\):
1. Agents control computation: 1. A NE of \(\mathcal{G}(\mathbf{\rho}_{\tau})\) \[([\operatorname{row}(M_{t,(a,b)}^{i*})_{(a,b)\in\mathcal{E},t\in\mathcal{T}}, \operatorname{row}({\rho_{t}^{i*}}^{\top})_{t\in\mathcal{T}^{+}}]^{\top})_{i\in \mathcal{I}}\] is computed using Algorithm 1, where each \((\Omega_{i})_{i\in\mathcal{I}}\) in (24b) is substituted with \(\{\omega_{i}\in\mathcal{Y}_{i}|(\text{300 holds})\}\), \(\lambda^{(1)}=\mathbf{0}\) and the dual update (25b), (25c) is ignored. 2. Each agent \(i\) computes \(\Pi_{i}^{*}\) as in (6).
2. Vehicles node update: 1. For all \(v\in\{1,...,V\}\), \(i\in\mathcal{I}\) draw \(a_{\tau+1}^{i,v}\in\mathcal{N}\) from the probability distribution \(\operatorname{col}([\Pi_{i}^{*}]_{(b,a_{\tau+1}^{i,v})})_{b\in\mathcal{N}}\)
3. Agents state update: 1. Each agent updates the empirical distribution: \[p_{n,i}=|\{v\in\{1,...,V\}\text{ s.t. }a_{\tau+1}^{i,v}=n\}|\text{ for all }n\in\mathcal{N}\] \[\rho_{r+1}^{i}=\operatorname{col}(p_{n,i}/
population size reduces the approximation error.
We then apply Algorithm 2 for \(\tau\in\{1,...,10\}\) with terminal cost \(f_{i}^{F}(x)=\gamma(k_{i}^{\text{kp}})^{\top}x\), \(\gamma\) as in the right hand side of (40) and \(V=10^{3}\). The results are compared to the pre-computed open loop solution of problem (16) without the constraint in (14). Figure 3 shows that the traversing time experienced is reduced with respect to the shortest path solution, and this advantage increases with the time horizon.
## VII Conclusion
Traffic routing of multiple vehicles can be modelled as an aggregative game with mixed strategies by performing a first-order approximation of the latency function. The approximation error decreases as the number of controlled vehicles increases. The particular structure of the road latency function guarantees the monotonicity of the game under mild conditions. Thus, the problem can be solved via existing equilibrium seeking algorithms for (non-strictly) monotone games. If the latency function is linear, then the game can be solved in receding horizon whenever the local objective functions satisfy a set of conditions inherited from the MPC literature.
**Lemma 11**.: _The only nonzero eigenvalues of a matrix_
\[A(\boldsymbol{y}):=2(\sigma+\zeta)\mathbf{1}_{N}\mathbf{1}_{N}^{\top}+ \frac{\xi}{N}(\boldsymbol{y}\mathbf{1}_{N}^{\top}+\mathbf{1}_{N}\boldsymbol{ y}^{\top}) \tag{42}\]
_where \(\boldsymbol{y}\in\mathbb{R}_{\geq 0}^{N}\), \(\sigma:=\frac{1}{N}\sum_{i}[y]_{i}\), \(\zeta\geq 0\), are \(\lambda_{-}:=\xi\sigma+\gamma_{-}\) and \(\lambda_{+}:=\xi\bar{\sigma}+\gamma_{+}\), where_
\[\gamma_{\pm} := N(\sigma+\zeta)\pm\sqrt{N^{2}(\sigma+\zeta)^{2}+2N\xi(\sigma+ \zeta)\sigma+\frac{\xi^{2}|y|^{2}}{N}}. \tag{43}\]
_Sketch of proof: \(A(\boldsymbol{y})\)_ is a sum of \(3\) rank-1 matrices, thus it is at most rank \(3\). We verify that \(\lambda_{+}\) and \(\lambda_{-}\) are the eigenvalues associated to the eigenvector \(\xi\boldsymbol{y}+\gamma_{+}\mathbf{1}_{N}\) and \(\xi\boldsymbol{y}+\gamma_{-}\mathbf{1}_{N}\), respectively. Finally, \(A(\boldsymbol{y})\) cannot have a third non-zero eigenvalue as \(\operatorname{trace}(A(\boldsymbol{y}))=\lambda_{-}+\lambda_{+}\).
**Lemma 12**.: _Let \(T=\bigtimes_{a\in\mathcal{A}}^{\text{op}}T_{a}\), where \(T_{a}:\mathbb{R}^{n_{a}}\rightrightarrows\mathbb{R}^{n_{a}}\) is \(L_{a}\)-Lipschitz [13, Def. 1.47] for all \(a\in\mathcal{A}\) and \(\mathcal{A}\) is a set of indexes. Then, \(T\) is \(L\)-Lipschitz, with \(L=\max_{a}(L_{a})\)._
We omit the proof of the latter statement.
### _Proofs of Section Iii_
#### Vi-A1 Proof of Lemma 1
We prove that that the equations (1) and (2) hold true for the matrices computed as in (6). We note that, if \([\rho_{t}^{i}]_{\bar{a}}=0\) for some \(\bar{a}\in\mathcal{N}\), by (5b) \(\sum_{b\in\bar{a},b)\in\mathcal{E}}M_{t,(\bar{a},b)}^{i}=0\) and, from (5c), \(M_{t,(\bar{a},b)}^{i}=0\) for all \(b\in\mathcal{N}\). Substituting in (6), we obtain (2):
\[[\Pi_{t}^{i}]_{(b,a)}[\rho_{t}^{i}]_{a}=\begin{cases}0&\text{if }[\rho_{t}^{i}]_{a}=0 \\ M_{t,(a,b)}^{i}&\text{if }[\rho_{t}^{i}]_{a}\neq 0\end{cases}=M_{t,(a,b)}^{i}\]
By expanding the matrix product and from (6) and (5a),
\[\Pi_{t}^{i}\rho_{t}^{i}=\operatorname{col}\left(\sum_{a:(a,b)\in\mathcal{E}} M_{t,(a,b)}^{i}\right)_{b\in\mathcal{N}}=\rho_{t+1}^{i}, \tag{44}\]
which implies (1). Finally, we sum both sides of (5a) and (5b) for all \(b\in\mathcal{N}\) and \(a\in\mathcal{N}\), respectively, to obtain:
\(\sum_{b\in\mathcal{N}}[\rho_{t+1}^{i}]_{b}=\sum_{(a,b)\in\mathcal{E}}M_{t,(a,b )}^{i}=\sum_{a\in\mathcal{N}}[\rho_{t}^{i}]_{a}\).
By induction, \(\rho_{t}^{i}\in\Delta^{|\mathcal{N}|}\) and \(\operatorname{col}(M_{t,(a,b)}^{i})_{(a,b)\in\mathcal{E}}\in\Delta^{|\mathcal{ E}|}\).
#### Vi-A2 Proof of Proposition 1
By the properties of the Bernoulli distribution, we have:
\[\sup_{i}\operatorname{Var}(\theta_{i})=\sup_{i}\mathbb{E}[\theta_{i}](1- \mathbb{E}[\theta_{i}])\leq 1/4\]
and \(\operatorname{Var}(\sigma_{n})=\frac{1}{n^{2}}\sum_{i}\operatorname{Var}( \theta_{i})\leq\frac{1}{4n}\). By the Chebyschev's inequality, for any \(\varepsilon>0\) and for \(K_{\varepsilon}=\frac{1}{\sqrt{\varepsilon}}\), we have
\[\mathbb{P}\left[(\sigma_{n}-\bar{\sigma})\geq\frac{K_{\varepsilon}}{2\sqrt{n}} \right]\leq\varepsilon.\]
The result then follows from [16, Theorem 6.2.3] by using \(r_{n}=\frac{1}{2\sqrt{n}}\) and \(\boldsymbol{a}=\bar{\sigma}\) (in the reference notation).
### _Proofs of Section Iv_
#### Vi-B1 Proof of Lemma 2 (sketch)
Compute \(J_{(a,b)}^{\prime\prime}(\cdot,\boldsymbol{M}_{(a,b),t)}^{-i}|_{M_{t,(a,b)}^{i}}\) for a generic \((a,b),t,i\) and note that it is non-negative using Assm. 2. The result then follows by [13, Prop. 8.14], Assm. 3 and [13, Prop. 8.17].
Fig. 1: \(\max_{t}\sigma_{(a,b)}^{t}/\overline{c}_{(a,b)}\), compared to the congestion obtained by the shortest path routing. The dotted line denotes \(c_{(a,b)}/\overline{c}_{(a,b)}\). The dots show the median values. The shaded area highlights the 95% confidence interval.
Fig. 3: Comparison of the total cost incurred by the agents, with respect to the shortest path without traffic information.
Fig. 2: Difference between approximated and empirical travel time with respect to \(V\), the number of vehicles per population.
2. Proof of Lemma 3
Let us compute \(F\):
\[\begin{array}{l}F(\mathbf{\omega})=\operatorname{col}\left(\nabla f_{i}(\omega_{i}) \right)_{i}+\\ \operatorname{col}\left(\begin{bmatrix}\operatorname{col}\left(J^{\prime}_{(a,b )}(\cdot,\mathbf{M}^{-i}_{t,(a,b)})|_{M^{i}_{t,(a,b)}}\right)_{(a,b),t}\\ \mathbf{0}_{|\mathcal{N}|(T+1)}\end{bmatrix}_{i\in I},\end{array} \tag{45}\]
where the zero vector appears because the cost function does not depend on \((\rho^{i}_{t})_{t,i}\). From Assumption 3 and [13, Example 20.3], \(\nabla f_{i}\) is monotone for each \(i\). Then, \(\operatorname{col}(\nabla f_{i})_{i}\) is monotone by [13, Prop. 20.23]. Let us denote the second addend in (45) as \(T(\mathbf{\omega})\). From [13, Prop. 20.10], \(F\) is monotone if \(T\) is monotone. Let us define the permutation matrix \(P\) such that
\[P\mathbf{\omega}=\begin{bmatrix}\operatorname{col}(\operatorname{col}(M^{i}_{t,( a,b)})_{i\in\mathcal{I}}(a,b)\in\mathcal{E},t\in\mathcal{T}\\ \operatorname{col}(\operatorname{col}(\rho^{i}_{t})_{t\in\mathcal{I}})_{t \in\mathcal{T}^{+}}\end{bmatrix}.\]
It holds, from the definition of \(F_{(a,b),t}\),
\[PT(\mathbf{\omega})=\begin{bmatrix}\operatorname{col}(F_{(a,b),t}((M^{i}_{t,(a,b) })_{i\in\mathcal{I}}))_{(a,b)\in\mathcal{E},t\in\mathcal{T}}\\ \mathbf{0}_{|N|\mathcal{N}|(T+1)}\end{bmatrix}. \tag{46}\]
As \(PP^{\top}=I\), for all \(\mathbf{\omega},\mathbf{y}\):
\[\begin{array}{l}\langle T(\mathbf{\omega})-T(\mathbf{y}),\mathbf{\omega}-\mathbf{y}\rangle =\langle PT(\mathbf{\omega})-PT(\mathbf{y}),P\mathbf{\omega}-P\mathbf{y}\rangle=\\ \sum_{(a,b),t}\langle F_{(a,b),t}|_{\mathbf{\omega}}-F_{(a,b),t}|_{\mathbf{y}},\mathbf{M}_{ t,(b,a)}|_{\mathbf{\omega}}-\mathbf{M}_{t,(b,a)}|_{\mathbf{y}}\rangle\end{array}\]
which is non negative if \(F_{(a,b),t}\) is monotone \(\forall(a,b),t\).
#### Iv-B3 Proof of Lemma 4
By [21, Proposition 12.3], \(T\) in (21) is monotone if \(DT(\mathbf{y})+DT(\mathbf{y})\succeq 0\)\(\forall\mathbf{y}\). Denote \(\sigma=\operatorname{avg}(\mathbf{y})\). We compute:
\[DT(\mathbf{y})=\tfrac{1}{N}\ell^{\prime}\sigma(\sigma)(I_{N}+\mathbf{1}\mathbf{1}^{\top})+ \tfrac{1}{N^{2}}\ell^{\prime\prime}\sigma(\sigma)(\mathbf{y}\mathbf{1}^{\top}). \tag{47}\]
As \(\ell^{\prime}(\sigma)=k(\sigma+\zeta)^{\xi}\), \(\ell^{\prime\prime}(\sigma)=k\xi(\sigma+\zeta)^{\xi-1}\), we compute
\[DT(\mathbf{y})+DT^{\top}(\mathbf{y})=\tfrac{2k}{N}(\sigma+\zeta)^{\xi}I _{N}+ \tag{48}\] \[\tfrac{k}{N}(\sigma+\zeta)^{\xi-1}(2(\sigma+\zeta)\mathbf{1}\mathbf{1}^{ \top}+\tfrac{\xi}{N}(\mathbf{y}\mathbf{1}^{\top}+\mathbf{1}\mathbf{y}^{\top})).\]
By Lemma 11, \(DT(\mathbf{y})+DT^{\top}(\mathbf{y})\succeq 0\) if
\[\tfrac{2k}{N}(\sigma+\zeta)^{\xi}+\tfrac{k}{N}(\sigma+\zeta)^{\xi-1}(\xi \sigma+\gamma_{-}(\mathbf{y}))\geq 0, \tag{49}\]
where \(\gamma_{-}\) is defined in (43). Excluding the trivial case \(\mathbf{y}=0,\zeta=0\), we divide by \(\tfrac{k}{N}(\sigma+\zeta)^{\xi}\) to obtain
\[\begin{array}{l}(\ref{eq:1})\Leftrightarrow 2+\tfrac{\xi\sigma}{ \sigma+\zeta}+\tfrac{\gamma_{-}(\mathbf{y})}{\sigma+\zeta}\geq 0\Leftrightarrow\\ \\ 2+\tfrac{\xi\sigma}{\sigma+\zeta}+N\geq\sqrt{N^{2}+2\tfrac{N\xi\sigma}{\sigma+ \zeta}+\tfrac{\xi^{2}\|\mathbf{y}\|^{2}}{N(\sigma+\zeta)^{2}}}\Leftrightarrow\\ \\ 4+\tfrac{\xi^{2}\sigma^{2}}{(\sigma+\zeta)^{2}}+\mathbf{\mathcal{N}}^{\mathbf{\sigma}}+ \tfrac{4\xi\sigma}{\sigma+4N}+2\tfrac{N\mathbf{\mathcal{E}}}{\sigma+\zeta}\geq \\ \\ \mathcal{N}^{\mathbf{\sigma}}+\tfrac{2N\mathbf{\mathcal{E}}}{\sigma+\zeta}+\tfrac{\xi^ {2}\|\mathbf{y}\|^{2}}{N(\sigma+\zeta)^{2}}\Leftrightarrow\\ \\ f(\mathbf{y}):=4(N+1)+\tfrac{\xi^{2}\sigma^{2}}{(\sigma+\zeta)^{2}}+\tfrac{4\xi \sigma}{\sigma+\zeta}-\tfrac{\xi^{2}\|\mathbf{y}\|^{2}}{N(\sigma+\zeta)^{2}}\geq 0.\end{array} \tag{50}\]
We look for the minimum of the left hand side of the latter inequality. Notice that \(\mathbf{y}_{\sigma}\sigma+\tfrac{1}{N}\mathbf{1}_{N}\). Then,
\[\begin{array}{l}\nabla f(\mathbf{y})=\tfrac{2\xi^{2}}{N}\tfrac{\sigma(\zeta+ \zeta)^{2}-\sigma^{2}(\sigma+\zeta)}{(\sigma+\zeta)^{4}}\mathbf{1}_{N}-\tfrac{4\xi} {N}\tfrac{\zeta}{(\sigma+\zeta)^{2}}\mathbf{1}_{N}\\ \\ \quad\quad\quad\quad-\xi^{2}\tfrac{2N\mathbf{y}(\sigma+\zeta)^{2}-2(\sigma+\zeta)\| \mathbf{y}\|^{2}}{N^{2}(\sigma+\zeta)^{4}}.\end{array}\]
Since \(\nabla f(\mathbf{y})\) contain either terms that multiply \(\mathbf{1}_{N}\) or \(\mathbf{y}\), it must be \(\mathbf{y}=\alpha\mathbf{1}_{N}\) for some \(\alpha\in(0,1]\) for \(\mathbf{y}\) to be a stationary point. Therefore, the minimum of \(f(\mathbf{y})\) is either obtained for \(\mathbf{y}=\alpha\mathbf{1}_{N}\) or at an extreme point of \([0,1]^{N}\), that is, \(\mathbf{y}=\sum_{i\in\mathcal{Q}}\mathbf{e}_{i}\), where \(\mathbf{e}_{i}\in\mathbb{R}^{N}\) with only non-zero element \([\mathbf{e}_{i}]_{i}=1\) and \(\mathcal{Q}\subset\{1,...,N\}\). Let us study the two cases separately:
_Case \(\mathbf{y}=\alpha\mathbf{1}_{N}\):_ In this case, \(\sigma=\alpha\) and \(\|\mathbf{y}\|^{2}=\alpha^{2}N\). We substitute these values in (50) to find
\[f(\mathbf{y})=4(N+1)+\tfrac{\xi^{2}\sigma^{2}\varkappa}{\varkappa\ell^{\prime}\zeta^{ \prime}}+\tfrac{4\xi\alpha}{\alpha+\zeta}-\tfrac{\xi^{2}\alpha^{2}\varkappa}{ \varkappa\varkappa}\varkappa}{\varkappa\mathcal{Q}(\alpha+\zeta)^{2}}\geq 0,\]
which is always true.
_Case \(\mathbf{y}=\sum_{i\in\mathcal{Q}}\mathbf{e}_{i}\):_ In this case, define \(q\)\(|\mathcal{Q}|\), we compute \(\sigma=\tfrac{q}{N}\) and \(\|\mathbf{y}\|^{2}=q\). We then substitute to find
\[f(\mathbf{y}):=4N+4+\tfrac{\xi^{2}\sigma^{2}}{(q+N\zeta)^{2}}+\tfrac{4\xi\sigma}{q+N }-\tfrac{\xi^{2}\sigma^{2}N}{(q+N\zeta)^{2}}\geq 0.\]
A sufficient condition for the latter is that the first addend is greater than the negative addend,which is true if
\[g(q):=4(q+\zeta N)^{2}-q\xi^{2}\geq 0.\]
Let us study the first derivative of \(g\):
\[g^{\prime}(q)=8\left(q+\zeta N\right)-\xi^{2}\leq 0\Leftrightarrow q\leq\tfrac{\xi ^{2}}{8}-\zeta N.\]
We conclude that \(g(q)\) has the minimum in \(q=1\) if \(\zeta\geq\tfrac{\xi^{2}-8N}{8N}\). We then note that \(g(1)\geq 0\) if \(\zeta\geq\tfrac{\xi-2}{2N}\). Therefore, \(g(q)\geq 0\) for all \(q\in\{1,...,N\}\) if (22) holds true, which in turns guarantees that (49) holds true for all \(\mathbf{y}\in[0,1]^{N}\).
#### Iv-B4 Proof of Proposition 2
As \(F_{(a,b),t}\) in (18) is in the form in (21) (see proof of Lemma 5), we follow the steps in Lemma 4 to find as in (48):
\[\begin{array}{l}(DF_{(a,b),t}+DF^{\top}_{(a,b),t})(\mathbf{y})\!=\!\tfrac{k_{(a,b)} }{N}[2(
#### V-A2 Derivation of (32)
Let us use the short-hand notation \(\phi_{t}^{i}=\phi^{i}(t;x_{\text{in}}^{i},(u_{\tau}^{i})_{\tau})\). From Assumptions 6 and 7, we rewrite \(J_{i}\) as:
\[J_{i}(\omega_{i})=f_{i}^{F}(\phi_{T+1}^{i})+\sum_{t}\Big{\{}f_{i} ^{S}(\phi_{t}^{i},u_{t}^{i})+\sum_{(a,b)}\bigl{[}(\tau_{(a,b)}\] \[+\tfrac{k_{(a,b)}}{N}\sum_{j}\left[S_{\text{edge}}^{(a,b)\top}(u_{t}^ {j}+u_{\text{eq}}^{i})\right]\bigr{)}S_{\text{edge}}^{(a,b)\top}(u_{t}^{i}+u_{ \text{eq}}^{i})\Big{]}\Big{\}}.\]
Using the definitions of \(C\) and \(\bar{\tau}\) and rearranging,
\[J_{i}= f_{i}^{F}(\phi_{T+1}^{i})+\sum_{t}f_{i}^{S}(\phi_{t}^{i},u_{t}^{i})+\] \[\bigl{(}\bar{\tau}^{\top}+\operatorname{avg}((u_{t}^{i}+u_{\text {eq}}^{i})_{j\in\mathcal{I}})^{\top}C)(u_{t}^{i}+u_{\text{eq}}^{i}).\]
From Assumption 6 and the definition of \(C\) and \(\bar{\tau}\), \(Cu_{\text{eq}}^{i}=\mathbf{0}\), \(\bar{\tau}^{\top}u_{\text{eq}}^{i}=0\) for any \(i\in\mathcal{I}\), thus (32) follows.
#### V-A3 Proof of Lemma 7
Let us denote \(\mathbf{\phi}_{t}=\mathbf{\phi}(t;x_{\text{in}},(u_{\tau})_{\tau})\), \(\phi_{t}^{i}=S_{\text{x}}^{i}\mathbf{\phi}_{t}\) and:
\[\bar{u}^{i}:=\operatorname{col}(u_{t}^{i})_{t\in\mathcal{T}};\quad\bar{\mathbf{u }}=\operatorname{col}(\bar{u}^{i})_{i\in\mathcal{I}}.\]
We further state the following, which are verified for any \(y_{i}\in\mathbb{R}^{m}\), \(\Gamma\in\mathbb{R}^{m\times m}\), \(i\in\{1,...,n\}\):
\[\sum_{i}\|y_{i}\|_{\mathcal{I}}^{\top}=\|\operatorname{col}(y_{i} )_{i}^{\top}\|_{I_{n}\otimes\Gamma}^{2}, \tag{52}\] \[\sum_{i}y_{i}=(\mathbf{1}_{\pi}^{\top}\otimes I_{m})\operatorname{col} (y_{i})_{i},\] (53) \[\mathbf{1}_{n}\mathbf{1}_{n}^{\top}\otimes\Gamma=(\mathbf{1}_{n}\otimes I_{m} )\Gamma(\mathbf{1}_{n}^{\top}\otimes I_{m}). \tag{54}\]
Let us write the agent cost in (32) as
\[J_{i}= f_{i}^{F}(\phi_{T+1}^{i})+\sum_{j}(\tfrac{\bar{y}^{j}}{N})^{\top}(I_{T} \otimes C)\bar{u}^{i}+\sum_{t}f_{i}^{S}(\phi_{t}^{i},u_{t}^{i})+\bar{\tau}^{ \top}u_{t}^{i}\]
The pseudo-gradient of (33) reads then as [11, Eq. 32]
\[F(\bar{\mathbf{u}}) =\operatorname{col}(\nabla_{\mathbf{u}_{t}}(f_{i}^{F}(\phi_{T+1}^{i}) +\sum_{t}f_{i}^{S}(\phi_{t}^{i},u_{t}^{i})+\bar{\tau}^{\top}u_{t}^{i}))_{i}\] \[+\tfrac{1}{N}(I_{N}+\mathbf{1}_{N}\mathbf{1}_{N}^{\top})\otimes(I_{T} \otimes C)\bar{\mathbf{u}}.\]
We now compute the gradient of \(p\). Let us first compute:
\[\sum_{t}\|\mathbf{u}_{t}\|_{I_{N}\otimes C}^{2}\overset{\eqref{eq:p_ {T+1}}}{=}\sum_{i,t}\|u_{t}^{i}\|_{C}^{2}\overset{\eqref{eq:p_ {T+1}}}{=}\|\bar{\mathbf{u}}\|_{I_{T\otimes C}}^{2};\]
\[\sum_{t}\|\mathbf{u}_{t}\|_{\mathbf{1}_{1}^{\top}\otimes C}^{\text{(54)}}\overset{ \eqref{eq:p_{T+1}}}{=}\sum_{t}\mathbf{u}_{t}^{\top}(\mathbf{1}\otimes I_{n_{n}})C(\bm {1}^{\top}\otimes I_{n_{n}})\mathbf{u}_{t}\] \[\overset{\eqref{eq:p_{T+1}}}{=}\sum_{t}\left\|\sum_{i\in\mathcal{ I}}u_{t}^{i}\right\|_{C}^{2}\overset{\eqref{eq:p_{T+1}}}{=}\|\sum_{i}\bar{u}^{i}\|_{I_{T \otimes C}}^{\text{(53)}}\overset{\eqref{eq:p_{T+1}}}{=}\] \[\bar{\mathbf{u}}^{\top}(\mathbf{1}_{N}\otimes I_{Tn_{n}})(I_{T}\otimes C )(\mathbf{1}_{N}^{\top}\otimes I_{Tn_{n}})\tilde{\mathbf{u}}\tilde{\mathbf{u}}^{\top}(\mathbf{ \|}\tilde{\mathbf{u}}^{\top}_{1\mathbf{N}_{N}^{\top}\otimes(I_{T}\otimes C)}.\]
We then rewrite \(p\) as:
\[p =\sum_{i}\left(f_{i}^{F}(\phi_{T+1}^{i})+\sum_{t}f_{i}^{S}(\phi_{t }^{i},u_{t}^{i})+\bar{\tau}^{\top}u_{t}^{i}\right)+\] \[+\tfrac{1}{2N}\|\bar{\mathbf{u}}\|_{(I_{N}+\mathbf{1}_{N}^{\top})\otimes(I _{T}\otimes C)}^{2}.\]
We apply [13, Prop. 16.9] to compute \(\nabla p\) and verify that it reads as \(F\).
#### V-A4 Proof of Lemma 8
We begin with a preliminary lemma:
**Lemma 13**.: _The following hold for all \((x,u)\in\mathbb{Z},i\in\mathcal{I}\):_
\[\sum_{a\neq b}(S_{\text{edge}}^{(a,b)\top})^{\top}S_{\text{u}}^{i}u =-(S_{\text{edge}}^{(d_{i},d_{i})})^{\top}S_{\text{u}}^{i}u; \tag{55a}\] \[[S_{\text{x}}^{i}x]_{d_{i}}\geq(S_{\text{edge}}^{(d_{i},d_{i})})^{ \top}S_{\text{u}}^{i}u\ \ ;\] (55b) \[-[S_{\text{x}}^{i}x]_{d_{i}}\geq\max_{a\in\mathcal{N}}[S_{\text{x} }^{i}x]_{a}. \tag{55c}\]
Proof.: (55a): From the definition of \(\mathbb{Z}\), \(PS_{\text{u}}^{i}u=S_{\text{x}}^{i}x\). Substituting the definition of \(P\) and summing each row,
\[\sum_{a}\sum_{b\in(a,b)\in E}S_{\text{edge}}^{(a,b)\top}S_{\text{u}}^{i}u=\sum_{a }[S_{\text{x}}^{i}x]_{a}=0 \tag{56}\]
where we used the definition of \(\mathbb{X}_{i}\) and \(\sum_{a\in\mathcal{N}}\rho_{\text{eq}}^{i}=1\). Using the definition of \(\mathbb{U}\), (55a) follows by noting
\[\sum_{a\neq b}S_{\text{edge}}^{(a,b)\top}S_{u}^{i}u =\sum_{a,b}\{S_{\text{edge}}^{(a,b)\top}S_{\text{u}}^{i}u\}-S_{\text{edge}}^{(d _{i},d_{i})\top}S_{\text{u}}^{i}u.\] (55b): By \(S_{u}^{i}u\in\mathbb{R}_{\geq 0}^{|\mathcal{E}|}-\{u_{\text{eq}}^{i}\}\) and \([u_{\text{eq}}^{i}]_{a}=0\ \forall\ a\neq d_{i}\), \((S_{\text{edge}}^{(d_{i},b)\top})^{\top}S_{\text{u}}^{i}u\geq 0\) for all \(b\in\mathcal{N}\setminus\{d_{i}\}\). As \(S_{\text{x}}^{i}x=PS_{\text{u}}^{i}u\), \([S_{\text{x}}^{i}x]_{d_{i}}=\sum_{b:(d_{i},b)\in E}(S_{\text{edge}}^{(d_{i},b)})^{ \top}S_{\text{u}}^{i}u\geq(S_{\text{edge}}^{(d_{i},d_{i})})^{\top}S_{\text{u}}^{i}u\).
(55c): From \(S_{\text{x}}^{i}x+\rho_{\text{eq}}^{i}\in\Delta^{|\mathcal{N}|}\) and \(\rho_{\text{eq}}^{i}\in\Delta^{|\mathcal{N}|}\),
\[\geq m_{F}((\mathbf{k}^{\text{kp}})^{\top}x-(\mathbf{k}^{\text{kp}})^{\top}x^{+})=m_{F}( \mathbf{\tau}^{\text{kp}})^{\top}x.\qed\]
### -6 Proof of Lemma 10
For compactness of notation, we drop the dependencies of \(u_{i}^{\text{kp}}\). From the definition of \(\bar{\tau}\) and from \(\tau_{(d_{i},d_{i})}=0\), \(\forall x\in\mathbb{X}\),
\[\bar{\tau}^{\top}u_{i}^{\text{kp}}=\sum_{(a,b)}\tau_{(a,b)}(S_{ \text{edge}}^{(a,b)})^{\top}u_{i}^{\text{kp}} \overset{\eqref{eq:m_F}}{=} \tag{60}\] \[\sum_{a\in\mathcal{N}}\tau_{(a,\text{KP}_{i}(a))}([S_{x}^{i}x]_{a }-\delta_{d_{i}}(a)) = (\tau_{i}^{\text{kp}})^{\top}S_{\text{x}}^{i}x.\]
Then, by Assumption (8) and denoting \(\bar{C}=(I_{N}+\mathbf{1}\mathbf{1}^{\top})\otimes C\)
\[\begin{split} p(x,\mathbf{u}^{\text{kp}})=\frac{\|\mathbf{u}^{\text{kp}} \|_{C}^{2}}{2N}+\sum_{i}f_{i}^{\text{S}}(S_{x}^{i}x,u_{i}^{\text{kp}})+(\tau_{ i}^{\text{kp}})^{\top}S_{\text{x}}^{i}x\\ \leq\sum_{i}(L_{S}+1)(\tau_{i}^{\text{kp}})^{\top}S_{\text{x}}^{i }x+\frac{1}{2N}\|\mathbf{u}^{\text{kp}}\|_{C}^{2}.\end{split} \tag{61}\]
From Lemma 9 and (61), then (41) holds if
\[(m_{F}-1-L_{S})(\mathbf{\tau}^{\text{kp}})^{\top}x\geq\frac{1}{2N}\|\mathbf{u}^{\text{kp }}\|_{C}^{2}. \tag{62}\]
Let us find a lower bound for the LHS of (62).
\[(\mathbf{\tau}^{\text{kp}})^{\top}x\!=\!\sum_{i,a}[\tau_{i}^{\text{kp}}]_{a}[S_{x} ^{i}x]_{a}^{\text{Ass.6}}\!\sum_{i}\sum_{a\neq d_{i}}[\tau_{i}^{\text{kp}}]_{a }[S_{x}^{i}x]_{a} \tag{63}\] \[\geq\tau_{\text{min}}\sum_{i}\sum_{a\neq d_{i}}[S_{x}^{i}x]_{a}^{ \text{(\ref{eq:m_F})}}\overset{\eqref{eq:m_F}}{=}\tau_{\text{min}}\sum_{i}(-[S _{x}^{i}x]_{d_{i}}).\]
We now rewrite the RHS of (62):
\[\|\mathbf{u}^{\text{kp}}\|_{C}^{2}=\sum_{i}\left((u_{i}^{\text{kp}})^{\top}Cu_{i} ^{\text{kp}}+\sum_{j}\left((u_{j}^{\text{kp}})^{\top}Cu_{i}^{\text{kp}}\right) \right). \tag{64}\]
We then note that for all \(i,j\in\mathcal{N}\), from the definition of \(C\):
\[(u_{j}^{\text{kp}})^{\top}Cu_{i}^{\text{kp}}=\!\sum_{(a,b)}k_{(a,b)}(u_{j}^{ \text{kp}})^{\top}S_{\text{edge}}^{(a,b)}(S_{\text{edge}}^{(a,b)})^{\top}u_{ i}^{\text{kp}}. \tag{65}\]
From (36), \((u_{i}^{\text{kp}})^{\top}S_{\text{edge}}^{(a,b)}\leq 1\) for all \((a,b)\) and \((S_{\text{edge}}^{(a,b)})^{\top}u_{i}^{\text{kp}}=0\) if \(b\neq\text{KP}_{i}(a)\). We continue from (65):
\[\begin{split}&\leq\sum_{a}k_{(a,\text{KP}_{i}(a))}S_{\text{edge}}^{ (a,\text{KP}_{i}(a))}u_{i}^{\text{kp}}\\ &=\sum_{a}k_{(a,\text{KP}_{i}(a))}([S_{x}^{i}x]_{a}-\delta_{d_{i} }(a))\\ &=\sum_{a\neq d_{i}}k_{(a,\text{KP}_{i}(a))}[S_{x}^{i}x]_{a}\\ \end{split}\]
where we noted \(k_{(d_{i},\text{KP}_{i}(d_{i}))}=k_{(d_{i},d_{i})}=0\) from Ass. 6. Then, we continue from the latter using (65):
\[(u_{i}^{\text{kp}})^{\top}Cu_{i}^{\text{kp}}\leq\bar{k}\sum_{a\neq d_{i}}[S_{ x}^{i}x]_{a}=-\bar{k}[S_{x}^{i}x]_{d_{i}}. \tag{66}\]
Substituting (66) in (64),
\[\|\mathbf{u}^{\text{kp}}\|_{C}^{2}\leq(N+1)\bar{k}\sum_{i}(-[S_{x}^{i}x]_{d_{i}}). \tag{67}\]
From (67) and (63), (62) holds true under (41).
### -7 Proof of Theorem 1
By [20, Theorem 2], for any \(\mathbf{x}\in\mathbb{X}\), a solution of \(\mathcal{G}(\mathbf{x})\) solves \(\mathcal{O}(\mathbf{x})\). Then, \(\operatorname{col}(\kappa_{i}(\mathbf{x}))_{i}\) is the first input of a sequence which solves (36) with initial state \(\mathbf{x}\). Problem (36) satisfies [19, Assm. 2.2, 2.3] under Assumptions 3 and 6. [19, Assm. 2.14a] follows from Lemma 10. By Assumption 3, \(p^{F}\) is Lipschitz continuous, thus \(p^{F}(x)\leq L\|x\|\) for some \(L>0\). Thus, [19, Assm. 2.14b] is satisfied by Lemma 8. The set \(\mathbb{X}\) is control invariant under \(\mathbf{u}^{\text{kp}}(\cdot)\), as verified by computing \((I\otimes B)\mathbf{u}^{\text{kp}}(x)\) for a generic \(x\in\mathbb{X}\). [19, Assm. 2.17] is then satisfied by applying [19, Prop. 2.16]. The thesis follows from Lemma [19, Thm. 2.19] with the control action \(\operatorname{col}(\kappa_{i}(\mathbf{x}))_{i}\).\(\blacksquare\)
| We examine the routing problem for self-interested vehicles using stochastic decision strategies. By approximating the road latency functions and an non-linear variable transformation, we frame the problem as an aggregative game. We characterize the approximation error and we derive a new monotonicity condition for a broad category of games that encompasses the problem under consideration. Next, we propose a semi-decentralized algorithm to calculate the routing as a variational generalized Nash equilibrium, and demonstrate the solution's benefits with numerical simulations. In the particular case of potential games, which emerges for linear latency functions, we explore a receding-horizon formulation of the routing problem, showing asymptotic convergence to destinations and analysing closed-loop performance dependence on horizon length through numerical simulations. |
2307.11366 | Many equiprojective polytopes | A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of
$P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$
vertices. In 1968, Geoffrey Shephard asked for a description of all
equiprojective polytopes. It has been shown recently that the number of
combinatorial types of $k$-equiprojective polytopes is at least linear as a
function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such
combinatorial types as $k$ goes to infinity. This relies on the
Goodman--Pollack lower bound on the number of order types and on new
constructions of equiprojective polytopes via Minkowski sums. | Théophile Buffière, Lionel Pournin | 2023-07-21T05:37:40 | http://arxiv.org/abs/2307.11366v2 | # Many Equiprojective Polytopes
###### Abstract
A \(3\)-dimensional polytope \(P\) is \(k\)-equiprojective when the projection of \(P\) along any line that is not parallel to a facet of \(P\) is a polygon with \(k\) vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of \(k\)-equiprojective polytopes is at least linear as a function of \(k\). Here, it is shown that there are at least \(k^{3k/2+o(k)}\) such combinatorial types as \(k\) goes to infinity. This relies on the Goodman-Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums.
## 1 Introduction
In 1968, Geoffrey Shephard asked a number of questions related to the combinatorics of Euclidean polytopes--convex hulls of finitely many points from \(\mathbb{R}^{d}\)[19, 20]. Among them, Question IX asks for a method to construct every _equiprojective polytope_. A \(3\)-dimensional polytope \(P\) is \(k\)-equiprojective when its orthogonal projection on any plane (except for the planes that are orthogonal to a facet of \(P\)) is a polygon with \(k\) vertices. A straightforward example of equiprojective polytopes is provided by prisms: a prism over a polygon with \(k-2\) vertices is \(k\)-equiprojective. Hallard Croft, Kenneth Falconer, and Richard Guy recall Shephard's question in their book about unsolved problems of geometry [3, Problem B10]. While a practical criterion (see Theorem 3.2 thereafter) and
an algorithm allowing to recognise equiprojective polytopes have been proposed by Masud Hasan and Anna Lubiw [11], the problem is still open.
Some equiprojective polytopes have been recently constructed by truncating Johnson solids (3-dimensional polytopes whose facets are regular polygons) and by gluing two well-chosen prisms along a facet [10]. The latter construction shows, in particular, that the number of different combinatorial types of \(k\)-equiprojective polytopes is at least a linear function of \(k\) (recall that two polytopes have the same combinatorial type when their face lattices are isomorphic). Here, this result is improved as follows.
Theorem 1.1: There are at least
\[k^{k\left(\frac{3}{2}+O\left(\frac{1}{\log k}\right)\right)}\]
different combinatorial types of \(k\)-equiprojective polytopes.
When \(k\) is an even integer, this is a consequence of the fact (already observed in [11]) that 3-dimensional zonotopes--Minkowski sums of line segments--are equiprojective. In turn, the number of combinatorial types of zonotopes can be estimated via the Goodman-Pollack bound on the number of order types [1, 9]. When \(k\) is odd, however, Theorem 1.1 requires a different construction that uses Minkowski sums. In order to analyse how equiprojectivity behaves under Minkowski sums, we rely on the notion of an _aggregated cone_ of a polytope at one of its edge directions. Roughly, an edge direction of a polytope \(P\) contained in \(\mathbb{R}^{3}\) is just a vector \(u\) in the unit sphere \(\mathbb{S}^{2}\) that is parallel to an edge of \(P\) and the aggregated cone of \(P\) at \(u\) is the union of the 2-dimensional normal cones of \(P\) that are contained in the plane \(u^{\perp}\) orthogonal to \(u\). In particular, we obtain the following characterization of equiprojectivity.
Theorem 1.2: A 3-dimensional polytope \(P\) is equiprojective if and only if, for every edge direction \(u\) of \(P\), either
1. the aggregated cone of \(P\) at \(u\) is equal to \(u^{\perp}\) or
2. the aggregated cone of \(P\) at \(u\) and the relative interior of the opposite of that cone form a partition of \(u^{\perp}\).
Since in practice, computing the faces of a Minkowski sum of polytopes is done via their normal fans, Theorem 1.2 provides a way to prove Theorem 1.1 in the case when \(k\) is odd. More generally, Theorem 1.2 allows to construct new classes of equiprojective polytopes. For instance, we prove the following.
Theorem 1.3: Consider a \(3\)-dimensional polytope \(P\) obtained as a Minkowski sum of finitely many polygons. If two of these polygons never share an edge direction, then \(P\) is an equiprojective polytope.
More generally, we will provide a condition under which a Minkowski sum of equiprojective polytopes, polygons, and line segments (that are allowed to share edge directions) is equiprojective (see Theorem 4.3). We will also explain how the value of \(k\) such that this Minkowski sum is \(k\)-equiprojective can be computed from the aggregated cones of the summands (see Theorem 4.4).
We provide bounds on the number of combinatorial types of zonotopes (see Theorem 2.4) and prove Theorem 1.1 in the case when \(k\) is even in Section 2. We introduce the aggregated cones of a polytope in Section 3, where we also prove Theorem 1.2. We give the announced Minkowski sum constructions of equiprojective polytopes and prove Theorem 1.3 in Section 4. We provide the proof of Theorem 1.1 in the case when \(k\) is odd in Section 5 and we conclude the article with Section 6, where some questions in the spirit of Shephard's are stated about the decomposability of equiprojective polytopes.
## 2. The combinatorial types of zonotopes
Up to translation, a zonotope is any subset of \(\mathbb{R}^{d}\) of the form
\[Z=\sum_{g\in\mathcal{G}}\operatorname{conv}\{0,g\}\]
where \(\mathcal{G}\) is a finite, non-empty set of pairwise non-collinear vectors from \(\mathbb{R}^{d}\backslash\{0\}\), which we refer to as a _set of generators_ of \(Z\). Note that \(Z\) admits several sets of generators, each obtained from \(\mathcal{G}\) by negating a part of the vectors it contains. In particular, \(Z\) has \(2^{|\mathcal{G}|}\) sets of generators, each of the same cardinality. We will refer to this common cardinality as the _number of generators_ of \(Z\).
Recall that the face lattice of a polytope is the set of its faces ordered by inclusion and that two polytopes have the same combinatorial type if their face lattices are isomorphic (here, by an isomorphism, we mean a bijection that preserves face inclusion). The goal of this section is to estimate the number of combinatorial types of zonotopes in terms of their number of generators. This will allow to prove Theorem 1.1 when \(k\) is even thanks to the following statement from [11], which we provide an alternative proof for.
Proposition 2.1: A \(3\)-dimensional zonotope \(Z\) is a \(k\)-equiprojective polytope, where \(k\) is twice the number of generators of \(Z\).
Proof. Consider a \(3\)-dimensional zonotope \(Z\) contained in \(\mathbb{R}^{3}\) and denote by \(\mathcal{G}\) a set of generators of \(Z\). Further consider a plane \(H\), also contained in \(\mathbb{R}^{3}\), that is not orthogonal to a facet of \(Z\) and denote by \(\pi:\mathbb{R}^{3}\to H\) the orthogonal projection on \(H\). By construction \(\pi(Z)\) can be expressed as
\[\pi(Z)=\sum_{g\in\pi(\mathcal{G})}\operatorname{conv}\{0,g\}\]
up to translation. As an immediate consequence, \(\pi(Z)\) is a Minkowski sum of line segments contained in \(H\) and, therefore, it is a \(2\)-dimensional zonotope (or in other words, a zonogon). However, since \(H\) is not orthogonal to a facet of \(Z\), the orthogonal projections on \(H\) of two distinct generators of \(Z\) cannot be collinear. Hence, \(Z\) and \(\pi(Z)\) have the same number of generators.
Finally, recall that the number of vertices of a zonogon is twice the number of its generators. Therefore, we have shown that the number of vertices of the orthogonal projection of \(Z\) on a plane that is not orthogonal to any of its facets is always twice the number of generators of \(Z\), as desired.
It is well known that the combinatorial types of zonotopes are determined by the _oriented matroids_ of their sets of generators [2]. Let us recall what the oriented matroids of a finite subset \(\mathcal{X}\) of \(\mathbb{R}^{d}\backslash\{0\}\) are. First pick a bijection
\[\sigma:\{1,\ldots,|\mathcal{X}|\}\to\mathcal{X}\]
whose role is to order the vectors of \(\mathcal{X}\) so that each of them corresponds to a coordinate of \(\mathbb{R}^{|\mathcal{X}|}\). A vector \(z\) from \(\{-1,0,1\}^{|\mathcal{X}|}\) is a _covector of \(\mathcal{X}\)_ with respect to \(\sigma\) when there exists a non-zero vector \(y\) in \(\mathbb{R}^{d}\) such that \(z_{i}\) is the sign of \(\sigma(i)\cdot y\) for every \(i\). The set \(M_{\sigma}(\mathcal{X})\) of all the covectors of \(\mathcal{X}\) with respect to \(\sigma\) is an _oriented matroid of \(\mathcal{X}\)_. Note that the other oriented matroids of \(\mathcal{X}\) can be obtained by varying \(\sigma\) or, equivalently, by letting the isometries of \(\mathbb{R}^{|\mathcal{X}|}\) that permute the coordinates act on \(M_{\sigma}(\mathcal{X})\). For more details on oriented matroids, see for instance [2] or [17]. Two finite subsets \(\mathcal{X}\) and \(\mathcal{X}^{\prime}\) of \(\mathbb{R}^{d}\backslash\{0\}\) are called _oriented matroid equivalent_ when they have at least one oriented matroid in common. It is easy to check that oriented matroid equivalence is indeed an equivalence relation on the finite subsets of \(\mathbb{R}^{d}\backslash\{0\}\).
The following statement is proven in [2] (see Corollary 2.2.3 therein). It provides the announced correspondence between the combinatorial type of a zonotope and the oriented matroids of its sets of generators.
Proposition 2.2: Two zonotopes \(Z\) and \(Z^{\prime}\) have the same combinatorial type if and only if for every set \(\mathcal{G}\) of generators of \(Z\), there exists a set \(\mathcal{G}^{\prime}\) of generators of \(Z^{\prime}\) such that \(\mathcal{G}\) and \(\mathcal{G}^{\prime}\) are oriented matroid equivalent.
According to Proposition 2.2, counting the number of combinatorial types of \(d\)-dimensional zonotopes with \(n\) generators amounts to counting the number of oriented matroids of \(d\)-dimensional sets of \(n\) vectors from \(\mathbb{R}^{d}\backslash\{0\}\). Estimates on these numbers have been given by Jacob Goodman and Richard Pollack [9] and by Noga Alon [1] in the terminology of order types [5, 6, 7, 8, 13]. Theorem 4.1 from [1] can be rephrased as follows in terms of oriented matroids. Observe that the lower bound in that statement is established in [9, Section 5].
Theorem 2.3: The number \(t(n,d)\) of oriented matroids of sets of \(n\) vectors that span \(\mathbb{R}^{d}\) and whose last coordinate is positive satisfies
\[\Big{(}\frac{n}{d-1}\Big{)}^{(d-1)^{2}n\big{(}1+O\big{(}\frac{\log(d-1)}{\log n }\big{)}\big{)}}\leq t(n,d)\leq\Big{(}\frac{n}{d-1}\Big{)}^{(d-1)^{2}n\big{(}1 +O\big{(}\frac{\log\log n/(d-1)}{d\log(n/(d-1))}\big{)}\big{)}}\]
when \(n/d\) goes to infinity.
Combining Proposition 2.2 and Theorem 2.3 makes it possible to provide estimates on the number of combinatorial types of zonotopes in terms of their dimension and number of generators. We prove the following theorem when \(d\) is fixed and \(n\) grows large instead of when \(n/d\) grows large as in the statement of Theorem 2.3 because we will only need it in the \(3\)-dimensional case.
Theorem 2.4: The number \(z(n,d)\) of combinatorial types of \(d\)-dimensional zonotopes with \(n\) generators satisfies
\[n^{(d^{2}-2d)n\big{(}1+O\big{(}\frac{1}{\log n}\big{)}\big{)}}\leq z(n,d)\leq n ^{(d-1)^{2}n\big{(}1+O\big{(}\frac{\log\log n}{\log n}\big{)}\big{)}}\]
when \(d\) is fixed and \(n\) goes to infinity.
Proof. It follows from Theorem 2.3 that
\[n^{(d-1)^{2}n\big{(}1+O\big{(}\frac{1}{\log n}\big{)}\big{)}}\leq t(n,d)\leq n ^{(d-1)^{2}n\big{(}1+O\big{(}\frac{\log\log n}{\log n}\big{)}\big{)}}\]
when \(d\) is fixed and \(n\) goes to infinity. Hence, it suffices to show that
\[\frac{t(n,d)}{2^{n}n!}\leq z(n,d)\leq t(n,d),\]
and use Stirling's approximation formula which implies
\[2^{n}n!=n^{n\left(1+O\left(\frac{1}{\log n}\right)\right)}\]
when \(n\) goes to infinity.
Let \(Z\) be a \(d\)-dimensional zonotope with \(n\) generators contained in \(\mathbb{R}^{d}\). Note that \(Z\) admits sets of generators that are contained in an open half-space of \(\mathbb{R}^{d}\) bounded by a hyperplane through the origin: these sets of generators can be obtained by appropriately negating some of the vectors in an arbitrary set of generators of \(Z\). Denote by \(\mathcal{M}(Z)\) the union of the equivalence classes modulo oriented matroid equivalence of all these sets of generators of \(Z\). Further note that any oriented matroid in any of these equivalence classes is the oriented matroid of a set of \(n\) vectors spanning \(\mathbb{R}^{d}\) and whose last coordinate is positive. This is due to the fact that isometries of \(\mathbb{R}^{d}\) do not change the oriented matroid of a set of vectors. It follows from Proposition 2.2 that a zonotope \(Z^{\prime}\) has the same combinatorial type as \(Z\) if and only if \(\mathcal{M}(Z^{\prime})\) coincides with \(\mathcal{M}(Z)\). This shows, in particular that \(z(n,d)\) is at most \(t(n,d)\).
By construction, the oriented matroid of a set \(\mathcal{G}\) of \(n\) vectors that span \(\mathbb{R}^{d}\) and whose last coordinate is positive is contained in \(\mathcal{M}(Z)\), where \(Z\) is any zonotope that admits \(\mathcal{G}\) as a set of generators. Moreover, \(\mathcal{M}(Z)\) is the union of at most \(2^{n}n!\) equivalence classes modulo oriented matroid equivalence because \(Z\) has \(2^{n}\) sets of generators and each of these sets has at most \(n!\) oriented matroids. Therefore, \(t(n,d)\) is at most \(2^{n}n!\) times \(z(n,d)\), as desired.
Theorem 1.1 in the special case of the \(k\)-equiprojective polytopes such that \(k\) is even follows from Proposition 2.1 and from Theorem 2.4.
## 3 Aggregated cones
In order to prove Theorem 1.1 when \(k\) is odd, we will use a construction of equiprojective polytopes via Minkowski sums. The main aim of this section is to study how equiprojectivity behaves with respect to Minkowski sums. In particular, we will prove Theorem 1.2. Our starting point is the article by Masud Hasan and Anna Lubiw [11]. Let us recall some of the terminology introduced in that article. In the following definition, given at the beginning
of Section 2 in [11], an _edge-facet incidence_ of a 3-dimensional polytope \(P\) is a pair \((e,F)\) such that \(F\) is a facet of \(P\) and \(e\) an edge of \(F\).
Definition 3.1: Two edge-facet incidences \((e,F)\) and \((e^{\prime},F^{\prime})\) of a 3-dimensional polytope \(P\)_compensate_ when \(e\) and \(e^{\prime}\) are parallel, and either
1. \(F\) and \(F^{\prime}\) coincide but \(e\) and \(e^{\prime}\) are distinct or
2. \(F\) and \(F^{\prime}\) are parallel and distinct facets of \(P\) whose relative interiors are on the same side of the plane that contains \(e\) and \(e^{\prime}\).
The following theorem is Theorem 1 from [11].
Theorem 3.2: A 3-dimensional polytope is equiprojective if and only if the set of its edge-facet incidences can be partitioned into compensating pairs.
Now recall that, given a polytope \(P\) contained in \(\mathbb{R}^{d}\) (of dimension possibly less than \(d\)) and a face \(F\) of \(P\), the _normal cone of \(P\) at \(F\)_ is defined as
\[N_{P}(F)=\bigl{\{}u\in\mathbb{R}^{d}:\forall\,(x,y)\in P\times F,\,u\cdot x \leq u\cdot y\bigr{\}}.\]
The normal cone of a \(j\)-dimensional face of \(P\) is a \((d-j)\)-dimensional closed polyhedral cone. In particular, if \(P\) is a polytope of any dimension contained in \(\mathbb{R}^{3}\), then the normal cones of \(P\) at its edges are 2-dimensional.
Definition 3.3: Consider a polytope \(P\) and a non-zero vector \(u\), both contained in \(\mathbb{R}^{3}\). The _aggregated cone_\(C_{P}(u)\) of \(P\) at \(u\) is the union of the 2-dimensional normal cones of \(P\) contained in the plane \(u^{\perp}\).
It should be noted that in the above definition, if there is no edge of \(P\) whose difference of vertices is a multiple of \(u\), then \(C_{P}(u)\) is empty. For this reason, we are only really interested in the vectors that model the edge directions of a polytope. In addition, it will be useful in the sequel to have just one vector for each possible edge direction and we propose the following definition.
Definition 3.4: Consider a polytope \(P\) contained in \(\mathbb{R}^{3}\) (of dimension possibly less than 3). A vector \(u\) from \(\mathbb{S}^{2}\) is an _edge direction_ of \(P\) when
1. the first non-zero coordinate of \(u\) is positive and
2. there is an edge of \(P\) whose difference of vertices is a multiple of \(u\).
Note that edge directions could be defined indifferently as a finite subset of the real projective plane \(\mathbb{RP}^{2}\). The notion of an aggregated cone at an edge direction is illustrated in Figure 1. This figure shows a regular dodecahedron
and two of its opposite edges \(e\) and \(e^{\prime}\) that correspond to a single edge direction \(u\). On the left of the figure, the intersection of \(P\) with \(u^{\perp}\) is shown in a darker color and outlined with dashed lines (\(P\) is thought of as centered at the origin of \(\mathbb{R}^{3}\) here). On the right of the figure, \(P\) is viewed from the edge direction \(u\) and \(C_{P}(u)\) is the two dimensional cone obtained by translating the striped cones in such a way that their apices coincide with the origin of \(\mathbb{R}^{3}\). In particular, this illustrates that \(C_{P}(u)\) is not always a convex cone. However, by definition it is always the union of finitely-many closed convex cones. We denote by \(\mathcal{E}_{P}(u)\) the smallest set of closed convex cones whose union is \(C_{P}(u)\). Alternatively, consider the connected components of \(C_{P}(u)\cap\mathbb{S}^{2}\). These connected components are circular arcs contained in \(u^{\perp}\) and the cones spanned by each of these arcs are precisely the elements of \(\mathcal{E}_{P}(u)\). It will be useful to keep in mind that one cone in \(\mathcal{E}_{P}(u)\) may still be the union of several of the normal cones of \(P\) contained in \(u^{\perp}\). Note that a facet of \(P\) can admit at most two edges whose difference of vertices is a multiple of \(u\). By construction, the normal cones of \(P\) at its facets that admit exactly one such edge are precisely the half lines that bound the cones contained in \(\mathcal{E}_{P}(u)\). On the other hand, a normal cone of \(P\) at any of its facets with two such edges is still contained in one of the cones from \(\mathcal{E}_{P}(u)\) but intersects the relative interior of that cone. Similarly, the closure of \(u^{\perp}\backslash C_{P}(u)\) is the union of finitely-many 2-dimensional closed convex cones contained in \(u^{\perp}\) and we denote by \(\mathcal{V}_{P}(u)\) the smallest such set of cones.
We are now ready to prove the announced Theorem 1.2 that characterizes equiprojectivity in terms of aggregated cones. The statement of this theorem is an equivalence and we shall prove each of the two implications separately.
Figure 1. A regular dodecahedron \(P\) (left) and the aggregated cone (striped on the right) at the edge direction corresponding to the opposite edges \(e\) and \(e^{\prime}\) of \(P\).
Lemma 3.5: Consider a 3-dimensional polytope \(P\). If \(P\) is equiprojective, then for any edge direction \(u\) of \(P\), either
Demonstration Proof: Assume that \(P\) is equiprojective and consider an edge direction \(u\) of \(P\) such that \(C_{P}(u)\) is not equal to \(u^{\perp}\). It suffices to show that a cone belongs to \({\Cal{E}}_{P}(u)\) if and only if its opposite belongs to \({\Cal{V}}_{P}(u)\).
First consider a cone \(E\) in \({\Cal{E}}_{P}(u)\). Note that \(E\) is bounded by two half-lines whose vertex is the origin (it may be that the union of these two half lines is a straight line through the origin in the case when \(E\) is a half-plane). Consider the two unit vectors \(v\) and \(w\) that span these half lines. As discussed above, \(v\) and \(w\) are normal vectors to two facets \(F\) and \(G\) of \(P\), respectively, such that both \(F\) and \(G\) admit a unique edge whose difference of vertices is a multiple of \(u\). Denote these edges of \(F\) and \(G\) by \(e_{F}\) and \(e_{G}\), respectively.
Since \(P\) is equiprojective, it follows from Theorem 3.2 that the edge-facet incidence \((e_{F},F)\) must be compensated, but since \(F\) doesn't have another edge parallel to \(e_{F}\), there must exist a facet \(F^{\prime}\) of \(P\) distinct from but parallel to \(F\) and an edge \(e_{F^{\prime}}\) of \(F^{\prime}\) parallel to \(e_{F}\) such that the relative interiors of \(F\) and \(F^{\prime}\) are on the same side of the plane that contains these two edges. Now observe that \(F^{\prime}\) cannot have another edge parallel to \(e_{F^{\prime}}\). Indeed the edge-facet incidence formed by such an edge with \(F^{\prime}\) couldn't be compensated by any other edge-facet incidence than \((e_{F^{\prime}},F^{\prime})\) but this one already compensates \((e_{F},F)\). It follows that the half-line spanned by \(-v\) is the boundary between a cone \(V\) contained in \({\Cal{V}}_{P}(u)\) and a cone in \({\Cal{E}}_{P}(u)\). In particular, \(E\) must be contained in a half-plane. Indeed, otherwise, \(-v\) would be in the relative interior of \(E\) and it couldn't belong to the boundary of \(V\) as shown on the left of Figure 2. The cone \(E\) is then as shown on the right of the figure.
Since the relative interiors of \(F\) and \(F^{\prime}\) are on the same side of the plane that contains \(e_{F}\) and \(e_{F^{\prime}}\), the cone in \({\Cal{E}}_{P}(u)\) that contains \(-v\) in its relative boundary must be on the same side than \(E\) of the line spanned by \(v\) and \(-v\) while \(V\) must be on the opposite side, as shown on the right of Figure 2 (where the cones that belong to \({\Cal{E}}_{P}(u)\) are colored green and the ones that belong to \({\Cal{V}}_{P}(u)\) yellow). As a consequence, \(-V\) either contains \(E\) or is contained in it. By the same argument, \(-w\) is contained in the relative boundary of a cone
from \(\mathcal{V}_{P}(u)\) that lies on the opposite side of the line spanned by \(w\) and \(-w\) than \(E\). It can be seen in Figure 2 that either \(V\), \(W\), and \(-E\) coincide or there is a cone \(E^{\prime}\) in \(\mathcal{E}_{P}(u)\) that lies between \(V\) and \(W\). However, in the latter case, \(-E^{\prime}\) would be contained in \(E\), which is impossible because we proved that such a cone must contain or be contained in the opposite of a cone from \(\mathcal{V}_{P}(u)\). This proves that \(-E\) must belong to \(\mathcal{V}_{P}(u)\).
Finally, observe that exchanging \(\mathcal{V}_{P}(u)\) and \(\mathcal{E}_{P}(u)\) in the argument shows that the opposite of every cone in \(\mathcal{V}_{P}(u)\) must belong to \(\mathcal{E}_{P}(u)\).
The following lemma states the other implication from Theorem 1.2. It will be proven via Theorem 3.2 by constructing an explicit partition of the set of all edge-facet incidences into compensating pairs.
**Lemma 3.6**: _If, for any edge direction \(u\) of a \(3\)-dimensional polytope \(P\),_
1. _the aggregated cone of_ \(P\) _at_ \(u\) _is equal to_ \(u^{\perp}\) _or_
2. _the aggregated cone of_ \(P\) _at_ \(u\) _and the relative interior of the opposite of that cone form a partition of_ \(u^{\perp}\)_,_
_then \(P\) is equiprojective._
Assume that the condition in the statement of the lemma holds for any edge direction \(u\) of a \(3\)-dimensional polytope \(P\). We will partition the edge-facet incidences of \(P\) into compensating pairs and the desired result will follow from Theorem 3.2. Consider a facet \(F\) of \(P\) and an edge \(e\) of \(F\). Denote by \(u\) the edge direction of \(P\) that is a multiple of the difference between the vertices of \(e\). We consider two mutually exclusive cases.
First, assume that \(e\) is the only edge of \(F\) whose difference of vertices is a multiple of \(u\). In that case, \(C_{P}(u)\) is not equal to \(u^{\perp}\) and \(N_{P}(F)\) is one of the
Figure 2. An illustration of the proof of Lemma 3.5.
half-lines bounding \(C_{P}(u)\). As \(C_{P}(u)\) and the relative interior of \(-C_{P}(u)\) form a partition of \(u^{\perp}\), the opposite of \(-N_{P}(F)\) is also one of the half lines bounding \(C_{P}(u)\). Therefore, \(P\) has a facet \(F^{\prime}\) parallel to and distinct from \(F\) that has a unique edge \(e^{\prime}\) whose difference of vertices is a multiple of \(u\). Moreover, the cones in \(\mathcal{E}_{P}(u)\) bounded by \(N_{P}(F)\) and \(-N_{P}(F)\) lie on the same side of the line \(N_{P}(F)\cup[-N_{P}(F)]\). As a consequence, the relative interiors of \(F\) and \(F^{\prime}\) lie on the same side of the plane that contains \(e\) and \(e^{\prime}\).
This shows that \((e^{\prime},F^{\prime})\) compensates \((e,F)\) and we assign these two edge-facet incidences to a pair in the announced partition. Observe that the same process starting with \((e^{\prime},F^{\prime})\) instead of \((e,F)\) would have resulted in the same pair of compensating edge-facet incidences of \(P\).
Now assume that \(F\) has an edge \(e^{\prime}\) other than \(e\) whose difference of vertices is a multiple of \(u\). In that case, \((e,F)\) and \((e^{\prime},F)\) compensate and we assign these two edge-facet incidences to a pair of the announced partition. Again, if we would have started with \((e^{\prime},F^{\prime})\) instead of \((e,F)\) this would have resulted in the same pair of edge-facet incidences of \(P\). Hence, repeating this process for all the edge-facet incidences of \(P\) allows to form a partition of the edge-facet incidences of \(P\) into compensating pairs, as desired.
Theorem 1.2 is obtained by combining Lemmas 3.5 and 3.6.
## 4. Equiprojectivity and Minkowski sums
In this section, we exploit Theorem 1.2 in order to study the equiprojectivity of Minkowski sums. We will also focus on the value of \(k\) for which a polytope or a Minkowski sum of polytopes is \(k\)-equiprojective.
Definition 4.1: Consider an edge direction \(u\) of a polytope \(P\) contained in \(\mathbb{R}^{3}\). The _multiplicity_\(\mu_{P}(u)\) of \(u\) as an edge direction of \(P\) is equal to \(2\) when \(C_{P}(u)\) coincides with \(u^{\perp}\) and to \(1\) when it does not.
Note that, in Definition 4.1, \(P\) can be a \(3\)-dimensional polytope, a polygon or a line segment. From now on, we denote by \(\kappa(P)\) the value of \(k\) such that an equiprojective polytope \(P\) is \(k\)-equiprojective.
Theorem 4.2: If \(P\) is an equiprojective polytope, then
\[\kappa(P)=\sum_{i=1}^{k}\mu_{P}(u)\]
where the sum is over the edge directions \(u\) of \(P\).
Proof. Consider an equiprojective polytope \(P\) contained in \(\mathbb{R}^{3}\) and a hyperplane \(H\) that is not orthogonal to any of the facets of \(P\). We can assume without loss of generality that \(H\) is through le origin of \(\mathbb{R}^{3}\) by translating it if needed. Denote by \(\pi:\mathbb{R}^{3}\to H\) the orthogonal projection on \(H\).
Since \(H\) is not orthogonal to a facet of \(P\), the edges of \(\pi(P)\) are the image by \(\pi\) of edges of \(P\). Moreover, the edges of \(P\) whose orthogonal projection on \(H\) are edges of \(\pi(P)\) are precisely the edges \(e\) of \(P\) such that the relative interior of \(N_{P}(e)\) intersects \(H\). Now consider an edge direction \(u\) of \(P\). According to Theorem 1.2, two cases are possible. In the first case, \(C_{P}(u)\) is equal to \(u^{\perp}\) and, in that case, exactly two of the normal cones of \(P\) contained in \(C_{P}(u)\) intersect \(H\) in their relative interiors. In the second case, \(C_{P}(u)\) and the relative interior of \(-C_{P}(u)\) form a partition of \(u^{\perp}\) and exactly one of the normal cones of \(P\) contained in \(C_{P}(u)\) intersects \(H\) in its relative interior. By the definition of the multiplicity of an edge direction, this proves the theorem.
Now recall that the faces of the Minkowski sum of two polytopes \(P\) and \(Q\) can be recovered from the normal cones of these polytopes (see for instance Proposition 7.12 in [22]): the faces of \(P+Q\) are precisely the Minkowski sums of a face \(F\) of \(P\) with a face \(G\) of \(Q\) such that the relative interiors of \(N_{P}(F)\) and \(N_{Q}(G)\) are non-disjoint. For this reason, Theorem 1.2 provides a convenient way to determine how equiprojectivity behaves under Minkowski sums.
Theorem 4.3: Let \(P\) and \(Q\) each be an equiprojective polytope, a polygon, or a line segment contained in \(\mathbb{R}^{3}\). The Minkowski sum \(P+Q\) is equiprojective if and only if for each edge direction \(u\) shared by \(P\) and \(Q\), either
* \(C_{P}(u)\) or \(C_{Q}(u)\) is equal to \(u^{\perp}\) or
* \(C_{P}(u)\) coincides with \(C_{Q}(u)\) or with \(-C_{Q}(u)\).
Proof. Pick an edge direction \(u\) of \(P+Q\). Note that \(u\) must then be an edge direction of \(P\) or \(Q\). According to Proposition 7.12 in [22],
\[C_{P+Q}(u)=C_{P}(u)\cup C_{Q}(u).\]
It will be important to keep in mind that the aggregated cones of a polygon or a line segment at their edge directions are always planes or half-planes.
If \(u\) is not an edge direction for both \(P\) and \(Q\), then by (1), \(C_{P+Q}(u)\) is equal to \(C_{P}(u)\) or to \(C_{Q}(u)\). Theorem 1.2 and the above remark on polygons and line
segments then imply that \(C_{P+Q}(u)\) is either equal to the plane \(u^{\perp}\) or forms a partition of that plane with the relative interior of \(-C_{P+Q}(u)\). Hence, if one of the assertions (i) or (ii) holds for every edge direction \(u\) shared by \(P\) and \(Q\), then \(P+Q\) must be equiprojective by Theorem 1.2.
Now assume that \(P+Q\) is equiprojective and assume that \(u\) is an edge direction shared by \(P\) and \(Q\) such that neither \(C_{P}(u)\) or \(C_{Q}(u)\) is equal to \(u^{\perp}\). According to Theorem 1.2, \(C_{P+Q}(u)\) is either equal to \(u^{\perp}\) or, together with the relative interior of \(-C_{P+Q}(u)\), it forms a partition of \(u^{\perp}\). By the same theorem and the above remark on polygons and line segments, the same holds for the aggregated cones of \(P\) and \(Q\) at \(u\). As \(C_{P}(u)\) and \(C_{Q}(u)\) are not equal to \(u^{\perp}\), the sum of the arc lengths of the circular arcs contained in \(C_{P}(u)\cap\mathbb{S}^{2}\) and \(C_{Q}(u)\cap\mathbb{S}^{2}\) is the circumference of the unit circle \(u^{\perp}\cap\mathbb{S}^{2}\). Moreover, the sum of the arc lengths of the circular arcs contained in \(C_{P+Q}(u)\cap\mathbb{S}^{2}\) is either equal to the circumference of \(u^{\perp}\cap\mathbb{S}^{2}\) or to half of it. Hence according to (1), \(C_{P}(u)\) necessarily coincides with or \(C_{Q}(u)\) or with \(-C_{Q}(u)\), as desired.
Recall that the multiplicity of an edge direction is defined indifferently for a 3-dimensional polytope, a polygon, or a line segment. From now on, if \(P\) is a polygon or a line segment contained in \(\mathbb{R}^{3}\), we denote
\[\kappa(P)=\sum\mu_{P}(u)\]
by analogy with Theorem 4.2, where the sum is over the edge directions \(u\) of \(P\). It should be noted that when \(P\) is a polygon, \(\kappa(P)\) is equal to the number of edges of \(P\) and when \(P\) is a line segment, \(\kappa(P)\) is always equal to 2.
If \(P\) and \(Q\) are two polytopes contained in \(\mathbb{R}^{3}\) that each can be an equiprojective polytope, a polygon, or a line segment, then we further denote
\[\lambda(P,Q)=k+2k^{\prime}\]
where \(k^{\prime}\) is the number of edge directions \(u\) common to \(P\) and \(Q\) such that both \(C_{P}(u)\) and \(C_{Q}(u)\) are equal to \(u^{\perp}\) while \(k\) is the number of edge directions \(u\) common to \(P\) and \(Q\) such that at least one of the cones \(C_{P}(u)\) or \(C_{Q}(u)\) is distinct from \(u^{\perp}\) and is contained in the other one.
Theorem 4.4: Consider two polytopes \(P\) and \(Q\) contained in \(\mathbb{R}^{3}\), that each can be an equiprojective polytope, a polygon, or a line segment. If the Minkowski sum \(P+Q\) is an equiprojective polytope, then
\[\kappa(P+Q)=\kappa(P)+\kappa(Q)-\lambda(P,Q).\]
Proof. Assume that \(P+Q\) is equiprojective and consider an edge direction \(u\) of \(P+Q\). Note that \(u\) is then an edge direction of \(P\) or an edge direction of \(Q\). As already mentioned in the proof of Theorem 4.3,
\[C_{P+Q}(u)=C_{P}(u)\cup C_{Q}(u).\]
Hence, if \(u\) is an edge direction of \(P\) but not one of \(Q\), then
\[\mu_{P+Q}(u)=\mu_{P}(u) \tag{2}\]
and if \(u\) is an edge direction of \(Q\) but not \(P\), then
\[\mu_{P+Q}(u)=\mu_{Q}(u). \tag{3}\]
If \(P\) and \(Q\) share \(u\) as an edge direction, we review the different possibilities given by Theorem 4.3 for how \(\mu_{P+Q}(u)\), \(\mu_{P}(u)\) and \(\mu_{Q}(u)\) relate to one another. If both the aggregated cones \(C_{P}(u)\) and \(C_{Q}(u)\) are equal to \(u^{\perp}\), then \(\mu_{P}(u)\), \(\mu_{Q}(u)\) and \(\mu_{P+Q}(u)\) are all equal to \(2\). Hence,
\[\mu_{P+Q}(u)=\mu_{P}(u)+\mu_{Q}(u)-2. \tag{4}\]
If \(C_{P}(u)\) and \(C_{Q}(u)\) are not both equal to \(u^{\perp}\), but one of these aggregated cones is contained in the other, then
\[\mu_{P+Q}(u)=\max\{\mu_{P}(u),\mu_{Q}(u)\}.\]
Moreover, the smallest value between \(\mu_{P}(u)\) and \(\mu_{Q}(u)\) is equal to \(1\). Hence,
\[\mu_{P+Q}(u)=\mu_{P}(u)+\mu_{Q}(u)-1. \tag{5}\]
Finally, if \(C_{P}(u)\) and \(C_{Q}(u)\) are opposite and not equal to \(u^{\perp}\), then \(\mu_{P}(u)\) and \(\mu_{Q}(u)\) are both equal to \(1\) while \(\mu_{P+Q}(u)\) is equal to \(2\). Therefore,
\[\mu_{P+Q}(u)=\mu_{P}(u)+\mu_{Q}(u). \tag{6}\]
The result is then obtained from Theorem 4.2 by summing (2), (3), (4), (5), and (6) when \(u\) ranges over the corresponding edge directions of \(P+Q\).
Observe that when two polytopes \(P\) and \(Q\) do not share an edge direction, \(\lambda(P,Q)\) vanishes. Hence, we obtain the following corollary from Theorems 4.3 and 4.4. In turn, that corollary immediately implies Theorem 1.3.
Corollary 4.5: Consider two polytopes \(P\) and \(Q\) contained in \(\mathbb{R}^{3}\), that each can be an equiprojective polytope, a polygon, or a line segment. If \(P\) and \(Q\) do not share an edge direction, then \(P+Q\) is equiprojective and
\[\kappa(P+Q)=\kappa(P)+\kappa(Q).\]
## 5. Many \(k\)-equiprojective polytopes when \(k\) is odd
The goal of this section is to build many \(k\)-equiprojective polytopes when \(k\) is a large enough odd integer. This will be done using the Minkowski sum between a zonotope with \((k-3)/2\) generators and a well-chosen triangle. In this section, all the considered polytopes are contained in \(\mathbb{R}^{3}\).
For any \(3\)-dimensional zonotope \(Z\), we will consider a triangle \(t_{Z}\) whose edge directions do not belong to any of the planes spanned by two edge directions of \(Z\) and such that the planes spanned by the edge directions of \(t_{Z}\) do not contain an edge direction of \(Z\). Note that such a triangle always exists because \(Z\) and \(t_{Z}\) have only finitely many edge directions. A consequence of our requirements on \(t_{Z}\) is that this triangle does not share any edge direction with \(Z\). Moreover, the normal cones of \(Z\) at its facets are never contained in the normal cone of \(t_{Z}\) at itself or in the plane spanned by the normal cone of \(t_{Z}\) at one of its edges (see Figure 3 for an illustration of the normal cones of a triangle at its edges and at itself). Inversely, the normal cone of \(t_{Z}\) at itself is not contained in the plane spanned by the normal cone of \(Z\) at any of its edges. It should be noted that there are many possible choices for \(t_{Z}\) but we fix that choice for each zonotope \(Z\) so that \(Z\mapsto t_{Z}\) defines a map that sends a zonotope to a triangle.
Figure 3. A triangle in \(\mathbb{R}^{3}\) (colored yellow) and its normal cones at edges (colored green) and at itself (vertical line).
It follows from the results of Section 4 that the Minkowski sum of \(Z\) with \(t_{Z}\) is always an equiprojective polytope.
Lemma 5.1: If \(Z\) is a \(3\)-dimensional zonotope with \(n\) generators, then its Minkowski sum with \(t_{Z}\) is a \((2n+3)\)-equiprojective polytope.
Proof. Recall that the edge directions of \(Z+t_{Z}\) are precisely the edge directions of \(Z\) and the edge directions of \(t_{Z}\). By Proposition 2.1, \(Z\) is \(2n\)-equiprojective where \(n\) is the number of generators of \(Z\).
Now recall that when \(P\) is a polygon, \(\kappa(P)\) is equal to the number of edges of \(P\). As a consequence, \(\kappa(t_{Z})\) is equal to \(3\) and since \(Z\) does not share an edge direction with \(t_{Z}\), it follows from Corollary 4.5 that the Minkowski sum of \(Z\) with \(t_{Z}\) is a \((2n+3)\)-equiprojective polytope.
Let us recall that the set of the normal cones of a polytope \(P\) ordered by reverse inclusion form a lattice \({\Cal{N}}(P)\), called the _normal fan of \(P\)_ and that
\[N_{P}:{\Cal{F}}(P)\to{\Cal{N}}(P)\]
is an isomorphism. This correspondence between the face lattice and the normal fan of a polytope will be useful in order to determine the combinatorial type of the Minkowski sum between a \(3\)-dimensional zonotope \(Z\) and its associated triangle \(t_{Z}\). Recall that a face \(F\) of \(Z+t_{Z}\) can be uniquely written as the Minkowski sum of a face of \(Z\) with a face of \(t_{Z}\). We will denote by \(\tau_{Z}(F)\) the face of \(t_{Z}\) that appears in this Minkowski sum.
Note that \(\tau_{Z}\) defines a morphism from the face lattice of \(Z+t_{Z}\) to that of \(t_{Z}\) in the sense that it preserves face inclusion. This is a consequence, for instance of Proposition 7.12 from [22]. In this context, an _isomorphism_ refers to a bijective morphism between the face lattice of two polytopes.
Lemma 5.2: Consider two \(3\)-dimensional zonotopes \(Z\) and \(Z^{\prime}\). If
\[\psi:{\Cal{F}}(Z+t_{Z})\to{\Cal{F}}(Z^{\prime}+t_{Z^{\prime}})\]
is an isomorphism, then there exists an isomorphism
\[\phi:{\Cal{F}}(t_{Z})\to{\Cal{F}}(t_{Z^{\prime}})\]
such that \(\phi\circ\tau_{Z}\) is equal to \(\tau_{Z^{\prime}}\circ\psi\).
Proof. Assume that \(\psi\) is an isomorphism from the face lattice of \({\Cal{F}}(t_{Z})\) to that of \({\Cal{F}}(t_{Z^{\prime}})\). First observe that \(Z+t_{Z}\) has exactly two parallel triangular facets
each of whose is a translate of \(t_{Z}\). Further observe that all the other facets of \(Z+t_{Z}\) are centrally-symmetric, since they are the Minkowski sum of a centrally-symmetric polygon with a line segment or a point. The same observations hold for the Minkowski sum of \(Z^{\prime}\) with \(t_{Z^{\prime}}\): that Minkowski sum has exactly two parallel triangular facets, each a translate of \(t_{Z^{\prime}}\) and its other facets all are centrally-symmetric. As \(\psi\) induces an isomorphism between the face lattices of any facet \(F\) of \(Z+t_{Z}\) and the face lattice of \(\psi(F)\), this shows that \(\psi\) sends the two triangular facets of \(Z+t_{Z}\) to the two triangular facets of \(Z^{\prime}+t_{Z^{\prime}}\). Moreover, two parallel edges of a centrally-symmetric facet of \(Z+t_{Z}\) are sent by \(\psi\) to two parallel edges of a centrally-symmetric facet of \(Z^{\prime}+t_{Z^{\prime}}\).
Recall that all the aggregated cones of \(t_{Z}\) are half-planes. As \(Z\) and \(t_{Z}\) do not have edge direction in common, the aggregated cones of \(Z+t_{Z}\) at the edge directions of \(t_{Z}\) are still half-planes. Note that the two half-lines that bound these half-planes are precisely the normal cones of \(Z+t_{Z}\) at its triangular facets. Similarly, the aggregated cones of \(Z^{\prime}+t_{Z^{\prime}}\) at the edge directions of \(t_{Z^{\prime}}\) are half-planes bounded by the normal cones of \(Z^{\prime}+t_{Z^{\prime}}\) at its triangular facets. Since two parallel edges of a centrally-symmetric facet of \(Z+t_{Z}\) are sent by \(\psi\) to two parallel edges of a centrally-symmetric facet of \(Z^{\prime}+t_{Z^{\prime}}\), this implies that for every edge \(e\) of \(t_{Z}\), there exists an edge \(\phi(e)\) of \(t_{Z^{\prime}}\) such that any face of \(Z+t_{Z}\) contained in \(\tau_{Z}^{-1}(\{e\})\) is sent to \(\phi(e)\) by \(\tau_{Z^{\prime}}\circ\psi\).
By the correspondence between face lattice and normal fans,
\[\overline{\psi}=N_{Z^{\prime}+t_{Z^{\prime}}}\circ\psi\circ N_{Z+t_{Z}}^{-1}\]
provides an isomorphism from \({\Cal{N}}(Z+t_{Z})\) to \({\Cal{N}}(Z^{\prime}+t_{Z^{\prime}})\). Consider two edges \(e\) and \(f\) of \(t_{Z}\) and denote by \(x\) the vertex they share. Let \({\Cal{P}}\) be the set of the normal cones of \(Z+t_{Z}\) at its two triangular faces and at the faces contained in \(\tau_{Z}^{-1}(\{e\})\) and in \(\tau_{Z}^{-1}(\{f\})\). Similarly, let \({\Cal{P}}^{\prime}\) denote the set made up of the normal cones of \(Z^{\prime}+t_{Z^{\prime}}\) at its triangular faces and at the faces from \(\tau_{Z^{\prime}}^{-1}(\{\phi(e)\})\) and \(\tau_{Z^{\prime}}^{-1}(\{\phi(f)\})\). By the above, \(\overline{\psi}({\Cal{P}})\) is equal to \({\Cal{P}}^{\prime}\). Moreover, it follows from Proposition 7.12 in [22] that \({\Cal{P}}\) and \({\Cal{P}}^{\prime}\) are polyhedral decompositions of the boundaries of the normal cone of \(t_{Z}\) at \(x\) and of the normal cone of \(t_{Z^{\prime}}\) at the vertex \(\phi(x)\) shared by \(\phi(e)\) and \(\phi(f)\). As \(\overline{\psi}\) is an isomorphism from \({\Cal{N}}(Z+t_{Z})\) to \({\Cal{N}}(Z^{\prime}+t_{Z^{\prime}})\), it must then send the normal cones of \(Z+t_{Z}\) at the faces from \(\tau_{Z}^{-1}(\{x\})\) to the normal cones of \(Z^{\prime}+t_{Z^{\prime}}\) at the faces from \(\tau_{Z^{\prime}}^{-1}(\{\phi(x)\})\). In other words, any face of \(Z+t_{Z}\) contained in \(\tau_{Z}^{-1}(\{x\})\) is sent to \(\phi(x)\) by \(\tau_{Z^{\prime}}\circ\psi\). Hence, setting \(\phi(t_{Z})\) to \(t_{Z^{\prime}}\) results in the desired isomorphism \(\phi\).
Lemma 5.2 states, in other words that, if there exists an isomorphism
\[\psi:\Cal{F}(Z+t_{Z})\to\Cal{F}(Z^{\prime}+t_{Z^{\prime}})\]
then there is a commutative diagram
where \(\phi\) is an isomorphism. Recall that a face \(F\) of \(Z+t_{Z}\), where \(Z\) is an arbitrary 3-dimensional zonotope is the Minkowski sum of a unique face of \(Z\) with \(\tau_{Z}(F)\). We will denote by \(\zeta_{Z}(F)\) the face of \(Z\) that appears in this sum. This defines a morphism \(\zeta_{Z}\) from \(\Cal{F}(Z+t_{Z})\) to \(\Cal{F}(Z)\). We derive the following from our requirements for the choice of \(t_{Z}\).
Proposition 5.3: Consider a 3-dimensional zonotope \(Z\) and two proper faces \(F\) and \(G\) of \(Z+t_{Z}\). If \(F\) is a facet of \(G\), then either
1. \(\zeta_{Z}(G)\) coincides with \(\zeta_{Z}(F)\) or
2. \(\tau_{Z}(G)\) coincides with \(\tau_{Z}(F)\).
Proof. Let us first assume that \(G\) is a facet of \(Z+t_{Z}\) and \(F\) an edge of \(G\). By our choice for \(t_{Z}\), all the polygonal faces of \(Z+t_{Z}\) are the Minkowski sum of a facet of \(Z\) with a vertex of \(t_{Z}\), of a vertex of \(Z\) with \(t_{Z}\) itself, or of an edge of \(Z\) with an edge of \(t_{Z}\). If \(\zeta_{Z}(G)\) is a polygon and \(\tau_{Z}(G)\) a vertex of \(t_{Z}\) then it is immediate that \(F\) is the Minkowski sum of an edge of \(\zeta_{Z}(G)\) with \(\tau_{Z}(G)\) and it follows that \(\tau_{Z}(F)\) coincides with \(\tau_{Z}(G)\). Similarly, if \(\zeta_{Z}(G)\) is a vertex of \(Z\) and \(\tau_{Z}(G)\) is equal to \(t_{Z}\), then \(\tau_{Z}(F)\) must be an edge of \(t_{Z}\) and \(\zeta_{Z}(F)\) is equal to \(\zeta_{Z}(G)\). Now if \(\zeta_{Z}(G)\) is an edge of \(Z\) and \(\tau_{Z}(G)\) an edge of \(t_{Z}\), then observe that \(G\) is a parallelogram. As \(F\) is an edge of \(G\), it must be a translate of either \(\zeta_{Z}(G)\) or \(\tau_{Z}(G)\). In the former case, \(\zeta_{Z}(F)\) is equal to \(\zeta_{Z}(G)\) and in the latter, \(\tau_{Z}(F)\) is equal to \(\tau_{Z}(G)\), as desired.
Finally, assume that \(G\) is an edge of \(Z+t_{Z}\) and \(F\) a vertex of \(G\). Recall that \(Z\) and \(t_{Z}\) do not share an edge direction. As a consequence, either \(\zeta_{Z}(G)\) is an edge of \(Z\) and \(\tau_{Z}(G)\) a vertex of \(t_{Z}\) or inversely, \(\zeta_{Z}(G)\) is a vertex of \(Z\) and \(\tau_{Z}(G)\) an edge of \(t_{Z}\). In the former case, \(\tau_{Z}(F)\) is equal to \(\tau_{Z}(G)\) and in the latter \(\zeta_{Z}(F)\) coincides with \(\zeta_{Z}(G)\), which completes the proof.
We can now prove the following statement similar to that of Lemma 5.2.
**Lemma 5.4**: Consider two 3-dimensional zonotopes \(Z\) and \(Z^{\prime}\). If
\[\psi:{\Cal{F}}(Z+t_{Z})\to{\Cal{F}}(Z^{\prime}+t_{Z^{\prime}})\]
is an isomorphism, then there exists an isomorphism
\[\theta:{\Cal{F}}(Z)\to{\Cal{F}}(Z^{\prime})\]
such that \(\theta\circ\zeta_{Z}\) is equal to \(\zeta_{Z^{\prime}}\circ\psi\).
Demonstration Proof: Consider a proper face \(F\) of \(Z\) and let us show that all the faces of \(Z+t_{Z}\) contained in \(\zeta_{Z}^{-1}(\{F\})\) have the same image by \(\zeta_{Z^{\prime}}\circ\psi\). Assume for contradiction that this is not the case and recall that by Proposition 7.12 from [22], the normal cone \(N_{Z}(F)\) is decomposed into a polyhedral complex by the normal cones of \(Z+t_{Z}\) it contains. Since \(N_{Z}(F)\) is convex and
\[N_{Z^{\prime}+t_{Z^{\prime}}}:{\Cal{F}}(Z^{\prime}+t_{Z^{\prime}})\to{\Cal{N }}(Z^{\prime}+t_{Z^{\prime}})\]
is an isomorphism, \(\zeta_{Z}^{-1}(\{F\})\) must contain two faces \(P\) and \(Q\) whose images by \(\zeta_{Z^{\prime}}\circ\psi\) differ and such that the normal cone of \(Z+t_{Z}\) at \(Q\) is a facet of the normal cone of \(Z+t_{Z}\) at \(P\). By the correspondence between the face lattice of a polytope and its normal fan, \(P\) is a facet of \(Q\). Now recall that
\[P=F+\tau_{Z}(P)\]
and
\[Q=F+\tau_{Z}(Q).\]
It follows that \(\tau_{Z}(P)\) must differ from \(\tau_{Z}(Q)\) as \(P\) and \(Q\) would otherwise be equal. Hence, by Lemma 5.2, \(\psi(P)\) and \(\psi(Q)\) have different images by \(\tau_{Z^{\prime}}\). As they also have different images by \(\zeta_{Z^{\prime}}\) and as \(\psi(P)\) is a facet of \(\psi(Q)\), this contradicts Proposition 5.3. As a consequence, all the faces of \(Z+t_{Z}\) contained in \(\zeta_{Z}^{-1}(\{F\})\) have the same image by \(\psi\circ\zeta_{Z^{\prime}}\). In other words, there exists a face \(\theta(F)\) of \(Z^{\prime}\) such that \(\psi\) sends \(\zeta_{Z}^{-1}(\{F\})\) to a subset of \(\zeta_{Z^{\prime}}^{-1}(\{\theta(F)\})\).
However, observe that
\[\Big{\{}\zeta_{Z}^{-1}(\{F\}):F\in{\Cal{F}}(Z)\Big{\}}\]
is a partition of \({\Cal{F}}(Z+t_{Z})\). Similarly, the sets \(\zeta_{Z^{\prime}}^{-1}(\{F\})\) where \(F\) ranges within \({\Cal{F}}(Z^{\prime})\) form a partition of \({\Cal{F}}(Z^{\prime}+t_{Z^{\prime}})\). As \(\psi\) is a bijection, it must then send \(\zeta_{Z}^{-1}(\{F\})\) precisely to \(\zeta_{Z^{\prime}}^{-1}(\{\theta(F)\})\). It follows that \(\theta\) is a bijection from \({\Cal{F}}(Z)\) to \({\Cal{F}}(Z^{\prime})\) such that \(\theta\circ\zeta_{Z}\) is equal to \(\zeta_{Z^{\prime}}\circ\psi\), as desired.
Finally, if \(G\) is a face of \(Z\) and \(F\) a face of \(G\), then observe that some face of \(Z+t_{Z}\) from \(\zeta_{Z}^{-1}(\{F\})\) must be contained in a face from \(\zeta_{Z}^{-1}(\{G\})\). As \(\zeta_{Z^{\prime}}\circ\psi\) is a morphism, this shows that \(\theta(F)\) is a face of \(\theta(G)\). Hence \(\theta\) is an isomorphism from the face lattice of \(Z\) to that of \(Z^{\prime}\).
Just like with Lemma 5.2 the statement of Lemma 5.4 can be rephrased using a commutative diagram: it tells that, if there exists an isomorphism \(\psi\) from \(\mathcal{F}(Z+t_{Z})\) to \(\mathcal{F}(Z^{\prime}+t_{Z^{\prime}})\), then there is a commutative diagram
where \(\theta\) is an isomorphism.
The following is an immediate consequence of Lemma 5.4.
Lemma 5.5: Consider two \(3\)-dimensional zonotope \(Z\) and \(Z^{\prime}\). If the Minkowski sums \(Z+t_{Z}\) and \(Z^{\prime}+t_{Z^{\prime}}\) have the same combinatorial type, then the zonotopes \(Z\) and \(Z^{\prime}\) have the same combinatorial type.
Consider an odd integer \(k\) greater than or equal to \(9\) (under this assumption, there exist zonotopes with \((k-3)/2\) generators). It follows from Lemmas 5.1 and 5.5 that the number of different combinatorial types of \(k\)-equivprojective polytopes is at least the number of zonotopes with \((k-3)/2\) generators. Hence, Theorem 1.1 in the case of the \(k\)-equivprojective polytopes such that \(k\) is odd follows from Theorem 2.4 and from the observations that
\[\frac{k-3}{\log\frac{k-3}{2}}=O\biggl{(}\frac{k}{\log k}\biggr{)}\]
and
\[\biggl{(}\frac{k-3}{2}\biggr{)}^{\frac{3}{2}(k-3)}=k^{k\bigl{(}\frac{3}{2}+O \bigl{(}\frac{1}{\log k}\bigr{)}\bigr{)}}\]
as \(k\) goes to infinity.
## 6. Equiprojective polytopes and decomposability
Our results show that Minkowski sums allow for a sequential construction of equiprojective polytopes, thus providing a partial answer to Shephard's original question. Two of the possible kinds of summands in these Minkowski sums, line
segments and polygons, are well understood. However, equiprojective polytopes can also appear as a summand, which leads to asking about the primitive building blocks of these sequential Minkowski sum constructions. More precisely, recall that a polytope \(P\) is _decomposable_ when it can be written as a Minkowski sum of two polytopes, none of whose is homothetic to \(P\)[4, 12, 14, 15, 16, 18, 21]. We ask the following question, in the spirit of Shephard's.
Question 6.1: Are there indecomposable equiprojective polytopes?
It is a consequence of Lemma 3.6 that any 3-dimensional polytope whose aggregated cones at all edge directions are planes or half-planes is necessarily equiprojective. These equiprojective polytopes form a natural superset of the zonotopes since the aggregated cones of 3-dimensional zonotopes at their edge directions are planes. It should be noted that this superset of the zonotopes does not contain all equiprojective polytopes. For instance, the equitruncated tetrahedron described in [10] has an aggregated cone at an edge direction that is neither a plane or a half-plane. Now observe that, if the aggregated cone of a 3-dimensional polytope at some of its edge directions is a plane, then that polytope is decomposable (see for example [4]). However, the above question on polytope decomposability is open for the 3-dimensional polytopes whose aggregated cones at all edge directions are half-planes.
Question 6.2: Does there exist an indecomposable 3-dimensional polytope whose aggregated cones at all edge directions are half-planes?
**Acknowledgements.** We are grateful to Xavier Goaoc for useful comments on an early version of this article. The PhD work of the first author is partially funded by the MathSTIC (CNRS FR3734) research consortium
| 3次元多面体 $P$ は $k$ 等方向投影多面体であるとき、$P$ の各線に対する投影は $k$ 頂点の多面体になる。1968年、ジェフリー・シェパードは $k$ 等方向投影多面体についての記述を求めた。近年、$k$ 等方向投影多面体の組合せタイプは $k$ における線形関数として少なくとも $k$ を超えることが示された。ここでは $k$ が無限大に発散する際に $k$ 等方向投影多面体の組合せタイプは $k$ の $3k/2 + o(k)$ 個以上であることが示される。これは Goodman--Pollack の最小化の定理と新しい等方向投影多面体の構築法である Minkowski の和を用いることで得られた。 |
2307.07504 | Gravitational partial-wave absorption from scattering amplitudes | We study gravitational absorption effects using effective on-shell scattering
amplitudes. We develop an in-in probability-based framework involving plane-
and partial-wave coherent states for the incoming wave to describe the
interaction of the wave with a black hole or another compact object. We connect
this framework to a simplified single-quantum analysis. The basic ingredients
are mass-changing three-point amplitudes, which model the leading absorption
effects and a spectral-density function of the black hole. As an application,
we consider a non-spinning black hole that may start spinning as a consequence
of the dynamics. The corresponding amplitudes are found to correspond to
covariant spin-weighted spherical harmonics, the properties of which we
formulate and make use of. We perform a matching calculation to
general-relativity results at the cross-section level and derive the effective
absorptive three-point couplings. They are found to behave as ${\cal
O}(G_\text{Newton}^{s+1})$, where $s$ is the spin of the outgoing massive
state. | Rafael Aoude, Alexander Ochirov | 2023-07-14T17:55:45 | http://arxiv.org/abs/2307.07504v3 | # Gravitational partial-wave absorption from scattering amplitudes
###### Abstract
We study gravitational absorption effects using effective on-shell scattering amplitudes. We develop an in-in probability-based framework involving plane- and partial-wave coherent states for the incoming wave to describe the interaction of the wave with a black hole or another compact object. We connect this framework to a simplified single-quantum analysis. The basic ingredients are mass-changing three-point amplitudes that model the leading absorption effects. As an application, we consider a non-spinning black hole that may start spinning as a consequence of the dynamics. The corresponding amplitudes are found to correspond to covariant spin-weighted spherical harmonics, the properties of which we formulate and make use of. We perform a matching calculation to general-relativity results at the cross-section level and derive the effective absorptive three-point couplings. They are found to behave as \(\mathcal{O}(G^{s+1})\), where \(s\) is the spin of the outgoing massive state.
## 1 Introduction
Since 2015 [1], the steady flow of gravitational-wave observations has been stimulating theorists to look for new computational methods for general relativity (GR). In addition to the constant improvement of the classical approaches to solve the two-body problem in gravity [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12], new theoretical results have also been obtained by
using on-shell scattering amplitudes, which encode the relevant, quantum or classical, physics in a gauge-invariant way. (See [13; 14; 15; 16] for recent reviews).
The conservative scattering of non-spinning compact bodies has been calculated up to fourth post-Minkowskian (PM) order using amplitude- [17; 18] and worldline-based methods [19; 20; 21; 22]. For the spinning case, the conservative scattering has been evaluated at second PM order and all-order in the angular momenta [23; 24; 25] with the help of Heavy-Particle Effective Theory [26; 27]. Higher PM orders have also been obtained, though limited to lower spin orders [28; 29; 30; 22; 31]. Progress on the spinning front has resulted in different and complementary on-shell approaches [32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. For the interesting case of black-hole (BH) dynamics, many of these works rely on the matching of three-point amplitudes to the Kerr multipole expansion [43]. An all-spin understanding of the relevant four-point Compton scattering amplitude, however, is still lacking, despite recent progress in the description of massive higher-spin particles [44; 45; 46], matching to the Teukolsky-equation solutions [38; 39] through sixth order in spin, and the availability of the conservative tree-level Compton with arbitrary coefficients [47]. The quantum-field-theoretic (QFT) program of gravitational dynamics has also seen impressive advances in methods for obtaining classical observables from amplitudes, such as the Kosower-Maybee-O'Connell (KMOC) formalism [48; 49; 50; 51; 52; 53], heavy-particle expansion [54; 55; 56; 27; 57; 58], eikonal ideas [59; 60; 61; 62; 63; 64; 65], worldline QFT [28; 29; 30], boundary-to-bound map [66; 67; 68; 69], and strong-field amplitudes [70; 71; 72].
Despite the successes in the conservative section, the progress in non-conservative effects has been slower, since those effects are naturally smaller. Within those, the absorption of mass and angular momentum is very small, especially for non-spinning bodies, and it is unlikely to be observed by ground-based detects, as shown in [73] for 5-to-50 solar masses black holes. However, for space-based detectors, the fraction of the radiated energy that is absorbed by the BH is around 5% [74]. These effects are especially important for rapidly rotating BHs, as shown in [75]. The change of mass and spin of a BH naturally leads to a change in the horizon by the second law of BH thermodynamics [76], which are already included in a few of the effective-one-body waveform templates [77; 78; 79] and will be needed for a future precision program.
In this paper, we initiate the study of absorption effects using modern on-shell methods for scattering amplitudes. In particular, we use mass-changing three-point amplitudes to describe leading absorption effects from a simplified single-quantum approach. We thus construct an in-in on-shell probability-based formalism for a partial wave impinging on a BH. Using this covariant effective-field-theory (EFT) description, we can match the microscopic cross-section calculation from GR literature and obtain the values of the relevant effective coupling coefficients. As a concrete application, we focus on absorption by a non-spinning BH, while leaving the more phenomenologically relevant spinning case for future work.
Absorption effects have been considered before in the literature, starting with Starobinsky, Churilov [80; 81] and Page [82; 83] and with later higher-order correc
tions in [84] and relatively recently in [85] by using traditional GR methods. The scattering and absorption cross-sections are obtained using a partial-wave expansion (in spin-weighted spherical harmonics) of the scattering phases and transmission factors. These factors are obtained by solving the Teukolsky equation, which describes perturbation around Kerr BHs. Absorption of mass and angular momentum by a BH was also computed in great detail [73; 74; 86; 87] in post-Newtonian theory.
From the worldline perspective, the study of absorption is more recent. They have been introduced in [88] for scalar BHs, with subsequent inclusion of spin effects [89; 90]. Furthermore, absorption has been combined with spontaneous emission to understand superradiance effects in [91]. The authors of [88; 89; 90; 91; 92] put EFT operators on a classical worldline to model the intricate behavior of a compact object. In particular, higher-derivative operators were included in [92] for the spinning case, which starts at 1.5 PN order, tackling the discrepancy in the literature of the horizon fluxes in the test-body limit. We propose to go further and consider the object itself as a quantum particle, but amenable to an appropriately defined classical limit. This lets us profit not just from QFT techniques, which have been available on the worldline, but also from the on-shell approach to scattering amplitudes.
Purely mass-changing absorption effects from on-shell scattering amplitudes were never studied to the best of our knowledge,1 although similar amplitudes have appeared in the context of quantum emission [93; 94]. The basic building blocks for modeling absorption effects are three-point amplitudes of two different massive particles and a graviton, in which the initial state absorbs the graviton, changing its mass and spin. Even before matching, the EFT cross-section reproduces known facts about Schwarzchild BHs: _(i)_ the cross-section does not depend on the magnetic quantum number \(m\), and _(ii)_ there is no absorption in the static limit \(\sigma_{\rm abs}(\omega_{\rm cl}\to 0)=0\).
Footnote 1: In [38; 39] the authors have introduced contact terms non-analytical in spin for the Compton amplitude to match the solutions of the Teulkosky equation. These terms are then suggested to model absorption effects, despite the masses of the initial and the final particles being equal. Here what we call absorption effects are strictly inelastic changing-mass interactions.
Properly modeling the interaction of a BH with a classical wave from amplitudes requires the use of massless coherent states. For that, we describe a covariant probability-based formalism for spherical coherent states, so as to substantiate the single-quantum leading-order calculation and to explain how one could improve the absorption description to higher orders and combine it with conservative effects.
This paper is organized as follows. In section 2 we describe the mass-changing and spin-changing amplitudes required for our description of the mechanics of compact objects absorbing a spherical wave in section 3. In section 4 we match to the microscopic cross-section from GR to make sense of the effective couplings. Finally, in section 5, we connect the single-quantum cross-section description to the framework involving massless spherical coherent states. In this section, we also introduce a diagrammatic expansion of the \(T\)-matrix, which allows for perturbations of the BH-wave
interaction that can be matched to higher orders of the cross-section. We conclude in section 6. Though we assume familiarity with the spinor-helicity formalism [95], we briefly explain it and its connection to spherical harmonics in appendix A.
## 2 Basic mass-changing amplitudes
Using scattering amplitudes to model absorption effects relies on EFT ideas, such as treating black holes as point particles. These concepts have been heavily used in the recent years to provide predictions for conservative dynamics and dissipation effects.
As in most of EFTs, the knowledge of the coefficients that parametrize the theory is either provided by experimental data or by performing a matching calculation to the underlying theory. In our case, the underlying theory is Einstein's GR, or more practically, the solution to the Teukolsky equation [96; 97; 98]. Given these two sides of the matching calculation, we will sometimes be referring to the EFT side of the calculation as macroscopic and to the solution to Teukolsky equation as microscopic.
On the EFT side, the building blocks to model absorption effects include mass-changing amplitudes (involving massless messenger particles and two particles with different masses), first explored in [99] and covariantized in [95]. In this section, we further reorganize the latter formulation, while also using coherent-spin eigenvalues [52], which saturate spin indices and thus serve as a book-keeping device [44].
Here and below we work in the massive spinor-helicity formalism [95], which is briefly reviewed in appendix A. Hence the amplitudes \(\mathcal{A}_{\{b\}^{\{a\}}}\) carry \(2s_{1}\) symmetrized little-group indices \(a_{1},\ldots,a_{2s_{1}}\) for the incoming massive particle \(1\) and \(2s_{2}\) such indices \(b_{1},\ldots,b_{2s_{2}}\) for the outgoing massive particle \(2\). We choose to use the chiral basis of massive spinors (angle brackets) for positive helicities and the antichiral basis (square brackets) for negative helicities. Since \(\det\{|1^{a}\rangle_{\alpha}\}=\det\{|1^{a}]_{\dot{\alpha}}\}=M_{1}\) and \(\det\{|2^{b}\rangle_{\beta}\}=\det\{|2^{b}\rangle_{\beta}\}=M_{2}\), we may proceed by stripping the spinors \(|1_{a}\rangle\) and \(|2^{b}\rangle\) for the positive messenger helicity and \(|1_{a}]\) and \(|2^{b}]\) for the negative helicity. For instance, for the positive-helicity case we write
\[\mathcal{A}_{\{b\}^{\{a\}}}(p_{2},s_{2}|p_{1},s_{1};k,h\geq 0)=:A(k,h\geq 0)^{ \{\alpha\},\{\beta\}}(|1_{a}\rangle_{\alpha})^{\odot 2s_{1}}(|2^{b}\rangle_{ \beta})^{\odot 2s_{2}}, \tag{1}\]
where \(\odot\) denotes the symmetrized tensor product [100]. In addition to the \(\mathcal{A}_{\{b\}^{\{a\}}}\) and \(A^{\{\alpha\},\{\beta\}}\) objects, a third perspective on the same amplitude is provided by contracting massive spinors are contracted with auxiliary SU(2)-spinor variables [44],
\[|\mathbf{1}\rangle:=|1_{a}\rangle\alpha^{a},\hskip 28.452756pt|\bar{\mathbf{2}} \rangle:=|2^{b}\rangle\tilde{\beta}_{b}, \tag{2}\]
which may serve as an extra handle on the spin quantization axis.2 We write the fully contracted amplitude in boldface as a scalar in terms of the spinor-stripped one:
Footnote 2: The auxiliary SU(2) spinors \(\alpha^{a}\) and \(\tilde{\beta}_{b}\) transform under the little groups of \(p_{1}\) and \(p_{2}\), respectively, and in this sense have an implicit dependence on their momenta.
\[\boldsymbol{\mathcal{A}}(p_{2},s_{2}|p_{1},s_{1};k,h\geq 0):=A(k,h\geq 0)^{\{ \alpha\},\{\beta\}}(|\mathbf{1}\rangle^{\otimes 2s_{1}})_{\{\alpha\}}(|\bar{ \mathbf{2}}\rangle^{\otimes 2s_{2}})_{\{\beta\}}, \tag{3}\]
where the index symmetrization is now entirely automatic. Incidentally, the SU(2) spinors in eq. (2) are also connected with massive coherent-spin states, the scattering of which is described by the coherent-spin amplitude [52]
\[\mathcal{A}(p_{2},\beta|p_{1},\alpha;k,h)=e^{-(||\alpha||^{2}+||\beta||^{2})/2} \overset{\infty}{\underset{2s_{1}=1}{\sum}}\overset{\infty}{\underset{2s_{2}=1 }{\sum}}\frac{1}{\sqrt{(2s_{1})!(2s_{2})!}}\boldsymbol{\mathcal{A}}(p_{2},s_{ 2}|p_{1},s_{1};k,h). \tag{4}\]
### Classifying mass-changing amplitudes
Going back to the stripped amplitude \(A(k,h)_{\{\alpha\},\{\beta\}}\) with two sets of symmetrized SL(2,\(\mathbb{C}\)) indices, we may decompose it in the chiral-spinor basis of \(|k\rangle\) and \(p_{1}|k]\). Unlike the equal-mass case, these two spinors are linearly independent (and there is no need for a helicity factor as in [95]), because
\[\langle k|p_{1}|k]=2p_{1}\cdot k=M_{2}^{2}-M_{1}^{2}\neq 0 \tag{5}\]
due to momentum conservation \(p_{2}=p_{1}+k\). This equation also tells us about the possible dimensionful scales entering the three-point process from an EFT perspective, which will have to be matched later. We can either use the mass pair \((M_{1},M_{2})\) or \((M_{1},2p_{1}\cdot k)\), and in this work we are going to favor the latter. For instance, we may use \(M_{1}\) to absorb the mass dimension of the amplitude and allow the EFT coefficients to depend on the dimensionless ratio
\[w:=\frac{2p_{1}\cdot k}{M_{1}^{2}}, \tag{6}\]
while expanding in terms of the dimensionless spinors of helicity \(-1/2\) and \(1/2\):
\[\lambda_{\alpha}:=M_{1}^{-1/2}|k\rangle_{\alpha},\qquad\quad\mu_{\alpha}:=M_{1 }^{-3/2}p_{1,\alpha\dot{\beta}}|k]^{\dot{\beta}}\qquad\Rightarrow\qquad\langle \lambda\mu\rangle=w. \tag{7}\]
Therefore, the most general stripped amplitude involving two unequal masses and one massless positive-helicity particle is schematically given by [95, 99]
\[A(k,h\geq 0)_{\{\alpha\},\{\beta\}}=M_{1}^{1-s_{1}-s_{2}}\overset{\infty}{ \underset{i}{\sum}}c_{(i),s_{1},s_{2}}^{h}(w)[\lambda^{s_{1}+s_{2}-h}\mu^{s_{ 1}+s_{2}+h}]^{(i)}_{\{\alpha\},\{\beta\}}. \tag{8}\]
Here \(i\) enumerates inequivalent tensor products with the given spinorial index structure, and their scalar coefficients \(c_{i,s_{1},s_{2}}^{h}(\omega)\) may depend on each spin and in the dimensionless ratio \(w\). In order to specify the relevant spinorial structures, note that there are natural constraints that follow already from the form of eq. (8), such as
\[s_{1}+s_{2}\pm h\,\in\,\mathbb{Z}_{\geq 0}\qquad\Rightarrow\qquad s_{1}+s_{2} \geq|h|. \tag{9}\]
Moreover, there can clearly be no three-point amplitude for one or three half-integer spins -- in QFT this standard fact is usually derived from the spin-statistics theorem.
We find it helpful to observe that the massless little-group dependence may be completely factored out (in the tensor-product sense), leaving a polynomial in \(\lambda\) and \(\mu\), which is independent of it:
\[\begin{split}[\lambda\mu\oplus\mu\lambda]^{n}_{\{\alpha\},\{ \beta\}}:=c_{0}(\lambda^{n})_{\alpha_{1}\dots\alpha_{n}}(\mu^{n})_{\beta_{1} \dots\beta_{n}}+c_{1}(\lambda^{n-1}\mu)_{\alpha_{1}\dots\alpha_{n}}(\mu^{n-1} \lambda)_{\beta_{1}\dots\beta_{n}}\\ +\dots+c_{n-1}(\lambda\mu^{n-1})_{\alpha_{1}\dots\alpha_{n}}(\mu \lambda^{n-1})_{\beta_{1}\dots\beta_{n}}+c_{n}(\mu^{n})_{\alpha_{1}\dots\alpha _{n}}(\lambda^{n})_{\beta_{1}\dots\beta_{n}},\end{split} \tag{10}\]
where we have also omitted the \(\otimes\) sign for brevity. The exponent \(n\) depends on the total-spin quantum numbers, and in the amplitude each such term may have its own coefficient. Without loss of generality, we consider \(s_{2}\geq s_{1}\), where we have two cases:
* \(s_{2}-s_{1}\geq h\), where we saturate the \(s_{1}\) indices by the above polynomial, while the remaining \(s_{2}\) indices are accounted for by the tensor product, which is unambiguously defined by the overall helicity weight. The corresponding spinorial structures belong to the following tensor power of a direct sum: \[[\lambda^{s_{1}+s_{2}-h}\mu^{s_{1}+s_{2}+h}]^{(i)}_{\{\alpha\},\{\beta\}}\ \in\ [ \lambda\mu\oplus\mu\lambda]^{2s_{1}}_{\{\alpha\},\{\beta\}}\,(\lambda^{s_{2} -s_{1}-h}\mu^{s_{2}-s_{1}+h})_{\{\beta\}};\] (11)
* \(s_{2}-s_{1}<h\), where the polynomial (10) saturates the number of \(\lambda\)'s, which is equal to \(s_{1}+s_{2}-h\), while the remaining \(2h\) of \(\mu\)'s are unambiguously distributed among the two massive particles. The spanning spinorial structure is thus \[[\lambda^{s_{1}+s_{2}-h}\mu^{s_{1}+s_{2}+h}]^{(i)}_{\{\alpha\},\{\beta\}}\ \in\ [ \lambda\mu\oplus\mu\lambda]^{s_{1}+s_{2}-h}_{\{\alpha\},\{\beta\}}\,(\mu^{s_{ 1}-s_{2}+h})_{\{\alpha\}}(\mu^{s_{2}-s_{1}+h})_{\{\beta\}}.\] (12) Note that in electromagnetism this case only occurs for \(s_{1}=s_{2}\), whereas in GR both \(s_{2}=s_{1}\) and \(s_{2}=s_{1}+1\) are possible.
In both cases, we have the polynomial with free coefficients and the additional factor, which carries the massless helicity. This factor completes the \(\mathrm{SL}(2,\mathbb{C})\) indices of either massive particle that are not accounted for by the polynomial, and of course all \(\alpha\)'s and all \(\beta\)'s are implicitly symmetrized.
This analysis should be repeated for \(s_{1}\leq s_{2}\), and the \(\mathrm{SL}(2,\mathbb{C})\) can then be contracted with the massive spinors (and auxiliary variables), for which the Dirac equations \(|p_{1}|\mathbf{1}\rangle=M_{1}|\mathbf{1}]\) and \(|p_{2}|\bar{\mathbf{2}}\rangle=M_{2}|\bar{\mathbf{2}}]\) hold. In this way, we arrive at
\[\boldsymbol{\mathcal{A}}(p_{2},s_{2}|p_{1},s_{1};k,h)=\begin{cases}\boldsymbol {F}^{h}_{s_{1},s_{2}}\,\langle\bar{\mathbf{2}}k\rangle^{s_{2}-s_{1}-h}[\bar{ \mathbf{2}}k]^{s_{2}-s_{1}+h},&s_{2}-s_{1}\geq|h|,\\ \boldsymbol{F}^{h}_{s_{1},s_{2}}\,[\bar{\mathbf{2}}k]^{s_{2}-s_{1}+h}[k \mathbf{1}]^{s_{1}-s_{2}+h},&|s_{2}-s_{1}|<h,\\ \boldsymbol{F}^{h}_{s_{1},s_{2}}\,\langle\bar{\mathbf{2}}k\rangle^{s_{2}-s_{1 }-h}\langle k\mathbf{1}\rangle^{s_{1}-s_{2}-h},&|s_{2}-s_{1}|<-h,\\ \boldsymbol{F}^{h}_{s_{1},s_{2}}\,\langle k\mathbf{1}\rangle^{s_{1}-s_{2}-h}[ k\mathbf{1}]^{s_{1}-s_{2}+h},&s_{1}-s_{2}\geq|h|.\end{cases}\] (13a) where the factor \[\boldsymbol{F}^{h}_{s_{1},s_{2}}\] contains free coefficients and can now be written as \[\boldsymbol{F}^{h}_{s_{1},s_{2}}=M_{1}^{1-2s_{1}-2s_{2}}\sum_{r=0}^{n}g^{h}_{ r,s_{1},s_{2}}(w)\,\langle\bar{\mathbf{2}}|k|\mathbf{1}]^{r}\,[\bar{\mathbf{2}}|k| \mathbf{1}\rangle^{n-r}. \tag{13b}\]
These coefficients \(g^{h}_{r,s_{1},s_{2}}(w)\) are a refined version of \(c^{h}_{(i),s_{1},s_{2}}(w)\) in eq. (8); the main difference between them is some degree of rescaling by \(M_{2}/M_{1}\). The polynomial degree \(n\) above is related to the maximal number of terms:
\[n+1\,=\,\begin{cases}\phantom{-}2s_{1}+1,&\phantom{-}s_{2}-s_{1}\geq|h|,\\ s_{1}+s_{2}-|h|+1,&\phantom{-}|s_{2}-s_{1}|<|h|,\\ \phantom{-}2s_{2}+1,&\phantom{-}s_{1}-s_{2}\geq|h|,\end{cases} \tag{13c}\]
This number matches the counting in [101]. For completeness, the above formulae (13) already include the result of the above analysis for the negative messenger helicity, in which case we used the anti-chiral basis, \(|k|\) and \(|p_{1}|k\rangle\).
Interestingly, the coupling counting (13c) obeys the bound
\[\#\text{ coeffs.}\,\leq\,2\text{min}(s_{1},s_{2})+1. \tag{14}\]
For instance, there is only one term for the case of the scalar massive incoming state \(s_{1}=0\). Indeed, the constraint (9) immediately implies \(s_{2}>|h|\), so we get a trivial polynomial of degree \(n(0,s_{2},h)=0\). In that case, the amplitude takes the form
\[\boldsymbol{\mathcal{A}}(p_{2},s_{2}|p_{1},s_{1}=0;k,h)=g^{|h|}_{0,0,s_{2}}(w )M_{1}^{1-2s_{2}}\langle\bar{\boldsymbol{2}}k\rangle^{s_{2}-h}[\bar{\boldsymbol {2}}k]^{s_{2}+h}, \tag{15}\]
where we have assumed parity and thus conflated the dimensionless coupling coefficients \(g^{\pm h}_{0,0,s_{2}}(w)\) into the single coupling \(g^{\pm|h|}_{0,0,s_{2}}(w)\), which still depends on the absolute helicity value of the messenger particle.3
Footnote 3: In the worldline formalism, the parity assumption is called “electric-magnetic” duality [88; 89].
### Minimal mass-changing amplitudes
As a minor digression, let us note that, for non-zero initial spin, the proliferation of possible effective couplings in the three-point mass-changing amplitude (13) may be reduced if we come up with some notion of minimality. Indeed, in a similar situation in the equal-mass case, \(M_{1}=M_{2}\), Arkani-Hamed, Huang and Huang [95] managed to single out the so-called "minimal" amplitudes by considering its massless limit. For positive helicity, these minimal amplitudes include, for instance,
\[\mathcal{A}(p_{2},s|p_{1},s;k,h\geq 0)=g^{h}_{0}(p_{1}\cdot\varepsilon_{k}^{ \pm})^{h}\langle\bar{\boldsymbol{2}}\boldsymbol{1}\rangle^{2s}, \tag{16}\]
where for simplicity we have assumed \(s_{1}=s_{2}=s\). In other words, the stripped amplitude is proportional to the tensor product of \(\text{SL}(2,\mathbb{C})\) Levi-Civita tensors \((\epsilon^{2s})_{\{\alpha\},\{\beta\}}\).
To expose a similar unique structure in the unequal-mass case, where the couplings correspond to the terms in the polynomial (10), we may change the basis inside of it to the antisymmetric and symmetric combinations of the basis spinors:
\[[\lambda\mu\oplus\mu\lambda]^{n}_{\{\alpha\},\{\beta\}}=[\epsilon\oplus \sigma]^{n}_{\{\alpha\},\{\beta\}},\qquad\epsilon_{\alpha\beta}=\frac{\lambda _{\alpha}\mu_{\beta}-\mu_{\alpha}\lambda_{\beta}}{\langle\lambda\mu\rangle}, \qquad\sigma_{\alpha\beta}:=\lambda_{\alpha}\mu_{\beta}+\mu_{\alpha}\lambda_{ \beta}. \tag{17}\]
Since of course \(\langle\mathbf{1}|^{\alpha}\langle\mathbf{\bar{2}}|^{\beta}\epsilon_{\alpha\beta}= \langle\mathbf{1\bar{2}}\rangle\) and the symmetric combination leads to
\[\langle\mathbf{1}|^{\alpha}\langle\mathbf{\bar{2}}|^{\beta}\sigma_{\alpha\beta }=\frac{M_{2}^{2}+M_{1}^{2}}{M_{1}^{2}}\langle\mathbf{1\bar{2}}\rangle+\frac{2 M_{2}}{M_{1}}[\mathbf{1\bar{2}}], \tag{18}\]
the main amplitude factor can simply be expanded in the angle and square brackets:
\[\mathbf{F}_{s_{1},s_{2}}^{h}=M_{1}^{1-2s_{1}-2s_{2}+n}\sum_{r=0}^{n}\tilde{g}_{r,s _{1},s_{2}}^{h}(w)\,\langle\mathbf{\bar{2}}\mathbf{1}\rangle^{n-r}[\mathbf{ \bar{2}}\mathbf{1}]^{r}. \tag{19}\]
So we propose to define the minimal mass-changing stripped amplitudes as those with highest power in \(\epsilon_{\alpha\beta}\), or, equivalently,
\[\mathcal{A}_{\rm min}(p_{2},s_{2}|p_{1},s_{1};k,h\geq 0) \tag{20}\] \[=\tilde{g}_{0,s_{1},s_{2}}^{+}(w)\begin{cases}M_{1}^{1-2s_{2}} \langle\mathbf{\bar{2}}\mathbf{1}\rangle^{2s_{1}}\langle\mathbf{\bar{2}}k \rangle^{s_{2}-s_{1}-h}[\mathbf{\bar{2}}k]^{s_{2}-s_{1}+h},&s_{2}-s_{1}\geq h \geq 0,\\ M_{1}^{1-s_{1}-s_{2}-h}\langle\mathbf{\bar{2}}\mathbf{1}\rangle^{s_{1}+s_{2}- h}[\mathbf{\bar{2}}k]^{s_{2}-s_{1}+h}[\mathbf{k1}]^{s_{1}-s_{2}+h},&|s_{2}-s_{1}|<h,\\ M_{1}^{1-2s_{2}}\langle\mathbf{\bar{2}}\mathbf{1}\rangle^{2s_{2}}\langle k \mathbf{1}\rangle^{s_{1}-s_{2}-h}[k\mathbf{1}]^{s_{1}-s_{2}+h},&s_{1}-s_{2} \geq h\geq 0.\end{cases}\]
It is clear that for \(s_{1}=0\) and \(s_{2}>|h|\), the minimal-coupling amplitude coincides with the previously defined amplitude (15). Moreover, let us note in passing that these amplitudes satisfy the double-copy prescription explored in the presence of massive spinning states in [102; 103].
We hope to explore these amplitudes in more detail elsewhere, whereas in the rest of this paper for the sake of simplicity we focus on the mass-changing amplitudes (15) with the non-spinning initial state, which we use to model the radiation absorption by a Schwarzschild black hole. In this context, it is important to note that if we assume locality of the EFT Lagrangian that implies the above amplitudes, the dimensionless coupling constants \(g_{0,s_{1},s_{2}}^{h}(w)\) may then be constrained to only have non-negative powers of \(w\). Unfortunately, a rigorous proof of this statement may be to technical and require dealing with all sorts of field redefinitions. So for the purposes of this paper, let us simply impose that \(g_{0,0,s_{2}}^{h}(w)\) have no poles in \(w\):
\[g_{0,0,s_{2}}^{h}(w)=\mathcal{O}(w^{0})\qquad\Rightarrow\qquad\mathbf{\mathcal{ A}}(p_{2},s_{2}|p_{1},s_{1}=0;k,h)=\mathcal{O}(w^{s_{2}}),\qquad w\to 0, \tag{21}\]
which constitutes is a non-trivial EFT modeling assumption.
## 3 Absorption mechanics of compact objects
In this section we describe our setup for obtaining classical absorption effects from the quantum on-shell scattering amplitudes. We focus on the simplest relevant process depicted in figure 1: a graviton spherical state impinging on a massive particle of mass \(M_{1}\) (for simplicity taken spinless), which absorbs the graviton and changes its mass to \(M_{2}\) and spin to \(s_{2}\). It is natural to think of the corresponding scattering
mplitude in terms of plane-wave states as described in section 2. However, GR methods give us results [80; 81; 82; 83; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105; 106] for spherical waves with fixed angular-momentum quantum numbers. Therefore, we start by translating between these two pictures -- with a focus on single-graviton states. In section 5 we will come back to justifying this setup further using classical coherent states, which are more appropriate for modeling classical waves.
### Spherical helicity amplitude
By definition (see e.g. [107]), spherical helicity states partially diagonalize the total angular momentum operator \(\mathbf{J}\), more specifically, \(\mathbf{J}^{2}\), \(J_{z}\) and helicity \((\mathbf{J}\cdot\mathbf{P})/\mathbf{P}^{2}\), as well as the Hamiltonian \(P^{0}\). Such states are labeled by energy \(\omega\), angular-momentum quantum numbers \(j\), \(m=-j,\ldots,j\) and helicity \(h=\pm 2\) (graviton) or \(\pm 1\) (photon):4
Footnote 4: Here and below, the hat notation [48] means \(\hat{d}^{n}p:=d^{n}p/(2\pi)^{n}\) and \(\hat{\delta}^{n}(...):=(2\pi)^{n}\delta^{n}(...)\). For the spherical helicity states, we also assume that masslessness: \(P^{2}|\omega,j,m,h\rangle=0\).
\[|\omega,j,m,h\rangle=a^{\dagger}_{j,m,h}(\omega)|0\rangle,\qquad\langle\omega^ {\prime},j^{\prime},m^{\prime},h^{\prime}|\omega,j,m,h\rangle=\hat{\delta}( \omega^{\prime}-\omega)\delta^{j}_{j^{\prime}}\delta^{m}_{m^{\prime}}\delta^{h }_{h^{\prime}}. \tag{3.1}\]
This is in contrast to the more familiar plane-wave states \(|k,h\rangle\), which diagonalize the four-momentum \(P^{\mu}\) in addition to the helicity \((\mathbf{J}\cdot\mathbf{P})/\mathbf{P}^{2}\):
\[|k,h\rangle:=a^{\dagger}_{h}(k)|0\rangle,\qquad\langle k^{\prime},h^{\prime}|k,h\rangle=2|\mathbf{k}|\hat{\delta}^{3}(\mathbf{k}^{\prime}-\mathbf{k})\delta^{h}_{h^{ \prime}}. \tag{3.2}\]
The two bases of one-particle states may be related by [91]
\[\langle k,h^{\prime}|\omega,j,m,h\rangle=\frac{4\pi}{\sqrt{2\omega}}\delta^{h} _{h^{\prime}}\hat{\delta}(|\mathbf{k}|-\omega)\,_{-h}Y_{jm}(\hat{\mathbf{k}}), \tag{3.3}\]
where the spin-weighted spherical harmonics \({}_{-h}Y_{jm}(\hat{\mathbf{k}})\) depend on the momentum direction \(\hat{\mathbf{k}}:=\mathbf{k}/|\mathbf{k}|\) and constitute a generalization [108; 109] of the usual (scalar) spherical harmonics. The corresponding completeness relations imply that the one-particle spinning spherical state can be written as
\[|\omega,j,m,h\rangle=\frac{4\pi}{\sqrt{2\omega}}{\int_{k}}\hat{ \delta}(k^{0}-\omega)\,_{-h}Y_{j,m}(\hat{\mathbf{k}})|k,h\rangle=\sqrt{2\omega}{ \int}\frac{d\Omega_{\hat{\mathbf{k}}}}{4\pi}{}_{-h}Y_{j,m}(\hat{\mathbf{k}})|k,h \rangle\big{|}_{|\mathbf{k}|=\omega}, \tag{3.4}\]
Figure 1: Wave impinging on a scalar black hole
where \(d\Omega_{\hat{\mathbf{k}}}\) denotes the spherical-angle integration measure over the directions of \(\mathbf{k}\). We have also defined a shorthand for the on-shell momentum integration measure
\[\int_{p}:=\int\!\!\frac{d^{4}p}{(2\pi)^{3}}\Theta(p^{0})\delta(p^{2}-M_{p}^{2})=: \int\!\!\frac{d^{4}p}{(2\pi)^{3}}\delta^{+}(p^{2}-M_{p}^{2}),\qquad\quad M_{k}=0. \tag{11}\]
In order to write the scattering matrix element for a spherical helicity state, we need to be careful with the massive particle at the origin, which, strictly speaking, cannot be a plane-wave state either. So instead we use a wavepacket
\[|\psi\rangle:=\int_{p_{1}}\!\!\psi_{\xi}(p_{1})|p_{1}\rangle: \langle\psi|\psi\rangle=1,\qquad\langle\psi|P^{\mu}|\psi\rangle=p _{1,\rm cl}^{\mu}:=(M_{1},\mathbf{0}), \tag{12}\] \[\langle\psi|P^{\mu}P^{\nu}|\psi\rangle=\langle\psi|P^{\mu}|\psi \rangle\langle\psi|P^{\nu}|\psi\rangle+\mathcal{O}(\xi),\]
where \(\xi:=\ell_{\rm C}^{2}/\ell_{\rm WP}^{2}\) is related to the dimensionless ratio of the Compton wavelength and the position-space spread of the wavepacket [48]. We will be focusing on the scale hierarchy
\[\ell_{\rm C}\ll\ell_{\rm WP}\ll\frac{2\pi\hbar c}{\omega}\qquad\Rightarrow \qquad\xi\ll 1, \tag{13}\]
relevant for classical scattering of a wave with frequency \(\omega/\hbar\). For concreteness, we may think of \(\psi_{\xi}(p_{1})\propto\exp\left(-\frac{p_{1}^{0}}{\xi M_{1}}\right)\), the Lorentz-invariant version of which is [48, 110]
\[\psi_{\xi}(p_{1})=\frac{1}{M_{1}}\biggl{[}\frac{8\pi^{2}}{\xi K_{1}(2/\xi)} \biggr{]}^{1/2}\exp\biggl{(}\!-\frac{p_{1}\!\cdot u_{1}}{\xi M_{1}}\biggr{)}, \tag{14}\]
where \(K_{1}\) denotes the modified Bessel function of the second kind.
We are now ready to express the \(S\)-matrix element for a spherical helicity state in terms of the conventional plane-wave scattering amplitude:
\[\langle X|S|\psi;\omega,j,m,h\rangle=\frac{4\pi i}{\sqrt{2\omega}}\!\int_{p_{ 1}}\!\!\psi_{\xi}(p_{1})\!\int_{k}\!\!\hat{\delta}(k^{0}\!-\!\omega)\hat{ \delta}^{4}(p_{1}\!+\!k\!-\!p_{X})\,_{-h}Y_{j,m}(\hat{\mathbf{k}})\mathcal{A}(X|p_ {1};k,h), \tag{15}\]
where we have ignored the no-scattering term in \(S=1+iT\). For the amplitude arguments, we choose to mimic the structure of the matrix elements and write the outgoing particles first separated from the incoming particles by a vertical line.
Unfortunately, the matrix element (15) by itself is too singular to handle unambiguously, which is due to the infinite norm \(\langle\omega,j,m,h|\omega,j,m,h\rangle=\hat{\delta}(0)\) of the massless spherical state (10). So we also smear its energy with a wavefunction:
\[|\gamma\rangle:=\int_{0}^{\infty}\!\!\hat{d}\omega\gamma_{\zeta}(\omega)| \omega,j,m,h\rangle: \langle\gamma^{\prime}|\gamma\rangle=\delta_{j^{\prime}}^{j} \delta_{m^{\prime}}^{m}\delta_{h^{\prime}}^{h},\qquad\langle\gamma|P^{0}| \gamma\rangle=\omega_{\rm cl}, \tag{16}\] \[\langle\gamma|P^{0}P^{0}|\gamma\rangle=\langle\gamma|P^{0}| \gamma\rangle\langle\gamma|P^{0}|\gamma\rangle+\mathcal{O}(\zeta).\]
The corresponding scattering-matrix element is
\[\langle X|S|\psi;\gamma\rangle=4\pi i\!\int_{p_{1}}\!\!\psi_{\xi}(p_{1})\!\int_ {k}\frac{\gamma_{\zeta}(k^{0})}{\sqrt{2k^{0}}}\hat{\delta}^{4}(p_{1}+k-p_{X}) \,_{-h}Y_{j,m}(\hat{\mathbf{k}})\mathcal{A}(X|p_{1};k,h). \tag{17}\]
### Covariant spherical states
Before we proceed to the absorption cross-section, it is rewarding to covariantize our spherical-helicity state setup. By covariantization we mean allowing for an arbitrary time direction \(u^{\mu}\), with \(u^{2}=1\), as well a spacelike spin quantization axis \(n^{\mu}\), with \(n^{2}=-1\) and \(n\cdot u=0\). (In section 3.1, these were set to \((1,\mathbf{0})\) and \((0,0,0,1)\), respectively.) The corresponding angular momentum operator is
\[J^{\mu}(u):=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}J_{\nu\rho}u_{\rho}\qquad \Rightarrow\qquad[J^{\mu}(u),J^{\nu}(u)]=i\epsilon^{\mu\nu\rho\sigma}u_{\rho}J _{\sigma}(u), \tag{3.12}\]
which is not to be confused with the Pauli-Lubanski spin vector \(W^{\mu}\). A covariant spherical helicity state \(|\omega,j,m,h\rangle\) is then an eigenstate of "energy" \(E(u):=u\cdot P\) and angular-momentum combinations \(-J(u)^{2}\), \(n\cdot J(u)\) and \(J(u)\cdot P=W\cdot P\). Similarly to eq. (3.4), we choose to construct them directly from the plane-wave states:
\[|\omega,j,m,h\rangle_{u,n}:=\frac{4\pi}{\sqrt{2\omega}}{\int}\hat{d}^{4}k\, \hat{\delta}^{+}(k^{2})\hat{\delta}(k\cdot u-\omega)\,_{-h}Y_{j,m}(k;u,n)|k,h\rangle. \tag{3.13}\]
The new ingredient here is the covariant spin-weighted spherical harmonic. We define these functions in terms of spinor products as follows:
\[{}_{h}\tilde{Y}_{j,m}(k;u,n):=\frac{1}{\langle k|u|k]^{j}}\left[\left[u_{a}k \right]^{\odot(j+h)}\odot\langle ku_{a}\rangle^{\odot(j-h)}\right]_{\{a\}= \underbrace{(1\dots 1}_{j-m}\underbrace{2\dots 2}_{j+m})}. \tag{3.14}\]
We have hereby followed [39, 111] in using the massive spinor-helicity formalism [95] to covariantize the spinorial construction dating back to Newman and Penrose [108]. We adopt the conjugation conventions \(\left(|p^{a}\rangle_{\alpha}\right)^{*}=[p_{a}|_{\dot{\alpha}},\,\left(|p^{a }|_{\dot{\alpha}}\right)^{*}=-|p_{a}\rangle_{\alpha}\) for \(p^{0}>0\), which imply
\[{}_{h}\tilde{Y}_{j,m}^{*}(k;u,n)=(-1)^{2j+m-h}{}_{-h}\tilde{Y}_{j,-m}(k;u,n). \tag{3.15}\]
The properly normalized functions seen in eq. (3.13) are written without the tildes:
\[{}_{h}Y_{j,m}(k;u,n):=(-1)^{m}(2j)!\sqrt{\tfrac{2j+1}{4\pi(j+m)!(j-m)!(j+h)!(j -h)!}}\,{}_{h}\tilde{Y}_{j,m}(k;u,n), \tag{3.16}\]
with the orthonormality statement being
\[\frac{2}{\omega}{\int}d^{4}k\,\delta^{+}(k^{2})\delta(k\cdot u- \omega)\,_{h}Y_{j^{\prime},m^{\prime}}^{*}(k;u,n)\,_{h}Y_{j,m}(k;u,n)=\delta_ {j}^{j^{\prime}}\delta_{m}^{m^{\prime}}. \tag{3.17}\]
The proof and a detailed exposition of the harmonics (3.14) are given in appendix A.
Let us point out the new important features of these harmonics. First of all, the harmonics are by definition (3.14) insensitive to the overall scale of both \(k^{\mu}\) and \(u^{\mu}\). Moreover, they are now clearly formulated in a convention-independent way -- in the sense that it is covariant with respect to the two little groups:
* the massless little-group U(1) of \(k^{\mu}\) may be used to change the phases of all spherical harmonics in a local but mutually consistent way. Namely, transforming \(|k\rangle\to e^{-i\phi(k)/2}|k\rangle\), \(|k]\to e^{i\phi(k)/2}|k]\) implies phase adjustments of the form \({}_{h}Y_{j,m}(k;u,n)\to e^{ih\phi(k)}{}_{h}Y_{j,m}(k;u,n)\), which connect between various possible definitions of spin-weighted spherical harmonics, e.g. via quaternions [112].
* the massive little group SU(2) of \(u^{\mu}\) may be used to change the meaning of the magnetic quantum number \(m\). For instance, the explicit spinor parametrizations (107) and (108) correspond to the \(m\)-quantization along \(\boldsymbol{u}\neq 0\) and the conventional \(z\)-axis for \(\boldsymbol{u}=0\), respectively. However, we may just as well apply transformations \(|u^{a}\rangle\to U^{a}{}_{b}(u)|u^{b}\rangle\), \(|u^{a}]\to U^{a}{}_{b}(u)|u^{b}]\) to the massive spinors, and this will rotate the spin quantization axis \[n^{\mu}:=\frac{1}{2}(\langle u_{2}|\sigma^{\mu}|u^{2}]+[u_{2}|\bar{\sigma}^{ \mu}|u^{2}\rangle)\qquad\Rightarrow\qquad n^{2}=-1,\quad u\cdot n=0.\] (117) Having this relation in mind, we henceforth compress our notation to \({}_{h}Y_{j,m}(k;u)\).
In addition, we can specify the general frame transformations of the covariant spherical harmonics (101). Indeed, it is shown in appendix B that under the time-direction change \(u^{\mu}\to v^{\mu}=L^{\mu}{}_{\nu}(v\!\leftarrow\!u)u^{\nu}\) the massive spinors are boosted as follows:
\[|v^{a}\rangle=\frac{\sqrt{\mu}}{\mu\!+\!1}|u\!+\!v|u^{a}],\qquad|v^{a}]=\frac{ \sqrt{\mu}}{\mu\!+\!1}|u\!+\!v|u^{a}\rangle,\qquad\mu:=u\cdot v+\sqrt{(u\cdot v )^{2}\!-\!1}. \tag{118}\]
Here we have assumed that the spin quantization axis for the resulting time direction \(v^{\mu}\) is automatically \(L^{\mu}{}_{\nu}(v\!\leftarrow\!u)n^{\nu}\), i.e. the boosted version of the original quantization axis \(n^{\mu}\). Of course, it can then be easily tweaked by an additional little-group transformation of the resulting spinors \(|v^{a}\rangle\to U^{a}{}_{b}(v)|v^{b}\rangle\), \(|v^{a}]\to U^{a}{}_{b}(v)|v^{b}]\).
Given this covariant formulation of the spherical states, we rewrite eq. (100) as
\[\langle X|S|\psi;\gamma\rangle=4\pi i\!\int_{0}^{\infty}\!\frac{ \hat{d}\omega}{\sqrt{2\omega}}\gamma_{\zeta}(\omega)\!\int\!\hat{d}^{4}p_{1} \hat{\delta}^{+}(p_{1}^{2}\!-\!M_{1}^{2})\psi_{\xi}(p_{1}) \tag{119}\] \[\qquad\times\!\int\!\hat{d}^{4}k\hat{\delta}^{+}(k^{2})\hat{ \delta}(k\cdot u_{1}-\omega)\hat{\delta}^{4}(p_{1}+k-p_{X})\,_{-h}Y_{j,m}(k;u _{1})\mathcal{A}(X|p_{1};k,h),\]
which is what we are going to use in the absorption cross-section calculation below.
### Mass-changing amplitudes as harmonics
It is tempting to notice that the amplitudes (16) are simply proportional to spin-weighted spherical harmonics defined in eq. (101), namely
\[\mathcal{A}_{\underbrace{1\dots 12\dots 2}_{s_{2}-m\ s_{2}+m}}(p_{2},s_{2}|p_{ 1};k,h)\!=M_{1}g_{0,0,s_{2}}^{|h|}(w)(-1)^{s_{2}-h}w^{s_{2}}{}_{h}\tilde{Y}_{ s_{2},m}(k;u_{2})\!=:\mathcal{A}_{s_{2},m}^{h}(p_{2}|p_{1};k). \tag{120}\]
However, the harmonics are defined with respect to \(u_{2}^{\mu}\), which unlike \(u_{1}^{\mu}\) involves the integration variable \(k^{\mu}\). So we wish to make the transition between the two velocity vectors, which are related by the boost
\[u_{2}^{\rho}=L_{\sigma}^{\rho}(u_{2}\gets u_{1})u_{1}^{\sigma}=\exp\Bigl{(} \frac{i\log(u_{1}\!\cdot\!u_{2}\!+\!\sqrt{(u_{1}\!\cdot\!u_{2})^{2}\!-\!1})}{ \sqrt{(u_{1}\!\cdot\!u_{2})^{2}\!-\!1}}u_{1}^{\mu}u_{2}^{\nu}\Sigma_{\mu\nu} \Bigr{)}^{\rho}_{\sigma}u_{1}^{\sigma}. \tag{3.22}\]
The corresponding spinor transformations, given by eq. (3.19), may be rewritten as
\[|u_{2}^{a}\rangle=\frac{\sqrt{M_{1}}}{\sqrt{M_{2}}}\biggl{(}|u_{1}^{a}\rangle+ \frac{|k|u_{1}^{a}|}{M_{1}\!+\!M_{2}}\biggr{)},\qquad|u_{2}^{a}|=\frac{\sqrt{ M_{1}}}{\sqrt{M_{2}}}\biggl{(}|u_{1}^{a}|+\frac{|k|u^{a}\rangle}{M_{1}\!+\!M_{2}} \biggr{)}, \tag{3.23}\]
where we have used that \(\mu:=u_{1}\cdot u_{2}+\sqrt{(u_{1}\cdot u_{2})^{2}-1}=M_{2}/M_{1}\). The net effect of this is that the projection of the massive spinors onto the directions \(|k\rangle\) and \(|k]\) is invariant under this boost, so the spherical harmonics are simply related by
\[\langle 2^{a}k\rangle=\langle 1^{a}k\rangle,\qquad[2^{a}k]=[1^{a}k]\qquad \Rightarrow\qquad{}_{h}\tilde{Y}_{s_{2},m}(k;u_{2})={}_{h}\tilde{Y}_{s_{2},m }(k;u_{1}). \tag{3.24}\]
(This is because we switch between rest frames of \(p_{1}\) and \(p_{2}=p_{1}+k\) inside the harmonics in the same direction \(k\).) The caveat here is that the spin of particle 2 is now quantized along \(L^{\mu}{}_{\nu}(u_{2}\gets u_{1})n_{1}^{\sigma}\), i.e. the boost of the spin quantization axis of particle 1, which may be arbitrary but has to be the same for every \(p_{2}=p_{1}+k\). With this restriction in mind, we may rewrite the three-point amplitude as
\[\mathcal{A}_{s_{2},m}^{h}(p_{2}|p_{1};k)=M_{1}\,g_{0,0,s_{2}}^{|h|}(w)(-1)^{s _{2}-h}w^{s_{2}}\,{}_{h}\tilde{Y}_{s_{2},m}(k;u_{1}). \tag{3.25}\]
Let us now introduce the spherical scattering amplitude5
Footnote 5: Note that the definition (3.26) ignores the delta function \(\delta^{4}(p_{1}+k-p_{2})\), which accompanies the scattering amplitude and imposes momentum conservation. Although it will play a role the cross-section calculation in the next section, the above definition can still be found useful.
\[\mathcal{A}_{\{b\}}(p_{2},s_{2}|p_{1};\omega,j,m,h)\!:=\!\frac{4\pi}{\sqrt{2 \omega}}\!\int_{k}\!\hat{\delta}(k\!\cdot\!u_{1}\!-\!\omega)\,{}_{-h}Y_{j,m}(k ;u_{1})\mathcal{A}_{\{b\}}(p_{2},s_{2}|p_{1};k,h) \tag{3.26}\]
in an analogous manner to eq. (3.13). Using the conjugation and orthogonality properties (3.15) and (3.17), we find
\[\mathcal{A}_{\underbrace{1\ldots 1}_{s_{2}-m^{\prime}}\! \underbrace{2\ldots 2}_{s_{2}+m^{\prime}}}(p_{2},s_{2}|p_{1};\omega,j,m,h)=\frac{(-1)^{-2j+m +h}}{\pi\sqrt{2\omega}}\!\int\!d^{4}k\,\delta^{+}(k^{2})\delta(k\cdot u_{1}- \omega)\\ \times{}_{h}Y_{j,-m}^{*}(k;u_{1})\mathcal{A}_{s_{2},m^{\prime}}^{ h}(p_{2}|p_{1};k) \tag{3.27}\]
This neatly expresses the angular-momentum conservation law thanks to our assumption that the quantum number \(m^{\prime}\) is defined with respect to the axis \(L^{\mu}{}_{\nu}(u_{2}\gets u_{1})n_{1}^{\sigma}\).
### Leading-order absorption cross-section
We are now ready to construct the leading absorption cross-section from the above three-point amplitude. The inclusive cross-section for the spherical scattering setup described in section 3.1 is [88; 91]
\[\sigma_{\rm inc}(\omega_{\rm cl},j,m,h)=\frac{\pi}{\omega_{\rm cl}^{2}}P_{\rm inc }(\omega_{\rm cl},j,m,h)=\frac{\pi}{\omega_{\rm cl}^{2}}\sum_{X}\frac{\left| \langle X|S|\psi;\gamma\rangle\right|^{2}}{\langle X|X\rangle\langle\psi| \psi\rangle\langle\gamma|\gamma\rangle}, \tag{3.28}\]
which is invariant under the basis choice for the outgoing states. The leading contribution due to absorption is then given by the 3-point process:
\[P_{\rm inc}^{\rm LO}(\omega_{\rm cl},j,m,h)=V{\int_{0}^{\infty}}dM_{2}^{2} \rho(M_{2}^{2})\int\!\hat{d}^{3}p_{2}\frac{\left|\langle p_{2}|S|\psi;\gamma \rangle\right|^{2}}{\langle p_{2}|p_{2}\rangle\langle\psi|\psi\rangle\langle \gamma|\gamma\rangle}. \tag{3.29}\]
Here \(V:=\langle p_{2}|p_{2}\rangle/(2p_{2}^{0})=\hat{\delta}^{3}(\mathbf{0})\) is the space volume, which immediately cancels against the normalization of the outgoing state, for which we have temporarily suppressed any quantized degrees of freedom. We have also been compelled to include the spectral density \(\rho(M_{2}^{2})\), which is positive and normalized to 1:
\[\rho(q^{2})\geq 0,\hskip 28.452756pt\int_{0}^{\infty}\!\!\rho(q^{2})dq^{2}=1. \tag{3.30}\]
In a conservative scenario, one may simply assume \(\rho(q^{2})=\delta(q^{2}-M_{1}^{2})\), and the relevant amplitude would be the same-mass three-point amplitude. More generally, it is allowed to contain suitably normalized delta-functions for the "elementary" particles and the continuous part due to multi-particle states. Since we are interested in modeling absorption effects, we are led to explore the continuous part of the spectrum for \(q^{2}>M_{1}^{2}\). In view of the normalization of the initial states, \(\langle\psi|\psi\rangle=\langle\lambda|\lambda\rangle=1\), the resulting probability is given by
\[P_{\rm inc}^{\rm LO}(\omega_{\rm cl},j,m,h)=\sum_{s_{2}}\!\int\!dM_{2}^{2} \rho_{s_{2}}(M_{2}^{2})\int_{p_{2}}\sum_{b_{1},\ldots,b_{s_{2}}}\!\left| \langle p_{2},s_{2},\{b\}|S|\psi;\gamma\rangle\right|^{2}, \tag{3.31}\]
where we have now made the spin degrees of freedom of the outgoing state explicit. The integration over masses of \(p_{2}\) different from \(M_{1}\) is what allows the three-point amplitude to exist on real kinematics and thus makes this cross-section meaningful. As we will see, momentum conservation will later fix this mass to
\[M_{2}^{2}=M_{1}^{2}+2M_{1}\omega_{\rm cl}. \tag{3.32}\]
After restoring \(\hbar\) in front of \(\omega_{\rm cl}\), it actually becomes sent back to \(M_{1}\) in the classical limit, so the spectral density will only by probed in the vicinity of the original BH mass. This, however, does not negate the crucial roles that the unequal masses and
the spectral density play in allowing for a non-singular construction of the cross-section from three-point amplitudes.
Coming back to the squared amplitude in the integrand of eq. (3.31), we have
\[\begin{split}\sum_{\{b\}}\big{|}\langle p_{2},s_{2},& \{b\}|S|\psi;\gamma\rangle\big{|}^{2}=8\pi^{2}\!\int_{0}^{\infty}\!\! \frac{\hat{d}\omega\hat{d}\omega^{\prime}}{\sqrt{\omega\omega^{\prime}}}\gamma _{\zeta}^{*}(\omega)\gamma_{\zeta}(\omega^{\prime})\!\int_{p_{1},p_{1}^{ \prime},k,k^{\prime}}\!\!\!\psi_{\xi}^{*}(p_{1})\psi_{\xi}(p_{1}^{\prime})\\ &\times\hat{\delta}(k\cdot u_{1}-\omega)\hat{\delta}(k^{\prime} \!\cdot u_{1}-\omega^{\prime})\hat{\delta}^{4}(p_{1}+k-p_{2})\hat{\delta}^{4}( p_{1}^{\prime}+k^{\prime}-p_{2})\\ &\times{}_{-h}Y_{j,m}^{*}(k;u_{1})\,_{-h}Y_{j,m}(k^{\prime};u_{1} )\,\mathcal{A}^{*\{b\}}(p_{2},s_{2}|p_{1};k,h)\,\mathcal{A}_{\{b\}}(p_{2},s_{ 2}|p_{1}^{\prime};k^{\prime},h),\end{split} \tag{3.33}\]
where the summation over the little-group indices \(\{b\}\) is now implicit. We may use \(\hat{\delta}^{4}(p_{1}+k-p_{2})\) to perform the integration over \(p_{2}\), which leaves the on-shell constraint \(\hat{\delta}((p_{1}+k)^{2}-M_{2}^{2})\). We then change the integration variables to
\[p_{\rm a}^{\mu}:=(p_{1}^{\mu}+p_{1}^{\prime\mu})/2,\qquad\quad q^{\mu}:=p_{1} ^{\prime\mu}-p_{1}^{\mu}, \tag{3.34}\]
and remove \(q\) with \(\hat{\delta}^{4}(q+k^{\prime}-k)\) originating from \(\hat{\delta}^{4}(p_{1}^{\prime}+k^{\prime}-p_{2})\). Thus we get
\[\begin{split}& P_{\rm inc}^{\rm LO}(\omega_{\rm cl},j,m,h)=8\pi^{2} \sum_{s_{2}}\!\int\!dM_{2}^{2}\rho_{s_{2}}(M_{2}^{2})\int_{0}^{\infty}\!\!\frac {\hat{d}\omega\hat{d}\omega^{\prime}}{\sqrt{\omega\omega^{\prime}}}\gamma_{ \zeta}^{*}(\omega)\gamma_{\zeta}(\omega^{\prime})\!\int_{k,k^{\prime}}\!\! \hat{\delta}(k\cdot u_{1}-\omega)\\ &\times\hat{\delta}(k^{\prime}\!\cdot u_{1}-\omega^{\prime})\,_{-h }Y_{j,m}^{*}(k;u_{1})\,_{-h}Y_{j,m}(k^{\prime};u_{1})\!\int\!\hat{d}^{4}p_{ \rm a}\,\hat{\delta}^{+}(p_{\rm a}^{2}-M_{1}^{2}-k^{\prime}\!\cdot\!k/2)|\psi_{ \xi}(p_{\rm a})|^{2}\\ &\times\hat{\delta}(2p_{\rm a}\!\cdot k-2p_{\rm a}\!\cdot k^{ \prime})\hat{\delta}(M_{1}^{2}+2p_{\rm a}\!\cdot k+k^{\prime}\!\cdot k-M_{2}^{2 })\,\mathcal{A}^{*\{b\}}(p_{\rm a}\!+\!\frac{k+k^{\prime}}{2},s_{2}|p_{\rm a} \!+\!\frac{k^{\prime}-k}{2};k,h)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times \mathcal{A}_{\{b\}}(p_{\rm a}\!+\!\frac{k+k^{\prime}}{2},s_{2}|p_{\rm a}\!+\! \frac{k-k^{\prime}}{2};k^{\prime},h),\end{split} \tag{3.35}\]
where we have also used the convenient property \(\psi_{\xi}^{*}(p_{\rm a}\!-\!\frac{q}{2})\psi_{\xi}(p_{\rm a}\!+\!\frac{q}{2}) =|\psi_{\xi}(p_{\rm a})|^{2}\) of the momentum wavepackets (3.8).
### Absorption cross-section in classical limit
So far no classical limit was taken, and eq. (3.35) still represents a quantum probability. To rectify that, we send \(\xi\to 0\) and evaluate the integral over \(p_{\rm a}\), which in the presence of the squared wavefunction \(|\psi_{\xi}(p_{\rm a})|^{2}\) and the mass-shell delta function has the effect of setting the momentum \(p_{\rm a}^{\mu}\) to its classical value \(u_{1}^{\mu}\sqrt{M_{1}^{2}+k^{\prime}\!\cdot k/2}=:M_{\rm a}u_{1}^{\mu}\). Subsequently, using the delta function \(\hat{\delta}(2p_{\rm a}\!\cdot k-2p_{\rm a}\!\cdot k^{\prime})\) becomes \(\hat{\delta}(\omega-\omega^{\prime})/(2M_{\rm a})\), which removes the integration over \(\omega^{\prime}\). In the integral over the remaining \(\omega\), we send \(\zeta\to 0\), so the squared wavefunction \(|\gamma_{\zeta}(\omega)|^{2}\) localizes it at the classical value \(\omega_{\rm cl}\). In this way, the above probability becomes
\[\begin{split}&\lim_{\zeta\to 0}\lim_{\xi\to 0}P_{\rm inc}^{\rm LO}( \omega_{\rm cl},j,m,h)=\frac{16\pi^{3}}{\omega_{\rm cl}}\sum_{s_{2}}\!\int_{k,k ^{\prime}}\!\frac{1}{2M_{\rm a}}\rho_{s_{2}}(M_{1}^{2}+2M_{\rm a}\omega_{\rm cl }+k^{\prime}\!\cdot k)\hat{\delta}(k\cdot u_{1}-\omega_{\rm cl})\\ &\times\hat{\delta}(k^{\prime}\!\cdot u_{1}-\omega_{\rm cl})\,_{-h }Y_{j,m}^{*}(k;u_{1})\,_{-h}Y_{j,m}(k^{\prime};u_{1})\,\mathcal{A}^{*\{b\}}(p_{ \rm a}\!+\!\frac{k+k^{\prime}}{2},s_{2}|p_{\rm a}\!+\!\frac{k^{\prime}-k}{2};k,h )\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\mathcal{A}_{ \{b\}}(p_{\rm a}\!+\!\frac{k+k^{\prime}}{2},s_{2}|p_{\rm a}\!+\!\frac{k-k^{ \prime}}{2};k^{\prime},h)\big{|}_{p_{\rm a}=M_{\rm a}u_{1}},\end{split} \tag{3.36}\]
where we have also taken the integral over \(M_{2}^{2}\) using \(\hat{\delta}(M_{1}^{2}+2M_{\rm a}\omega+k^{\prime}\!\cdot k-M_{2}^{2})\).
Even though we have simplified the probability expression considerably, the integrals over \(k^{\mu}\) and \(k^{\prime\mu}\) are still intertwined, in particular because the spectral density and \(M_{\rm a}\) both depend on \(k\cdot k^{\prime}\). Note, however, that the two massless momenta are constrained to have the energy projection \(\omega_{\rm cl}\), so \(|k\cdot k^{\prime}|\leq 2\omega_{\rm cl}^{2}\), as most easily seen in the rest frame of \(u_{1}^{\mu}\). The basic classical-limit assumption \(\omega_{\rm cl}\ll M_{1}\) then implies
\[|k^{\mu}|,|k^{\prime\mu}|\ \ll\ M_{1}\qquad\Rightarrow\qquad|k\cdot k^{\prime}| \ \ll\ M_{1}u_{1}\!\cdot k=M_{1}u_{1}\!\cdot k^{\prime}=M_{1}\omega_{\rm cl}. \tag{3.37}\]
Therefore, we may define the classical limit of the above probability as
\[\begin{split} P^{\rm LO}_{\rm inc,\,cl}=\frac{8\pi^{3}}{M_{1} \omega_{\rm cl}}\sum_{s_{2}}\rho_{s_{2}}(M_{1}^{2})&\!\int_{k,k ^{\prime}}\!\hat{\delta}(k\!\cdot\!u_{1}-\omega_{\rm cl})\,_{-h}Y^{*}_{j,m}(k ;u_{1})\,{\cal A}^{*\{b\}}(p_{2},s_{2}|p_{1};k,h)\\ &\times\hat{\delta}(k^{\prime}\!\cdot\!u_{1}-\omega_{\rm cl})\,_{ -h}Y_{j,m}(k^{\prime};u_{1})\,{\cal A}_{\{b\}}(p_{2},s_{2}|p^{\prime}_{1};k^{ \prime},h),\end{split} \tag{3.38}\]
where for brevity we have now used the momenta
\[p_{1}=M_{1}u_{1}+\tfrac{k^{\prime}\!-\!k}{2},\qquad p^{\prime}_{1}=M_{1}u_{1} +\tfrac{k\!-\!k^{\prime}}{2},\qquad p_{2}=M_{1}u_{1}+\tfrac{k\!+\!k^{\prime}}{ 2}=:M_{2}u_{2} \tag{3.39}\]
not as independent integration variables but to denote their classical values. Note that in the expression above, we have already assumed that the outgoing states are described by a sufficiently smooth spectral-density function, which makes sense because our EFT is meant to describe absorption of classical waves of arbitrary frequency (provided it is small). Therefore, \(\rho_{s_{2}}\) can be expanded in \(\omega_{\rm cl}/M_{1}\), for which \(2M_{1}\omega_{\rm cl}\) and \(k^{\prime}\!\cdot k\) provide linear and quadratic terms, respectively, and both may be dropped, leaving only the leading term \(\rho_{s_{2}}(M_{1}^{2})\) in the classical limit.
Let us now deal with the momentum dependence of the amplitudes, which, as we have noticed in eq. (3.21), are proportional to the covariant spin-weighted spherical harmonics \({}_{h}\tilde{Y}_{s_{2},m^{\prime}}(k;u_{2})\), while their prefactors depend on the dimensionless ratio
\[w:=\frac{2p_{1}\!\cdot k}{M_{1}^{2}}\ \simeq\ \frac{2\omega_{\rm cl}}{M_{1}}\ \simeq\ \frac{2p^{\prime}_{1}\!\cdot k^{\prime}}{M_{1}^{2}}=:w^{\prime}. \tag{3.40}\]
Moreover, just as we did in section 3.3, we may boost the time direction \(u_{2}^{\mu}\) of either harmonic to our preferred \(u_{1}^{\mu}\), with their difference now being equal to \((k+k^{\prime})^{\mu}/2\), but the result still being6\({}_{h}\tilde{Y}_{s_{2},m^{\prime}}(k;u_{2})\simeq{}_{h}\tilde{Y}_{s_{2},m^{ \prime}}(k;u_{1})\). Therefore, the squared
amplitude is
\[\begin{split}\mathcal{A}^{*\{b\}}&(p_{2},s_{2}|p_{1};k,h )\mathcal{A}_{\{b\}}(p_{2},s_{2}|p_{1}^{\prime};k^{\prime},h)\\ &\simeq M_{1}^{2}\,|g_{0,0,s_{2}}^{|h|}(w)|^{2}w^{2s_{2}}\!\!\! \sum_{m^{\prime}=-s_{2}}^{s_{2}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
and recently generalized to arbitrary \(j\) in [104; 105; 106]. However, the dynamics of non-spinning BSs under small perturbations date back to Regge and Wheeler [113], who proved linear stability of Schwarzschild BHs. From the point of view of the EFT amplitudes, in which the BH is treated as a particle, the GR results serve as the microscopic computation, to which the effective couplings should be matched.
### Classical absorption cross-section
In the general case of wave of spin \(|h|\) scattering off a spinning BH, the transmission and scattering coefficients are usually obtained by solving the Teukolsky equation [96; 97; 98]. In this work, we focus on the simpler case of non-spinning BHs. Let the Schwarzschild radius be \(r_{\rm S}:=2GM_{1}\) and \(\omega\) the frequency of the classical spin-\(|h|\) wave, which obey \(r_{\rm S}\omega\ll 1\). Then the absorption cross-section is given by [106]8
Footnote 8: We have dropped the prefactor \((2j+1)\) from the expressions in the literature, which comes from summing over \(m=-j,\ldots,j\).
\[\sigma^{\rm Schw}_{\rm abs}(\omega,j,m,h)=\frac{(-1)^{h}2\pi}{\omega^{2}} \frac{(j+h)!(j-h)!}{(2j)!(2j+1)!}(2r_{\rm S}\omega)^{2j+1}{\rm Im}F^{\rm Schw} _{-hjh}(\omega). \tag{4.1}\]
Here \(F^{\rm Schw}_{hjm}\) is the harmonic near-zone response function
\[F^{\rm Schw}_{hjm}(\omega)=i(-1)^{h}\,r_{\rm S}\omega\,\frac{(j+h)!(j-h)!}{(2 j)!(2j+1)!}\prod_{l=1}^{j}\big{[}l^{2}+(2r_{\rm S}\omega)^{2}\big{]}, \tag{4.2}\]
which does not depend on the quantum number \(m\), since we wrote it for a non-spinning black hole. We have followed the GR literature [104; 106] in writing the cross-section (4.1) using the response function so as to point out that it is the latter that contains the expansion in \(\omega\), whereas the outside power of \(\omega\) is fixed to be \(2j-1\), combined from the \(\pi/\omega^{2}\) dimensionful prefactor and \(2j+1\) powers of a dimensionless frequency combination. This factorization mimics the structure of the corresponding EFT cross-section (3.46), that we are going to match to next.
Our focus, however, is on the leading powers in \(\omega\) for each \(j\), which amounts to replacing the complicated product in the response function (4.2) by \((j!)^{2}\). We obtain
\[\sigma^{\rm Schw}_{\rm abs,\,LO}(\omega,j,m,h)=4\pi r_{\rm S}^{2}\left[\frac {j!(j+h)!(j-h)!}{(2j)!(2j+1)!}\right]^{2}(2r_{\rm S}\omega)^{2j}. \tag{4.3}\]
where of course \(|m|,|h|\leq j\), and otherwise it vanishes.
### Scales and effective couplings
In order to properly compare the classical and EFT results, it is helpful to restore \(\hbar\) (while leaving \(c=1\) throughout this paper). This introduces the distinction between frequencies/lengths and masses:
\[[\hbar]=L\times M,\quad[M_{1}]=[\omega_{\rm cl}]=M,\quad[\omega]=L^{-1}, \quad[r_{\rm S}]=L,\quad[G]=L\times M^{-1}, \tag{4.4}\]
where we have insisted on the new meaning of \(\omega:=\omega_{\rm cl}/\hbar\) as the wave frequency. We should also multiply the right-hand side of the cross-section given from (3.46) by \(\hbar^{2}\), so as to switch its dimensionality from \(M^{-2}\) to \(L^{2}\), which gives
\[\sigma^{\rm LO}_{\rm inc,\,cl}(\omega,j,m,h)=\frac{\pi}{4\omega^{2}}\frac{(j+h)!(j-h)!}{(2j+1)!}M_{1}^{2}\rho_{j}(M_{1}^{2})\,|g^{|h|}_{0,0,j}(\omega)|^{2} \!\left(\!\frac{2\hbar\omega}{M_{1}}\!\right)^{2j+1}, \tag{4.5}\]
Here we have left the effective couplings \(g_{0,0,s_{2}}(\omega)\) fully dimensionless. Note, however, that in view of the presence of multiple scales, they are now allowed to depend on \(\omega\) through more than just the \(\hbar\omega/M_{1}\) ratio.
Now let us discuss the two basic assumptions underlying the EFT- and GR-based computations, i.e. \(\hbar\omega\ll M_{1}\) and \(r_{\rm S}\omega\ll 1\). The point is that the latter is much stronger than the former, as the Schwarzschild radius must of course be assumed to be many orders of magnitude larger than the Compton wavelength of the black hole:
\[\omega\ \ll\ \frac{1}{r_{\rm S}}\ \ll\ \frac{1}{\lambda_{\rm C}}:=\frac{M_{1} }{2\pi\hbar}, \tag{4.6}\]
otherwise we would be in the realm of quantum gravity and not GR. It is then clear that in the context of comparing the classical and amplitude-based results, which both constitute frequency expansions, we should then retain only the leading order in \(\hbar\omega/M_{1}\), but classical frequency dependence may still be present in the form of \(r_{\rm S}\omega\).
Therefore, matching the leading-order cross-sections (4.3) and (4.5) directly, we obtain
\[M_{1}^{2}\rho_{j}(M_{1}^{2})\,|g^{|h|}_{0,0,j}(\omega)|^{2}=\frac{8[j!]^{2}(j+h )!(j-h)!}{[(2j)!]^{2}(2j+1)!}\!\left(\!\frac{M_{1}r_{\rm S}}{\hbar}\!\right)^{ 2j+1}\!r_{\rm S}\omega. \tag{4.7}\]
It is perhaps more aesthetically pleasing to rephrase this relationship in terms of the classical response function:
\[M_{1}^{2}\rho_{j}(M_{1}^{2})\,|g^{|h|}_{0,0,j}(\omega)|^{2}=\frac{8(-1)^{h}}{ (2j)!}\!\left(\!\frac{M_{1}r_{\rm S}}{\hbar}\!\right)^{2j+1}\!\mathrm{Im}F^{ \rm Schw}_{-hjh,\,{\rm LO}}(\omega). \tag{4.8}\]
In other words, we have related the \(j\)-th effective absorptive coupling squared to the imaginary part of the response function, resembling a dispersion relation. It might seem awkward to keep \(\hbar\) in the now classically meaningful cross-section expression (4.5), as well as eqs. (4.7) and (4.8). However, the effective couplings are a priori arbitrary, and we are free to make convenient modelling assumptions about them, so nothing prevents us from absorbing the Planck constants into them as9
Footnote 9: Recalling the form of the three-point amplitude (2.15), we see that the effective-coupling rescaling (4.9) amounts to replacing massless momenta \(k^{\mu}\) with wavevectors \(\bar{k}^{\mu}:=k^{\mu}/\hbar\), which is commonplace in the KMOC formalism [48], plus an additional overall \(\hbar^{-1/2}\).
\[\bar{g}^{|h|}_{0,0,s_{2}}(\omega):=\hbar^{s_{2}+1/2}g^{|h|}_{0,0,s_{2}}(\omega). \tag{4.9}\]
Comparing the macroscopic and microscopic formulae (4.1) and (4.5), there are a number of things to observe.
* Both cross-sections are consistent in that neither depends on the magnetic quantum number \(m\) of the spherical wave.
* The EFT cross-section (4.5) reproduces the static limit \(\sigma^{\text{LO}}_{\text{inc,cl}}(\omega\!=\!0,j,m,h)=0\) for electromagnetism and gravity (\(|h|=1\) and \(2\), respectively) because of the locality assumption (2.21) that the Wilson coefficients have no negative powers of \(\omega\). This can be considered as an EFT prediction, i.e. it holds prior to the matching of the three-point couplings.
* As previously mentioned, the growth of the superficial leading power in \(\omega\) with \(j\) is the same in both cross-sections, where by superficial we mean excluding the \(\omega\) dependence in the response function and the three-point couplings. In other words, the matching (4.7) contains that same leading power of \(\omega\) for any \(j\), and the cleaner matching (4.8) between the response functions and the three-point couplings does not involve \(\omega\) at all.
* In the EFT cross-section (4.5), every three-point coupling \(|g^{|h|}_{0,0,s_{2}}(\omega)|^{2}\) comes accompanied by the dimensionless combination \(M_{1}^{2}\rho_{s_{2}}(M_{1}^{2})\) involving the spectral density. Its appearance is very sensible from the QFT point of view, as the probability that a massive particle absorbs a lower-energy massless boson is necessarily proportional to the number of possible resulting states with nearly identical mass. However, since it always accompanies the couplings, one may regard the complete expression \(M_{1}\sqrt{\rho_{s_{2}}(M_{1}^{2})}\,g^{|h|}_{0,0,s_{2}}(\omega)\) as a kind of effective coupling. Alternatively, if one's focus is on modeling classical effects that are guaranteed to be insensitive to the difference between spectral densities for different masses and spins, one could consider disregarding the normalization constraint (3.30) altogether and make a modeling assumption \(\rho_{s_{2}}(M_{1}^{2})=1/M_{1}^{2}\).
* Perhaps most importantly, we observe that the matching (4.7) means that \[g^{|h|}_{0,0,s_{2}}(\omega)=\mathcal{O}(G^{s_{2}+1}),\] (4.10) in the post-Minkowskian expansion, since \(r_{\text{S}}=GM\). In other words, the amplitude that the scalar particle which models a Schwarzschild black hole absorbs a spherical wave with total angular momentum \(j\) is a \((j\!+\!1)\)-PM object.
* For gravity (\(|h|=2\)), the PM behavior (4.10) means that the Wilson coefficient starts at \(s_{2}=2\) and scales as \(\mathcal{O}(G^{3})\), whereas the resulting leading absorption cross-section is at 6PM for a \(j=2\) spherical wave, and higher harmonics are suppressed in the PM expansion.
In view of the classical cross-section (4.1) being a polynomial in \(\omega\) spanned by \(\{\omega^{2j},\ldots,\omega^{4j}\}\), one might hope that higher orders in \(r_{\text{S}}\omega\) could be retained, as long as they are captured by the response function (4.2) in a perturbation scheme [106] that is consistent classically. Unfortunately, this is not the case in the present three-point
setup, because going to higher orders requires a more subtle matching. Indeed, the higher orders in \(r_{\rm S}\omega\) in the EFT cross-sections(4.5) are subject to interference from higher-multiplicity amplitudes. More specifically, the next order in the cross-section is \(\mathcal{O}(G^{2j+4})\), for which the EFT treatment must, for instance, include amplitudes with two additional conservative couplings to the graviton, each \(\mathcal{O}(\!\sqrt{G})\). Furthermore, double-graviton absorption or even the mass-changing contact terms contribution to the Compton amplitude might contribute to this matching. These matters will be further discussed in sections 5.4 and 5.5.
Improving this result to spinning objects is another story. In the non-spinning case, the coupling constant \(G\) only enters in the Schwarzschild radius \(r_{\rm S}\), whereas in the Kerr case where the dimensionless spin ratio \(a_{*}=a/GM\) also contains negative powers in \(G\). This shows that for Schwarzschild black holes, the first contribution to such amplitudes is at 6PM (as can be reproduced by off-shell EFT methods [88; 89]), while it comes at a lower order for Kerr black holes due to the negative power of \(G\) in \(a_{*}\). For instance, the authors of [92] consider four-point contact interactions where such effects come at spin-5 in \(\mathcal{O}(G)\) amplitudes. Nevertheless, the general formalism presented in this paper does allow to go to higher orders in spin, and we leave this for future work.
In this purely on-shell approach, we have modelled the absorption effects by allowing a changing-mass amplitude from \(s_{1}=0\) to a spinning degree of freedom and the leading order corresponds to a \(s_{2}=2\) particle. We have observed some similarities with the worldline EFT approach [88; 89; 114], where the point-particle action coupled to the Weyl tensor is not enough to model absorption. One then has to introduce electric and magnetic composite operators \(Q^{E}_{ab}\) and \(Q^{B}_{ab}\) representing new degrees of freedom, which carry two indices and couple to electric and magnetic components of the Weyl tensor \(E^{ab}\) and \(B^{ab}\), respectively. While in our approach higher orders require considering \(s_{2}\geq 2\) particles and higher-multiplicity amplitudes, on the worldline higher-derivative operators acting on the Weyl tensor and multi-index composite operators are needed to improve the calculation beyond \(\omega^{4}\), which is explored e.g. in [92].
## 5 Coherent-state cross-section
A proper description of the interaction between a gravitational wave and a compact object using scattering amplitudes requires the use of a coherent-state formalism to model the incoming and outgoing wave [51; 91; 115]. In section 3, we have circumvented it by using a single-graviton state with a wavefunction peaked at the classical frequency \(\omega_{\rm cl}\). The point of this section is two-fold:
* substantiate the leading-order calculation via the coherent-state framework,
* explain how higher-order calculations may be done in a similar fashion.
of an observable-based one [48]. We start with a quantum description and make gradual assumptions relevant to the classical limit.
### Elastic-inelastic separation
The initial state for our absorption process consists of a heavy non-spinning particle \(|\psi_{1}\rangle\) and a wave of helicity \(h\) modeled by a massless coherent state \(|\gamma^{h}\rangle\).
\[|\text{in}\rangle:=|\psi_{1};\gamma^{h}\rangle=\int_{p_{1}}\psi_{\xi}(p_{1})e^{ ib\cdot p_{1}/\hbar}|p_{1};\gamma^{h}\rangle, \tag{5.1}\]
where the relativistic momentum-space wavefunction \(\psi_{\xi}(p_{1})\) peaks at the classical momenta \(p_{1,\text{cl}}^{\mu}=M_{1}u_{1}^{\mu}\), as discussed in section 3. We have also allowed for an impact parameter. For the final state, we should distinguish two cases:
* a different coherent state \(|\tilde{\gamma}^{\tilde{h}}\rangle\), but the heavy particle's mass is preserved;
* a different coherent state \(|\tilde{\gamma}^{\tilde{h}}\rangle\) and an unspecified particle \(|X\rangle\) with \(M_{2}\neq M_{1}\).
The two cases are depicted in figure 2, and we need to integrate over the possible final states. Despite these assumptions, the formalism easily allows for initial spinning states, and we delay the specification of the massless coherent-state type (plane-wave or partial-wave) to later on. It is also worth commenting that even though case (c) has the same mass as the initial state, intermediate mass transitions are allowed (e.g. Compton scattering with different masses in the factorization channels).
The need to separate these two cases on the quantum side comes from the discontinuous nature of basic scattering-amplitude building blocks at \(M_{2}=M_{1}\), as discussed in section 2, and on the classical side from the usual separation between conservative and non-conservative effects. The total probability will then include the following mass-preserving and mass-changing probabilities
\[P_{\gamma\rightarrow\tilde{\gamma}}=P_{\gamma\rightarrow\tilde{\gamma}}^{( \text{c})}+P_{\gamma\rightarrow\tilde{\gamma}}^{(\text{nc})}. \tag{5.2}\]
Figure 2: Gravitational diagrams in a non-spinning black-hole-wave interaction
For the first one, we may write
\[\begin{split} P^{\rm(c)}_{\gamma\to\tilde{\gamma}}&=\sum_ {2s_{2}=0}^{\infty}\sum_{b_{1},\ldots,b_{2s_{2}}=1,2}\,\int_{p_{2}}\langle{\rm in }|S^{\dagger}|p_{2},s_{2},\{b\};\tilde{\gamma}^{\tilde{h}}\rangle\langle p_{2},s_{2},\{b\};\tilde{\gamma}^{\tilde{h}}|S|{\rm in}\rangle\\ &=\int_{p_{2}}\!\int\!\frac{d^{4}\beta}{\pi^{2}}\langle{\rm in}|S^ {\dagger}|p_{2},\beta;\tilde{\gamma}^{\tilde{h}}\rangle\langle p_{2},\beta; \tilde{\gamma}^{\tilde{h}}|S|{\rm in}\rangle.\end{split} \tag{5.3}\]
where in the second line we have used the coherent-spin states mentioned around eq. (2.4). They rely on Schwinger's construction [116] for massive spin states, which are obtained from the zero-spin state by acting with two kinds of creation operators distinguished by an SU(2) index, see [52] for more details. As long as the integration over the SU(2) spinors \(\beta_{b}\) appears in the final-state summation, one may regard and use it as a shorthand for the bulkier spin sum.
In the second case, we are interested in the probability of all different configurations \(X\) involving a heavy particle of mass \(M_{2}\neq M_{1}\):
\[P^{\rm(nc)}_{\gamma\to\tilde{\gamma}}=\sum_{X\ni M_{2}\neq M_{1}}\!\!|\langle X ;\tilde{\gamma}^{\tilde{h}}|S|{\rm in}\rangle|^{2}=\sum_{X\ni M_{2}\neq M_{1} }\langle{\rm in}|S^{\dagger}|X;\tilde{\gamma}^{\tilde{h}}\rangle\langle X; \tilde{\gamma}^{\tilde{h}}|S|{\rm in}\rangle. \tag{5.4}\]
The crucial point now is to determine what part of the Hilbert space contributes to the problem at hand. We are going to assume that all relevant configurations contain only one heavy particle; in other words, in the classical limit no new black holes are created in this \(S\)-matrix evolution. Let us also exclude decay of the heavy particle, i.e. black-hole evaporation, from current consideration. In other words, we assume that the spectral density of the heavy-particle states has a non-trivial continuous part only for \(M_{2}>M_{1}\) (alongside the delta-function responsible for case (c)):10
Footnote 10: In the classical limit, the SU(2) spinors \(\beta_{b}\) determine the resulting classical angular momentum of the compact object [52], so one could trade the \(s_{2}\)-dependence of the spectral density for the perhaps more appropriate dependence on \(\hbar\|\beta\|^{2}=2\sqrt{-S_{\rm cl}^{2}}\) and use the coherent-spin final-state integration, as shown in eq. (5.3). Modifying the subsequent formulae in this way is straightforward.
\[1^{\rm(nc)}=\sum_{X_{\rm rad}}\sum_{s_{2}}\sum_{\{b\}}\int_{M_{1}^{2}}^{\infty }\!\!dM_{2}^{2}\rho_{s_{2}}(M_{2}^{2})\int_{p_{2}}|p_{2},s_{2},\{b\};X_{\rm rad }\rangle\langle p_{2},s_{2},\{b\};X_{\rm rad}|. \tag{5.5}\]
The above "completeness" relation should normally also include a sum over possible emitted radiation
\[|X_{\rm rad}\rangle\langle X_{\rm rad}|=\sum_{n=0}^{\infty}\sum_{h_{1},\cdots, h_{n}}\int_{k_{1},\cdots,k_{n}}|k_{1}^{h_{1}};\cdots;k_{n}^{h_{n}}\rangle \langle k_{1}^{h_{1}};\cdots;k_{n}^{h_{n}}|. \tag{5.6}\]
However, we choose to make another assumption that all the outgoing radiation belongs coherently to the wave \(\tilde{\gamma}\), and there is no extra scattered photons/gravitons. In other words, the final state is given by \(|p_{2},\beta;\tilde{\gamma}^{\tilde{h}}\rangle\) and not \(|p_{2},\beta;\tilde{\gamma}^{\tilde{h}},k_{1}^{h_{1}};k_{2}^{h_{2}};\cdots\rangle\),
which was also assumed for the mass-preserving case (5.3). This assumption relies on the expectation that radiated quanta are not classically significant unless they belong to a classical wave modeled by a coherent state, see e.g. [53]. Therefore, remembering the meaning of the incoming state, we can write the absorption probability as
\[P^{\rm(nc)}_{\gamma\rightarrow\tilde{\gamma}}= \int_{p_{1},p^{\prime}_{1}}\!\!\psi^{*}_{\xi}(p_{1})\psi_{\xi}(p^{ \prime}_{1})e^{ib\cdot(p^{\prime}_{1}-p_{1})} \tag{5.7}\] \[\times\sum_{s_{2}}\int_{M_{1}^{2}}^{\infty}\!\!dM_{2}^{2}\rho_{s_ {2}}(M_{2}^{2})\int_{p_{2}}\!\sum_{\{b\}}\langle p_{1};\gamma^{h}|S^{\dagger}| p_{2},s_{2},\{b\};\tilde{\gamma}^{\tilde{h}}\rangle\langle p_{2},s_{2},\{b\}; \tilde{\gamma}^{\tilde{h}}|S|p^{\prime}_{1};\gamma^{h}\rangle.\]
The building block \(\langle p_{2},s_{2},\{b\};\tilde{\gamma}^{\tilde{h}}|S|p_{1};\gamma^{h}\rangle\) involves a transition of a scalar heavy state into a possibly spinning one along with the incoming and outgoing massless coherent states. Since the latter states contain an infinite number of photons/gravitons, the matrix elements of \(S=1+iT\) should be expanded in perturbation theory.
### \(T\)-matrix perturbative expansion
The massless coherent states (plane or spherical) are sensitive to all orders in perturbation theory, and their matrix elements are non-trivial [51]. However, we can expand operators in terms of annihilation and creation operators, plane or spherical. We are going to perform the \(T\)-matrix expansion in the following way:11
Footnote 11: We thank Donal O’Connell for valuable discussions on the expansion (5.8).
\[T= \sum_{m,n=0}^{\infty}\left(T^{\rm(c)}_{(m|n)}+T^{\rm(nc)}_{(m|n)} \right)=T^{\rm(nc)}_{(0|1)}+T^{\rm(nc)}_{(1|0)}\] \[+T^{\rm(c)}_{(1|1)}+T^{\rm(c)}_{(0|2)}+T^{\rm(c)}_{(2|0)}+T^{\rm (nc)}_{(1|1)}+T^{\rm(nc)}_{(0|2)}+T^{\rm(nc)}_{(2|0)}+\cdots,\]
where the superscripts (c) and (nc) represent mass-preserving and mass-changing elements, respectively, while the subscript \((m|n)\) corresponds to \(n\) incoming and \(m\) outgoing photons/gravitons, and each \(T\)-matrix element will generate an \((m+n+2)\)-point amplitude. In the first line of eq. (5.8), we have isolated the leading non-conservative effects due to absorption, \(T^{\rm(nc)}_{(0|1)}\), and emission, \(T^{\rm(nc)}_{(1|0)}\). Both terms are mass-changing three-point amplitudes and non-zero even on real kinematics, while the mass-preserving counterparts vanish, \(T^{\rm(c)}_{(1|0)}=T^{\rm(c)}_{(0|1)}=0\).12 In this paper, we have been studying the leading-order in absorption term \(T^{\rm(nc)}_{(0|1)}\), but the above expansion allows to also systematically understand higher orders.
Footnote 12: See [117] for a discussion of large gauge effects, where such amplitudes contribute.
In the second line, we have four-point terms that lead to the usual conservative Compton amplitude \(T^{\rm(c)}_{(1|1)}\) and its non-conservative counterpart \(T^{\rm(nc)}_{(1|1)}\). The former has been vastly studied recently, but the latter has been unexplored to the best of our knowledge. Furthermore, we have double-emission \((2|0)\) and double-absorption
(\(0|2\)) both on the conservative and non-conservative sides. Together with the non-conservative Compton, double-absorption would give the naive next-to-leading order (NLO) terms to our leading-order analysis.
The \(T\)-matrix elements can be written in terms of scattering amplitudes:
\[\begin{split} T_{(m|n)}=&\sum_{2s_{1},2s_{2}=0}^{\infty} \int_{p_{1},p_{2}}\!\!\!\!\!\!\sum_{\begin{subarray}{c}h_{1},\dots,h_{n}\\ \tilde{h}_{1},\dots,\tilde{h}_{m}\end{subarray}}\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
Setting \(\langle\gamma^{h}|\gamma^{h}\rangle=1\) gives the normalization prefactor as
\[{\cal N}_{\gamma}=\exp\biggl{[}-\frac{1}{2}\sum_{j,m}\!\int_{0}^{\infty}\!\!\hat{ d}\omega\,|\gamma_{j,m}(\omega)|^{2}\biggr{]}. \tag{111}\]
The waveshape \(\gamma_{j,m}(\omega)\) of these coherent states describes the contribution of each \((j,m)\) component to the total wave, and we expect that in the classical limit \(\gamma_{j,m}(\omega)\) is peaked at the frequency \(\omega_{\rm cl}\). We can simplify the problem further by studying the incoming wave \(|\gamma_{j,m}^{h}\rangle\) with just a particular \((j,m)\) component, in which case the spherical waveshape reduces to \(\gamma_{j^{\prime},m^{\prime}}(\omega)=\delta_{j^{\prime}}^{j}\delta_{m^{ \prime}}^{m^{\prime}}\gamma(\omega)\), such that
\[a_{j,m,h}(\omega)|\gamma_{j^{\prime},m^{\prime}}^{h^{\prime}}\rangle=\gamma( \omega)\delta_{j^{\prime}}^{j^{\prime}}\delta_{m}^{m^{\prime}}\delta_{h}^{h^ {\prime}}|\gamma_{j,m}^{h}\rangle. \tag{112}\]
Coming back to the initial state \(|{\rm in}\rangle\) given in eq. (109), which describes a scalar black hole and a partial wave as a wavepacket superposition of \(|p_{1};\gamma_{j,m}^{h}\rangle\). The \(S\)-matrix determines the probability amplitude of its evolution into a final massive state \(X\) and another partial wave \(|\tilde{\gamma}^{\tilde{h}}\rangle\) with perhaps more than one \((\tilde{j},\tilde{m})\) components. Let us write the leading absorption term \(T_{(0|1)}^{({\rm nc})}\) to such a process, by switching the states on the left-hand side of eq. (100) from plane to spherical waves:
\[\langle p_{2},s_{2},\{b\};\tilde{\gamma}^{\tilde{h}}|S|p_{1};\gamma_{j,m}^{h} \rangle\simeq i\!\int_{k}\!\!\hat{\delta}^{4}(p_{1}+k-p_{2}){\cal A}_{\{b\}}(p _{2},s_{2}|p_{1};k,h^{\prime})\langle\tilde{\gamma}^{\tilde{h}}|a_{h^{\prime} }(k)|\gamma_{j,m}^{h}\rangle. \tag{113}\]
The main difference is that to evaluate the matrix element of a plane-wave annihilation operator between two spherical coherent states, we need to summon the decomposition of the plane-wave operator into partial waves:
\[a_{h}(k)=4\pi\sum_{j=|h|}^{\infty}\sum_{m=-j}^{j}\int_{0}^{\infty}\!\frac{\hat {d}\omega}{\sqrt{2\omega}}\hat{\delta}(k\cdot u_{1}-\omega)\,_{-h}Y_{j,m}(k; u_{1})a_{j,m,h}(\omega), \tag{114}\]
and hence
\[a_{h^{\prime}}(k)|\gamma_{j,m}^{h}\rangle=\frac{4\pi\delta_{h^{\prime}}^{h}} {\sqrt{2k\cdot u_{1}}}\gamma_{j,m}(k\!\cdot\!u_{1})\,_{-h}Y_{j,m}(k;u_{1})| \gamma_{j,m}^{h}\rangle. \tag{115}\]
Therefore, we compute the leading mass-changing matrix element as
\[\langle p_{2},s_{2},\{b\};\tilde{\gamma}^{\tilde{h}}|S|p_{1};\gamma_{j,m}^{h }\rangle\simeq 4\pi i\langle\tilde{\gamma}^{\tilde{h}}|\gamma_{j,m}^{h} \rangle\!\int_{0}^{\infty}\!\frac{\hat{d}\omega}{\sqrt{2\omega}}\gamma_{j,m}(\omega) \tag{116}\]
\[\times\!\int_{k}\!\!\hat{\delta}(k\cdot u_{1}-\omega)\,_{-h}Y_{j,m}(k;u_{1}) \hat{\delta}^{4}(p_{1}+k-p_{2}){\cal A}_{\{b\}}(p_{2},s_{2}|p_{1};k,h).\]
The leading contribution to the absorption probability (113) is then given by
\[P_{\gamma\to\tilde{\gamma}}^{({\rm nc})}\simeq 8\pi^{2}\big{|} \langle\tilde{\gamma}^{h}|\gamma_{j,m}^{h}\rangle\big{|}^{2}\sum_{s_{2}}\!\int _{M_{1}^{2}}^{\infty}\!\!dM_{2}^{2}\rho_{s_{2}}(M_{2}^{2})\!\int_{p_{1},p_{1} ^{\prime},k,k^{\prime},p_{2}}\!\psi_{\xi}^{*}(p_{1})\psi_{\xi}(p_{1}^{\prime} )e^{ib\cdot(p_{1}^{\prime}-p_{1})} \tag{117}\] \[\times\!\int_{0}^{\infty}\!\frac{\hat{d}\omega\hat{d}\omega^{ \prime}}{\sqrt{\omega\omega^{\prime}}}\gamma^{*}(\omega)\gamma(\omega^{\prime}) \hat{\delta}(k\cdot u_{1}\!-\!\omega)\hat{\delta}(k^{\prime}\cdot u_{1}\!-\! \omega^{\prime})\hat{\delta}^{4}(p_{1}+k-p_{2})\hat{\delta}^{4}(p_{1}^{ \prime}\!+k^{\prime}\!-p_{2})\] \[\times\,_{-h}Y_{j,m}^{*}(k;u_{1})\,_{-h}Y_{j,m}(k^{\prime};u_{1}) \,{\cal A}^{*\{b\}}(p_{2},s_{2}|p_{1};k,h)\,{\cal A}_{\{b\}}(p_{2},s_{2}|p_{1} ^{\prime};k^{\prime},h).\]
Note that apart from the overlap between the two spherical coherent states and the impact-parameter exponent, we have landed exactly on the single-quantum absorption cross-section given in eqs. (3.31) and (3.33) -- with the \((j,m)\) waveshape \(\gamma(\omega)\) as the single-particle energy wavefunction. In other words, we observe that the waveshape \(\gamma(\omega)\) acts as a one-dimensional wavefunction, which smears the energy spectrum but is peaked at the classical frequency \(\omega_{\text{cl}}\). This observation was also made in [91], where single quanta and coherent states gave the same results.
Regarding the seeming discrepancies between the leading-order cross-sections (3.31) and (5.19), for a spherical wave defined in the rest-frame of (the classical momentum of) the compact body and centered at it, the impact parameter should of course be set to zero. Moreover, eqs. (3.31) and (3.33) were written for an inclusive probability, let us rename it to \(P_{(0|1)}^{\text{(nc)}}:=P_{\text{inc}}^{\text{LO}}(\omega_{\text{cl}},j,m,h)\), whereas retaining the dependence on the outgoing waveshape in eq. (5.19) is actually an enhancement of the single-quantum formulae:
\[P_{\gamma\rightarrow\tilde{\gamma}}^{\text{(nc)}}=\big{|}\langle\tilde{ \gamma}^{\tilde{h}}|\gamma_{j,m}^{h}\rangle\big{|}^{2}P_{(0|1)}^{\text{(nc)}}+ \ldots, \tag{5.20}\]
where the dots denote the higher-orders to be briefly discussed below. In the limit where the outgoing classical wave changes very little, the above prefactor may furthermore disappear, \(\langle\tilde{\gamma}^{\tilde{h}}|\gamma_{j,m}^{h}\rangle\approx 1\).
### Higher-order diagrammatics
In this section, we use diagrams to help us understand all the effects relevant for BH-wave interactions. Having a diagrammatic realization of the expressions from the previous sections will guide us for the NLO corrections. However, this diagrammatic approach is general enough to be also applicable to any order in perturbation theory, as well as such processes as emitted radiation and superradiance.
Figure 3: \(T\)-matrix operator expansion
Let us take a brief moment to explain the diagrammatic expansion of \(T\)-matrix in figure 3, which represents eq. (100). The operator nature of this diagram is represented by the "vertical line" after the wavy graviton line, and the double lines, which will "act" on a ket quantum state, e.g. the massless coherent state \(|\gamma^{h}\rangle\) or the black-hole \(|p_{1}\rangle\). In this diagram, we then have
* \(n\) incoming graviton annihilation operators shown by wavy lines and labeled by \(\{k_{1},\cdots,k_{n}\}\);
* \(m\) outgoing graviton creation operators shown by wavy lines and labeled by \(\{\tilde{k}_{1},\cdots,\tilde{k}_{m}\}\);
* incoming and outgoing double line, labeled by \(p_{1}\) and \(p_{2}\). The two lines of different thickness inside of the double line represents the fact that this diagram contains mass-preserving and mass-changing transitions.
* vertical lines at the end of graviton/BH lines represent the operator nature of these diagrams. For instance, the double-line part of the operator will act on \(|p_{1}\rangle\), while the wavy line will act on the coherent state \(|\gamma^{h}\rangle\).
* Evaluating this operator with outgoing states on the left and incoming states on the right will result in scattering amplitudes, waveshapes, and coherent-state overlap. Due to the operator-action convention, time flows from right to left in the resulting amplitude.
Let us now apply these diagrams to the evaluation of the leading-order contribution to absorption given in eq. (104). We take the first term \(T^{\rm(nc)}_{(0|1)}\) on the right-hand side of figure 3 and take its matrix element \(\langle p_{2},s_{2},\{b\};\tilde{\gamma}^{h}|T^{\rm(nc)}_{(0,1)}|p_{1};\gamma ^{h}\rangle\). The result is the overlap between the coherent states, a scattering amplitude, and the waveshape \(\gamma(k)\), represented in figure 4. Note that the integrated scattering amplitude is a single-graviton amplitude smeared by the waveshape.
Similarly, figure 5 shows how this diagrammatic technique applies to the NLO non-conservative contributions. They contains double absorption and the mass-changing Compton amplitude, which both involve two photons/gravitons, now integrated with two waveshapes coming from the coherent states.
Figure 4: \(T^{\rm(nc)}_{(0|1)}\)-matrix operator acting on the quantum states. Time flows right to left.
### PM absorption analysis
In the previous section, we have explained how to include higher orders in multiplicity into the BH-wave interaction modeling by expanding the \(T\)-matrix. The PM expansion, however, enters into the mass-changing amplitudes in a rather intricate way. Indeed, as we have seen from eq. (4.7), even the three-point absorptive amplitudes must behave \(\mathcal{O}(G^{s_{2}+1})\). Let us now explore the mass-changing \((m+n+2)\)-point amplitude \(\mathcal{A}_{\{b\}}{}^{\{a\}}(p_{2},s_{2};\tilde{k}_{1},\tilde{h}_{1};\ldots; \tilde{k}_{m},\tilde{h}_{m}|p_{1},s_{1};k_{1},h_{1};\ldots;k_{n},h_{n})\) in eq. (5.9). For brevity, we compress the notation to \(\mathcal{A}_{\text{abs}(m|n)}^{(s_{2}|s_{1})}\), emphasizing its distinction from the mass-conserving counterparts \(\mathcal{A}_{(m|n)}^{(s_{2}|s_{1})}\). In particular, at three points we have
\[\mathcal{A}_{\text{abs}(0|1)}^{(s_{2},0)}\propto G^{s_{2}+1}, \qquad\quad\mathcal{A}_{\text{3,min}}^{(s)}\propto\sqrt{G}, \tag{5.21}\]
where the second one is the usual three-point same-mass amplitude [95] of the minimal form (2.16), which are known to correspond to Kerr BHs at 1PM [32; 33].
To obtain higher multiplicities, we can now naively multiply the powers of the Newton constant of these three-point amplitudes, assuming that they scale uniformly in \(G\), and any subleading orders at three points should come from higher loop orders.13 At four points, we have two incoming gravitons and a mass-changing heavy particle. We then have three types of contributions: a contact four-point term, two successive three-point absorptions, and one absorption together with one minimal-coupling amplitude. These terms be written respectively as
Footnote 13: See [118] for loop corrections to Love numbers in the worldline EFT framework. For quantum corrections to Love numbers due to emission see [93], which we also ignore in the above analysis.
\[\mathcal{C}_{\text{abs}(0|2)}^{(s_{2},0)}\,+\,\underbrace{\mathcal{A}_{\text {abs}(0|2)+0}^{(s_{2},0)}}_{\propto G^{2s_{2}+2}}\,+\,\underbrace{\mathcal{A}_ {\text{abs}(0|1)+1}^{(s_{2},0)}}_{\propto G^{s_{2}+3/2}}=:\mathcal{A}_{\text{ abs}(0|2)}^{(s_{2},0)}, \tag{5.22}\]
where the subscript notation \((0|r)+n-r\) means that we have \(n\) gravitons, \(r\) out which couple via an absorptive three-point amplitude and \((n-r)\) via the mass-preserving
Figure 5: Next-to-leading order contributions to mass-changing absorption effects
minimal coupling. More generally, for \(n\)-graviton absorption we thus have
\[\mathcal{A}^{(s_{2},0)}_{\text{abs}(0|n)}\!=\sum_{r=1}^{n}\mathcal{A}^{(s_{2},0)}_ {\text{abs}(0|r)+n-r}\,+\,\mathcal{C}^{(s_{2},0)}_{\text{abs}(0|n)},\qquad \mathcal{A}^{(s_{2},0)}_{\text{abs}(0|r)+n-r}\!\propto G^{r(s_{2}+1)+(n-r)/2}. \tag{101}\]
In section 4, we have seen that, on the GR side, the PM expansion of the near-zone response function (100) suggests that the leading-order absorption cross-section scales as \(G^{2j+2}\), whereas the NLO does as \(G^{2j+4}\).14 Now from squaring the amplitudes (101), we see that we obtain terms that scale as \(G^{2j+3}\), \(G^{3j+7/2}\) and \(G^{4j+4}\) for \(s_{2}=j\) (as follows from spin conservation seen in eq. (102)). Therefore, it is not possible to obtain the NLO \(G^{2j+4}\) expected on the GR side from the tree-level counting on the EFT side, unless the contact term is artificially introduced to account for this counting. However, a more natural way to obtain the expected behavior in \(G\) is from the amplitude with three incoming gravitons, which is expanded as
Footnote 14: Tail effects may modify the NLO to \(\mathcal{O}(G^{2j+2})\)[92; 118], but we expect them to arise from loops.
\[\mathcal{A}^{(s_{2},0)}_{\text{abs}(0|3)}=\underbrace{\mathcal{A}^{(s_{2},0)}_ {\text{abs}(0|1)+2}}_{\propto\,G^{s_{2}+2}}+\underbrace{\mathcal{A}^{(s_{2},0) }_{\text{abs}(0|2)+1}}_{\propto\,G^{2s_{2}+5/2}}+\underbrace{\mathcal{A}^{(s_ {2},0)}_{\text{abs}(0|3)+0}}_{\propto\,G^{3s_{2}+3}}+\mathcal{C}^{(s_{2},0)}_ {\text{abs}(0|3)}. \tag{102}\]
Indeed, we see that the first contribution squared induces the desired NLO \(G^{2j+4}\) correction to the absorption cross-section.
## 6 Summary and discussion
In this work, we have initiated exploring classical absorption effects for compact bodies using quantum scattering amplitudes. Central to this program are the mass-changing three-point scattering amplitudes [95; 99] that entail new degrees of freedom modeling non-conservative effects, which may change the mass and spin of the heavy particle (representing the compact object) due to the incoming wave.
We have made use of these amplitudes and their connection to covariantized spin-weighted spherical harmonics to describe leading gravitational absorption effects from a macroscopic/EFT point of view. Since this is an effective description, matching to the underlying theory was required to obtain the values of the EFT coupling coefficients. We have chosen to match at the cross-section level to the GR calculation dating back to Starobinsky, Churilov [80; 81] and Page [82; 83]. Although we have performed a leading-order match, this probability-based formalism can accommodate higher orders in the PM expansion and incoming spinning BHs and neutron stars as well. For the latter case, absorption effects were considered via tidal heating [119; 120], and it would be interesting to understand how the effective couplings \(g_{r,s_{1},s_{2}}\) deviate from the BH values. We leave this for future work.
Having made sense of the effective couplings, we have explored how the used single-quantum framework fits into a more general and consistent description of classical waves using massless coherent states. In particular, we were able to connect the frequency wavefunction used in the former with the coherent-state waveshape, i.e. the eigenvalue of the annihilation operator. An interesting feature of this analysis is the diagrammatic approach for expanding the \(T\)-matrix and systematically introducing higher-order terms in the coherent cross-section. Crucial to this analysis was the separation of the probabilities into conservative and absorptive, which is motivated by the intrinsically distinct nature of the quantum amplitudes building blocks. Although the classical limit sends \(M_{2}\to M_{1}\), the form of resulting cross-section follows from the amplitudes constructed on \(M_{2}\neq M_{1}\) kinematics, which are qualitatively different from their same-mass counterparts.
The natural next step is to include spin effects for the initial black hole with the end goal of modeling a Kerr BH absorption cross-section purely from on-shell amplitudes. According to the microscopic calculation from the GR side, such leading-order non-spinning effects come at \(\mathcal{O}(G^{3})\) at the cross-section level, suggesting that the effective coupling in the amplitude should start at \(\mathcal{O}(G^{3/2})\). From the EFT side, in this more general case of \(s_{1}\neq 0\), we have observed the proliferation of possible effective couplings in the three-point mass-changing amplitude (13), making the matching a harder task. However, the proposed definition of the mass-changing minimal amplitudes (20) might streamline the calculation and perhaps even correspond to the Kerr BH in the same way as the same-mass "minimal coupling" [95] of the form (16) are known to [32; 33].
Another direction that we have not explored is the study of observables from amplitudes, in particular using the KMOC formalism [48; 49; 50; 51; 52; 53]. With the obtained absorption effective coefficients, many interesting local and global observables could be already be explored at leading or higher PM orders using the presented formalism. Perhaps the most interesting ones are the change in mass and spin induced by absorption, where one could naturally use such quantum operators as \(\mathbb{P}^{2}=\mathbb{P}^{\mu}\mathbb{P}_{\mu}\) to obtain \(\Delta M^{2}\) and \(\mathbb{S}^{2}=\mathbb{S}^{\mu}\mathbb{S}_{\mu}\) to obtain \(\Delta S^{2}\). Moreover, one could imagine probing the change in the area of the BH due to absorptive effects. In classical GR, the area is defined as
\[A_{\rm H}:=8\pi(GM)^{2}\biggl{[}1+\sqrt{1-\chi^{2}}\biggr{]},\qquad\chi=\frac{ \mathfrak{a}}{GM}, \tag{171}\]
and \(\mathfrak{a}=S/M\) is the Kerr ring radius. To obtain the change in this quantity from amplitudes, one would like to define a QFT operator for the area and try to compute \(\Delta A_{\rm H}\) in a scattering process. For that, one could substitute \((S^{2},M^{2})\to(\mathbb{S}^{2},\mathbb{P}^{2})\), which imples the following proposal for the area operator:
\[\mathbb{A}_{\rm H}=8\pi\left[G^{2}\,\mathbb{P}^{2}+\sqrt{(G^{2}\,\mathbb{P}^{ 2})^{2}-G^{2}\,\mathbb{S}^{2}}\right], \tag{172}\]
which mixes PM orders. The simplicity of this proposal also comes from the fact that the two operators commute \([\mathbb{S}^{2},\mathbb{P}^{2}]=0\). The mixing between orders in the expansion brings an interesting interplay between the \(\mathbb{S}^{2}\) and the \(\mathbb{P}^{2}\) calculation. We leave the exploration of such an operator for future work.
We hope that this work may open these and other avenues to include absorption effects in the on-shell amplitude approach to gravitational waves. In particular, the work [39] on matching Teukolsky-equation solutions to the gravitational Compton scattering amplitudes suggests that absorption effects could be included into them in relation to horizon effects. It is tempting to consider these effects from a purely on-shell perspective, as the four-point amplitudes are likely to be related to the leading-order absorption cross-section by dispersion relations.
Another direction is to explore in more detail the role of the spectral density function that we were forced to introduce in our formalism. For instance, it would be interesting to see if it appears in a similar way in the context of the Heavy Particle Effective Theory [26; 27], which streamlines the classical limit. We also leave this for future work.
## Acknowledgements
We are grateful to Fabian Bautista, Kays Haddad, Andreas Helset, Yu-tin Huang, Jung-Wook Kim, Nathan Moynihan, Donal O'Connell and M.V.S. Saketh for valuable conversations, especially to Andreas, Donal and Jung-Wook for the comments of an early draft of this paper. RA's research was supported by the F.R.S.-FNRS project no. 40005600 and the FSR Program of UCLouvain.
## Appendix A Spherical harmonics and spinors
Here we discuss the spinorial construction for the spin-weighted spherical harmonics.
Spherical harmonics in 3d.The original construction due to Newman and Penrose [108] may be neatly formulated (see e.g. [121]) in terms of \(\mathrm{SU}(2)\) spinors on the sphere \(S^{2}=\{\hat{\mathbf{k}}=(\cos\varphi\sin\theta,\sin\varphi\sin\theta,\cos\theta) \}\subset\mathbb{R}^{3}\):
\[\kappa^{a}_{+}\!=\!\left(\begin{array}{cc}e^{-\frac{i\varphi}{2}}\!\cos \frac{\theta}{2}\\ e^{\frac{i\varphi}{2}}\!\sin\frac{\theta}{2}\end{array}\right)\!,\ \ \kappa^{a}_{-}\!=\!\left(\begin{array}{cc}-e^{-\frac{i\varphi}{2}}\!\sin \frac{\theta}{2}\\ e^{\frac{i\varphi}{2}}\!\cos\frac{\theta}{2}\end{array}\right)\ \ \Rightarrow\ \ \begin{cases}\hat{\mathbf{k}}\cdot\mathbf{\sigma}^{a}_{\,b}\,\kappa^{b}_{\pm}=\pm \kappa^{a}_{\pm},\\ \hat{k}^{i}\!=-\frac{1}{2}\sigma^{i,a}_{\,b}(\kappa^{a}_{+}\kappa_{-b}+\kappa ^{a}_{-}\kappa_{+b}),\end{cases} \tag{100}\]
where \(\mathbf{\sigma}^{a}{}_{b}\) is the concatenation of the three standard Pauli matrices. We then define
\[{}_{h}\tilde{Y}_{j,m}(\hat{\mathbf{k}}):=\underbrace{\overbrace{\kappa^{(1}_{+} \cdots\kappa^{1}_{+}}^{j-m}\overbrace{\kappa^{2}_{+}\cdots\kappa^{2}_{+}}^{j +m}\underbrace{\kappa^{2}_{-}\cdots\kappa^{2}_{-}}^{j+m}}_{j-h}. \tag{101}\]
Up to normalization, these functions are directly related to the conventional angle-dependent harmonics [109] via the spinor parametrization (A.1):
\[{}_{h}Y_{j,m}(\theta,\varphi):=(-1)^{m}\sqrt{\frac{(2j+1)(j+m)!(j-m)!} {4\pi(j+h)!(j-h)!}}\big{(}\sin\tfrac{\theta}{2}\big{)}^{2j}\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
This is ambiguous for \(\mathbf{p}=0\), so one may choose e.g.
\[p^{\mu}=(M,\mathbf{0})=0\qquad\Rightarrow\qquad\langle p^{a}|^{\alpha}=\sqrt{M} \epsilon^{\alpha a},\qquad[p^{a}|_{\dot{\alpha}}=\sqrt{M}\epsilon_{\dot{a}a}. \tag{111}\]
The SU(2) little-group rotations, \(|p^{a}\rangle\to U^{a}{}_{b}(p)|p^{a}\rangle\), \(|p^{a}|\to U^{a}{}_{b}(p)|p^{b}]\), leave momentum \(p^{\mu}\) invariant and correspond to choosing different spin quantization axes \(n^{\mu}\). (More details may be found in [52; 95; 131]). The parametrization (110) picks \(n^{\mu}=(\rho,\varepsilon\cos\varphi\sin\theta,\varepsilon\sin\varphi\sin \theta,\varepsilon\cos\theta)/M\), i.e. quantization along the momentum, while eq. (111) chooses the conventional \(z\)-axis.
The momentum spinors serve as basic building blocks for scattering amplitudes. For massless particles, the spin is always quantized along the momentum, and is thus counted by helicity weights: \(-1/2\) for each \(|k\rangle\) and \(+1/2\) for \(|k]\). Moreover, each massive spin-\(s\) particle is represented by \(2s\) symmetrized SU(2) indices. We denote the corresponding symmetrized tensor product of spinors by \(\odot\), following [100].
Spherical harmonics in 4d.Returning to the spherical harmonics, we may now embed the 3d construction in 4d. Namely, we regard it as corresponding to the default choice of the time direction \(u^{\mu}=(1,\mathbf{0})\) and the celestial sphere swept by a massless momentum \(k^{\mu}=\omega\;(1,\hat{\mathbf{k}}(\theta,\varphi))\) and parametrized by the spinors \(|k\rangle_{\alpha}=\sqrt{2\omega}\kappa_{-}^{a=\alpha}\) and \(|k]^{\dot{\alpha}}=\sqrt{2\omega}\kappa_{+}^{a=\dot{\alpha}}\). Lorentz boosts change the time direction and induce Mobius transformations on the celestial sphere.
For a general time direction \(u^{\mu}\) (such that \(u^{2}=1\) and \(u^{0}>0\)), we choose to parametrize the celestial sphere by the massless spinors \(|k\rangle_{\alpha}\) and \(|k]^{\dot{\alpha}}\). Of course, the quantum numbers of a spherical harmonic must be the same as in the rest frame of \(u^{\mu}\). The massive spinors \(\langle u^{a}|^{\alpha}\) and \([u^{a}]_{\dot{\alpha}}\) provide a perfect transformation device between the current inertial frame and the rest frame of \(u^{\mu}\). This brings us to eq. (19), i.e.
\[{}_{h}\tilde{Y}_{j,m}(k;u,n):=\frac{1}{\langle k|u|k]^{j}}\underbrace{ \overbrace{[u_{(}1k]\cdots[u_{1}k]}^{j-m}\overbrace{[u_{(}2k]\cdots[u_{2}k] \cdots[u_{2}k]}^{j-m}\overbrace{[u_{(}2k]\cdots[u_{2}k]}^{j+m}\overbrace{j- h}^{j-m}}. \tag{112}\]
Here the subscripts \(1\) and \(2\) are the explicitly symmetrized little-group indices, and the prefactor involving \(\langle k|u|k]=2k\cdot u\xrightarrow[\mathbf{u}\to 0]2k^{0}\) serves to cancel out the mass dimension. Together with eq. (111), it guarantees the consistency with the rest-frame definition (104) -- up to the functional U(1) transformation of the form (103) in view of the differences in the \(\varphi\)-dependence between eqs. (103) and (104). This is an example of acceptable convention discrepancies, which maybe caused by switching between different spinor parametrizations. The validity of the harmonics (112) as representations of the spin algebra follows from the properties of massive spinors, see e.g. [52; 34]. Note that the dependence on the spin-quantization axis \(n^{\mu}\) enters via the choice of the massive spinors, as discussed around eq. (20). In other words, the SU(2) little-group transformations \(|u^{a}\rangle\to U^{a}{}_{b}(p)|u^{a}\rangle\), \(|u^{a}]\to U^{a}{}_{b}(p)|u^{b}]\) induce the
SO(3) rotations of \(n^{\mu}\) orthogonally to the time direction given by \(u^{\mu}\). Since the choice of spinors for \(u^{\mu}\) defines \(n^{\mu}\), the notation may as well be compressed to \({}_{h}Y_{j,m}(k;u)\).
Let us now discuss the orthonormality property (3.17). It is valid for the normalized versions of the covariant harmonics, rescaled from those in eq. (A.11) analogously to their non-covariant counterparts in eq. (A.3). It can be easily seen that in the rest frame of \(u^{\mu}\) the covariant integration measure reduces to the solid-angle one:
\[\frac{2}{\omega}{\int}d^{4}k\,\delta^{+}(k^{2})\delta(k\cdot u- \omega)\ \xrightarrow[u\to 0]{}\ \ {\int}d\Omega_{\hat{\mathbf{k}}},\qquad k^{0}=|\mathbf{k}|=\omega.\] (A.12)
So eq. (3.17) clearly holds for \(\mathbf{u}=0\), and what we need is to extend it to any \(u^{\mu}\).
Spinor integration.To expose the properties of the measure (A.12) in a neat way, we first rewrite it using a null basis [132]:
\[k^{\mu}\!=t\Big{(}r^{\mu}\!+\!\gamma q^{\mu}\!+\!\frac{z}{2}[r| \bar{\sigma}^{\mu}|q)\!+\!\frac{\bar{z}}{2}[q|\bar{\sigma}^{\mu}|r)\Big{)}\ \ \Rightarrow\ \ \int\!d^{4}k=\frac{i(r+q)^{4}}{4}{\int}t^{3}dt\!\wedge\!d\gamma\! \wedge\!dz\!\wedge\!d\bar{z},\] (A.13)
where \(\bar{\sigma}^{\mu}=(1,-\mathbf{\sigma})\), and the massless vectors \(r^{\mu}\) and \(q^{\mu}\) are not collinear but otherwise arbitrary. Adding the masslessness condition eliminates \(\gamma\) from the measure:
\[{\int}d^{4}k\,\delta^{+}(k^{2})=\frac{i(r+q)^{2}}{4}{\int}_{0}^{ \infty}tdt\int\!dz\!\wedge\!d\bar{z},\qquad k^{\mu}\!=\frac{t}{2}\big{(}(r|\!+ \!z\langle q|\big{)}\sigma^{\mu}\big{(}|r|\!+\!\bar{z}|q|\big{)}.\] (A.14)
(Here for concreteness one may assume \(r^{0},q^{0}>0\) so that \(k^{0}>0\).) However, this massless measure may now be rewritten using spinor integration [133; 134; 135]
\[{\int}d^{4}k\,\delta^{+}(k^{2})=-\frac{i}{4}\int_{0}^{\infty} tdt\int_{\tilde{\lambda}=\tilde{\lambda}}\langle\lambda d\lambda\rangle \wedge[\tilde{\lambda}d\tilde{\lambda}],\qquad k^{\mu}\!=\frac{t}{2}\langle \lambda|\sigma^{\mu}|\tilde{\lambda}],\] (A.15)
such that the dependence on \(r^{\mu}\) and \(q^{\mu}\) has entirely canceled out due to
\[(r+q)^{2}dz\wedge d\bar{z}=-\big{(}\langle r|+z\langle q|\big{)}|q \rangle dz\wedge\big{(}[r|+\bar{z}[q]\big{)}|q\big{)}d\bar{z}=-\langle\lambda d \lambda\rangle\wedge[\tilde{\lambda}d\tilde{\lambda}].\] (A.16)
Now introducing the second delta function let us fix the energy scale of \(k^{\mu}\) and get
\[\frac{1}{\omega}{\int}d^{4}k\,\delta^{+}(k^{2})\delta(k\!\cdot\!u- \omega)=-i{\int}_{\tilde{\lambda}=\tilde{\lambda}}\frac{\langle \lambda d\lambda\rangle\wedge[\tilde{\lambda}d\tilde{\lambda}]}{\langle \lambda|u|\tilde{\lambda}]^{2}},\qquad k^{\mu}\!=\!\omega\frac{\langle\lambda| \sigma^{\mu}|\tilde{\lambda}]}{\langle\lambda|u|\tilde{\lambda}]}.\] (A.17)
This measure allows us to reformulate the orthonormality property (3.17) of the spin-weighted spherical harmonics in the following way:
\[{\int}_{\tilde{\lambda}=\tilde{\lambda}}\frac{\langle\lambda d \lambda\rangle\wedge[\tilde{\lambda}d\tilde{\lambda}]}{\langle\lambda|u| \tilde{\lambda}]^{2}}\,{}_{h}Y^{*}_{j^{\prime},m^{\prime}}(\lambda,\tilde{ \lambda};u)\,{}_{h}Y_{j,m}(\lambda,\tilde{\lambda};u)=\frac{i}{2}\delta^{j^{ \prime}}_{j}\delta^{m^{\prime}}_{m},\] (A.18)
where the notation \({}_{h}Y_{j,m}(\lambda,\tilde{\lambda};u):={}_{h}Y_{j,m}(k;u)\) serves to emphasize their independence of the energy scale. Then the validity of eq. (3.17) for \(\mathbf{u}\neq 0\) follows from
the fact that the entire left-hand side is independent of \(\omega=k\cdot u\). Indeed, for any spinor conventions and in any frame, we can rewrite it as the same integral over the complex plane by parametrizing \(|\lambda\rangle=|u^{1}\rangle+z|u^{2}\rangle\) and \(|\tilde{\lambda}|=|u_{1}|+\bar{z}|u_{2}|\), so that the left-hand side of eq. (102) will exclusively involve the following ingredients:
\[\begin{split}\langle u_{a}\lambda\rangle&=-\delta_ {a}^{1}-\delta_{a}^{2}z,\qquad\quad\langle\lambda d\lambda\rangle\wedge[\tilde {\lambda}d\tilde{\lambda}]&=-dz\wedge d\bar{z}:=2i\,d\Re z\wedge d \Im z,\\ [u_{a}\tilde{\lambda}]&=\epsilon_{1a}+\epsilon_{2a} \bar{z},\qquad\qquad\qquad\langle\lambda|u|\tilde{\lambda}]&=1+z \bar{z}.\end{split} \tag{103}\]
Therefore, it only depends on the quantum numbers \(h,j,j^{\prime},m\) and \(m^{\prime}\), and may only produce a combinatorial result, which may as well be fixed at \(u^{\mu}=(1,\mathbf{0})\).
## Appendix B Frame transformations of harmonics
Here we derive the spinor transformations (102), which induce the relationship between covariant spin-weighted spherical harmonics \({}_{h}\tilde{Y}_{j,m}(k;u)\) and \({}_{h}\tilde{Y}_{j,m}(k;v)\).
These harmonics correspond to two different unit timelike vectors \(u^{\mu}\) and \(v^{\mu}\), with a relative Lorentz factor
\[\gamma:=u\cdot v=:\frac{1}{\sqrt{1-\nu^{2}}},\qquad\quad 0\leq\nu<1. \tag{104}\]
These vectors can be Lorentz-transformed into each other using the minimal boost
\[L^{\rho}{}_{\sigma}(v\!\leftarrow\!u):=\delta_{\sigma}^{\rho}+2v^{\rho}u_{ \sigma}-\frac{(u+v)^{\rho}(u+v)_{\sigma}}{1+u\cdot v}=\exp\Bigl{(}\frac{i\log( \gamma+\sqrt{\gamma^{2}-1})}{\sqrt{\gamma^{2}-1}}u^{\mu}v^{\nu}\Sigma_{\mu \nu}\Bigr{)}^{\rho}_{\sigma}, \tag{105}\]
written in terms of the spin-1 Lorentz generators \((\Sigma^{\mu\nu})^{\rho}{}_{\sigma}:=i[\eta^{\mu\rho}\delta_{\sigma}^{\nu}- \eta^{\nu\rho}\delta_{\sigma}^{\mu}]\). The spinors may be boosted using the corresponding \(\mathrm{SL}(2,\mathbb{C})\) transformations, namely
\[S^{\alpha}{}_{\beta}(v\!\leftarrow\!u)=\exp\bigl{(}\tfrac{i\log\mu}{\gamma\nu} u^{\mu}v^{\nu}\sigma_{\mu\nu}\bigr{)}^{\alpha}{}_{\beta},\qquad\qquad\mu:= \gamma+\sqrt{\gamma^{2}-1}, \tag{106}\]
written in terms of the chiral spin-1/2 generators \(\sigma^{\mu\nu}:=\tfrac{i}{2}\sigma^{[\mu}\bar{\sigma}^{\nu]}\). Using the Clifford-algebra property \(\sigma^{(\mu}\bar{\sigma}^{\nu)}=\eta^{\mu\nu}\), it is easy to derive
\[\begin{split}\bigl{(}\tfrac{i\log\mu}{\gamma\nu}u^{\mu}v^{\nu} \sigma_{\mu\nu}\bigr{)}^{2n}|u^{a}\rangle&=(\log\sqrt{\mu})^{2n} |u^{a}\rangle,\\ \bigl{(}\tfrac{i\log\mu}{\gamma\nu}u^{\mu}v^{\nu}\sigma_{\mu\nu} \bigr{)}^{2n+1}|u^{a}\rangle&=(-\log\!\sqrt{\mu})^{2n+1}\Bigl{(} \tfrac{1}{\nu}|u^{a}\rangle-\tfrac{1}{\gamma\nu}|v|u^{a}]\Bigr{)}.\end{split} \tag{107}\]
This lets us sum the matrix exponent, whose action simplifies to
\[S^{\alpha}{}_{\beta}(v\!\leftarrow\!u)|u^{a}\rangle=\frac{\sqrt{\mu}}{\mu+1} \Bigl{(}|u^{a}\rangle+|v|u^{a}]\Bigr{)}. \tag{108}\]
We thus arrive at the following massive-spinor transformations:
\[|v^{b}\rangle=\frac{\sqrt{\mu}}{\mu+1}U^{b}{}_{a}(v\!\leftarrow\!u)|u\!+\!v|u^ {a}],\qquad\quad|v^{b}]=\frac{\sqrt{\mu}}{\mu+1}U^{b}{}_{a}(v\!\leftarrow\!u)|u \!+\!v|u^{a}\rangle. \tag{109}\]
Here we have allowed for the SU(2) matrix \(U^{b}{}_{a}(v\!\leftarrow\!u)\). Its purpose is to fix the misalignment between what we get from the minimal boost (100) and the desired spin quantization axis for the resulting time direction, which generically do not coincide:
\[n^{\mu}_{v}:=\frac{1}{2}(\langle v_{2}|\sigma^{\mu}|v^{2}]+[v_{2}|\bar{\sigma}^{ \mu}|v^{2}\rangle) \neq L^{\mu}{}_{\nu}(v\!\leftarrow\!u)n^{\nu}=n^{\mu}-\frac{n\cdot v}{1+u \cdot v}(u+v)^{\mu}. \tag{102}\]
In fact, unitary matrices like \(U^{b}{}_{a}(v\!\leftarrow\!u)\) represent the SO(3) rotations of the spin quantization axis even in the absence of Lorentz-frame boosts. Therefore, the spinor transformations (100) induce the most general frame transformations of the covariant spherical harmonics.
| ```
私たちは、有効なオンシェル散乱振幅を用いて重力吸収効果を研究しています。私たちは、入射波とブラックホールまたはコンパクトなオブジェクトとの相互作用を記述するために、平面波と部分波相関状態を伴う確率ベースの枠組みを構築しています。この枠組みを、単純な1量子分析に接続しています。基本的な材料は、質量変化の3点振幅で、それはブラックホールの主な吸収効果を模倣し、ブラックホールのスペクトル密度関数です。アプリケーションとして、回転しないブラックホールを考慮し、その回転はブラックホールのダイナミクスによって引き起こされる可能性があります。対応する振幅は、可変スピンウェイトの球対称関数に対応しており、その性質を構成し、活用しています。私たちは、クロスセクションレベルの一般相対性結果とのマッチング計算を行い、有効吸収の3点結合を導出しました。それは |
2306.10351 | Bkd-FedGNN: A Benchmark for Classification Backdoor Attacks on Federated
Graph Neural Network | Federated Graph Neural Network (FedGNN) has recently emerged as a rapidly
growing research topic, as it integrates the strengths of graph neural networks
and federated learning to enable advanced machine learning applications without
direct access to sensitive data. Despite its advantages, the distributed nature
of FedGNN introduces additional vulnerabilities, particularly backdoor attacks
stemming from malicious participants. Although graph backdoor attacks have been
explored, the compounded complexity introduced by the combination of GNNs and
federated learning has hindered a comprehensive understanding of these attacks,
as existing research lacks extensive benchmark coverage and in-depth analysis
of critical factors. To address these limitations, we propose Bkd-FedGNN, a
benchmark for backdoor attacks on FedGNN. Specifically, Bkd-FedGNN decomposes
the graph backdoor attack into trigger generation and injection steps, and
extending the attack to the node-level federated setting, resulting in a
unified framework that covers both node-level and graph-level classification
tasks. Moreover, we thoroughly investigate the impact of multiple critical
factors in backdoor attacks on FedGNN. These factors are categorized into
global-level and local-level factors, including data distribution, the number
of malicious attackers, attack time, overlapping rate, trigger size, trigger
type, trigger position, and poisoning rate. Finally, we conduct comprehensive
evaluations on 13 benchmark datasets and 13 critical factors, comprising 1,725
experimental configurations for node-level and graph-level tasks from six
domains. These experiments encompass over 8,000 individual tests, allowing us
to provide a thorough evaluation and insightful observations that advance our
understanding of backdoor attacks on FedGNN.The Bkd-FedGNN benchmark is
publicly available at https://github.com/usail-hkust/BkdFedGCN. | Fan Liu, Siqi Lai, Yansong Ning, Hao Liu | 2023-06-17T13:51:33 | http://arxiv.org/abs/2306.10351v1 | # Bkd-FedGNN: A Benchmark for Classification Backdoor Attacks on Federated Graph Neural Network
###### Abstract
Federated Graph Neural Network (FedGNN) has recently emerged as a rapidly growing research topic, as it integrates the strengths of graph neural networks and federated learning to enable advanced machine learning applications without direct access to sensitive data. Despite its advantages, the distributed nature of FedGNN introduces additional vulnerabilities, particularly backdoor attacks stemming from malicious participants. Although graph backdoor attacks have been explored, the compounded complexity introduced by the combination of GNNs and federated learning has hindered a comprehensive understanding of these attacks, as existing research lacks extensive benchmark coverage and in-depth analysis of critical factors. To address these limitations, we propose Bkd-FedGNN, a benchmark for backdoor attacks on FedGNN. Specifically, Bkd-FedGNN decomposes the graph backdoor attack into trigger generation and injection steps, and extending the attack to the node-level federated setting, resulting in a unified framework that covers both node-level and graph-level classification tasks. Moreover, we thoroughly investigate the impact of multiple critical factors in backdoor attacks on FedGNN. These factors are categorized into global-level and local-level factors, including data distribution, the number of malicious attackers, attack time, overlapping rate, trigger size, trigger type, trigger position, and poisoning rate. Finally, we conduct comprehensive evaluations on 13 benchmark datasets and 13 critical factors, comprising 1,725 experimental configurations for node-level and graph-level tasks from six domains. These experiments encompass over 8,000 individual tests, allowing us to provide a thorough evaluation and insightful observations that advance our understanding of backdoor attacks on FedGNN. The Bkd-FedGNN benchmark is publicly available at [https://github.com/usail-hkust/BkdFedGCN](https://github.com/usail-hkust/BkdFedGCN).
## 1 Introduction
The Federated Graph Neural Network (FedGNN) has emerged as a fast-evolving research area that combines the capabilities of graph neural networks and federated learning. Such integration allows for advanced machine learning applications without requiring direct access to sensitive data [1, 2, 3, 4, 5, 6, 7, 8, 9]. However, despite its numerous advantages, the distributed nature of FedGNN introduces additional vulnerabilities, particularly related to backdoor attacks originating from malicious participants. In particular, these adversaries have the ability to inject graph backdoor triggers into their training data, thereby undermining the overall trustworthiness of the system [10, 11, 12, 13, 14].
Although considerable research efforts have explored graph backdoor attacks on FedGNN [15, 16, 17, 18], a comprehensive understanding of these attacks is hindered by the compounded complexity introduced by the combination of Graph Neural Networks (GNNs) and Federated Learning (FL). Existing studies suffer from a lack of extensive benchmark coverage and in-depth analysis of critical factors. **(1) Lack of Extensive Benchmark Coverage**. Specifically, the lack of extensive benchmark coverage poses challenges in fairly and comprehensively comparing graph backdoor attacks on FedGNN across different settings. These settings can be categorized into two levels: the graph backdoor attack level and the FedGNN task level. At the graph backdoor attack level, trigger generation and injection steps are involved. Additionally, the classification tasks in FedGNN encompass both node and graph classification tasks. However, there is still a dearth of comprehensive exploration of graph backdoor attacks on FedGNN under these various settings. **(2) Insufficient Exploration of Multiple Factors.** Furthermore, there has been the insufficient exploration of multiple factors that impact FedGNN. The combination of GNN with FL introduces various factors that affect backdoor attacks, such as trigger type, trigger size, and data distribution. The insufficient exploration and analysis of these multiple factors make it difficult to understand the influence of key factors on the behavior of FedGNN.
To address these limitations, we propose a benchmark for graph backdoor attacks on FedGNN, called Bkd-FedGNN. As far as we are aware, our work is the first comprehensive investigation of graph backdoor attacks on FedGNN. Our contributions can be summarized as follows.
* **Unified Framework**: We propose a unified framework for classification backdoor attacks on FedGNN. Bkd-FedGNN decomposes the graph backdoor attack into trigger generation and injection steps and extends the attack to the node-level federated setting, resulting in a unified framework that covers both node-level and graph-level classification tasks.
* **Exploration of Multiple Critical Factors**: We thoroughly investigate the impact of multiple critical factors on graph backdoor attacks in FedGNN. We systematically categorize these factors into two levels: global level and local level. At the global level, factors such as data distribution, the number of malicious attackers, the start time of backdoor attacks, and the overlapping rate play significant roles. In addition, the local level factors involve factors such as trigger size, trigger type, trigger position, and poisoning rate.
* **Comprehensive Experiments and Analysis**: We conduct comprehensive experiments on both benchmark experiments and critical factor analysis. For the benchmark experiments, we consider combinations of trigger types, trigger positions, datasets, and models, resulting in 315 configurations for the node level and 270 configurations for the graph-level tasks. Regarding the critical factors, we consider combinations of factors, datasets, and models, resulting in 672 configurations for the node-level tasks and 468 configurations for the graph-level tasks. Each configuration is tested five times, resulting in approximately 8,000 individual experiments in total. Based on these experiments, we thoroughly evaluate the presented comprehensive analysis and provide insightful observations that advance the field.
## 2 Federated Graph Neural Network
In this section, we provide an introduction to the preliminary aspects of FedGNN. Currently, FedGNN primarily focuses on exploring common classification tasks, which involve both node-level and graph-level classification. The FedGNN consists of two levels: client-level local training and server-level federated optimization. We will begin by providing an overview of the notations used, followed by a detailed explanation of the client-level local training, which encompasses message passing and readout techniques. Lastly, we will introduce server-level federated optimization.
### Notations
Assume that there exist \(K\) clients denoted as \(\mathcal{C}=\{c_{k}\}_{k=1}^{K}\). Each client, \(c_{i}\), possesses a private dataset denoted as \(\mathcal{D}^{i}=\{(\mathcal{G}_{j}^{i},\mathcal{Y}_{j}^{i})\}_{j=1}^{N_{i}}\), wherein \(\mathcal{G}_{j}^{i}=(\mathcal{V}_{j}^{i},\mathcal{E}_{j}^{i})\) is the graph, where \(\mathcal{V}^{i}=\{v_{t}\}_{t=1}^{n_{i}}\) (\(n_{i}\) denotes the number of nodes) is the set of nodes, and \(\mathcal{E}^{i}=\{e_{tk}\}_{t,k}\) is the set of edges (for simplicity, we exclude the subscript \(j\) that indicates the index of the \(j\)-th dataset in the dataset \(\mathcal{D}^{i}\)). \(N_{i}=\left|\mathcal{D}^{i}\right|\) denotes the total number of data samples in the private dataset of client \(c_{i}\). We employ the
notation \(\mathbf{A}_{j}^{i}\) to denote the adjacency matrix of graph \(\mathcal{G}_{j}^{i}\) belonging to client \(c_{i}\) within the set of clients \(\mathcal{C}\). \(\mathbf{X}_{j}^{i}\) represents the node feature set, and \(\mathbf{Y}_{j}^{i}\) corresponds to the label sets.
### Client-level Local Training
To ensure versatility and inclusiveness, we employ the message passing neural network (MPNN) framework [19, 20], which encompasses a diverse range of spectral-based GNNs, such as GCN [21], as well as spatial-based GNNs including GAT [22] and GraphSage [23], _etc._ Each client possesses a GNN model that collaboratively trains a global model. The local graph learning process can be divided into two stages: message passing and readout.
**Message Passing.** For each client \(c_{i}\), the \(l\)-th layer in MPNN can be formulated as follows,
\[\mathbf{h}_{j}^{l,i}=\sigma(w^{l,i}\cdot(\mathbf{h}_{j}^{l-1,i},\textit{Agg}( \{\mathbf{h}_{k}^{l-1,i}|v_{k}\in\mathcal{N}(v_{j})\}))), \tag{1}\]
where \(\mathbf{h}_{j}^{l,i}\) (\(l=0,\cdots,L-1\)) represents the hidden feature of node \(v_{j}\) in client \(c_{i}\) and \(\mathbf{h}_{j}^{0,i}=\mathbf{x}_{j}\) denotes the node \(v_{j}\)'s raw feature. The \(\sigma\) represents the activation function (e.g., ReLU, sigmoid). The parameter \(w^{l,i}\) corresponds to the \(l\)-th learnable parameter. The aggregation operation _Agg_ (e.g., mean pooling) combines the hidden features \(\mathbf{h}_{k}^{l-1,i}\) of neighboring nodes \(v_{k}\in\mathcal{N}(v_{j})\) for node \(v_{j}\), where \(\mathcal{N}(v_{j})\) represents the set of neighbors of node \(v_{j}\). Assume that the \(\mathbf{w}^{i}=\{w^{l,i}\}_{l=0}^{L-1}\) is the set of learn able parameters for client \(c_{i}\).
**Readout.** Following the propagation of information through \(L\) layers of MPNN, the final hidden feature is computed using a readout function for subsequent tasks.
\[\hat{y}_{I}^{i}=R_{\theta^{i}}(\{\mathbf{h}_{j}^{L,i}|v_{j}\in\mathcal{V}_{I}^ {i}\}), \tag{2}\]
where \(\hat{y}_{I}^{i}\) represents the prediction for a node or graph. Specifically, \(I\) serves as an indicator, where \(I=v_{j}\) denotes the prediction for node \(v_{j}\), and \(I=\mathcal{G}^{i}\) denotes the prediction for the graph \(\mathcal{G}^{i}\). The readout function \(R_{\theta^{i}}(\cdot)\) encompasses methods such as mean pooling or sum pooling _etc._, where \(\theta^{i}\) is the parameter for readout function.
### Server-level Federated Optimization
Let us consider that \(\mathbf{w}^{i}=\{w^{l,i}\}_{l=0}^{L-1}\) represents the set of trainable parameters within the MPNN framework associated with client \(c_{i}\). Consequently, we define the overall model parameters as \(\mathbf{W}^{i}=\{\mathbf{w}^{i},\theta^{i}\}\) for each client \(c_{i}\in\mathcal{C}\). The GNNs, which constitute a part of this framework, can be represented as \(f_{i}(\mathbf{X}_{j}^{i},\mathbf{A}_{j}^{i};\mathbf{W}^{i})\). The objective of FL is to optimize the global objective function while preserving the privacy of local data on each individual local model. The overall objective function can be formulated as follows,
\[\min_{\{\mathbf{W}^{i}\}}\sum_{i\in\mathcal{C}}\frac{N_{i}}{N}F_{i}(\mathbf{ W}^{i}),\quad F_{i}(\mathbf{W}^{i})=\frac{1}{N_{i}}\sum_{j\in\mathcal{D}^{i}} \mathcal{L}((f_{i}(\mathbf{X}_{j}^{i},\mathbf{A}_{j}^{i};\mathbf{W}^{i}), \mathbf{Y}_{j}^{i}), \tag{3}\]
where \(F_{i}(\cdot)\) denotes the local objective function, and \(\mathcal{L}(\cdot)\) denote the loss function (_e.g._, cross-entropy _etc._), and \(N=\sum_{i=1}^{K}N_{i}\) represent the total number of data samples encompassing all clients.
We illustrate the process of federated optimization, aimed at achieving a generalized model while ensuring privacy preservation, by utilizing a representative federated algorithm, FedAvg [24]. Specifically, in each round denoted by \(t\), the central server transmits the global model parameter \(\mathbf{W}_{t}\) to a subset of clients that have been selected for local training. Subsequently, each chosen client \(c_{i}\) refines the received parameter \(\mathbf{W}_{t}\) using an optimizer operating on its private dataset \(\mathcal{D}^{i}\). Following this, the selected clients upload the updated model parameter \(\mathbf{W}_{t}^{i}\), and the central server aggregates the local model parameters to obtain the enhanced global model parameter \(\mathbf{W}_{t+1}\).
In FedGNN setting, there exist diverse scenarios involving distributed graphs that are motivated by real-world applications. In these scenarios, classification tasks can be classified into two distinct settings based on how graphs are distributed across clients. **Node-level FedGNN**. Each client is equipped with a subgraph, and the prevalent tasks involve node classification. Real-world applications,
such as social networks, demonstrate situations where relationships between nodes can span across different clients, and each node possesses a unique label. **Graph-level FedGNN**. Each client possesses a set of graphs, and the primary focus lies on graph classification tasks. Real-world applications, such as protein discovery, exemplify instances where each institution holds a limited graph along with associated labels.
## 3 A Unified Framework for Classification Backdoor Attack on FedGNN
This section presents a unified framework for classification backdoor attacks on federated GNNs. Our primary focus is on graph-based backdoor attacks, where malicious entities strategically insert triggers into graphs or subgraphs to compromise the trustworthiness of FedGNN. A comprehensive illustration of our unified framework for classification backdoor attacks on FedGNN can be found in Figure 1. In detail, we first introduce the dataset and models and then give the evaluation metric, then introduce the threat model. Next, we introduce the federated graph backdoor attack, which involves the formulation of the attack goal and a two-step attack process: trigger generation and trigger injection. Finally, we explore various critical factors at both global and local levels.
### Datasets and Models
In this study, we have considered six distinct domains comprising a total of thirteen datasets, along with three widely used GNNs. _Node-level Datasets:_ For node-level analysis, we have included three extensively studied citation graphs, such as Cora, CiteSeer, and PubMed. Additionally, we have incorporated the Co-authorship graphs (CS and Physics), along with the Amazon Co-purchase graphs (Photo and Computers). _Graph-level Datasets:_ For graph-level analysis, we have utilized molecular graphs such as AIDS and NCI1. Furthermore, bioinformatics graphs, including PROTEINS-full, DD, and ENZYMES, have been incorporated. Lastly, a synthetic graph, COLORS-3, has also been employed. _Models:_ We have employed three widely adopted GNNs: GCN, GAT, and
Figure 1: A unified framework for classification backdoor attack on FedGNN.
GraphSage, which have been demonstrated effective in various graph-based tasks. For detailed statistical information about the graphs used, please refer to Appendix A.1.
### Evaluation Metrics
To assess the effectiveness of the graph backdoor attack on FedGNN, three metrics are employed: the average clean accuracy (ACC) across all clients, the average attack success rate (ASR) on malicious clients, and the transferred attack success rate (TAST) on normal clients. The ACC metric evaluates the performance of federated GNNs when exposed to clean examples from all clients. The ASR metric measures the performance of the graph backdoor attack specifically on the malicious clients. Lastly, the TAST metric gauges the vulnerability of normal clients to the graph backdoor attack. For the detailed equations corresponding to these metrics, please refer to Appendix A.2.
### Threat Model
**Attack Objective.** Assuming there are a total of \(K\) clients, with \(M\) (\(M\leq K\)) of them being malicious, each malicious attacker independently conducts the backdoor attack on their own models. The primary goal of a backdoor attack is to manipulate the model in such a way that it misclassifies specific pre-defined labels (known as target labels) only within the poisoned data samples. It is important to ensure that the model's accuracy remains unaffected when processing clean data. **Attack Knowledge.** In this setting, we assume that the malicious attacker has complete knowledge of their own training data. They have the capability to generate triggers. It should be noted that this scenario is quite practical since the clients have full control over their own data. **Attacker Capability.** The malicious client has the ability to inject triggers into the training datasets, but this capability is limited within predetermined constraints such as trigger size and poisoned data rate. The intention is to contaminate the training datasets. However, the malicious client lacks the ability to manipulate the server-side aggregation process or interfere with other clients' training processes and models.
### Federated Graph Backdoor Attack
Mathematically, the formal attack objective for each malicious client \(c_{i}\) during round \(t\) can be defined as follows,
\[\begin{split}&\mathbf{W}_{t}^{i*}=\arg\min_{\mathbf{W}_{t}} \frac{1}{N_{i}}\left[\sum_{j\in\mathcal{D}_{p}^{i}}\mathcal{L}((f_{i}( \mathbf{X}_{j}^{i},g_{\tau}\circ\mathbf{A}_{j}^{i};\mathbf{W}_{t-1}^{i}),\tau )+\sum_{j\in\mathcal{D}_{c}^{i}}\mathcal{L}((f_{i}(\mathbf{X}_{j}^{i},\mathbf{ A}_{j}^{i};\mathbf{W}_{t-1}^{i}),\mathbf{Y}_{j}^{i})\right],\\ &\forall j\in\mathcal{D}_{p}^{i},N_{\tau}=|g_{\tau}|\leq\bigtriangle _{g}\quad\text{and}\quad\rho=\frac{|\mathcal{D}_{p}^{i}|}{|\mathcal{D}^{i}|} \leq\bigtriangleup_{p},\end{split} \tag{4}\]
where \(\mathcal{D}_{p}^{i}\) refers to the set of poisoned data and \(\mathcal{D}_{c}^{i}\) corresponds to the clean dataset. Noted that \(\mathcal{D}_{p}^{i}\sqcup\mathcal{D}_{c}^{i}=\mathcal{D}^{i}\) and \(\mathcal{D}_{p}^{i}\sqcap\mathcal{D}_{c}^{i}=\phi\), indicating the union and intersection of the poisoned and clean data sets, respectively. \(g_{\tau}\circ\mathbf{A}_{j}^{i}\) represents the poisoned graph resulting from an attack. \(g_{\tau}\) represents the trigger generated by the attacker, which is then embedded into the clean graph, thereby contaminating the datasets. Additionally, \(\tau\) denotes the target label. \(N_{\tau}=|g_{\tau}|\) denotes the trigger size and \(\bigtriangleup_{g}\) represents the constrain to ensures that the trigger size remains within the specified limit. \(\rho=\frac{|\mathcal{D}_{p}^{i}|}{|\mathcal{D}^{i}|}\) represents the poisoned rate, and \(\bigtriangleup p\) denotes the budget allocated for poisoned data.
In the federated graph backdoor attack, to generate the trigger and poisoned data sets, the graph backdoor attack can be divided into two steps: trigger generation and trigger injection. The term "trigger" (a specific pattern) has been formally defined as a subgraph in the work by Zhang _et al._ (2021), providing a clear and established framework for its characterization [25].
**Trigger Generation.** The process of trigger generation can be defined as the function \(\varphi(\mathbf{X}_{j}^{i},\mathbf{A}_{j}^{i})\), which yields the generated trigger \(g_{\tau}\) through \(\varphi(\mathbf{X}_{j}^{i},\mathbf{A}_{j}^{i})=g_{\tau}\).
**Trigger Injection.** The process of trigger injection can be defined as the function \(a(g_{\tau},\mathbf{A}_{j}^{i})\), which generates the final poisoned graph \(g_{\tau}\circ\mathbf{A}_{j}^{i}\) by incorporating the trigger \(g_{\tau}\) into the pristine graph \(\mathbf{A}_{j}^{i}\).
### Factors in Federated Graph Backdoor
The graph backdoor attack framework in FedGNN encompasses various critical factors that warrant exploration. These factors can be categorized into two levels: the global level and the local level. At the global level, factors such as data distribution, the number of malicious attackers, the start time of backdoor attacks, and overlapping rate play significant roles. On the other hand, the local level involves parameters like trigger size, trigger type, trigger position, and poisoning rate. Notably, the overlapping rate holds particular importance in node-level FedGNN, as it involves cross-nodes across multiple clients.
_Global Level Factors:_ **Data Distribution.** The data distribution encompasses two distinct types: independent and identically distributed (IID) and non-independent and identically distributed (Non-IID). In detail, IID refers to data distribution among clients remaining constant, while Non-IID (L-Non-IID [26, 27], PD-Non-IID [28], N-Non-IID [29]) refers that the data distribution among clients exhibiting variations. **Number of Malicious Attackers.** The concept of the number of malicious attackers, denoted as \(M\), can be defined in the following manner. Let us assume that the set of malicious clients is denoted as \(\mathcal{C}_{m}\), and the set of normal clients is denoted as \(\mathcal{C}_{n}\). It can be inferred that \(\mathcal{C}_{m}\sqcup\mathcal{C}_{m}=\mathcal{C}\) and \(\mathcal{C}_{m}\sqcap\mathcal{C}_{c}=\phi\). **Attack Time.** In the context of FL, the attack time denotes the precise moment when a malicious attack is launched. The attack time can be denoted by \(t^{*}\). **Overlapping Rate (specific to Node-level FedGNN).** The overlapping rate, represented by the variable \(\alpha\), pertains to the proportion of additional samples of overlapping data that across clients. This phenomenon arises in node-level FedGNN, where cross-client nodes exist, resulting in the sharing of common data samples between different clients.
_Local Level Factors:_ **Trigger Size.** The size of the trigger can be quantified by counting the number of nodes within the corresponding graph. The trigger size is denoted by \(N_{\tau}\). **Trigger Type.** Based on the methods used to generate triggers(_e.g._, Renyi [25], WS [30], BA [31], RR [32], GTA [33], and UGBA [34]_etc._), the categorization of trigger types can be refined into two categories: universal triggers and adaptive triggers. Universal triggers are pre-generated through graph generation techniques, such as the Erdos-Renyi (ER) model [35], which are agnostic to the underlying graph datasets. On the other hand, adaptive triggers are specifically designed for individual graphs using optimization methods. **Trigger Position.** The trigger position refers to the specific location within a graph or sub-graph where the trigger is injected. Typically, the trigger position can be categorized into two types: random position and important indicator position. In the case of the random position, the trigger is injected into the graph in a random manner without any specific consideration. Conversely, the important indicator position entails injecting the trigger based on certain crucial centrality values, such as the degree or cluster-based scores, that indicate the significance of specific nodes within the graph. **Poisoning Rate.** The concept of poisoning rate, denoted as \(\rho\), can be defined as the ratio of the cardinality of the set of poisoned data samples, \(\mathcal{C}_{p}^{i}\), to the total number of data samples, denoted as \(\mathcal{D}^{i}\). Mathematically, this can be expressed as \(\rho=\frac{|\mathcal{D}_{p}^{i}|}{|\mathcal{D}^{i}|}\), where \(\forall c_{i}\in\mathcal{C}\) signifies that the cardinality calculations are performed for every client \(c_{i}\) belonging to the set \(\mathcal{C}\).
## 4 Experimental Studies
In this section, we present the experimental studies conducted to investigate classification backdoor attacks on FedGNN. Our main objective is to evaluate the impact of graph backdoor attacks on FedGNN covering both the node and graph level tasks. Additionally, we aim to explore the critical
\begin{table}
\begin{tabular}{c c|c|c} \hline \hline & Factors & Symbol & Node Level & Graph Level \\ \hline \multirow{4}{*}{Global Level} & Data Distribution & - & \{\(\text{IID}^{*},\text{L-Non-IID}\) \} & \{\(\text{IID}^{*},\text{PD-Non-IID},\text{N-Non-IID}\) \\ \cline{2-4} & \# of Malicious Attack & \(M\) & \(\{1^{*},2,3,4,5\}\) \\ \cline{2-4} & Attack Time & \(t^{*}\) & \(T*\{0.0^{*},0.1,0.2,0.3,0.4,0.5\}\) \\ \cline{2-4} & Overlapping Rate & \(\alpha\) & \(\{0.1^{*},0.2,0.3,0.4,0.5\}\) \\ \hline \multirow{4}{*}{Local Level} & Trigger Size & \(N_{\tau}\) & \(\{3^{*},4,5,6,7,8,9,10\}\) & \(N_{\tau}*\{0.1^{*},0.2,0.3,0.4,0.5\}\) \\ \cline{2-4} & Trigger Type & \(g_{\tau}\) & \(\{\text{Rensyi}^{*},\text{WS},\text{BA},\text{GTA},\text{UGBA}\) \} & \{ Renyi\({}^{*},\text{WS},\text{BA},\text{RR},\text{GTA}\) \} \\ \cline{1-1} & Trigger Position & - & \(\{\text{Random}^{*},\text{Degree},\text{Cluster}\) \} \\ \cline{1-1} & Poisoning Rate & \(\rho\) & \(\{0.1^{*},0.2,0.3,0.4,0.5\}\) \\ \hline \hline \end{tabular}
* marks the default value and # represents the Number.
\(T\) represents the total training round time and \(N_{\tau}\) represents the average number of graph nodes.
\end{table}
Table 1: Critical factors in federated graph backdoor.
factors that influence the effectiveness of graph backdoor attacks on FedGNN, considering aspects from both the global and local levels.
### Experimental Settings
**Factors Settings.** We present the detailed factors setup considered in our study. It is important to note that the first value presented represents the default setting. To assess the individual impact of each factor, we keep the remaining factors fixed while systematically varying the corresponding values in our experiments. The factors range is shown in Table 1. For the detailed setting for factor, please refer to Appendix A.3.
**Federated Graph Backdoor Attack.** The federated graph backdoor attack can be characterized by the combination of trigger generation techniques (Renyi [25], WS [30], BA [31], RR [32], GTA [33], and UGBA [34]) and trigger position strategies (Random, Degree, and Cluster). For instance, the attack method Renyi-Random refers to the utilization of the ER model to generate the trigger, which is then randomly injected into the graph.
**Implementation Details.** Our implementation of the backdoor attack on FedGNN is based on the PyTorch framework. The experiments were carried out on two server configurations: three Linux Centos Servers, each with 4 RTX 3090 GPUs, and two Linux Ubuntu Servers, each with 2 V100 GPUs. In both node-level and graph-level tasks, we adopt the inductive learning settings as outlined in [16, 34]. For each dataset, we ensure consistent experimental conditions by employing the same training and attack settings. We set the total number of clients to \(5\), and all clients participate in the training process at each round. Each experiment is repeated five times. For a detailed description of the training and attack settings, please refer to Appendix A.4.
### Benchmark Results of Graph Backdoor Attack on FedGNN
The results of the benchmark for the graph backdoor attack on FedGNN are presented in Figure 2. The observations are summarized as follows. (1) The node-level task exhibits higher vulnerability to attacks compared to the graph-level task at a relatively small trigger size. Specifically, a significant majority of graph backdoor attacks achieve an ASR (Attack Success Rate) exceeding \(90\%\), while the highest ASR recorded at the graph level is \(82.24\%\). (2) Despite not being intentionally poisoned by malicious attackers, the normal clients are still susceptible to graph backdoor attacks. For instance, in the node-level task, there is a TASR (Transferred Attack Success Rate) of \(24.52\%\), while the graph-level task exhibits even higher vulnerability with a TASR of \(61.86\%\). This observation suggests that the weights uploaded by the malicious clients can inadvertently influence the normal clients when they download the global model's weights. 3). The combination of trigger size and trigger position significantly influences the attack performance on the graph-level task compared to the node-level
Figure 2: Graph backdoor attack on both node and graph level tasks for GCN. (Color intensity corresponds to value magnitude)
task. For instance, the attack WS-Cluster achieves an ASR of approximately \(82.24\%\), while the GTA-Random achieves only about \(13.87\%\). Due to the page limit, the benchmark results on other datasets and models please refer to Appendix A.5.1.
### Factors in Federated GNN
The overall results of factors can be shown in Figures 3-4. _Global Level Factors:_**Data Distribution (DD).** For node-level tasks, there models trained on IID data are more vulnerable than models trained on Non-IID data. For graph-level tasks, the GCN trained on IID data are more vulnerable than models trained on Non-IID data (PD-Non-IID and N-Non-IID), while GAT and GraphSagr trained on Non-IID data are more vulnerable than models trained on IID data. **Number of Malicious Attackers (NMA).** For node-level tasks, an increase in NMA leads to an increase in ASR for both GCN and GAT models. Conversely, an increase in NMA results in a decrease in ASR for both GraphSage. Concerning graph-level tasks, the ASR demonstrates an increase with the increase of NMA in the case of GAT and GraphSage. However, in the scenario of GCN, the ASR shows a decrease with the increase of NMA. **Attack Time (AT).** For both node-level and graph-level tasks, an increase in AT results in a decrease in ASR for three models. **Overlapping Rate (OR).** The ASR demonstrates an upward trend as the overlapping rate increases. This correlation can be attributed to the possibility that overlapping nodes facilitate the backdooring of normal clients, primarily through the presence of cross-edges.
_Local Level Factors:_**Trigger Size (TS).** For node-level tasks, an increase in TS leads to an increase in ASR for GCN. However, in the case of GAT and GraphSage, the ASR demonstrates a decrease with the increase of TS. Concerning the graph-level task, the ASR shows an increase with the increase of TS across all three GNNs. **Trigger Types (TT).** In the node-level task, the adaptive trigger demonstrates a higher ASR on most models. Conversely, in the graph-level task, the universal trigger exhibits higher ASR. **Trigger Position (TP).** In node-level tasks, we observed a significantly large ASR when using importance-based positions (Degree and Cluster) compared to random positions.
Figure 4: Graph-level task factors.
Figure 3: Node-level task factors.
However, for the graph-level task, while importance-based positions showed higher ASR for GCN, random positions yielded higher ASR for GAT and GraphSage. **Poisoning Rate (PR).** On node classification, an increase in PR results in a slight decrease in ASR. However, graph classification exhibits an upward trend in ASR. Due to the page limit, the results on other datasets and metrics, please refer to Appendix A.5.2.
## 5 Related Works
**FedGNN.** FedGNN present a distributed machine learning paradigm that facilitates collaborative training of GNNs among multiple parties, ensuring the privacy of their sensitive data. In recent years, extensive research has been conducted on FedGNN, with a particular focus on addressing security concerns [15, 16, 17, 18]. Among these concerns, poisoning attacks have garnered significant attention, encompassing both data poisoning attacks and model poisoning attacks. Data poisoning attacks occur when an adversary employs tainted data to train the local model, while model poisoning attacks involve manipulation of either the training process or the local model itself. Currently, the majority of attacks on FedGNN primarily concentrate on data poisoning attacks. Chen _et al._[15] proposed adversarial attacks on vertical federated learning, utilizing adversarial perturbations on global node embeddings based on gradient leakage from pairwise nodes. Additionally, Xu _et al._[16] investigated centralized and distributed backdoor attacks on FedGNN.
**Graph Backdoor Attacks.** Backdoor attacks on GNNs have received significant attention in recent years [25, 36, 37, 38, 33, 39, 34]. Regarding graph backdoor attacks, they can be classified into two types based on the employed trigger: universal graph backdoor attacks and adaptive backdoor attacks. In universal graph backdoor attacks, Zhang _et al._[25] generated sub-graphs using the Erdos-Renyi (ER) model as triggers and injected them into the training data. Additionally, Xu _et al._[33] observed that the position of the trigger injection into the graph can also affect the attack's performance. As for adaptive trigger backdoor attacks, Xi _et al._[33] developed an adaptive trigger generator that optimizes the attack's effectiveness for both transductive and inductive tasks. In our benchmark, we focus primarily on data poisoning attacks. While model poisoning attacks can be effective, data poisoning attacks may be more convenient because they do not require tampering with the model learning process, and they allow non-expert actors to participate [40].
## 6 Conclusions and Open Problems
**Conclusions.** In this paper, we proposed a unified framework for classification backdoor attacks on FedGNN. We then introduced the critical factors involved in graph backdoor attacks on FedGNN, including both global and local level factors. Along this line, we performed approximately 8,000 experiments on the graph backdoor attacks benchmark and conduct critical factor experiments to provide a comprehensive analysis.
**Open Problems.** (1) Enhancing the success rate of transferred attacks: Our findings reveal that malicious attackers can also backdoor normal clients through the FL mechanism. However, there is a need to explore methods that can identify and exploit the worst vulnerabilities under these circumstances. (2) Evaluating the defense method under backdoor attack: We demonstrate that FedGNN can be compromised by malicious attackers. However, assessing the effectiveness of defense mechanisms against such attacks still requires further exploration. (3) Cooperative malicious attackers: Currently, the majority of malicious attackers operate independently during the attack process, neglecting the potential benefits of collaboration. An intriguing research direction lies in investigating the utilization of collaboration to enhance attack performance.
## Acknowledgments and Disclosure of Funding
This research was supported in part by the National Natural Science Foundation of China under Grant No.62102110, Guangzhou Science and Technology Plan Guangzhou-HKUST(GZ) Joint Project No. 2023A03J0144, and Foshan HKUST Projects (FSUST21-FYTRI01A, FSUST21-FYTRI02A). | フェデレーテッドグラフネURALネットワーク（FedGNN）は、グラフネURALネットワークの強みと federated learning を統合することで、敏感なデータを直接アクセスせずに高度な機械学習アプリケーションを可能にする急速に注目を集めている研究テーマです。その利点に加え、FedGNN の分布的な性質は、悪意のある参加者からのバックドア攻撃などの追加的な脆弱性を導入しています。バックドア攻撃は探索されてきましたが、GNN と federated learning の組み合わせによる複雑性が、これらの攻撃の包括的な理解を阻害しており、既存の研究は広範なベンチマークの coverage と重要な要因の深度分析に欠けているのが現状です。この課題に対処するために、私たちは Bkd-FedGNN を提案しました。これは、FedGNN のバックドア攻撃のためのベンチマークです。具体的には、Bkd-FedGNN はグラフバックドア攻撃をトリガー生成と注入のステップに分解し、攻撃 |
2305.09350 | Two-species reaction-diffusion system in the presence of random velocity
fluctuations | We study random velocity effects on a two-species reaction-diffusion system
consisting of three reaction processes $A + A \rightarrow (\varnothing, A),A+B
\rightarrow A$. Using the field-theoretic perturbative renormalization group we
analyze this system in the vicinity of its upper critical dimension $d_c = 2$.
Velocity ensemble is generated by means of stochastic Navier-Stokes equations.
In particular, we investigate the effect of thermal fluctuations on reaction
kinetics. The overall analysis is performed to the one-loop approximation and
possible macroscopic regimes are identified. | Michal Hnatič, Matej Kecer, Tomáš Lučivjanský | 2023-05-16T11:08:10 | http://arxiv.org/abs/2305.09350v1 | # Two-species reaction-diffusion system in the presence of random velocity fluctuations
###### Abstract
We study random velocity effects on a two-species reaction-diffusion system consisting of three reaction processes \(A+A\to(\emptyset,A)\), \(A+B\to A\). Using the field-theoretic perturbative renormalization group we analyze this system in the vicinity of its upper critical dimension \(d_{c}=2\). Velocity ensemble is generated by means of stochastic Navier-Stokes equations. In particular, we investigate the effect of thermal fluctuations on reaction kinetics. The overall analysis is performed to the one-loop approximation and possible macroscopic regimes are identified.
\({}^{1}\) Institute of Physics, Faculty of Science, P. J. Safarik University, Park Angelinum 9, 040-01 Kosice, Slovakia
\({}^{2}\) Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Kosice, Slovakia
\({}^{3}\) Joint Institute for Nuclear Research, 141980 Dubna, Russia
## 1 Introduction
Diffusion-limited reactions constitute prominent models in non-linear statistical physics [1]. Theoretical study of such systems attracted a lot of attention in the past [2, 3]. A straightforward approach to theoretical analysis of such systems is based on kinetic rate equations, which might be regarded as a simple mean-field-like approximation [3, 4]. However, reaction systems are known to exhibit non-trivial behavior, especially in low space dimensions [5], where density fluctuations become especially pronounced. There the kinetic rate equations approach is not adequate and more sophisticated approaches are called for. In this paper, we study a multi-species reaction-diffusion system [6, 7, 8, 9], which consists of the following three reaction processes
\[A+A\to\begin{cases}A&\text{coalescence,}\\ \emptyset&\text{annihilation,}\\ \end{cases} \tag{1}\] \[A+B\to A\qquad\text{trapping,}\]
where coalescence process occurs with probability \(p\,(0\leq p\leq 1)\), and annihilation process with a complementary probability \(1-p\). The model becomes even more intricate when additional effects are taken into account. Their investigation is especially important, as they naturally arise in many practical circumstances. For instance, the majority of chemical reactions in typical experimental settings occur in some fluid environment. Various aspects of such a problem have already been studied recently [7, 8, 9]. Here, our aim is to investigate the influence of thermal fluctuations of a surrounding environment on the kinetics of reaction-scheme (1). We model the environment as a fluid at a constant temperature using a well-known approach based on the stochastic Navier-Stokes equation [10, 11].
A powerful tool for analyzing the asymptotic behavior of stochastic systems is provided by the renormalization group (RG) method [12, 13]. It allows us to determine the long-time and large-scale - or infrared (IR) - asymptotic regimes of the system and also is a very efficient tool for the calculation of various universal physical quantities, e.g. critical exponents. The aim of this paper is to address the possible IR behavior of the reaction-diffusion process (1) under the influence of advecting velocity fluctuations and to determine their IR regimes.
The paper is organized as follows. In Sec. 2 we give a field-theoretic formulation of the model and specify the main ingredients of the perturbation theory. Sec. 3 is devoted to the analysis of ultraviolet divergences and renormalization of the model in one-loop order of perturbation scheme. The analysis of fixed points (FP) and their regions of stability are discussed in Sec. 4. Conclusions are drawn in Sec. 5.
## 2 Field-theoretic formulation of the model
The field theory for the reaction-diffusion system described by the scheme (1) can be constructed from the master equation by means of Doi-Peliti formalism [4, 14, 15, 16]. For brevity, we omit the derivation as it can be easily found elsewhere (see e.g., [3]). We start our analysis with the field-theoretic action for the reaction scheme (1) augmented with diffusion processes
\[\mathcal{S}_{r}[\Psi] =\psi_{A}^{\dagger}(-\partial_{t}+\nu_{0}u_{A0}\partial^{2})\psi _{A}+\psi_{B}^{\dagger}(-\partial_{t}+\nu_{0}u_{B0}\partial^{2})\psi_{B}-\nu_ {0}u_{A0}\lambda_{0}\psi_{A}^{\dagger}\psi_{A}^{2}\] \[-\nu_{0}u_{A0}\lambda_{0}\psi_{A}^{\dagger 2}\psi_{A}^{2}- \lambda_{0}^{\prime}Q\nu_{0}u_{A0}\psi_{B}^{\dagger}\psi_{A}\psi_{B}-\nu_{0}u _{A0}\lambda_{0}^{\prime}\psi_{A}^{\dagger}\psi_{B}^{\dagger}\psi_{A}\psi_{B}, \tag{2}\]
where \(\Psi\equiv\{\psi_{A},\psi_{A}^{\dagger},\psi_{B},\psi_{B}^{\dagger}\}\) are bosonic-like coherent fields arising in taking a continuum limit in the Doi-Peliti approach [3], \(\partial^{2}=\partial_{i}\partial_{i}\) denotes Laplace operator in \(d\)-dimensions and diffusion parameters are expressed through respective Prandtl numbers \(u_{A0}\), \(u_{B0}\) and viscosity \(\nu_{0}\) (see below Eq. (7)). The parameters \(\lambda_{0},\lambda_{0}^{\prime}\) denote reaction constants, and parameter \(Q=1/(2-p)\) is related to the probability of whether annihilation or coagulation process takes place. In this work, we employ the RG method, which introduces two different kinds of variables - bare (unrenormalized) quantities and their renormalized counterparts. Therefore we denote the former ones with the subscript "0", whereas the latter will be written without the subscript "0".
Reaction process (1) is an example of a genuine non-equilibrium system and, therefore, we have to specify its initial conditions. We choose them in the following form
\[\mathcal{S}_{init}[\Psi]=(a_{0}\psi_{A}^{\dagger}+b_{0}\psi_{B}^{\dagger}) \delta(t), \tag{3}\]
where \(a_{0},b_{0}\) are appropriately rescaled initial average densities [4, 7].
In writing actions (2) and (3) we have employed a condensed notation, in which integrations over space and time variables in the expressions for action functionals are implied. For instance, the first term in the action (2) corresponds to
\[-\psi_{A}^{\dagger}\partial_{t}\psi_{A}=-\int\mathrm{d}x\,\psi_{A}^{\dagger}(x )\partial_{t}\psi_{A}(x), \tag{4}\]
where we have written coordinates compactly as \(x=(t,\mathbf{x})\) and integration measure as \(\mathrm{d}x=\mathrm{d}t\mathrm{d}^{d}x\).
The aim of this paper is to study the case where chemical particles are advected within the fluid environment with random fluctuations. We introduce advection processes into the formalism by the inclusion of convective derivative [17]. This corresponds to the replacement of the time derivative as follows
\[\partial_{t}\to\partial_{t}+\mathbf{v}\cdot\mathbf{\nabla}=\partial_{t}+v_{j}\partial _{j}, \tag{5}\]
where summation over repeated indices is implied in the last term. Let us stress that the advection for both particle types is considered to be passive, i.e., the velocity field itself is not affected by the particles or reactions processes, respectively. Corresponding advective terms to the action (2) take the form
\[\mathcal{S}_{adv}[\Psi,\mathbf{v}]=-\psi_{A}^{\dagger}v_{j}\partial_{j}\psi_{A}- \psi_{B}^{\dagger}v_{j}\partial_{j}\psi_{B}. \tag{6}\]
To finalize the model construction we need to specify velocity field \(\mathbf{v}\). Here, we assume that velocity field \(\mathbf{v}(t,\mathbf{x})\) is a random variable with zero mean, whose dynamics is governed by stochastic Navier-Stokes equation [10, 18].
\[\partial_{t}v_{i}+(v_{j}\partial_{j})v_{i}=\nu_{0}\partial^{2}v_{i}-\partial_ {i}P+f_{i}, \tag{7}\]
where \(P=P(x)\) is the pressure field, and \(f_{i}=f_{i}(x)\) denotes \(i\)-th component of an external random force \(\mathbf{f}\). Following earlier works [10, 18, 19] we assume the force \(\mathbf{f}\) is a random Gaussian variable with zero mean and correlation function of the prescribed form
\[\langle f_{i}(t,\mathbf{x})f_{j}(0,\mathbf{0})\rangle=\int\frac{\mathrm{d}^{d}k}{(2 \pi)^{d}}D_{ij}(t,\mathbf{k})\mathrm{e}^{i\mathbf{k}\cdot\mathbf{x}}. \tag{8}\]
We consider the case of an incompressible fluid, which implies transversality of the field \(\mathbf{v}\) (\(\partial_{i}v_{i}=0\)). Using this condition it is possible to express pressure in terms of velocity field [18]. This is equivalent to work in transversal space by taking the following replacement for velocity field \(\mathbf{v}\) in the momentum representation
\[v_{i}(\mathbf{k})\to P_{ij}(\mathbf{k})v_{j}(\mathbf{k}), \tag{9}\]
where \(P_{ij}(\mathbf{k})=\delta_{ij}-k_{i}k_{j}/k^{2}\) with \((k=|\mathbf{k}|)\) is transverse projection operator.
The incompressibility condition implies that the kernel \(D_{ij}\) in the momentum representation is proportional to transverse projector \(P_{ij}(\mathbf{k})\). In fact, it can be readily shown that for incompressible medium \(D_{ij}\sim\delta_{ij}\) is sufficient. However, we follow the traditional notation in previous works and keep \(P_{ij}\) in the expression for kernel \(D_{ij}\). Using a specific
choice for the momentum dependence of \(D_{ij}\) term it is possible to generate fluctuations of the velocity field near thermal equilibrium.
These considerations finally lead to
\[D_{ij}(t,\mathbf{k})=\delta(t)D_{0}k^{2}P_{ij}(\mathbf{k}), \tag{10}\]
where \(\delta=\delta(t)\) is Dirac delta function. It can be shown that delta correlations in time of the kernel \(D_{ij}\) ensures that the present model possesses the Galilean symmetry [11, 20].
In hindsight, this particular form (10) is convenient for the application of RG method, because both velocity fluctuations and reaction processes of the original reaction-diffusion system become simultaneously marginal in the critical space dimension \(d=d_{c}=2\). The stochastic problem (7)-(10) can be recast into a field theory with the doubled set of fields \(\Phi=\{\mathbf{v},\mathbf{v}^{\prime}\}\) described by the De Dominicis-Janssen action functional [12, 11],
\[\mathcal{S}_{v}[\Phi]=\frac{1}{2}v^{\prime}_{i}D_{ij}v^{\prime}_{j}+v^{\prime }_{i}\left(-\partial_{t}v_{i}-v_{j}\partial_{j}v_{i}+\nu_{0}\partial^{2}v_{i} \right), \tag{11}\]
where response field \(v^{\prime}_{i}\) is incompressible, and again condensed notation in the sense of Eq. (4) is assumed. Let us note that quadratic term in the response field \(\mathbf{v}^{\prime}\) in the action (11) actually stands for
\[v^{\prime}_{i}D_{ij}v^{\prime}{}_{j}=\int\mathrm{d}x\int\mathrm{d}x^{\prime}v ^{\prime}_{i}(x)D_{ij}(x-x^{\prime})v^{\prime}_{k}(x^{\prime}), \tag{12}\]
where \(D_{ij}\) corresponds to the inverse Fourier transform of the kernel (10).
The sum of action functionals (2), (3), (6), and (11), respectively, then gives us a final field-theoretic action
\[\mathcal{S}=\mathcal{S}_{r}+\mathcal{S}_{v}+\mathcal{S}_{adv}+\mathcal{S}_{ init}. \tag{13}\]
Expectation values of some physical observable \(A=A(t,\mathbf{x})\) can be, in principle, calculated as a functional integral [4, 12]
\[\langle A(t,\mathbf{x})\rangle=\mathcal{N}^{-1}\int\mathcal{D}\Psi\mathcal{D}\Phi \,A(t,\mathbf{x})\mathrm{e}^{S}, \tag{14}\]
where \(\mathcal{N}\) is a normalization constant.
In what follows we analyze field-theoretic action (13) using the field-theoretic renormalization group. This technique was employed in the past on similar problems as well [3, 21, 22, 23, 24, 25, 26]. We apply it here in a perturbation setting, which is based on expressing Green functions as a series in coupling constants of a theory. The perturbation theory of the model is then constructed using well-known Feynman diagrammatic rules [3, 12, 13]. The part of the action (13) quadratic in fields determines the bare propagators, which in frequency-momentum representation take form
\[\langle\psi_{A}\psi^{\dagger}_{A}\rangle_{0} =\frac{1}{-i\omega+\nu_{0}u_{A0}k^{2}}, \langle\psi_{B}\psi^{\dagger}_{B}\rangle_{0} =\frac{1}{-i\omega+\nu_{0}u_{B0}k^{2}}, \tag{15}\] \[\langle v_{i}v_{j}\rangle_{0} =\frac{D_{0}k^{2}P_{ij}(\mathbf{k})}{\omega^{2}+\nu_{0}^{2}k^{4}}, \langle v_{i}v^{\prime}_{j}\rangle_{0} =\frac{P_{ij}(\mathbf{k})}{-i\omega+\nu_{0}k^{2}}. \tag{16}\]
The nonlinear terms determine interaction vertices with associated vertex factors [12]. They can be calculated with the help of the formula
\[V_{N}(x_{1},\ldots,x_{N};\varphi)=\frac{\delta^{N}\mathcal{S}_{\rm int}}{\delta \varphi(x_{1})\ldots\delta\varphi(x_{N})},\quad\varphi\in\{\psi_{A},\psi_{A}^{ \dagger},\psi_{B},\psi_{B}^{\dagger},\mathbf{v},\mathbf{v}^{\prime}\}, \tag{17}\]
where \(\mathcal{S}_{\rm int}\) corresponds to the non-linear terms of the action (13). In a straightforward manner, we get the following bare vertices without an inclusion of velocity field
\[V_{\psi_{A}^{\dagger}\psi_{A}\psi_{A}} =-2\lambda_{0}\nu_{0}u_{A0}, V_{\psi_{B}^{\dagger}\psi_{B}\psi_{A}} =-\lambda_{0}^{\prime}\nu_{0}u_{A0}Q,\] \[V_{\psi_{A}^{\dagger}\psi_{A}^{\dagger}\psi_{A}\psi_{A}} =-4\lambda_{0}\nu_{0}u_{A0}, V_{\psi_{A}^{\dagger}\psi_{B}^{\dagger}\psi_{A}\psi_{B}} =-\lambda_{0}^{\prime}\nu_{0}u_{A0}. \tag{18}\]
On the other hand, there are three additional vertices that include the velocity field
\[V_{\psi_{A}^{\dagger}(\mathbf{k})\psi_{A}v_{j}}=ik_{j},\quad V_{\psi_{B}^{\dagger }(\mathbf{k})\psi_{B}v_{j}}=ik_{j},\quad V_{\psi_{i}^{\prime}(\mathbf{k})v_{i}v_{j}}=i (k_{l}\delta_{ij}+k_{j}\delta_{il}). \tag{19}\]
First, two describe advection processes and the latter vertex is responsible for interactions between velocity fluctuations. Also, we have explicitly written, the momentum of which field enters a given interaction vertex. For instance, in expression for the vertex factor \(V_{\psi_{i}^{\prime}(\mathbf{k})v_{i}v_{j}}\) the momentum \(k_{j}\) is carried by the response field \(v_{i}^{\prime}\)[11, 12].
## 3 Renormalization of the model
The analysis of UV divergences starts with a determination of the canonical dimensions for model parameters. In dynamical models, there are two independent scales that need to be considered [3, 12]. These are frequency and momentum scales (time and length). Then any quantity \(F\) is characterized with both frequency dimension \(d_{F}^{\omega}\) and a momentum dimension \(d_{F}^{k}\), respectively. Canonical dimensions are determined from normalization conditions
\[d_{k}^{k}=-d_{x}^{k}=1,\ d_{\omega}^{\omega}=-d_{t}^{\omega}=1,\ d_{k}^{\omega }=d_{\omega}^{k}=0, \tag{20}\]
and the fact that the action functional has to be a dimensionless quantity [12]. The total canonical dimension of any \(F\) is then given as \(d_{F}=d_{F}^{k}+2d_{F}^{\omega}\) (because of \(\partial_{t}\propto\partial^{2}\) proportionality in quadratic part of the action functional). Canonical dimensions of all the fields and parameters of model (13) are listed in Tab. 1.
There are altogether five charges (coupling constants) of the theory
\[g_{0}=\frac{D_{0}}{\nu_{0}^{3}},\ u_{A0},\ u_{B0},\ \lambda_{0},\ \lambda_{0}^{\prime}. \tag{21}\]
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \(F\) & \(\psi_{A}\), \(\psi_{B}\) & \(\psi_{A}^{\dagger}\), \(\psi_{B}^{\dagger}\) & \(\mathbf{v}\) & \(\mathbf{v}^{\prime}\) & \(\lambda_{0}\), \(\lambda_{0}^{\prime}\), \(g_{0}\) & \(Q\) & \(a_{0}\), \(b_{0}\) & \(\nu_{0}\) & \(D_{0}\) & \(u_{A0}\), \(u_{B0}\) \\ \hline \(d_{F}^{k}\) & \(d\) & \(0\) & \(-1\) & \(d+1\) & \(2-d\) & \(0\) & \(d\) & \(-2\) & \(-d-4\) & \(0\) \\ \hline \(d_{F}^{k}\) & \(0\) & \(0\) & \(1\) & \(-1\) & \(0\) & \(0\) & \(0\) & \(1\) & \(3\) & \(0\) \\ \hline \(d_{F}\) & \(d\) & \(0\) & \(1\) & \(d-1\) & \(2-d\) & \(0\) & \(d\) & \(0\) & \(2-d\) & \(0\) \\ \hline \end{tabular}
\end{table}
Table 1: Canonical dimensions of fields and parameters.
In the space dimension \(d=2\), all of these charges become simultaneously dimensionless and the model becomes logarithmic. Therefore this dimension is identified as an upper critical dimension \(d_{c}\) of the model. In dimensional regularisation, the UV divergences manifest themselves as poles in expansion parameter \(\varepsilon=2-d\), whereas the IR divergences are regulated by the sharp cutoff at \(k=m\), which is an analog of the inverse of integral turbulence scale \(L=1/m\). Let us note that the latter divergences do not affect renormalization constants [12].
Probably the most economical way to renormalize the translationally invariant model is through the renormalization of its one-particle irreducible (1PI) Green functions. This is a restricted class of Feynman diagrams that consists of such diagrams that remain connected even after one internal line is cut off [3, 12]. An arbitrary one-particle irreducible (1PI) Green's function will be denoted as \(\Gamma_{\{\varphi\}}=\langle\varphi\ldots\varphi\rangle_{1PI}\), where \(\varphi\in\Psi\cup\Phi\) denotes an arbitrary field from the full set of fields of the model (13). Its total canonical dimension is given by a general formula [12, 13]
\[d_{\Gamma}=d+2-\sum_{\varphi}N_{\varphi}d_{\varphi}, \tag{22}\]
where the sum runs through all the types of fields \(\varphi\), \(N_{\varphi}\) denotes the number of times the given field appears in the particular 1PI function and \(d_{\varphi}\) is its canonical dimension. Following the standard approach [12] the task is to identify superficial divergences in 1PI functions and construct renormalized action, in which introduced additional counter-terms ensure the removal of these divergences in the given order of perturbation theory.
The UV divergences, which require further treatment, are identified with those 1PI Green functions, which possess a non-negative formal index of divergence \(\delta_{\Gamma}=d_{\Gamma}|_{\varepsilon=0}\). However, for the present case, this statement is to be adjusted based on the following considerations. First, the 1PI functions not involving any of the response functions \(\psi_{B}^{\dagger},\psi_{A}^{\dagger},\mathbf{v}^{\prime}\) as external fields vanish as they necessarily contain closed cycles of causal propagators [12]. Since vertex factor \(V_{\mathbf{v}^{\prime}\mathbf{v}\mathbf{v}}\) is proportional to the momentum carried by field \(\mathbf{v}^{\prime}\) (see the corresponding expression in (19)), every instance of \(\mathbf{v}^{\prime}\) appearing as external field lowers the overall index of divergence. Thus the real index of divergence is defined as
\[\tilde{\delta}_{\Gamma}=\delta_{\Gamma}-N_{\mathbf{v}^{\prime}}. \tag{23}\]
Second, the number of counter-terms is further reduced because of the invariance property of generating functional of model (13) with respect to Galilean transformations. This symmetry implies that the function \(\langle\mathbf{v}^{\prime}\mathbf{v}\mathbf{v}\rangle_{1PI}\) does not diverge (for further discussions on the subject see e.g. [11, 12, 27]). Taking these into account along with available diagrammatic elements and transversality of the velocity field we can identify the following irreducible functions with superficial UV divergences
\[\langle\mathbf{v}^{\prime}\mathbf{v}^{\prime}\rangle_{1PI}, \langle\psi_{A}^{\dagger}\psi_{A}\psi_{A}\rangle_{1PI},\] \[\langle\mathbf{v}^{\prime}\mathbf{v}\rangle_{1PI}, \langle\psi_{B}^{\dagger}\psi_{B}\psi_{A}\rangle_{1PI},\] \[\langle\psi_{A}^{\dagger}\psi_{A}\rangle_{1PI}, \langle\psi_{A}^{\dagger}\psi_{A}^{\dagger}\psi_{A}\psi_{A} \rangle_{1PI},\] \[\langle\psi_{B}^{\dagger}\psi_{B}\rangle_{1PI}, \langle\psi_{B}^{\dagger}\psi_{A}^{\dagger}\psi_{B}\psi_{A} \rangle_{1PI}. \tag{24}\]
All of these have the form that is already present in the bare action functional. This implies that the model is multiplicatively renormalizable.
The total renormalized action takes the form
\[S_{R} =\psi_{A}^{\dagger}\left(-\partial_{t}+Z_{1}u_{A}\nu\partial^{2} \right)\psi_{A}+\psi_{B}^{\dagger}\left(-\partial_{t}+Z_{2}u_{B}\nu\partial^{2} \right)\psi_{B}+\frac{v_{i}^{\prime}\mu^{\epsilon}Z_{3}D_{ij}v_{j}^{\prime}}{2}\] \[+v_{i}^{\prime}(-\partial_{t}+Z_{4}\nu\partial^{2})v_{i}-u_{A} \nu\lambda\mu^{\epsilon}Z_{5}\left[\psi_{A}^{\dagger}+\psi_{A}^{\dagger 2}\right]\psi_{A}^{2}- \lambda^{\prime}u_{A}\nu\mu^{\epsilon}Z_{6}\left[Q\psi_{A}+\psi_{A}^{\dagger} \psi_{A}\right]\psi_{B}^{\dagger}\psi_{B}\] \[-v_{i}^{\prime}(\mathbf{v}\cdot\mathbf{\nabla})v_{i}-\left[\psi_{A}^{ \dagger}(\mathbf{v}\cdot\mathbf{\nabla})\psi_{A}+\psi_{B}^{\dagger}(\mathbf{v}\cdot\mathbf{ \nabla})\psi_{B}\right]+\delta(t)\left(\psi_{A}^{\dagger}\;a_{0}+\psi_{B}^{ \dagger}\;b_{0}\right), \tag{25}\]
and was obtained from the bare action (13) by introducing the following renormalization of fields and parameters of the model
\[\varphi \to Z_{\varphi}\varphi, u_{A0} \to Z_{u_{A}}u_{A}, u_{B0} \to Z_{u_{B}}u_{B}, \nu_{0} \to Z_{\nu}\nu,\] \[g_{0} \to\mu^{\varepsilon}Z_{g}g, \lambda_{0} \to\mu^{\varepsilon}Z_{\lambda}\lambda, \lambda^{\prime}_{0} \to\mu^{\varepsilon}Z_{\lambda^{\prime}}\lambda^{\prime}. \tag{26}\]
Here, \(\varphi\in\{\psi_{A},\psi_{A}^{\dagger},\psi_{B},\psi_{B}^{\dagger},\mathbf{v}, \mathbf{v}^{\prime}\}\), \(\mu\) is an arbitrary momentum scale and \(Z_{F}\) denotes the corresponding renormalization constant.
By direct inspection, we get relations between renormalization constants in the renormalized action (25) and RG constants (26)
\[Z_{u_{A}} =Z_{1}Z_{4}^{-1}, Z_{u_{B}} =Z_{2}Z_{4}^{-1}, Z_{\lambda} =Z_{5}Z_{1}^{-1}, Z_{\lambda} ^{\prime} =Z_{6}Z_{1}^{-1},\] \[Z_{g} =Z_{3}Z_{4}^{-3}, Z_{\nu} =Z_{4}, Z_{\Psi} =Z_{Q}=1. \tag{27}\]
The explicit form of RG constants \(Z_{1}-Z_{6}\) is calculated from the one-loop 1PI Feynman diagrams using dimensional regularisation and a minimal subtraction scheme. The final expressions read
\[Z_{1} =1-\frac{\hat{g}}{4u_{A}(u_{A}+1)\varepsilon}, Z_{2} =1-\frac{\hat{g}}{4u_{B}(u_{B}+1)\varepsilon}, Z_{3} =Z_{4} =1-\frac{\hat{g}}{16\varepsilon},\] \[Z_{5} =1+\frac{\hat{\lambda}}{\varepsilon}, Z_{6} =1+\frac{\hat{\lambda}^{\prime}u_{A}}{(u_{A}+u_{B})\varepsilon}, \tag{28}\]
where \(\hat{F}\equiv FS_{d}/(2\pi)^{d}\), \(S_{d}=2\pi^{d/2}/\Gamma(d/2)\) is the area of unit \(d\)-dimensional sphere, and \(\Gamma(x)\) is Euler's gamma function.
## 4 RG functions and scaling regimes
Once the calculation of RG constants is successfully accomplished, it is possible to analyze the asymptotic behavior of the system. A fundamental equation that governs the behavior of renormalized Green functions is expressed with the help of the RG operator, which can be expressed in the present case as
\[D_{RG}=\mu\partial_{\mu}+\sum_{e}\beta_{e}\partial_{e}-\gamma_{\nu}\nu \partial_{\nu}, \tag{29}\]
where the given sum runs through all charges of the theory \(e=\{g,u_{A},u_{B},\lambda^{\prime},\lambda\}\). The coefficient functions are defined as
\[\beta_{e}=\mu\frac{\partial e}{\partial\mu}\bigg{|}_{0},\quad\gamma_{F}=\frac{ \partial\ln Z_{F}}{\partial\ln\mu}\bigg{|}_{0}, \tag{30}\]
for any parameter, \(F\) and \(|_{0}\) means that bare parameters are held constant during evaluation. For the model (13), we have altogether five beta functions
\[\beta_{g}= -g(\varepsilon+\gamma_{g}), \beta_{u_{A}}=-u_{A}\gamma_{u_{A}}, \beta_{u_{B}}=-u_{B}\gamma_{u_{B}},\] \[\beta_{\lambda}= -\lambda(\varepsilon+\gamma_{\lambda}), \beta_{\lambda^{\prime}}=-\lambda^{\prime}(\varepsilon+\gamma_{ \lambda^{\prime}}), \tag{31}\]
with corresponding anomalous dimensions (by definition Eq. (30))
\[\gamma_{g}= -\frac{\hat{g}}{8}, \gamma_{u_{i}}=\hat{g}\bigg{(}\frac{1}{4u_{i}(1+u_{i})}-\frac{1 }{16}\bigg{)};i\in\{A,B\},\] \[\gamma_{\lambda}= -\hat{\lambda}-\frac{\hat{g}}{4u_{A}(1+u_{A})}, \gamma_{\lambda^{\prime}}=-\hat{\lambda^{\prime}}\frac{u_{A}}{u_ {A}+u_{B}}-\frac{\hat{g}}{4u_{A}(1+u_{A})}, \tag{32}\]
where the higher order corrections \(\hat{g}^{2},\hat{\lambda}^{2}\) are neglected in the one-loop approximation.
The long-time asymptotic behavior of the model is governed by the IR stable fixed points (FP) [12, 13] of beta functions. These are such points \(e^{*}=(g^{*},u_{A}^{*},u_{B}^{*},\lambda^{*},\lambda^{\prime*})\) in coupling constant space that satisfy
\[\beta_{g}(e^{*})=\beta_{u_{A}}(e^{*})=\beta_{u_{B}}(e^{*})=\beta_{\lambda}(e^ {*})=\beta_{\lambda^{\prime}}(e^{*})=0. \tag{33}\]
IR stability is determined by the eigenvalues of the matrix of the first derivatives
\[\Omega_{ij}=\frac{\partial\beta_{i}}{\partial e_{j}}\bigg{|}_{e^{*}}, \tag{34}\]
where index \(i\) and charge \(e_{j}\) belong to the set \(\{g,u_{A},u_{B},\lambda^{\prime},\lambda\}\). For IR stable regimes the eigenvalues of the matrix (34) have to have positive real parts. We have found eight FPs, however, only two of them are IR stable (see Tab. 2). These are
1. Gaussian fixed point (FP1): \(g^{*}=0\), \(u_{A}^{*}=\) arbitrary, \(u_{B}^{*}=\) arbitrary, \(\lambda^{*}=0\), \(\lambda^{\prime*}=0\). IR stable for \(\varepsilon<0\).
2. Thermal fixed point (FP8): \(g^{*}=8\varepsilon\), \(u_{A}^{*}=u_{B}^{*}=(\sqrt{17}-1)/2\), \(\lambda^{*}=\varepsilon/2\), \(\lambda^{\prime*}=\varepsilon\). IR stable for \(\varepsilon>0\).
Let us note that in non-trivial (thermal) FP both velocity fluctuations and reaction interactions are simultaneously IR-relevant. RG predicts also an FP for which only reaction processes are relevant (FP4). However, even though it would have been stable without the velocity field [16], it can never be truly IR stable in the presence of thermal fluctuations, which are inevitable in practice. A similar conclusion was obtained in the past for different reaction-diffusion model [24]. On the borderline of the two regimes, i.e. for the case \(\varepsilon=0\), couplings of the theory become marginally irrelevant, and logarithmic corrections are expected to appear in expressions for Green's functions. Based on the standard analysis [12] we predict that these corrections will be different from the ones realized if the velocity field was not present (and FP4 would be stable). Therefore the behavior in two-dimensional systems is also expected to be affected by the presence of velocity field fluctuations. The proof of this statement is deferred to future work.
## 5 Conclusion
We have investigated the influence of thermal fluctuations on a reaction-diffusion system with reactions \(A+A\rightarrow(\emptyset,A)\), \(A+B\to A\). Using the field-theoretic formulation of the model we have analyzed possible macroscopic behavior utilizing the renormalization group approach. In particular, we have renormalized the model to the one-loop order of the perturbation scheme. The RG analysis revealed the existence of two IR-stable FPs which govern the long-time behavior of the system.
**Conflict of Interest**
The authors declare that they have no conflicts of interest.
## Acknowledgment
The work was supported by VEGA grant No. 1/0535/21 of the Ministry of Education, Science, Research and Sport of the Slovak Republic.
Conflict of Interest: The authors declare that they have no conflicts of interest.
| |
2307.10078 | A Dual Formulation for Probabilistic Principal Component Analysis | In this paper, we characterize Probabilistic Principal Component Analysis in
Hilbert spaces and demonstrate how the optimal solution admits a representation
in dual space. This allows us to develop a generative framework for kernel
methods. Furthermore, we show how it englobes Kernel Principal Component
Analysis and illustrate its working on a toy and a real dataset. | Henri De Plaen, Johan A. K. Suykens | 2023-07-19T15:51:25 | http://arxiv.org/abs/2307.10078v1 | # A Dual Formulation for Probabilistic Principal Component Analysis
###### Abstract
In this paper, we characterize _Probabilistic Principal Component Analysis_ in Hilbert spaces and demonstrate how the optimal solution admits a representation in dual space. This allows us to develop a generative framework for kernel methods. Furthermore, we show how it englobes _Kernel Principal Component Analysis_ and illustrate its working on a toy and a real dataset.
Machine Learning, Probabilistic Principal Component Analysis
## 1 Introduction
Classical datasets often consist of many features, making dimensionality reduction methods particularly appealing. _Principal Component Analysis_ (PCA) is one of the most straightforward frameworks to that goal and it is hard to find a domain in machine learning or statistics where it has not proven to be useful. PCA considers new decorrelated features by computing the eigendecomposition of the covariance matrix.
Probabilistic models on another side participate to the building of a stronger foundation for machine learning models. By considering models as probability distributions, we are able to natively access notions such as variance or sampling, _i.e._ generation. A probabilistic approach to PCA, known as _Probabilistic Principal Component Analysis_ (Prob. PCA), has been formulated by (Tipping & Bishop, 1999). Its principles can be visualized in the primal part of Table 1.
Even endowed with a probabilistic interpretation, PCA remains restricted to linear relations between the different features. _Kernel Principal Component Analysis_ (KPCA) (Mika et al., 1998; Scholkopf et al., 1998) was an attempt to give a non-linear extension to (non-probabilistic) PCA by decomposing a kernel matrix instead of the covariance matrix. An earlier attempt to give a probabilistic formulation of KPCA has been done by (Zhang et al., 2004). As developed further, the latter model does not consist in a kernel equivalent of the Prob. PCA, but rather in another model based on similar principles.
More recently, _Restricted Kernel Machines_(Suykens, 2017) opened a new door for a probabilistic version of PCA both in primal and dual. They essentially use the Fenchel-Young inequality on a variational formulation of KPCA (Suykens et al., 2003; Alaiz et al., 2018) to obtain an energy function, closely resembling to _Restricted Boltzmann Machines_. The framework has been further extended to generation (Schereurs & Suykens, 2018; Winant et al., 2020), incorporating robustness (Pandey et al., 2020), multi-view models (Pandey et al., 2021), deep explicit feature maps (Pandey et al., 2022b) or times-series (Pandey et al., 2022a).
### Contributions
1. We characterize the Prob. PCA framework in Hilbert spaces and give a dual interpretation to the model.
2. We develop a new extension of KPCA incorporating a noise assumption on the explicit feature map.
3. We give a probabilistic interpretation of the generation in KPCA.
4. We illustrate how the dual model works on a toy and a real dataset and show its connections to KPCA2. Footnote 2: Resources: [https://hdeplaen.github.io/kppca](https://hdeplaen.github.io/kppca).
## 2 Primal and Dual Spaces
The key idea behind the duality in PCA is that outer and inner products share the same eigenvalues. The consequence is that instead of decomposing the covariance matrix of any given feature map, we can decompose the associated Gram matrix, _i.e._ the kernel matrix. The former is considered as the _primal_ formulation and the latter as the _dual_ formulation and they are both equivalent. Extending Prob. PCA to a dual formulation is however not straightforward: if all feature maps have an associated kernel, the converse is trickier. Some kernels correspond to feature maps in infinite dimensional spaces, where probability distributions cannot be properly defined. We therefore need to choose well defined finite subspaces to work in and consider linear operators instead of matrices. All formal definitions, propositions and proofs are provided in Appendix A.
### Primal Spaces
**Feature Space \(\mathcal{H}\).** Given an input space \(\mathcal{X}\), we first consider any feature map \(\varphi:\mathcal{X}\to\mathcal{H}\). Following (Alaiz et al., 2018), we will consider a separable, possibly infinite dimensional, Hilbert space \((\mathcal{H},\langle\cdot,\cdot\rangle_{\mathcal{H}})\). By \(\boldsymbol{\varphi}\), we denote an element of \(\mathcal{H}\) and its adjoint by \(\boldsymbol{\varphi}^{*}=\langle\boldsymbol{\varphi},\cdot\rangle\in\mathcal{ H}^{*}\), with \(\mathcal{H}^{*}\sim\mathcal{H}\) its Frechet-Riesz dual space. Essentially, it corresponds to the transpose \(\boldsymbol{\varphi}^{\top}\) in real, finite dimensional spaces as \(\boldsymbol{\varphi}_{1}^{\top}\boldsymbol{\varphi}_{2}=\langle\boldsymbol{ \varphi}_{1},\boldsymbol{\varphi}_{2}\rangle_{\mathcal{H}}\), but generalizes it for the possibly infinite dimensional spaces that will be necessary for the introduction of kernels. Furthermore, we assume our space to be defined over the reals such that \(\langle\cdot,\cdot\rangle_{\mathcal{H}}:\mathcal{H}\times\mathcal{H}\to \mathbb{R}\) and its inner product is symmetric \(\langle\boldsymbol{\varphi}_{1},\boldsymbol{\varphi}_{2}\rangle_{\mathcal{H}}= \langle\boldsymbol{\varphi}_{2},\boldsymbol{\varphi}_{1}\rangle_{\mathcal{H}}\). If \(\mathcal{H}\) is of finite dimension \(d\), we can therefore identify its canonical basis \(\boldsymbol{u}_{1},\ldots,\boldsymbol{u}_{d}\) with the canonical basis of \(\mathbb{R}^{d}\).
**Finite Feature Space \(\mathcal{H}_{\mathcal{E}}\).** Considering a set of \(N\) observations \(\{\boldsymbol{x}_{i}\in\mathcal{X}\}_{i=1}^{N}\), the idea is to work directly in \(\mathcal{H}\) by considering instead the feature map of the datapoints \(\boldsymbol{\varphi}_{i}=\varphi\left(\boldsymbol{x}_{i}\right)\). We can however not define a normal distribution onto the full \(\mathcal{H}\) yet as it is possibly infinite dimensional. We therefore have to consider a finite subspace \(\mathcal{H}_{\mathcal{E}}\subset\mathcal{H}\). A natural choice would be \(\mathcal{H}_{\mathcal{E}}=\operatorname{span}\left\{\boldsymbol{\varphi}_{1}, \ldots,\boldsymbol{\varphi}_{N}\right\}\). We now first have to find an orthonormal basis for \(\mathcal{H}_{\mathcal{E}}\).
### Dual Spaces
**Kernels.** For each feature map, there is an induced positive semi-definite kernel \(k:\mathcal{X}\times\mathcal{X}\to\mathbb{R}:k\left(\boldsymbol{x},\boldsymbol {y}\right)=\langle\varphi(\boldsymbol{x}),\varphi(\boldsymbol{y})\rangle_{ \mathcal{H}}=\varphi(\boldsymbol{x})^{*}\varphi(\boldsymbol{y})\). Inversely, to each positive semi-definite kernel corresponds a, possibly infinite dimensional, feature map, even if not explicitly defined. This follows from the theory of _Reproducing Kernel Hilbert Spaces_. We refer to (Scholkopf & Smola, 2001) for further info.
**Kernel Space \(\mathcal{E}\).** We now consider a finite dimensional Hilbert space \((\mathcal{E},\langle\cdot,\cdot\rangle_{\mathcal{E}})\) of dimension \(N\), the number of observations. It is defined similarly as above, with orthonormal basis \(\boldsymbol{e}_{1},\ldots,\boldsymbol{e}_{N}\). The basis also defines the identity over \(\mathcal{E}\) as \(\boldsymbol{I}_{\mathcal{E}}=\sum_{i=1}^{N}e_{i}\boldsymbol{e}_{i}^{*}\). The goal for \(\mathcal{E}\) is to represent the space of the kernel representations. We therefore define the linear operator \(\boldsymbol{\Phi}:\mathcal{E}\to\mathcal{H}:\sum_{i=1}\boldsymbol{\varphi}_{i }\boldsymbol{e}_{i}^{*}\) and its adjoint \(\boldsymbol{\Phi}^{*}:\mathcal{H}\to\mathcal{E}:\sum_{i=1}^{N}\boldsymbol{e} _{i}\boldsymbol{\varphi}_{i}^{*}\). Essentially, \(\boldsymbol{\Phi}^{*}\) returns the kernel value with each datapoint: \(\boldsymbol{\Phi}^{*}\varphi(\boldsymbol{x})=\sum_{i=1}^{N}\boldsymbol{e}_{i }\left(\boldsymbol{\varphi}_{i}^{*}\varphi(\boldsymbol{x})\right)=\sum_{i=1}^ {N}\boldsymbol{e}_{i}k\left(\boldsymbol{x}_{i},\boldsymbol{x}\right)\) for any \(\boldsymbol{x}\in\mathcal{X}\). Similarly, \(\boldsymbol{\Phi}\) projects this value back as a linear combination of the different \(\boldsymbol{\varphi}_{i}\)'s, thus mapping back to \(\mathcal{H}_{\mathcal{E}}\subset\mathcal{H}\). For this reason, the covariance \(\boldsymbol{\Phi}\circ\boldsymbol{\Phi}^{*}=\sum_{i=1}^{N}\boldsymbol{\varphi}_ {i}\boldsymbol{\varphi}_{i}^{*}\) acts as a projector from \(\mathcal{H}\to\mathcal{H}_{\mathcal{E}}\). Its eigenvectors therefore form an orthonormal basis of the finite feature space \(\mathcal{H}_{\mathcal{E}}\)
\begin{table}
\begin{tabular}{l l l l}
**Distribution** & **Interpretation** & **Primal (features)** & **Dual (kernels)** \\ latent \(|\) observation \(|\) latent projection & \(\boldsymbol{h}|\boldsymbol{\phi}\sim\mathcal{N}\big{(}\boldsymbol{\Sigma}_{ \boldsymbol{h}|\boldsymbol{\phi}}^{-1}\circ\boldsymbol{W}_{\mathrm{ML}}^{*}( \boldsymbol{\phi}-\boldsymbol{\phi}_{c}),\sigma^{2}\boldsymbol{\Sigma}_{ \boldsymbol{h}|\boldsymbol{\phi}}^{-1}\big{)}\) & \(\boldsymbol{h}|\boldsymbol{k}_{c}\sim\mathcal{N}\big{(}\boldsymbol{\Sigma}_{ \boldsymbol{h}|\boldsymbol{k}_{c}}^{-1}\circ\boldsymbol{A}_{\mathrm{ML}} \boldsymbol{k}_{c},\boldsymbol{\Sigma}_{\boldsymbol{h}|\boldsymbol{k}_{c}}^{-1} \big{)}\) \\ observation \(|\) latent & latent-based generation & \(\boldsymbol{\phi}|\boldsymbol{h}\sim\mathcal{N}\big{(}\boldsymbol{W}_{\mathrm{ ML}}h-\boldsymbol{\phi}_{c},\sigma^{2}\boldsymbol{I}_{\mathcal{H}_{\mathcal{E}}}\big{)}\) & \(\boldsymbol{k}_{c}|\boldsymbol{h}\sim\mathcal{N}\big{(}(\boldsymbol{\Phi}_{c}^{*} \boldsymbol{\Phi}_{c})\circ\boldsymbol{A}_{\mathrm{ML}}h,\sigma^{2}\boldsymbol {\Phi}_{c}^{*}\circ\boldsymbol{\Phi}_{c}\big{)}\) \\ latent & latent prior & \(\boldsymbol{h}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{I}_{\mathcal{E}})\) & \(\boldsymbol{h}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{I}_{\mathcal{E}})\) \\ observation & absolute generation & \(\boldsymbol{\phi}\sim\mathcal{N}(\boldsymbol{\mu},\boldsymbol{W}_{\mathrm{ML}} \circ\boldsymbol{W}_{\mathrm{ML}}^{*}+\sigma^{2}\boldsymbol{I}_{\mathcal{H}_{ \mathcal{E}}})\) & \(\boldsymbol{k}_{c}\sim\mathcal{N}\big{(}\boldsymbol{0},\boldsymbol{A}_{\mathrm{ML}} \circ\boldsymbol{A}_{\mathrm{ML}}+\sigma^{2}\left(\boldsymbol{\Phi}_{c}^{*} \circ\boldsymbol{\Phi}_{c}\right)^{-1}\big{)}\) \\ \end{tabular}
\end{table}
Table 1: Interpretation of the different distributions of the Prob. PCA framework after training, in both primal and dual formulations. The covariance operators are given by \(\boldsymbol{\Sigma}_{\boldsymbol{h}|\boldsymbol{\phi}}=\left(\boldsymbol{W}_{ \mathrm{ML}}^{*}\circ\boldsymbol{W}_{\mathrm{ML}}+\sigma^{2}\boldsymbol{I}_{ \mathcal{E}}\right)^{-1}\) and \(\boldsymbol{\Sigma}_{\boldsymbol{h}|\boldsymbol{k}_{c}}=\left(\boldsymbol{A}_{ \mathrm{ML}}^{*}\circ(\boldsymbol{\Phi}_{c}^{*}\circ\boldsymbol{\Phi}_{c}) \circ\boldsymbol{A}_{\mathrm{ML}}+\sigma^{2}\boldsymbol{I}_{\mathcal{L}} \right)^{-1}\), with maximum likelihood estimators for the primal and dual interconnection operators \(\boldsymbol{W}_{\mathrm{ML}}\) and \(\boldsymbol{A}_{\mathrm{ML}}\).
Figure 1: Global overview of the Probabilistic Principal Component Analysis in both primal and dual formulations. The primal spaces, or feature \(\mathcal{H}\), \(\mathcal{H}_{\mathcal{E}}\) and \(\mathcal{H}_{\mathcal{L}}\) are in blue. The dual, or kernel and latent spaces \(\mathcal{E}\) and \(\mathcal{L}\) are in brown. The input space \(\mathcal{X}\) is in green. The color or the applications (arrows) is just for the readability and has nothing to do with the color of the spaces.
which acts as the primal equivalent of the kernel space \(\mathcal{E}\).
**Centered Kernels.** In most applications however, we prefer to work with the centered feature map, which we define as \(\varphi_{c}(\cdot)=\varphi(\cdot)-\mathbf{\varphi}_{c}\) with \(\mathbf{\varphi}_{c}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{\varphi}_{i}\). We denote the associated kernel associated centered kernel \(k_{c}:\mathcal{X}\times\mathcal{X}\to\mathbb{R}:k_{c}(\mathbf{x}_{1},\mathbf{x}_{2})= \varphi_{c}(\mathbf{x}_{1})^{*}\varphi_{c}(\mathbf{x}_{2})\). This leads to the definition of a new centered operator \(\mathbf{\Phi}_{c}=\sum_{i=1}(\mathbf{\varphi}_{i}-\mathbf{\varphi}_{c})\mathbf{e}_{i}^{*}= \mathbf{\Phi}\left(\mathbf{I}_{\mathcal{E}}-\frac{1}{N}\mathbf{1}_{\mathcal{E}\times \mathcal{E}}\right)\), with \(\mathbf{1}_{\mathcal{E}\times\mathcal{E}}=\sum_{i,j=1}^{N}\mathbf{e}_{i}\mathbf{e}_{j}^{*}\). As always, we also consider its adjoint \(\mathbf{\Phi}_{c}^{*}\). Considering the dual operator, we have \(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}=\sum_{i=1}^{N}(\mathbf{\varphi}_{i}-\mathbf{ \varphi}_{c})^{*}(\mathbf{\varphi}_{i}-\mathbf{\varphi}_{c})\mathbf{e}_{i}\mathbf{e}_{j}^{*}= \sum_{i=1}^{N}k_{c}(\mathbf{x}_{i},\mathbf{x}_{j})\mathbf{e}_{i}\mathbf{e}_{j}^{*}\). We notice now that \(\mathcal{H}_{\mathcal{E}}=\mathrm{span}\{\mathbf{\varphi}_{1},\dots,\mathbf{\varphi}_ {N}\}=\mathrm{span}\{\mathbf{\varphi}_{1}-\mathbf{\varphi}_{c},\dots,\mathbf{\varphi}_{N}- \mathbf{\varphi}_{c}\}\) because \(\mathbf{\varphi}_{c}\) is a linear combination of the elements of the basis. Therefore, the primal operator \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}=\sum_{i=1}^{N}(\mathbf{\varphi}_{i}-\mathbf{ \varphi}_{c})(\mathbf{\varphi}_{i}-\mathbf{\varphi}_{c})^{*}\) also acts as a projector from \(\mathcal{H}\to\mathcal{H}_{\mathcal{E}}\) and we can choose its eigenvectors instead as an orthonormal basis of \(\mathcal{H}_{\mathcal{E}}\).
**Covariance and Kernels**. We now consider the key idea behind the duality in PCA: the operators \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\) and \(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\) are self-adjoint, positive semi-definite and share the same non-zero eigenvalues. We have \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}=\sum_{i=1}^{N}\lambda_{i}\mathbf{v}_{i}\mathbf{v}_ {i}^{*}\) and \(\mathcal{H}_{\mathcal{E}}=\mathrm{span}\{\mathbf{v}_{1},\dots,\mathbf{v}_{N}\}\). Similarly, we have \(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}=\sum_{i=1}^{N}\lambda_{i}\mathbf{\epsilon}_{i} \mathbf{\epsilon}_{i}^{*}\) and \(\mathcal{E}=\mathrm{span}\{\mathbf{\epsilon}_{1},\dots,\mathbf{\epsilon}_{N}\}\). The identity over the (primal) finite feature space \(\mathcal{H}_{\mathcal{E}}\) can now be defined as \(\mathbf{I}_{\mathcal{H}_{\mathcal{E}}}=\sum_{i=1}^{N}\mathbf{v}_{i}\mathbf{v}_{i}^{*}\) and the identity over the (dual) kernel space \(\mathcal{E}\) as \(\mathbf{I}_{\mathcal{E}}=\sum_{i=1}^{N}\mathbf{\epsilon}_{i}\mathbf{\epsilon}_{i}^{*}\). This is synthetized in the two first columns of Table 2. The identity over \(\mathcal{H}\) reads \(\mathbf{I}_{\mathcal{H}}=\mathbf{I}_{\mathcal{H}_{\mathcal{E}}}+\mathbb{P}_{\mathbf{ \mathcal{H}}_{\mathcal{E}}^{\perp}}\), with \(\mathbb{P}_{\mathbf{\mathcal{H}}_{\mathcal{E}}^{\perp}}\) a projector over the null space of \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\). It most be noted that it may happen that these basis may contain too much basis vectors if the two operators \(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\) and \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\) are not of full rank. In particular, this is the case when \(\dim(\mathcal{H})=d\) is finite and \(d<N\). In this particular case, we would also have \(\dim(\mathcal{H}_{\mathcal{E}})=\dim(\mathcal{E})=d\). Without loss of generality, we will assume that this is not the case. Similarly, we will neglect the case \(N>d\) as we could just neglect the null space of \(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\).
**Notations**. We can now define our probabilistic model over \(\mathcal{H}_{\mathcal{E}}\). We will therefore use the notation \(\phi\) instead of \(\varphi\) to consider the feature map in our finite dimensional subspace \(\mathcal{H}_{\mathcal{E}}\). More formally, we have \(\phi:\mathcal{X}\to\mathcal{H}_{\mathcal{E}}:\mathbf{I}_{\mathcal{H}_{\mathcal{E }}}\circ\varphi\) and following from that \(\phi_{c}:\mathcal{X}\to\mathcal{H}_{\mathcal{E}}:\mathbf{I}_{\mathcal{H}_{\mathcal{E }}}\circ\varphi_{c}\). In particular, we have the observations \(\mathbf{\phi}_{i}=\phi(\mathbf{x}_{i})=\mathbf{\varphi}_{i}\) and \(\mathbf{\phi}_{c}=\mathbf{\varphi}_{c}\), as those are linear combinations of the basis. For the sake of readability, we will write \(\mathbf{\phi}=\phi(\mathbf{x})\), the image of a random variable \(\mathbf{x}\in\mathcal{X}\) and refer to it as a _feature_ observation or representation. Given any Hilbert space, \(\mathbf{a}\) an element of it and a linear operator \(\mathbf{\Sigma}\) from and to that space, we consider the _multivariate normal distribution_\(\mathbf{a}\sim\mathcal{N}\big{(}\mathbf{b},\mathbf{\Sigma}\big{)}\) as the distribution with density \(\frac{1}{Z}\exp\bigl{(}-\frac{1}{2}(\mathbf{a}-\mathbf{b})^{*}\mathbf{\Sigma}^{-1}(\mathbf{a}- \mathbf{b})\bigr{)}\). It is well defined if \(Z\) is non-zero and finite.
## 3 Primal Model
We will now essentially follow the work of (Tipping & Bishop, 1999) and redefine the model distributions. This section corresponds to the primal formulation and we only consider the feature representations. It does not yet introduce the kernel representations, which will appear in the dual formulation (Section 4).
### Model and Latent Space
**Factor Analysis.** The starting point is to consider a _factor analysis_ relationship (Bartholomew et al., 2011; Basilevsky, 2009) between the feature observations \(\mathbf{\phi}\) and the latent variables \(\mathbf{h}\). In particular, we consider
\[\mathbf{\phi}=\mathbf{W}\mathbf{h}+\mathbf{\mu}+\mathbf{\zeta}. \tag{1}\]
The observations \(\mathbf{\phi}\) live in the primal space \(\mathcal{H}_{\mathcal{E}}\) of dimension \(N\). We consider an isotropic normal noise \(\mathbf{\zeta}\sim\mathcal{N}\big{(}\mathbf{0},\sigma^{2}\mathbf{I}_{\mathcal{H}_{ \mathcal{E}}}\big{)}\) of variance \(\sigma^{2}\in\mathbb{R}_{>0}\) and a mean \(\mathbf{\mu}\in\mathcal{H}_{\mathcal{E}}\).
**Latent Space \(\mathcal{L}\).** The latent variables \(\mathbf{h}\) on the other hand live in a latent dual space \(\mathcal{L}\subset\mathcal{E}\) of dimension \(q\leq N\). They are related by a primal _interconnection linear operator_\(\mathbf{W}\). As it was the case before with \(\mathbf{\Phi}\), the interconnection operator does not project to the full space \(\mathcal{H}_{\mathcal{E}}\) because of its reduced dimensionality. It therefore projects to yet another feature space \(\mathcal{H}_{\mathcal{L}}\subset\mathcal{H}_{\mathcal{E}}\), which acts as the primal equivalent of the latent space \(\mathcal{L}\). The equality of these two spaces only holds if \(q=N\). We will therefore consider the mappings \(\mathbf{W}^{*}:\mathcal{H}_{\mathcal{E}}\to\mathcal{L}\) and \(\mathbf{W}:\mathcal{L}\to\mathcal{H}_{\mathcal{L}}\). The identity over \(\mathcal{L}\) can be written as \(\mathbf{I}_{\mathcal{L}}=\sum_{p=1}^{q}\mathbf{r}_{p}\mathbf{r}_{p}^{*}\), over \(\mathcal{H}_{\mathcal{L}}\) as \(\mathbf{I}_{\mathcal{H}_{\mathcal{L}}}=\sum_{p=1}^{q}\mathbf{\varrho}_{p}\mathbf{\varrho}_{p}^{*}\) and finally the identity over \(\mathcal{H}_{\mathcal{E}}\) rewritten as \(\mathbf{I}_{\mathcal{H}_{\mathcal{E}}}=\mathbf{I}_{\mathcal{H}_{\mathcal{L}}}+\mathbb{P}_ {\mathcal{H}_{\mathcal{L}}^{\perp}}\), with \(\mathbb{P}_{\mathcal{H}_{\mathcal{L}}^{\perp}}\) as a projector over the null space of \(\mathbf{W}^{*}\circ\mathbf{W}\). This is summarized in the last column of Table 2.
### Feature Distributions
**Latent-Based Generation.** The relation between the feature observations and the latent variables being set up
\begin{table}
\begin{tabular}{l l c c c} \hline \hline
**Dimension** & \(d\) & \(N\) & \(q\) \\ \hline \multirow{4}{*}{**Centered Kernels**} & **Space** & \(\mathcal{
(Eq. (1)), we can derive the conditional probability of the feature observations given a latent variable:
\[\mathbf{\phi}|\mathbf{h}\sim\mathcal{N}\left(\mathbf{W}\mathbf{h}-\mathbf{\mu},\sigma^{2}\mathbf{I}_{ \mathcal{H}_{\mathcal{E}}}\right). \tag{2}\]
As discussed earlier, we see that the latent variables do not participate to the full scope of the observations in \(\mathcal{H}_{\mathcal{E}}\), but only to their component in \(\mathcal{H}_{\mathcal{L}}\). The rest is only constituted from the isotropic normal noisy mean. This distribution can be interpreted as a generative one: given a latent variable, we can sample a variable in feature space.
**Absolute Generation**. Considering the latent prior \(\mathbf{h}\sim\mathcal{N}\big{(}\mathbf{0},\mathbf{I}_{\mathcal{L}}\big{)}\), we can derive the marginal distribution of the observations in feature space:
\[\mathbf{\phi}\sim\mathcal{N}\left(\mathbf{\mu},\mathbf{W}\circ\mathbf{W}^{*}+\sigma^{2}\mathbf{I}_ {\mathcal{H}_{\mathcal{E}}}\right). \tag{3}\]
It can be considered as the data distribution of the model. Sampling from it also means generating feature representations in a more absolute way, _i.e._, without considering any latent variable, or more precisely considering a random latent variable according to its prior. As a consequence of Eq. (2) and the isotropic aspect of the latent prior, we see that the observations are only non-isotropically distributed in \(\mathcal{H}_{\mathcal{L}}\). Again, the rest is only the isotropically normally noisy mean. In other words, this means that the model parameter \(\mathbf{W}\) only influences \(\mathbf{\phi}\) for its components in \(\mathcal{H}_{\mathcal{L}}\).
### Training the Model
**Maximum Likelihood.** As we now have the marginal distribution of the model (Eq. (3)), the goal is to find the optimal hyperparameters \(\mathbf{W}\) and \(\mathbf{\mu}\) to match the set of observations \(\{\mathbf{\phi}_{i}\}_{i=1}^{N}\). One way to determine them is by maximizing the likelihood of our observations. The _Maximum Likelihood_ (ML) estimator for the hyperparameters is given by:
\[\mathbf{\mu}_{\rm ML} = \mathbf{\phi}_{c}, \tag{4}\] \[\mathbf{W}_{\rm ML} = \sum_{p=1}^{q}\sqrt{\lambda_{p}/N-\sigma^{2}}\mathbf{v}_{p}\mathbf{r}_{p} ^{*}, \tag{5}\]
with \(\{(\lambda_{p},\mathbf{v}_{p})\}_{p=1}^{q}\) the \(q\) dominant eigenpairs of \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\) (\(\lambda_{1}\geq\cdots\geq\lambda_{q}\geq\cdots\lambda_{N}\)), and \(\{\mathbf{r}_{p}\}_{p=1}^{q}\) and arbitrary orthonormal basis of the latent space \(\mathcal{L}\). The choice for the latter basis is arbitrary and makes the model rotational invariant in latent space. An additional condition is that \(\sigma^{2}\leq\lambda_{q}/N\). It is not surprising to see that the optimal mean \(\mathbf{\mu}_{\rm ML}\) corresponds to the mean of the observations \(\mathbf{\phi}_{c}\). We observe that \(\mathbf{W}_{\rm ML}\) corresponds to the eigendecomposition of the centered covariance, at the exception that the noise assumption is substracted from its spectrum. By looking back at Eq. (1), it makes sense to avoid the noise in \(\mathbf{W}_{\rm ML}\) as it is still going to be added by the term \(\mathbf{\zeta}\).
**Noise Variance.** Maximizing the likelihood as a function of \(\sigma^{2}\) leads to
\[\sigma_{\rm ML}^{2}=\frac{1}{N(N-q)}\sum_{p=q+1}^{N}\lambda_{p}. \tag{6}\]
The eigenvalue \(\lambda_{p}\) corresponds to the variance for each component \(\mathbf{v}_{p}\) of the covariance \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\). The total variance of the data, noise included, is equal to \(\frac{1}{N}\sum_{p=1}^{N}\lambda_{p}\) and the variance learned by the model through the primal interconnection operator to \(\frac{1}{N}\sum_{p=1}^{q}\lambda_{p}\). Hence, the maximum likelihood estimator for the noise variance \(\sigma_{\rm ML}^{2}\) can be interpreted as the mean of the variance that is discarded by the model. It also verifies the earlier condition that \(\sigma^{2}\leq\lambda_{q}/N\), as the eigenvalues are taken in descending order. It can be interpreted as the normalized mean variance of the left over eigendirections, _i.e._ the orthogonal space of the latent space: \(\mathcal{L}^{\perp}=\mathcal{E}\backslash\mathcal{L}\). By consequence, we may decide to choose the latent dimension \(q=\dim(\mathcal{L})\) and deduct \(\sigma_{\rm ML}^{2}\). In the opposite, we may also decide to set an arbitrary \(\sigma^{2}\) and deduct the latent dimension \(q\) instead. We therefore can consider either \(\sigma^{2}\) or \(q\) as an additional hyperparameter. We must however keep in mind that this is strongly going to be influenced by the distribution of the eigenvalues and that the latent dimension \(q\) for the same \(\sigma^{2}\) may heavily vary from application to application.
**Uncentered Features.** We may also consider not to consider the mean as an optimizable hyperparameter and set it arbitrarily to \(\mathbf{\mu}=\mathbf{0}\). In this case, Eq. (5) would be the same at the difference that the \(\mathbf{W}_{\rm ML}\) would be constructed from the dominant eigenpairs of the uncentered covariance \(\mathbf{\Phi}\circ\mathbf{\Phi}^{*}\) instead of its centered counterpart \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\).
### Dimensionality Reduction in Feature Space
**Latent Projection.** Up to now, we only considered the distribution of the feature variables \(\mathbf{\phi}\). We can also calculate the posterior distribution of the latent variable \(\mathbf{h}\) given the primal feature variable \(\mathbf{\phi}\):
\[\mathbf{h}|\mathbf{\phi}\sim\mathcal{N}\left(\mathbf{\Sigma}_{\mathbf{h}|\mathbf{\phi}}^{-1} \circ\mathbf{W}^{*}(\mathbf{\phi}-\mathbf{\mu}),\sigma^{2}\mathbf{\Sigma}_{\mathbf{h}|\mathbf{\phi}}^ {-1}\right), \tag{7}\]
with \(\mathbf{\Sigma}_{\mathbf{h}|\mathbf{\phi}}=\big{(}\mathbf{W}^{*}\circ\mathbf{W}+\sigma^{2}\mathbf{I}_{ \mathcal{L}}\big{)}^{-1}\). The mean of the distribution can be considered as a pseudo-inverse of the observation \(\mathbf{\phi}\), but regularized by \(\sigma^{2}\). This regularization ensures to avoid the noise. If the prior of the latent variables was isotropic, this is not the case anymore for the posterior. If we consider the maximum likelihood estimator for the primal interconnection operator \(\mathbf{W}_{\rm ML}\), the variance becomes \(\sigma^{2}\mathbf{\Sigma}_{\mathbf{h}|\mathbf{\phi}}^{-1}=N\sigma^{2}\sum_{p=1}^{q}\lambda _{p}^{-1}\mathbf{r}_{p}\mathbf{r}_{p}^{*}\). It can be interpreted as the uncertainty for each component of the latent variable \(\mathbf{h}\) (w.r.t. the eigendirection \(\mathbf{r}_{p}\)), due to the noise assumption. By consequence, the greater the explained variance \(\lambda_{p}\) for the eigendirection \(\mathbf{v}_{p}\) of the covariance \(\mathbf{\Phi}_{c}\circ\mathbf{\Phi}_{c}^{*}\), the smaller
the corresponding uncertainty on the component \(\mathbf{r}_{p}\) of the latent variable \(\mathbf{h}\). For each observation in feature space \(\mathbf{\phi}\), this returns a distribution for the latent variable \(\mathbf{\phi}\) and can therefore be considered as a sort of probabilistic projection in latent space \(\mathcal{L}\).
**Maximum A Posteriori.** Up to now, we were only considering distributions. The only way to go from a feature representation to a latent variable or the opposite was probabilistic. In order to have a deterministic approach, we need proper mappings. One way is to consider the _Maximum A Posteriori_ (MAP) of \(\mathbf{h}\) given \(\mathbf{\phi}\). It maps the feature observation \(\mathbf{\phi}\in\mathcal{H}_{\mathcal{E}}\) to latent variable \(\mathbf{h}_{\mathrm{MAP}}\in\mathcal{L}\), hence reducing the dimensionality of any input to that of the latent space. To allow it to work for any input \(\mathbf{\varphi}\in\mathcal{H}\), we may again consider the projection \(\mathbf{\phi}=\mathbf{I}_{\mathcal{H}_{\mathcal{E}}}\mathbf{\varphi}\). As \(\mathbf{W}_{\mathrm{ML}}^{*}\circ\mathbf{I}_{\mathcal{H}_{\mathcal{E}}}=\mathbf{W}_{ \mathrm{ML}}^{*}\):
\[\mathbf{h}_{\mathrm{MAP}}= \left(\mathbf{W}_{\mathrm{ML}}^{*}\circ\mathbf{W}_{\mathrm{ML}}+\sigma^{ 2}\mathbf{I}_{\mathcal{L}}\right)^{-1} \tag{8}\] \[\circ\mathbf{W}_{\mathrm{ML}}^{*}\left(\mathbf{\varphi}-\mathbf{\varphi}_{c }\right).\]
To map back to the feature space \(\mathcal{H}_{\mathcal{L}}\), we may consider the _maximum a posteriori_ of \(\mathbf{\phi}\) given \(\mathbf{h}\) (Eq. (3)). This gives
\[\mathbf{\phi}_{\mathrm{MAP}}=\mathbf{W}_{\mathrm{MAP}}\mathbf{h}+\mathbf{\phi}_{c}. \tag{9}\]
The final projection reads
\[\mathbf{\phi}_{\mathrm{MAP}}= \mathbf{W}_{\mathrm{ML}}\circ\left(\mathbf{W}_{\mathrm{ML}}^{*}\circ\bm {W}_{\mathrm{ML}}+\sigma^{2}\mathbf{I}_{\mathcal{L}}\right)^{-1} \tag{10}\] \[\circ\mathbf{W}_{\mathrm{ML}}^{*}\left(\mathbf{\varphi}-\mathbf{\varphi}_{c }\right)+\mathbf{\phi}_{c}.\]
**No Noise.** We may also decide not to consider \(\sigma^{2}\) as a parameter to optimize and set it to an arbitrary value. The latent dimensions \(q\) could also be set an arbitrary value, without it to be related to the latent dimension \(q\) according to Eq. (6). We notice that in the limit of \(\sigma^{2}\to 0\), we recover the classical Principal Component Analysis reconstruction scheme. Indeed the conditional probability distributions become exact relations. We also notice that the condition \(\sigma^{2}\leq\lambda_{q}/N\) (Prop. 3) is then always satisfied. Furthermore, when \(q=\dim(\mathcal{H}_{\mathcal{E}})\), the reconstruction is perfect in \(\mathcal{H}_{\mathcal{E}}\) and in particular for our original observations \(\{\mathbf{\varphi}_{i}\}_{i=1}^{N}\) and \(\mathbf{\varphi}_{c}\) (as we have \(\mathbf{\phi}_{i}=\mathbf{\varphi}_{i}\)). Indeed, we would have
\[\mathbf{h}_{\mathrm{MAP}}=\mathbf{W}_{\mathrm{ML}}^{+}\left(\mathbf{\varphi}-\mathbf{\varphi} _{c}\right), \tag{11}\]
with \(\mathbf{W}_{\mathrm{ML}}^{+}\) the Moore-Penrose pseudo-inverse of \(\mathbf{W}_{\mathrm{ML}}\). We note here the symmetry with Eq. (9). If the maximum likelihood estimator for \(\sigma^{2}\) is to be respected (Eq. (6)), this would mean that all components are kept (\(\mathcal{L}=\mathcal{E}\)) and the model reconstructs the full feature variance. In this case, the primal interconnection operator would become \(\mathbf{W}_{\mathrm{ML}}=\sum_{p=1}^{N}\sqrt{\lambda_{p}/N}\mathbf{v}_{p}\mathbf{r}_{p}^{*}\) and be invertible. Its Moore-Penrose pseudo-inverse would become an exact inverse. Eqs. (9) and (11) would become exact opposites and there would be no loss due to the dimensionality reduction as there would be no noise to discard. By consequence, the reduction would become an identity over \(\mathcal{H}_{\mathcal{E}}\): \(\mathbf{\phi}_{\mathrm{MAP}}-\mathbf{\phi}_{c}=\mathbf{I}_{\mathcal{H}_{\mathcal{L}}} \left(\mathbf{\varphi}-\mathbf{\varphi}_{c}\right)\).
## 4 Dual Model
**Kernels without Dual.** In (Zhang et al., 2004), the authors made the kernel matrix appear by considering the new observations \(\left\{\sum_{i=1}^{d}\mathbf{u}_{i}\mathbf{u}_{j}^{*}\phi(\mathbf{x}_{i})\right\}_{j=1}^{N}\). In other words, each new datapoint consists in one particular feature of the feature map, for each original datapoint. If the original datapoints were organized as a matrix in \(\mathbb{R}^{N\times d}\), this would correspond to taking its transpose as new datapoints. The outer product of the covariance matrix is transformed to the inner product of the kernel matrix. If indeed this formulation makes the kernel appear, it is not a dual formulation of the original problem, but another problem. In this section, we show how the spaces defined hereabove help us build an equivalent dual formulation of the problem.
**Dual Formulation.** While keeping an equivalence with the primal model, we will now see that we can directly work in dual spaces \(\mathcal{E}\) and \(\mathcal{L}\) without considering the feature spaces at all, _i.e._ resorting to the primal space \(\mathcal{H}\) and its subsets. As we did for the primal feature variable \(\mathbf{\phi}\), we will consider \(\mathbf{k}_{c}=\mathbf{\Phi}_{c}^{*}(\mathbf{\phi}-\mathbf{\phi}_{c})=\sum_{i=1}^{N}k_{c}(\mathbf{x },\mathbf{x}_{i})\mathbf{e}_{i}\) to represent the image in \(\mathcal{E}\), of a random variable \(\mathbf{x}\in\mathcal{X}\). We will refer to it as a _dual feature variable_.
### Representation
Considering the dual spaces, we can always express the interconnection operator \(\mathbf{W}\) in the (non-orthonormal) basis \(\{\mathbf{\phi}_{1}-\mathbf{\phi}_{c},\dots,\mathbf{\phi}_{N}-\mathbf{\phi}_{c}\}\). As a consequence, we can always write
\[\mathbf{W}=\mathbf{\Phi}_{c}\circ\mathbf{A}, \tag{12}\]
with \(\mathbf{A}:\mathcal{L}\rightarrow\mathcal{L}\), the dual interconnection operator. Given the maximum likelihood estimator for the primal interconnection operator \(\mathbf{W}_{\mathrm{ML}}\), we can directly deduce the dual one:
\[\mathbf{A}_{\mathrm{ML}}=\sum_{p=1}^{q}\sqrt{1/N-\sigma^{2}\lambda_{p}^{-1}}\mathbf{ \epsilon}_{p}\mathbf{r}_{p}^{*}, \tag{13}\]
with \(\{(\lambda_{p},\mathbf{\epsilon}_{p})\}_{p=1}^{q}\) the \(q\) dominant eigenpairs of \(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\) and \(\{\mathbf{r}_{p}\}_{p=1}^{q}\) an arbitrary orthonormal basis of the latent space \(\mathcal{L}\). The rotational invariance of the dual interconnection operator \(\mathbf{A}_{\mathrm{ML}}\) is inherited from its primal counterpart \(\mathbf{W}_{\mathrm{ML}}\). Again, if we consider an optimized mean \(\mathbf{\mu}=\mathbf{0}\), we would have the relation \(\mathbf{W}_{\mathrm{ML}}=\mathbf{\Phi}\circ\mathbf{A}_{\mathrm{ML}}\) with \(\mathbf{A}_{\mathrm{ML}}\) then based on the eigenpairs of the non-centered \(\mathbf{\Phi}^{*}\circ\mathbf{\Phi}\) instead. Using the same structure for \(\mathbf{A}_{\mathrm{ML}}\), the optimal (primal) interconnection operator \(\mathbf{W}_{\mathrm{ML}}\) could be expressed in the (non-orthonormal) basis \(\{\mathbf{\phi}_{1},\dots,\mathbf{\phi}_{N}\}\).
### Kernel Distributions
**Projection and Generation**. We can also consider the dual counterparts of the distributions of the primal model (Eqs. (2) and (7)). For the sake of simplicity and to avoid
heavier equations with non-centered kernels, we will only consider here the equations of the trained model, in particular with \(\mathbf{\mu}_{\mathrm{ML}}=\mathbf{\phi}_{c}\) leading to centered kernels:
\[\mathbf{k}_{c}|\mathbf{h} \sim \mathcal{N}\big{(}(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c})\circ\mathbf{A }_{\mathrm{ML}}\mathbf{h},\sigma^{2}\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\big{)}, \tag{14}\] \[\mathbf{h}|\mathbf{k}_{c} \sim \mathcal{N}\left(\mathbf{\Sigma}_{\mathbf{h}|\mathbf{k}_{c}}^{-1}\circ\mathbf{A }_{\mathrm{ML}}\mathbf{k}_{c},\mathbf{\Sigma}_{\mathbf{h}|\mathbf{k}_{c}}^{-1}\right), \tag{15}\]
with \(\mathbf{\Sigma}_{\mathbf{h}|\mathbf{k}_{c}}=\big{(}\mathbf{A}_{\mathrm{ML}}^{*}\circ\left(\bm {\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\right)\circ\mathbf{A}_{\mathrm{ML}}+\sigma^{2}\bm {I}_{\mathcal{L}}\big{)}^{-1}\).
### Dimensionality Reduction in Kernel Space
**Maximum A Posteriori.** This now allows us to consider the dimensionality reduction in kernel space in a similar fashion as in Section 3.4. Again we consider the MAP of the latent variable \(\mathbf{h}\) given the kernel representation \(\mathbf{k}_{c}\):
\[\begin{split}\mathbf{h}_{\mathrm{MAP}}=&\left(\mathbf{A}_{ \mathrm{ML}}^{*}\circ\left(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\right)\circ \mathbf{A}_{\mathrm{ML}}+\sigma^{2}\mathbf{I}_{\mathcal{L}}\right)^{-1}\\ &\circ\mathbf{A}_{\mathrm{ML}}\mathbf{k}_{c},\end{split} \tag{16}\]
and similarly with the MAP of the kernel representation \(\mathbf{k}_{c}\) given the latent variable \(\mathbf{h}\):
\[\left(\mathbf{k}_{c}\right)_{\mathrm{MAP}}=\left(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi} _{c}\right)\circ\mathbf{A}_{\mathrm{ML}}\mathbf{h}. \tag{17}\]
As for the primal model, the dimensionality reduction in dual is computed as \(\left(\mathbf{k}_{c}\right)_{\mathrm{MAP}}=\left(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi} _{c}\right)\circ\mathbf{A}_{\mathrm{ML}}\mathbf{h}_{\mathrm{MAP}}\).
**No Noise.** Again, considering \(\sigma^{2}\to 0\) makes both dual conditional distributions become exact relations. In a ML context for \(\sigma^{2}\) (Eq. (6)), this would imply that \(q=\dim(\mathcal{E})\) and we would recover an identity \(\left(\mathbf{k}_{c}\right)_{\mathrm{MAP}}=\mathbf{k}_{c}\), _i.e._ no reduction. Without considering a ML context for \(\sigma^{2}\to 0\) and choosing an arbitrary \(q\leq\dim(\mathcal{E})\), the reduction become exactly the reconstruction done in KPCA.
### Kernel Sampling
**Probabilistic Sampling.** The dual counterpart of Eq. (3) after training is given by
\[\mathbf{k}_{c}\sim\mathcal{N}\left(\mathbf{0},\mathbf{A}_{\mathrm{ML}}^{*}\circ\mathbf{A}_{ \mathrm{ML}}+\sigma^{2}\left(\mathbf{\Phi}_{c}^{*}\circ\mathbf{\Phi}_{c}\right)^{-1} \right). \tag{18}\]
The covariance \(\mathbf{A}_{\mathrm{ML}}^{*}\circ\mathbf{A}_{\mathrm{ML}}+\sigma^{2}\left(\mathbf{\Phi}_{ c}^{*}\circ\mathbf{\Phi}_{c}\right)^{-1}\) can be decomposed as \(\mathbf{B}\circ\mathbf{B}^{*}\), with \(\mathbf{B}:\mathcal{E}\to\mathcal{E}:N^{-1/2}\sum_{p=1}^{q}\lambda_{p}\mathbf{\epsilon }_{p}\mathbf{\epsilon}_{p}^{*}+\sum_{p=p+1}^{N}\sigma\lambda_{p}^{1/2}\mathbf{ \epsilon}_{p}\mathbf{\epsilon}_{p}^{*}\) and \(\left\{\mathbf{\epsilon}_{i}\right\}_{i=1}^{N}\) any arbitrary orthonormal basis of the latent space \(\mathcal{E}\). This decomposition allows us to sample \(\mathbf{k}_{c}\) on the trained model with \(\mathbf{k}_{c}=\mathbf{B}\mathbf{\xi}\) with \(\mathbf{\xi}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{\mathcal{E}})\). We see that \(\mathbf{B}\) is rotational invariant, which is not surprising as this is also the case for the distribution from which \(\mathbf{\xi}\) is sampled. In practice and for simplicity, we may decide too choose the canonical basis for \(\left\{\mathbf{\epsilon}_{i}\right\}_{i=1}^{N}\) as any choice would be identified to the same covariance and to the same sampling of \(\mathbf{k}_{c}\). We will therefore assume that \(\mathbf{\epsilon}_{i}=\mathbf{e}_{i}\) for all \(i=1,\dots,N\). In that particular case, \(\mathbf{B}\) is self-adjoint and by consequence corresponds to the matrix square root of \(\mathbf{A}_{\mathrm{ML}}^{*}\circ\mathbf{A}_{\mathrm{ML}}+\sigma^{2}\left(\mathbf{\Phi}_{ c}^{*}\circ\mathbf{\Phi}_{c}\right)^{-1}\).
**KPCA Sampling** The classical sampling done by KPCA (Schreurs and Suykens, 2018) corresponds to the limit of \(\sigma^{2}\to 0\) for an arbitrary latent dimension \(q\). Unless the latent dimension is chosen as \(q=\dim(\mathcal{E})\), the sampling in that case can never cover \(\mathcal{E}\) fully, but rather \(\mathcal{L}\), as \(\mathbf{B}\) is not a bijection. The second term of \(\mathbf{B}\) (\(\sum_{p=q+1}^{N}\sigma\lambda_{p}^{1/2}\mathbf{\epsilon}_{p}\mathbf{\epsilon}_{p}^{*}\)) allows \(\mathbf{B}\) to be a bijection no matter what is the choice of the latent dimension \(q\), as long as \(\sigma^{2}>0\). We thus always sample in the full \(\mathcal{E}\). This can be observed at Fig. 2.
## 5 Experiments
**Hilbert Spaces to Matrices.** Working in Hilbert spaces is helpful to treat possibly infinite dimensional feature maps, but not very useful for practical applications. Matrix representations are possible in primal if \(d\) is finite and in dual if \(N\) is finite. It suffices to consider the different canonical basis. For the latent space \(\mathcal{L}\), this enforces a unique representation for \(\mathbf{W}_{\mathrm{ML}}\) and \(\mathbf{A}_{\mathrm{ML}}\), but we must keep in mind that they are rotational invariant. All the operators and elements described before are then represented in matrix or vector format (Table 3). We will use the tilde to denote these matrices and use software-like notation by denoting with \((\cdot)_{i_{1}:i_{2},j_{1}:j_{2}}\) the matrix truncated to its \(i_{1}\) to \(i_{2}\) rows and \(j_{1}\) to \(j_{2}\) columns.
**Preimage.** Given a dual representation, we will also consider the _kernel smoother_ preimage method, as suggested by (Schreurs and Suykens, 2018):
\[\hat{\mathbf{x}}=\frac{\sum_{i=1}^{N}(\tilde{\mathbf{k}})_{i}\mathbf{x}_{i}}{\sum_{i=1}^{N}( \tilde{\mathbf{k}})_{i}}. \tag{19}\]
In practice, as we work with centered feature maps and kernels, it may be that the kernel smoother may be unstable due to its normalization term. We therefore may consider to add a stabilization term.
Figure 2: Schematic overview of the dual sampling in Prob. PCA compared to the generation in KPCA.
Figure 3: Visualisation of the Probabilistic PCA reconstruction (in blue) the classical KPCA (in red). Samples generated by are also given (in grey). The dataset contains \(N=20\) points (in black).
### Model
The direct application of the theoretical discussions of the previous sections leads to the decompositions \(\tilde{\mathbf{K}}_{c}=\tilde{\mathbf{E}}\tilde{\mathbf{\Lambda}}\tilde{\mathbf{E}}^{\top}\), \(\tilde{\mathbf{C}}_{c}=\tilde{\mathbf{V}}\tilde{\mathbf{\Lambda}}\tilde{\mathbf{V}}^{\top}\), \(\tilde{\mathbf{\Phi}}_{c}=\tilde{\mathbf{V}}\tilde{\mathbf{\Lambda}}^{1/2}\tilde{\mathbf{E}}^{\top}\). The value of the operators after training are given in Table 4. Once the model is trained, we can verify that \(\tilde{\mathbf{W}}=\tilde{\mathbf{\Phi}}_{c}\tilde{\mathbf{A}}\).We can also have a look at the hidden variables. A way to do it is to consider the MAP of \(\mathbf{h}\) given \(\mathbf{\phi}\) or \(\mathbf{k}\). We have
\[\mathbf{h}_{\rm MAP} =N\tilde{\mathbf{\Lambda}}_{1:q,1:q}^{-1}\tilde{\mathbf{A}}^{\top}\tilde{ \mathbf{k}}_{c}\qquad\text{(if $\mathrm{rank}(\tilde{\mathbf{K}}_{c})\geq q$)}, \tag{20}\] \[=N\tilde{\mathbf{\Lambda}}_{1:q,1:q}^{-1}\tilde{\mathbf{W}}^{\top}\big{(} \tilde{\mathbf{\phi}}-\tilde{\mathbf{\phi}}_{c}\big{)}\quad\text{(if $\mathcal{H}$ is finite)}, \tag{21}\]
and
\[\big{(}\mathbf{k}_{c}\big{)}_{\rm MAP} =\tilde{\mathbf{K}}_{c}\tilde{\mathbf{A}}\tilde{\mathbf{h}}\qquad\quad\text{ (if $\mathrm{rank}(\tilde{\mathbf{K}}_{c})\geq q$)}, \tag{22}\] \[\mathbf{\phi}_{\rm MAP} =\tilde{\mathbf{W}}\tilde{\mathbf{h}}+\tilde{\mathbf{\phi}}_{c}\qquad\qquad \quad\text{(if $\mathcal{H}$ is finite)}. \tag{23}\]
As developed in Section 4, we can easily generate samples in both feature and kernel representations. For the latter and in canonical basis, it becomes
\[\tilde{\mathbf{k}}_{c}=\tilde{\mathbf{B}}\tilde{\mathbf{u}},\qquad\text{with $\tilde{\mathbf{u}}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_{N})$}. \tag{24}\]
### Examples
As the primal case is already treated by (Tipping & Bishop, 1999), we consider here the model in its dual formulation. A toy example can by found in Fig. 3. We use an RBF kernel \(k(\mathbf{x},\mathbf{y})=\exp\bigl{(}-\|\mathbf{x}-\mathbf{y}\|_{2}^{2}/(2\gamma^{2})\bigr{)}\) with bandwidth \(\gamma=2\). As the number of components increases, the mean variance of the \(N-q\) unused components \(\sigma^{2}\) becomes smaller and the model tends to the classical KPCA model. Another way the reduce \(\sigma^{2}\) is to increase the number of components \(q\), with \(\sigma^{2}\to 0\) when \(q\to N\). This can be observed in Fig. 2(c): the Probabilistic PCA model resembles closely the KPCA model, whereas more variance is left over, _i.e._ not projected back, in Fig.s 2(a) and 2(b). The results of the generation is Gaussian, which is a consequence of the linearity of the preimage method chosen (Eq. (19)). Here again, as the number of components increases and \(\sigma^{2}\) decreases, the model is allowed to project back more variance and the distribution becomes wider. Another example on the MNIST dataset (LeCun & Cortes, 2010) with the RBF kernel with \(\gamma=4\) is given at Fig. 4.
## 6 Conclusion
**Probabilistic Interpretation.** By reformulating the Prob. PCA model in Hilbert space, we were able to define a formulation of it. Likewise Prob. PCA in primal was englobing classical PCA (with \(\sigma^{2}\to 0\)), Prob. PCA in dual is also englobing KPCA in the same limit. Furthermore, we are now able to sample in dual space, enhancing the understanding of the generation done with KPCA.
**Limitations.** As most kernel methods, the model is still limited by the need of a preimage method to go back to the input space once a sample is projected or generated. Furthermore, training the model in dual required to find the \(q\) first eigenvalues of the kernel matrix, which may become expensive as the number of datapoints \(N\) increases. Generating renders the problem even worse as it requires the computation of all eigenvalues. The model also requires to determine a \(\sigma^{2}\) or alternatively a latent dimension \(q\).
## Acknowledgements
EU: The research leading to these results has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program / ERC Advanced Grant E-DUALITY (787960). This paper reflects only the authors' views and the Union is not liable for any use that may be made of the contained information. Research Council KUL: Tensor Tools for Taming the Curse iBOF/23/064, Optimization frameworks for deep kernel machines C14/18/068. Flemish Government: FWO projects: GOA4917N (Deep Restricted Kernel Machines: Methods and Foundations), PhD/Postdoc grant. This research received funding from the Flemish Government (AI Research Program). Henri De Plaen and Johan A. K. Suykens are also affiliated to Leuven.AI - KU Leuven institute for AI, B-3000, Leuven, Belgium.
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Name** & **Space** & **Trained** \\ \hline \(\tilde{\mathbf{W}}\) & \(\mathbb{R}^{d\times q}\) & \(\tilde{\mathbf{V}}_{1:N,1:q}\big{(}\tilde{\mathbf{\Lambda}}_{1:q,1:q}/N-\sigma^{2}\mathbf{ I}_{q}\big{)}^{1/2}\) \\ \(\tilde{\mathbf{A}}\) & \(\mathbb{R}^{N\times q}\) & \(\tilde{\mathbf{E}}_{1:N,1:q}\big{(}\mathbf{I}_{q}/N-\sigma^{2}\big{(}\tilde{\mathbf{ \Lambda}}_{1:q,1:q}\big{)}^{-1}\big{)}^{1/2}\) \\ \(\tilde{\mathbf{B}}\) & \(\mathbb{R}^{N\times q}\) & \(\tilde{\mathbf{E}}\tilde{\mathbf{\Lambda}}^{1/2}\left[\begin{array}{c}(N)^{-1/2} \tilde{\mathbf{\Lambda}}_{1:q,1:q}^{1/2}&\mathbf{0}\\ \mathbf{0}&\sigma\mathbf{I}_{N-q}\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Value of the different operators in the canonical basis, after training.
\begin{table}
\begin{tabular}{l l l l} \hline \hline & **Name** & **Space** & **Values** \\ \hline \multirow{5}{*}{\begin{tabular}{} \end{tabular} } & \(\tilde{\mathbf{K}}_{c}\) & \(\mathbb{R}^{N\times N}\) & \((\tilde{\mathbf{k}}_{c})_{i,j}=k_{c}(\mathbf{x}_{i},\mathbf{x}_{j})\) \\ & \(\tilde{\mathbf{E}}\) & \(\mathbb{R}^{N\times N}\) & \(\big{(}\tilde{\mathbf{E}}\big{)}_{i,j}=\mathbf{e}_{i}^{*}\mathbf{\epsilon}_{j}\) \\ & \(\tilde{\mathbf{R}}\) & \(\mathbb{R}^{q\times q}\) & \(\tilde{\mathbf{R}}=\mathbf{I}_{q}\) \\ & \(\tilde{\mathbf{h}}\) & \(\mathbb{R}^{q}\) & \(\big{(}\tilde{\mathbf{h}}\big{)}_{p}=\mathbf{e}_{p}^{*}\mathbf{h}\) \\ & \(\tilde{\mathbf{k}}_{c}\) & \(\mathbb{R}^{N}\) & \(\big{(}\tilde{\mathbf{k}}_{c}\big{)}_{i}=\mathbf{e}_{i}^{*}\mathbf{k}_{c}\) \\ \hline \multirow{2}{*}{\begin{tabular}{} \end{tabular} } & \(\tilde{\mathbf{\Lambda}}\) & \(\mathbb{R}_{\geq 0}^{N\times N}\) & \(\tilde{\mathbf{\Lambda}}=\mathrm{diag}(3_{1},\ldots,\lambda_{N})\) \\ & \(\tilde{\mathbf{S}}\) & \(\mathbb{R}_{\geq 0}^{\times q}\) & \(\tilde{\mathbf{S}}=\mathrm{diag}(s_{1},\ldots,s_{q})\) \\ \hline \multirow{5}{*}{
\begin{tabular}{} \end{tabular} } & \(\tilde{\mathbf{C}}_{c}\) & \(\mathbb{R}^{d\times d}\) & \(\big{(}\tilde{\mathbf{C}}_{c}\big{)}_{i,j}=\big{(}\mathbf{u}_{i}^{*}\mathbf{\Phi}_{c} \big{)}\circ\big{(}\mathbf{u}_{j}^{*}\mathbf{\Phi}_{c}\big{)}^{*}\) \\ & \(\tilde{\mathbf{\Phi}}_{c}\) & \(\mathbb{R}^{d\times N}\) & \(\big{(}\tilde{\mathbf{\Phi}}_{c}\big{)}_{i,j}=\mathbf{u}_{i}^{*}\mathbf{\Phi}_{c}\mathbf{e}_{j}\) \\ \(\tilde{\mathbf{V}}\) & \(\mathbb{R}^{d\times N}\) & \(\big{(}\tilde{\mathbf{V}}\big{)}_{i,j}=\mathbf{u}_{i}^{*}\mathbf{v}_{j}\) \\ & \(\tilde{\mathbf{P}}\) & \(\mathbb{R}^{d\times q}\) & \(\big{(}\tilde{\mathbf{P}}\big{)}_{i,p}=\mathbf{v}_{i}^{*}\mathbf{\phi}_{p}\) \\ & \(\tilde{\mathbf{\phi}}\) & \(\mathbb{R}^{d}\) & \(\big{(}\mathbf{\phi}\big{)}_{i}=\mathbf{v}_{i}^{*}\mathbf{\phi}\) \\ & \(\tilde{\mathbf{\phi}}_{c}\) & \(\mathbb{R}^{d}\) & \(\big{(}\mathbf{\phi}_{c}\big{)}_{i}=\mathbf{v}_{i}^{*}\mathbf{\phi}_{c}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Representation of the various operators and elements in their respective canonical basis, as matrices and vectors. The primal | この論文では、確率主成分分析をヒルベルト空間で特徴づけ、最適解は対偶空間で表現されることを示しています。これにより、カーネル手法の生成的フレームワークを開発することができました。さらに、カーネル主成分分析を包含し、玩具データセットと実データセットでの動作を示しました。
Please let me know if you have any other questions. |
2305.06063 | Enhancing Quantum Support Vector Machines through Variational Kernel
Training | Quantum machine learning (QML) has witnessed immense progress recently, with
quantum support vector machines (QSVMs) emerging as a promising model. This
paper focuses on the two existing QSVM methods: quantum kernel SVM (QK-SVM) and
quantum variational SVM (QV-SVM). While both have yielded impressive results,
we present a novel approach that synergizes the strengths of QK-SVM and QV-SVM
to enhance accuracy. Our proposed model, quantum variational kernel SVM
(QVK-SVM), leverages the quantum kernel and quantum variational algorithm. We
conducted extensive experiments on the Iris dataset and observed that QVK-SVM
outperforms both existing models in terms of accuracy, loss, and confusion
matrix indicators. Our results demonstrate that QVK-SVM holds tremendous
potential as a reliable and transformative tool for QML applications. Hence, we
recommend its adoption in future QML research endeavors. | Nouhaila Innan, Muhammad Al-Zafar Khan, Biswaranjan Panda, Mohamed Bennai | 2023-05-10T11:30:43 | http://arxiv.org/abs/2305.06063v2 | # Enhancing Quantum Support Vector Machines through Variational Kernel Training
###### Abstract
Quantum machine learning (QML) has witnessed immense progress recently, with quantum support vector machines (QSVMs) emerging as a promising model. This paper focuses on the two existing QSVM methods: quantum kernel SVM (QK-SVM) and quantum variational SVM (QV-SVM). While both have yielded impressive results, we present a novel approach that synergizes the strengths of QK-SVM and QV-SVM to enhance accuracy. Our proposed model, quantum variational kernel SVM (QVK-SVM), leverages the quantum kernel and quantum variational algorithm. We conducted extensive experiments on the Iris dataset and observed that QVK-SVM outperforms both existing models in terms of accuracy, loss, and confusion matrix indicators. Our results demonstrate that QVK-SVM holds tremendous potential as a reliable and transformative tool for QML applications. Hence, we recommend its adoption in future QML research endeavors.
Quantum Machine Learning, Quantum Support Vector Machine, Kernel, Quantum Variational Algorithm, Classification.
## I Introduction
Quantum computing is an exciting and quickly growing field that could change many areas of science and technology. Machine learning (ML) is one of the most promising
quantum computing applications, where quantum algorithms can potentially provide exponential speedups over classical algorithms. This field is known as Quantum machine learning (QML). Quantum machine learning is an emerging field of research that combines the principles of quantum computing and machine learning. QML algorithms can solve complex problems more efficiently and cost-effectively than classical machine learning algorithms. One of the most promising QML algorithms is the quantum support vector machine (QSVM), an extension of the classical support vector machine (SVM) to the quantum realm.
The classical SVMs are a powerful class of ML algorithms for classification and regression analysis. The development of SVMs can be traced back to the early 1960s when Vladimir Vapnik and his colleagues began working on a new approach to pattern recognition [1, 2]. However, only in the 1990s did SVMs gain widespread attention in the ML community [3], thanks to Corinna Cortes and Vladimir Vapnik's pioneering work at AT&T Bell Labs. They introduced the idea of maximum-margin hyperplanes, decision boundaries that separate data points from different classes with the most significant possible margin [4].
This approach allowed SVMs to perform excellent generalization, even with small training datasets. Since then, SVMs have become one of the most extensively used and popular machine learning models and have been successfully applied to various fields, including image recognition, text classification, and bioinformatics. However, as the size of the dataset increases, the computational complexity of SVM also increases, making it difficult to handle large datasets. Still, the QSVM aims to overcome this limitation by leveraging the principles of quantum computing to accelerate the SVM algorithm. Over the years, there has been significant research in the field of QSVM, exploring various theoretical and practical aspects of the algorithm. Researchers have developed several techniques to enhance the performance of QSVM, including the development of quantum kernel methods, quantum feature maps, and quantum optimization techniques.
One of the early works in QSVM was proposed by Rebentrost _et al._ in 2014 [5], which introduced a quantum algorithm for SVM classification that provides an exponential speedup over classical algorithms. Another essential aspect of QSVM is its robustness to noise. In 2015, Li _et al._ demonstrated a QML algorithm for handwriting recognition on a four-qubit nuclear magnetic resonance (NMR) test bench [6]; the authors argued that quantum speedup would be highly attractive for tackling significant data challenges. However, this algorithm was specific to NMR-based systems and could not be easily applied to other QML platforms.
And after different interesting works, in 2019, Havlicek _et al._ demonstrated that supervised quantum machine learning models, including QSVM [7], can be robust to noise, increasing their practicality for real-world applications. Subsequently, several studies have been conducted to improve the performance of QSVM, including using quantum feature maps for kernel-based learning, as proposed by Park _et al._ in 2020 [9]. Another interesting research explores the potential use of quantum state encoding as a nonlinear feature map, enabling efficient computations in a large Hilbert space efficiently and pro
poses two approaches for building a quantum model for classification, illustrated with mini-benchmark datasets [8].
In contrast, Liu _et al._ established a rigorous quantum speedup for supervised classification using a general-purpose quantum learning algorithm that only requires classical access to data. This algorithm represents a significant advancement [10]. In the same year, Schuld _et al._ investigated the impact of quantum data encoding strategies on the efficacy of parametrized quantum circuits in approximating functions [11], and the authors showed that quantum models could realize all possible sets of Fourier coefficients. Therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators. This result has significant implications for developing QML algorithms to tackle complex data challenges.
In another 2021 paper [12], Schuld, M explored the theoretical foundations of the link between quantum computing and kernel methods in machine learning, systematically rephrased supervised QML models as kernel methods, replacing many near-term and fault-tolerant QML models with a general SVM whose kernel computes distances between data-encoding quantum states. This approach has the potential to significantly reduce the complexity of QML algorithms and improve their performance.
In 2022, Zhang _et al._ proposed a new quantum optimization algorithm for QSVM that can improve the efficiency and scalability of QSVM on large datasets [13]. These advancements in QSVM and its related techniques have made it a promising candidate for solving complex problems in various fields, including bioinformatics, finance, image recognition, and material physics. One of the recent works in QSVM was proposed by Jiang _et al._ in 2023 [14], which introduced a quantum algorithm for SVM classification that leverages the quantum phase estimation algorithm to estimate the kernel matrix. This approach leads to significant speedup compared to classical SVM algorithms, making QSVM a more efficient choice for large-scale datasets.
In this paper, we build upon these studies and suggest a new QML model for classification that merges the two more accurate approaches identified by Schuld, M and the different works mentioned above. Our model leverages the expressive power of parametrized quantum circuits as function approximators and uses a kernel method to compute distances between data-encoding quantum states. We present theoretical analyses and numerical simulations to demonstrate the potential of our model for tackling classification tasks. Our results suggest that our model outperforms existing QML algorithms, highlighting its potential for future real-world problems and applications.
This paper is divided as follows:
In SSII, We provide an overview of classical support vector machines, highlighting their principal features and limitations that motivate the exploration of more accurate implementations in quantum machine learning.
In SSIII, we describe the quantum model for support vector machines and explain the three implementations, including our proposed approach, the quantum variational kernel SVM.
In SSIV, we present the results obtained using Pennylane by comparing the accuracy,
loss, and confusion matrix indicators of the three quantum SVM models on the Iris dataset.
In SSV, we discuss our findings' implications and highlight future research directions in this field.
## II Classical support vector machine
In classical machine learning, SVMs are used in supervised models to analyze data for classification and regression. By using this algorithm, we can also perform binary classification and multi-classification. To understand this, we are taking an example. For simplicity, we are taking a binary classification example. Suppose we have a collection of circles and rectangles in a 2D plane. Our job is to classify the circles and rectangles. This problem has two types, (a) linear and (b) non-linear, as shown in Fig.1.
### Linear SVMs
First of all, we are discussing linear SVMs. We take a dataset of \(n\) points of the form \((x_{1},y_{1}),(x_{2},y_{2}),...(x_{n},y_{n})\). Here \(y_{i}\) are either 1 or \(-1\), and each \(x_{i}\) is a p-dimensional real vector. We have to draw the positive hyperplane \((H_{+})\), negative hyperplane \((H_{-})\), and margin as shown in Fig.2. We can find the margin using the formula \(=H_{+}+H_{-}\).
Given a \(D\)-dimensional vector \(\mathbf{X}_{0}\in\mathbb{R}^{D\times 1}\), and a \((D-1)\)-dimensional linear hyperplane \(\mathcal{H}:\mathbf{W}^{T}\mathbf{X}+\mathbf{B}-\mathbf{Y}=\mathbf{0}\), where \(\mathbf{W}=(w_{1},w_{2},\ldots,w_{n})\) is the weights vector, \(\mathbf{B}\) is the bias vector, and \(\Phi(\mathbf{X}_{n})\) is the projection of the point \(\mathbf{X}_{n}\) into the nonlinear feature space. The goal is to ascertain the hyperplane that optimally separates the vectorial points into
Figure 1: Graphical representation of linear and non-linear SVMs problems.
classes while maximizing the margin between the hyperplane and the closest datapoints from each class. Mathematically, we translate this as a quadratic programming problem with linear constraints, whereby our goal is to determine the value of the weights that will maximize the margin
\[\mathbf{W}^{*}=\underset{\mathbf{W}}{\text{arg max}}\ \frac{1}{||\mathbf{W}||_{2}} \left\{\min_{n}\ \mathbf{Y}_{n}\left[\mathbf{W}^{T}\Phi(\mathbf{X}_{n})+\mathbf{B}\right] \right\}, \tag{1}\]
where \(||\mathbf{W}||_{2}=\left(\sum_{i=1}^{n}w_{i}^{2}\right)^{1/2}\). Mathematically, we can translate this into the primal form SVM optimization problem \(\min_{n}\ \frac{1}{2}||\mathbf{W}||_{2}\) subject to \(\mathbf{Y}_{n}\left[\mathbf{W}^{T}\Phi(\mathbf{X}_{n})+\mathbf{B}\right]\geq \mathbf{1}\) for every \(n\).
### Non-linear SVMs
As shown in Fig.2, the support vectors are the vectors utilized to generate both the positive and negative hyperplanes. Maximizing the margin length in this specific model is imperative to achieve precise classification and high accuracy.
In order to effectively tackle the non-linear problem we are facing, the kernel trick presents a compelling solution. This technique involves using a kernel function with data points, acquiring higher-dimensional vector points in our feature space. A plethora of kernel functions exists, each tailored to solve different problems. Below, we present a comprehensive list of some of these kernel functions:
* Polynomial (Homogeneous): denoted as \(K(a_{i},a_{j})=(a_{i}\cdot a_{j})^{d}\), where \(d\) is a positive integer that determines the degree of the polynomial. By setting \(d\) to \(1\), it becomes a linear kernel that is particularly useful for linearly separable data.
Figure 2: Geometric components of support vector machines.
* Polynomial (Inhomogeneous): which incorporates a constant term \(r\) to the dot product of the input vectors, resulting in \(K(a_{i},a_{j})=(a_{i}\cdot a_{j}+r)^{d}\). This kernel is well-suited for capturing nonlinear relationships between the data.
* Sigmoid function (Hyperbolic tangent): based on the hyperbolic tangent function, takes the form \(K(a_{i},a_{j})=\tanh(ka_{i}\cdot a_{j}+c)^{d}\), where \(k\) and \(c\) are kernel parameters. This kernel can be used to model data that exhibits sigmoidal behavior and has been applied in various applications such as image classification and text mining.
After applying the kernel function to our data points, we have to do the same operation as in linear. Then we can complete the classification successfully. We modify the primal form of the linear SVM to include the slack variables \(\boldsymbol{\xi}\geq\mathbf{0}\,\min_{n}\,\frac{1}{2}||\mathbf{W}||_{2}+ \mathbf{C}\sum_{n}\boldsymbol{\xi}_{n}\quad\text{subject to}\,\,\,\mathbf{Y}_{n} \left[\mathbf{W}^{T}\Phi(\mathbf{X}_{n})+\mathbf{B}\right]\,\geq 1-\boldsymbol{ \xi}_{n}\,\,\text{for every}\,\,\,n.\) In addition to quadratic programming, numerous alternative techniques exist for resolving this problem. These include the approaches of Lagrange multipliers, sequential minimal optimization, interior point methods, gradient descent (GD), stochastic gradient descent (SGD), and kernel methods.
### Disadvantages of Classical SVMs
Despite the popularity of classical SVMs, they have certain limitations that constrain their optimal performance. One of the significant limitations is handling high-dimensional feature spaces, which can result in slow training times and overfitting problems. Another area for improvement is the dependence on kernel functions, which may not effectively capture complex data relationships. Furthermore, classical SVMs are not easily scalable to large datasets and demand extensive parameter tuning for accurate results. Researchers have turned to quantum machine learning to overcome these limitations and explore more precise and efficient alternatives. In the next section, we examine how quantum support vector machines can effectively tackle these challenges and provide a promising solution for enhancing classification performance.
## III Quantum Support Vector Machine
Quantum Support Vector Machine is a burgeoning area of research in quantum machine learning that offers promising potential for enhanced computational performance in classification and regression tasks. While classical SVM has been widely utilized in machine learning, QSVM exploits the unique properties of quantum mechanics to outperform classical SVM in specific applications. The QSVM algorithm involves mapping input data onto a quantum state, which is subsequently subjected to quantum circuit processing to generate the classification outcome. The described circuit comprises a sequence of quantum gates that manipulate the quantum state and execute the SVM algorithm. The classification outcome is obtained by measuring the circuit's output.
In previous works, as mentioned in the introduction sections (I), QSVMs have been implemented using various approaches, such as the quantum kernel method, the quantum matrix inversion method, and the quantum feature mapping method. Nevertheless, these methodologies possess certain constraints, such as high error rates, enormous computational resources, and scalability issues. This section will focus on three recent approaches to QSVM: the quantum kernel approach, the quantum variational approach, and a novel hybrid approach that combines the quantum kernel approach with the quantum variational circuit. These approaches have shown promising results and offer potential accuracy, scalability, and robustness improvements. The following subsections will describe each approach, highlighting the steps and circuits used to develop the QSVM models.
In each approach, the first step in our methodology involves the conversion of our classical datapoints into quantum states. To achieve this, we begin by encoding the data points using a quantum circuit, as depicted in Fig.3. Subsequently, we establish our data set and opt for the iris data set for simplicity. Our selection of qubits is based on the features outlined in our data set. We utilize a quantum model represented as follows:
\[f(x)=\langle\phi(x)|M|\phi(x)\rangle. \tag{2}\]
Here \(|\phi(x)\rangle\) is prepared using an encoding circuit, and \(M\) is the measurement operator. \(M\) is observable that is defined as:
\[M(\theta)=G^{\dagger}(\theta)\sigma_{z}^{0}G(\theta). \tag{3}\]
Figure 3: Quantum circuit architecture for QSVMS: Generalized model description.
### Quantum Kernel Support Vector Machine
We propose implementing support vector machines with a kernel computed by a quantum circuit in this approach. Specifically, we utilize the angle-embedding template in conjunction with a SWAP test to evaluate the quantum kernel; this method reduces the number of required qubits by half, rendering it more viable for real-world implementations.
The kernel function is a central concept in SVMs, which are a popular class of ML algorithms for classification and regression tasks; this kernel function is a measure of similarity between two data points in a high-dimensional feature space and is used to map the data into a space where a linear classifier can separate the classes. This method can be used with any kernel function, like the linear kernel and radial basis function (RBF) kernel, computed using a quantum circuit, and we call it the quantum kernel. Mathematically, the quantum kernel is represented by the following equation:
\[k(x_{1},x_{2})=|\langle\phi(x_{1})|\phi(x_{2})\rangle|^{2}, \tag{4}\]
Where \(x_{1}\) and \(x_{2}\) are the input feature vectors, and \(\phi(x_{i})_{i=1,2}\) denotes the quantum embedding of \(x\) into a quantum state with the angle encoding routines \(S(x_{1})\) and \(S(x_{2})\), we then apply the inverse embedding to one of the states and compute the overlap between the two states using a SWAP test, and the SWAP test is a simple quantum protocol that measures the overlap between two quantum states, we can represent this step by the following equation:
\[\langle SWAP\rangle=|\langle\phi(x_{1})\otimes\phi(x_{2})|SWAP|\phi(x_{1}) \otimes\phi(x_{2})\rangle|^{2}, \tag{5}\]
\(SWAP\) is the swap gate, and \(|\langle SWAP\rangle|^{2}\) represents the probability of measuring the two quantum embeddings in the same state. Finally, we use the Hermitian observable to measure the projector onto the initial state \(|0...0\rangle\langle 0...0|\), and Fig.4 present this circuit.
The advantage of this approach is that it has the potential to scale to larger datasets by utilizing quantum hardware with more qubits. As we mentioned, it also requires only half the number of qubits as the number of features, and this is because we can prepare
Figure 4: QK-SVM circuit.
the two data points on the same set of qubits using the angle-embedding template and then apply the inverse embedding to one of the states, as shown in Fig.5 using Pennylene [15].
### Quantum Variational Support Vector Machine
In this method, we propose a novel approach for training data directly by utilizing an ansatz for the variational circuit. This ansatz, a quantum operation applied in multiple layers, enhances expressivity. Although the variational circuit cannot optimize the exact cost, similar to SVMs, we have incorporated a bias term termed hinge loss in our quantum model to minimize the gap between the two. And in the quantum node, we explicitly apply the parameter shift differentiation method. The variational quantum circuit is given in Fig.3. We have given our method's encoding, processing, and measurement steps in this circuit. The quantum variational method is a key concept in quantum machine learning, a rapidly growing field that aims to leverage quantum computing to develop robust machine learning algorithms. This answer will discuss the quantum variational method and its applications in quantum machine learning.
Mathematically, the quantum variational method can be described as follows:
Suppose we have a parameterized quantum circuit that can be represented by the unitary operator \(U(\theta)\), where \(\theta\) is a vector of parameters. Given a set of training data (x\({}_{1}\), y\({}_{1}\)), (x\({}_{2}\), y\({}_{2}\)),..., (x\({}_{n}\), y\({}_{n}\)), where x\({}_{i}\) is an input and y\({}_{i}\) is the desired output. We want to find the values of \(\theta\) that minimize the cost function:
\[f(\theta)=\frac{1}{n}\sum_{i=1}^{n}L(y_{i},U(\theta)x_{i}). \tag{6}\]
Figure 5: QK-SVM circuit using Pennlyne.
Here, \(L(y,y^{\prime})\) measures the difference between the desired output y and the actual output y\({}^{\prime}\) produced by the quantum circuit. This cost function is typically chosen to be a function that can be efficiently computed on a classical computer. We use an iterative optimization algorithm such as gradient descent to find the optimal values of \(\theta\) that minimize the cost function. Starting from an initial guess for \(\theta\), we compute the gradient of the cost function concerning each parameter and update the parameters in the direction of the negative gradient. This process is repeated until the cost function converges to a minimum. The circuit structure is given below in Fig.6.
### Quantum Variational Kernel Support Vector Machine
This study proposes a new approach for quantum support vector machines, which we call Quantum Variational Kernel Support Vector Machine (QVK-SVM). It combines two distinct methods to enhance the performance of quantum kernels and variational circuits. The first method utilizes the angle-embedding template to prepare the quantum states used to compute the kernel, as we explained in subsection III.1. The overlap between the two states is measured using a SWAP test, which requires only half the qubits.
The second method involves utilizing a variational circuit trained through the variational training principle, as outlined in Subsection III.2. The ansatz of the circuit can be improved by adding more layers, thereby enhancing its ability to express itself. Additionally, a bias term is incorporated into the quantum model to facilitate training on the hinge loss. The quantum node utilizes the parameter-shift differentiation method, which is very efficient on hardware.
The proposed circuit of the new approach consists of three main components: AngleEmbedding, the adjoint of AngleEmbedding, and StronglyEntanglingLayers, as shown
Figure 6: QV-SVM circuit using Pennlyne.
in Fig.7.
The AngleEmbedding is used to prepare the quantum states of the data points, which are then fed into the adjoint of the AngleEmbedding to prepare the inverse embedding. The StronglyEntanglingLayers component is used to apply the variational circuit, which is trained using the hinge loss.
The proposed approach has several advantages. First, it combines the strengths of both methods to enhance the performance of QSVMs. Second, it utilizes the variational training principle, allowing greater control over the training process. Third, it uses the parameter-shift differentiation method, which works well on hardware. Finally, the proposed circuit is simple and easy to implement, making it suitable for practical applications.
The proposed approach for quantum SVMs combines the angle-embedding kernel and the variational circuit to enhance the performance of QSVMs. The proposed approach has several advantages over the existing methods, including greater control over the training process, better hardware compatibility, and ease of implementation. Future research could explore the application of the proposed approach to other datasets and investigate the potential of the approach for solving more complex problems.
## IV Results and Discussion
The results and discussion of the three models in this research demonstrate the potential of quantum machine learning in enhancing binary classification tasks, particularly the quantum support vector machine. The first model, QK-SVM, employed a quantum kernel approach and delivered an impressive overall accuracy of 96.34% on the test set. The second model, QV-SVM, utilized variational training with a quantum support vector
Figure 7: QVK-SVM circuit using Pennlyne.
machine and achieved a maximum accuracy of 95.43% on the test set. The third and final model, QVK-SVM, combined quantum kernel and variational training to yield the most promising results, with an accuracy of 98.48% on the test set, as evidenced by the data presented in Table 1.
Table 1 displays the performance metrics for the three models: QK-SVM, QV-SVM, and QVK-SVM. Our findings indicate that QK-SVM achieved high precision and recall rates, indicating the robustness of the model in correctly identifying the different classes of Iris flowers. QV-SVM achieved high specificity and F1 score values, further validating the effectiveness of the quantum support vector machine using the variational algorithm. QVK-SVM achieved high precision, recall, and specificity; these results confirm the efficacy of QVK-SVM and highlight its potential as a reliable tool for ML applications.
The QK-SVM model achieved a maximum accuracy of 96.34%, high precision and recall rates, and a corresponding specificity and F1 score. Fig.9 and Fig.9 illustrate the loss and accuracy curves for both the training and testing sets, demonstrating a clear improvement in the model's performance during training.
The QV-SVM model achieved a steady improvement in performance as the number of iterations increased, with training losses ranging from 1.00 to 0.49 and testing losses ranging from 0.98 to 0.47. The model achieved an absolute accuracy of 95.43%, with high precision and recall rates. Fig.10 and Fig.11 illustrate the loss and accuracy plots, respectively, highlighting the model's optimization over the iterations.
Our suggested model represents a novel approach that combines the quantum kernel and variational algorithm used in QV-SVM and QK-SVM, respectively. The results of our QVK-SVM model indicate that this combined approach is highly effective for solving classification problems, even on relatively small datasets.
The proposed QVK-SVM model achieved an impressive accuracy of 98.48% on the test set, with a corresponding F1 score of 91.64%. Fig.12 shows the convergence of the training and testing losses throughout the experiment, demonstrating that the model could optimize the loss function to achieve high accuracy. Fig.13 displays the corresponding training and testing accuracies, showing that the model's accuracy improved steadily throughout the experiment, reaching a peak of 98% accuracy on the test set.
Furthermore, the superior performance of QVK-SVM, compared to QV-SVM and QK-SVM, suggests that the combination of these approaches can provide a more robust and reliable solution to binary classification tasks.
The results show the potential of quantum machine learning in enhancing binary classification tasks, especially the quantum support vector machine. As demonstrated in our novel method, combining the quantum kernel and variational algorithm represents a promising approach that could be extended to other datasets and classification problems.
## V Conclusion
This study delved into applying quantum support vector machines for binary classification tasks. Specifically, two pre-existing methods, the quantum kernel support vector machine and the quantum variational support vector machine, were compared and evaluated for their respective strengths and limitations. In addition, a novel approach was developed by combining these two methods, resulting in the quantum variational kernel support vector machine. The proposed QVK-SVM approach demonstrated exceptional accuracy and loss reduction performance, offering a promising solution for binary classification tasks in quantum machine learning. These findings hold significant potential for advancing the field of quantum machine learning and its diverse applications. The QVK-SVM approach represents a noteworthy contribution to this field's development and has clear implications for future research endeavors.
The proposed method presents an opportunity for further research to explore its efficacy in resolving intricate issues across various datasets. Advanced optimization techniques and the development of new quantum algorithms can enhance the efficiency and scalability of the approach. Furthermore, the potential of quantum machine learning can be investigated by extending the proposed method to other machine learning models, such as neural networks and decision trees. Through these efforts, the proposed approach can advance the field of quantum machine learning and unlock new opportunities for addressing complex real-world problems. Its potential to do so is significant
and warrants further academic investigation.
| 量子機械学習 (QML) は最近大きな進歩を遂げており、量子サポートベクターマシン (QSVMs) が期待されるモデルとして登場しました。この論文では、既存の QSVM 方法である量子kernel SVM (QK-SVM) と量子variational SVM (QV-SVM) を焦点に当てて議論しています。両方が素晴らしい結果を出していますが、既存の QK-SVM と QV-SVM の強みを調和させて精度を高める新たなアプローチを提案しました。我々は量子variational kernel SVM (QVK-SVM)という新しいモデルを提案し、量子kernel と量子variationalアルゴリズムを活用しています。Iris データセット üzerinde 多くの実験を行い、QVK-SVM が既存のモデルよりも精度、損失、混同マトリックス指標において優れていることを観察しました。私たちの研究結果は、QVK-SVM が QML アプリケーションのための信頼性の高い and 変革的なツールであることを示しています。した |
2307.08637 | LearnedSort as a learning-augmented SampleSort: Analysis and
Parallelization | This work analyzes and parallelizes LearnedSort, the novel algorithm that
sorts using machine learning models based on the cumulative distribution
function. LearnedSort is analyzed under the lens of algorithms with
predictions, and it is argued that LearnedSort is a learning-augmented
SampleSort. A parallel LearnedSort algorithm is developed combining LearnedSort
with the state-of-the-art SampleSort implementation, IPS4o. Benchmarks on
synthetic and real-world datasets demonstrate improved parallel performance for
parallel LearnedSort compared to IPS4o and other sorting algorithms. | Ivan Carvalho, Ramon Lawrence | 2023-07-17T16:53:22 | http://arxiv.org/abs/2307.08637v1 | # LearnedSort as a learning-augmented SampleSort: Analysis and Parallelization
###### Abstract.
This work analyzes and parallelizes LearnedSort, the novel algorithm that sorts using machine learning models based on the cumulative distribution function. LearnedSort is analyzed under the lens of algorithms with predictions, and it is argued that LearnedSort is a learning-augmented SampleSort. A parallel LearnedSort algorithm is developed combining LearnedSort with the state-of-the-art SampleSort implementation, IPS4o. Benchmarks on synthetic and real-world datasets demonstrate improved parallel performance for parallel LearnedSort compared to IPS4o and other sorting algorithms.
sorting, machine learning for systems, algorithms with predictions +
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the RMI, the PGM, and the RadixSpline always outperform state-of-the-art traditional indexes on look-up time and size, losing just on build time.
The two-layer RMI is used by LearnedSort. Mathematically, it is described by:
\[F(x)=f_{2}^{[B\times f_{1}(x)]}(x)\]
The RMI consists of the root model \(f_{1}\) and of \(B\) second-level models \(f_{2}^{(i)}\) for \(0\leq i<B\). The root model can be interpreted as an initial approximation of the CDF function that selects one of the \(B\) models in the next level. The second-level models by consequence can be seen as models specializing on a specific region of the CDF.
The RMI architecture is extremely flexible. \(f_{1}\) and \(f_{2}^{(i)}\) can have arbitrary model types such as linear, cubic or radix models. The number of second-level models \(B\) can also be configured. LearnedSort uses a RMI with linear models and \(B=1000\).
### Sorting with Machine Learning Models
Sorting with machine learning models goes beyond applying a single pass of \(A[F(x)]=x\) for all elements. To engineer a practical implementation, many details need to be resolved.
The first detail is that \(A[F(x)]=x\) has a hostile memory-access pattern to modern CPUs, as it performs mostly random accesses to the memory. Kristo et al. reported that even with a perfect model, applying the model directly was slower than optimized versions of RadixSort. This prompted the authors to try other approaches such as using buckets.
Another key detail is that the model is imperfect and inversions can happen i.e. there are \(x,y\) such that \(x<y\) but \(F(x)>F(y)\). Although uncommon for good models, the implementation needs to handle those cases to guarantee that the output is sorted.
Moreover, collisions between elements can happen i.e. there are \(x,y\) such that \(F(x)=F(y)\). Since it is possible to have only one element at position \(F(x)\), the implementation must handle collisions. Collisions are exacerbated by duplicates in the input, as all duplicate values \(x\) will collide at \(F(x)\). Duplicates are very common when sorting.
Kristo et al. improved the algorithm handling of these challenges and produced LearnedSort 2.0 (Kristo et al., 2017). LearnedSort 2.0 consists of four routines: training the model, two rounds of partitioning, model-based Counting Sort, and a correction step with Insertion Sort.
Training the model is the first routine of LearnedSort and requires the most empirical data for good performance. It is necessary to select a model type and sample size to train the CDF model. Kristo et al. chose the two-layer RMI as the model. Since producing the optimal RMI is computationally more expensive than sorting an array with Quicksort (Kristo et al., 2018), the authors fixed the root and second-level model types to be linear models. They also picked a sample size of 1% of \(N\) to train the RMI. These choices yield excellent results in practice. The model can be trained quickly and its predictions are accurate enough such that the sorting performance can outperform other state-of-the-art sorting algorithms.
The partitioning routine is in-place and uses the model to split the data into \(B=1000\) buckets. For each element, LearnedSort calculates its corresponding bucket using \(b_{i}=[B\times P(A\leq x)]\) and adds the element to the buffer associated with \(b_{i}\). When a buffer gets full, LearnedSort flushes the buffer. After processing all elements, the fragments of each bucket \(b_{i}\) are scattered across the input. To solve this, LearnedSort implements a defragmentation pass that makes the buckets contiguous. LearnedSort applies the partitioning routine twice, splitting the data into 1000 buckets and then splitting each of those buckets into 1000 sub-buckets.
To handle duplicates, Learned Sort 2.0 performs a homogeneity check after partitioning: if all elements within a bucket are equal, the bucket is left as is because it is already sorted. This condition handles the collision case that reduced the performance of the original LearnedSort.
The base case for LearnedSort is a Model-Based Counting Sort that uses the CDF to predict the final position of the keys in the sub-buckets. Lastly, Insertion Sort is executed to correct the possible mistakes from the RMI and guarantee that the output is sorted. Since the sequence is almost sorted, Insertion Sort is cheap to execute in practice.
### Quicksort
Although Quicksort is asymptotically optimal, engineering a good implementation of Quicksort can drastically improve its efficiency. This requires avoiding Quicksort's worst case with bad pivots and squeezing all the performance available by modern hardware.
IntroSort is a hybrid Quicksort algorithm (Kristo et al., 2017) that avoids the \(\Theta(N^{2})\) worst-case by switching to HeapSort (Kristo et al., 2018) when the recursion depth exceeds \(O(\log N)\). IntroSort has been chosen by some popular libraries, such as the GNU C++ library, to be their default sorting algorithm.
Pattern-defeating Quicksort (pdqsort) is an enhanced version of IntroSort (Kristo et al., 2017). It incorporates many improvements on partitioning and sorts in \(O(N\min(\log N,K))\) where \(K\) is the number of distinct elements on the input. pqdsort also leverages the contributions of BlockQuicksort (Kristo et al., 2018), which processes the elements in blocks to avoid branch mispredictions. pqdsort is currently the algorithm implemented by the Rust Standard Library for unstable sorting.
Vectorized Quicksort is a new implementation of Quicksort that uses Single Instruction, Multiple Data (SIMD) to exploit the parallelism available in modern CPUs (Wassenberg et al., 2017). Wassenberg et al. managed to vectorize each individual step of Quicksort: pivot selection, partitioning, and the sorting networks for the base case. By building on top of a high-level SIMD library, the authors were also able to port their implementation to seven distinct instruction sets, which is uncommon as previous implementations were generally not portable.
A takeaway from advancements in Quicksort is that engineering is a core part of high-performance sorting and that implementation details matter. Implementation optimizations improved performance in Learned Sort 2.0, and such optimizations are important for high parallel performance.
### SampleSort
SampleSort is a generalization of Quicksort to \(k\) pivots (Brandrands et al., 2017). The increased number of pivots pushes the number of comparisons of the algorithm closer to the \(\log_{2}n!\) theoretical bound, giving it an edge over Quicksort. It also makes the algorithm suitable for parallel processing, as SampleSort creates \(k+1\) perfectly parallel sub-problems.
Similar to Quicksort, engineering a good implementation of SampleSort can significantly boost performance. Sanders and Winkel introduced the Super Scalar SampleSort in (Sanders and Winkel, 2017). Their implementation of SampleSort exploits instruction-level parallelism available in modern CPUs. Sanders and Winkel organize the pivots into a branchless decision-tree that is friendly to optimization techniques such as pipelining and loop unrolling. This made their implementation competitive on single-core sequential settings.
Avtmann et al. take a step further in (Astrmann et al., 2017), introducing the In-place Parallel Super Scalar SampleSort (IPS4o). IPS4o is the state-of-the-art SampleSort implementation incorporating many improvements.
One key improvement of IPS4o is the in-place partitioning. Previous SampleSort implementations allocated \(\mathcal{O}(N)\) memory to copy elements of the input. IPS4o instead uses buffers of size \(b\) for each of the \(k\) buckets. It allocates \(\mathcal{O}(kb)\) total memory and when a buffer is full it flushes the buffer and overwrites some of the data of the original input that has already been processed. This initial pass creates \(\mathcal{O}(N/b)\) blocks. Afterwards, IPS4o permutes the blocks such that each bucket is contiguous in memory using a routine similar to defragmentation. Conceptually, the blocking strategy adopted by IPS4o shares many ideas with those adopted by LearnedSort, BlockQuicksort, and pdqsort.
Other improvements of IPS4o include the parallelization and the equality buckets. IPS4o uses atomic fetch-and-add operations to parallelize the block partitioning and leverages a custom task scheduler to manage threads when the sub-problems become small. IPS4o also gracefully handles inputs with many duplicates with equality buckets. It detects skewed inputs on sampling and creates a separate bucket for the duplicates when doing the partitioning. As a sequence where all elements are equal is already sorted, IPS4o avoids having to process the duplicate elements in the equality buckets.
It is also worth highlighting the ability to use IPS4o as a framework for building other sorting algorithms. Axtmann et al. also introduced the In-place Parallel Super Scalar Radix Sort (IPS2Ra) (Astrmann et al., 2017). IPS2Ra combines the qualities of IPS4o with the most-significant-digit radix sort strategy, resulting in another high-performance sorting algorithm. IPS4o has also been used to parallelize Vectorized Quicksort (Sanders and Winkel, 2017) and to test the efficiency of sorting networks as base cases for sorting algorithms (Brands and Goyal, 2017).
This work reuses the IPS4o framework to parallelize LearnedSort. This allows the combination of the engineering efforts of IPS4o with the best qualities of LearnedSort.
### Algorithms with Predictions
The area of algorithms with predictions (Sanders and Winkel, 2017) goes beyond worst-case analysis and considers algorithms augmented with machine learning models. For each algorithm, we can think of a prediction and a quality metric \(\eta\) for the prediction that depends on an error specified by the problem type. In case \(\eta\) is good, the algorithm proceeds to use the outputs from the model to solve the problem instance. Otherwise, it has a fallback mechanism that uses a traditional, prediction-less algorithm when the machine learning models fail. We expect that for real-world workflows, the outputs from the model will generally be used due to patterns found in the data.
A prominent example of the area is caching with predictions (Levy et al., 2017). Lykouris and Vassilvitskii solve the online caching problem with a machine learning model trained to predict the furthest time in the future an element will come back to the cache. Their model is inspired by the offline solution to the problem, the greedy Furthest-In-Future algorithm that out of all elements removes the one that appears the latest in the future. To prevent the worst-case that happens when the model is sub-optimal, they fall back to the classic Marker algorithm.
Algorithms with predictions share many similarities with LearnedSort. Both implement machine learning models and avoid the worst-case due to the quality of the predictions. Thus, it is natural to ask if LearnedSort is an algorithm with predictions. The next section discusses how LearnedSort is analogous to a SampleSort in which the pivots were learned.
## 3. Analyzing LearnedSort
To analyze LearnedSort under the lens of algorithms with predictions, it is important to determine what LearnedSort is trying to predict and what makes for a good prediction for a sorting algorithm.
From a high-level perspective, ignoring implementation details, what makes Quicksort an efficient algorithm is the quality of its pivots. The BFPRT algorithm, also known as median of medians, is a method to find an element that is guaranteed to be between the 30th and 70th percentile of the input (Brands and Goyal, 2017). It is possible to combine Quicksort with the BFPRT to produce a deterministic Quicksort with worst-case complexity of \(\Theta(N\log N)\)(Gurthest and Goyal, 2017). Hence, the quality of the pivots can avoid the worst-case of randomized Quicksort.
Inspired by the deterministic Quicksort, the analysis of LearnedSort in split into three parts. The first part introduces Quicksort with Learned Pivots, a variation of Quicksort where the CDF model selects the pivot. That section shows that training a CDF model is akin to other pivot selection techniques such as applying the BFPRT algorithm. The second part analyzes Learned Quicksort, a simplified LearnedSort with \(B=2\) buckets. It turns out that Learned Quicksort is in fact analogous to a Quicksort with Learned Pivots but with implicit pivots. Lastly, the third section considers \(B>2\) and the connections between LearnedSort and SampleSort.
### Quicksort with Learned Pivots
The analysis starts with the pseudocode of our Quicksort variant shown in Algorithm 1. For simplicity, assume that all elements on the input \(A\) are distinct. The algorithm is identical to many other Quicksort implementations with the exception of the partitioning call.
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 1**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 2**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 3**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 4**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 4**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 5**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 5**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 6**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 6**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 7**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 8**Quicksort(A, \(1\), \(r\))
```
ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\), \(r\)); return; q \(\leftarrow\) PartitionWithLearnedPivot(A, \(1\), \(r\)); Quicksort(A, \(1\), \(q\)-\(1\)); Quicksort(A, \(q+1\), \(r\)); return;
```
**Algorithm 8**Quicksort(A, \(1\), \(r\))
### Quicksort with Learned Pivots
The analysis starts with the pseudocode of our Quicksort variant shown in Algorithm 1. For simplicity, assume that all elements on the input \(A\) are distinct. The algorithm is identical to many other Quicksort implementations with the exception of the partitioning call.
``` ifdistance(\(l\), \(r\)) \(\leq\) BASECASE_SIZEthen InsertionSort(A, \(1\
Algorithm 2 describes how to use the CDF models to select an optimal pivot. Essentially, our goal is to find the median of the input. To do so, we select the largest element \(A[t]\) such that the predicted CDF is smaller than or equal to the true CDF of the median.
```
S \(\leftarrow\) Sample(A,1,r); HeapSort(S,0,S.size(-1); F \(\leftarrow\) TrainCPMedMed(S,0,S.size(0-1);//Functionthat calculatesP(A \(\leftarrow\)x)in[0,1] /* Select the largest element fromA that has the predicted CDF less than the true median */ t \(\leftarrow\) -1; forw\(\leftarrow\)l tordo ifF[A[w]] \(\leq\) 0.5 and(t < 0 orA[w] >A[t])then \(t\)\(\leftarrow\) w; swap(A[t],A[r]); /* After selecting the pivot with the CDF model, we can use any classic partition scheme */ pivot \(\leftarrow\) A[r]; i \(\leftarrow\) 1 - 1; forj\(\leftarrow\)l tor\(-\)l do ifA[j] \(\leq\) pivotthen i \(\leftarrow\) i + 1; swap(A[i],A[j]); swap(A[i],A[r]); swap(A[i],A[r]); returni + 1;
```
**Algorithm 2**PartitionWithLearnedPivot(A,1,r)
The TrainCPMed function is arbitrary such that any type of CDF model could work e.g. RMI, PLEX, RadixSpline. However, for the CDF model to be useful, some properties should hold.
The first is monotonicity: \(x\leq y\implies F(x)\leq F(y)\). This property is necessary to ensure that the selected pivot is indeed closest to the median and that the model contains no incorrect inversions.
The second is that the model needs to require a small number of samples. This follows from the fact that to train a CDF model you need a sorted input and sorting the samples with HeapSort takes \(\mathcal{O}(S\log S)\) (although any algorithm with the same complexity would work).
The third is that computing the predictions of the model for a key should take \(\mathcal{O}(1)\) time. Since we need to make a prediction for each of the \(N\) keys, if the time to compute a prediction is not constant it would lead to an algorithm slower than the traditional Quicksort.
Given these properties, Algorithm 2 takes \(\mathcal{O}(N)\) and its run time is dominated by the loop applying the model predictions and the Lomuto partitioning step.
The time complexity of Algorithm 1 depends on the quality of the learned pivot. In the best case, the complexity is modelled by \(T(N)=\mathcal{O}(N)+2T(N/2)\) which happens when the learned pivot is the median. Hence, the lower bound of Algorithm 1 is \(\Omega(N\log N)\).
The worst-case complexity is modelled by \(T(N)=\mathcal{O}(N)+T(N-1)\) and happens when the learned pivot is the smallest element in the sequence. Thus, the worst-case of the algorithm is \(\Theta(N^{2})\) just like the original Quicksort. However, if the chosen model is a good model, reaching the worst-case is unlikely. The average-case analysis is much closer to the best case in practice.
Let \(\eta\) be the error from finding the perfect partitioning as:
\[\eta=\max(P(A\leq pivot),1-P(A\leq pivot))-1/2\]
where \(P(A\leq pivot)\) is the true CDF of the learned pivot. \(\eta=0\) in case the CDF model always predicts the median. \(\eta=1/2\) in case the CDF model always predicts the smallest element. The complexity is then modelled by:
\[T(N)=\mathcal{O}(N)+T((\eta+1/2)N)+T((-\eta+1/2)N)\]
The value \(\eta\) is not known ahead of time, as it depends on the sample size and CDF model. However, we may assume that the model has better predictions than a random pick \(\eta_{\text{learned}}\leq\eta_{\text{random}}\) (otherwise we would fallback to a random pick). This implies that the Quicksort with Learned Pivots runs as fast as the Randomized Quicksort. Thus \(T(N)\in\mathcal{O}(N\log N)\).
Quicksort with Learned Pivots is not efficient to outperform IntroSort or pdosort. However, it is conceptually useful to show that training a CDF model is a step towards finding better pivots.
### Learned Quicksort
Progressing towards analyzing LearnedSort, we introduce Learned Quicksort. Learned Quicksort, shown in Algorithm 3, is a simpler version of LearnedSort that contains only \(B=2\) buckets.
```
ifdistance(l,r) \(\leq\) BASECASE_SIZEthen InsertionSort(A,l,r); return; S \(\leftarrow\) Sample(A,l,r); HeapSort(S,0,S.size(0-1); F \(\leftarrow\) TrainCPMed(S,0,S.size(0-1); /* Using the predictions directly is equivalent to using the learned pivot */ i \(\leftarrow\) 1; j \(\leftarrow\) r; while\(i\) - \(j\)do ifF(A[j]) \(\leq\) 0.5then i \(\leftarrow\) i + 1; else swap(A[i],A[j]); j \(\leftarrow\) j - 1; LearnedQuicksort(A,1,i); LearnedQuicksort(A,i + 1,r); return;
```
**Algorithm 3**LearnedQuicksort(A,l,r)
Similar to LearnedSort, Learned Quicksort partitions the data using machine learning models. Since there are only two buckets, the partitioning can be done such that elements with \(F(A[i])\leq 0.5\) are put in the initial section of the input starting from the first index and elements with \(F(A[i])>0.5\) are put at the end of the input starting from the last index.
The partitioning done by Quicksort with Learned Pivots and Learned Quicksort is almost identical. The only exception is for the learned pivot, which is in the last position of the first half in the former. Hence, the algorithms have the same time complexity which means that Learned Quicksort has the complexity of \(O(N\log N)\).
The interesting fact about Learned Quicksort is that it does not compute the pivot explicitly. Instead, it relies solely on the results of the model \(F\). Computationally, this is advantageous as Learned Quicksort always performs less operations than Quicksort with Learned Pivots.
We may interpret Learned Quicksort as a Quicksort variant that circumvents the bounds on the theoretical number of comparisons by embracing the numerical properties of the CDF. This is a hint to why LearnedSort is so efficient.
### LearnedSort
We now consider the general case of LearnedSort when \(B>2\). If Learned Quicksort is analogous to a Quicksort with Learned Pivots, LearnedSort is analogous to a SampleSort with \(B-1\) learned pivots.
```
B \(\leftarrow\) numberOfBuckets(A.size()); p \(\leftarrow\) Array(B, -1); // indexes of the pivots for the i-th bucket S \(\leftarrow\) Sample(A, 1, r); HeapSort(S, 0, S.size( 0 - 1); F \(\leftarrow\) TrainCDFModel(S, 0, S.size() - 1); /* Select for each i-th percentile the largest element from A that has predicted CDF less than that percentile */ for\(w\gets l\) to rdo g \(\leftarrow\)\(\lfloor F(A[w])\times B\rfloor\) ; if[p_g_] < 0 or A[w] > A[p[g]] then p[g] \(\leftarrow\) w; pivots \(\leftarrow\) p.filter(v \(\geq\) 0).map(v \(\rightarrow\) A[v]); return pivots;
```
**Algorithm 4**LearnedPivotsForSampleSort(A, l, r)
Algorithm 4 details the process to compute the learned pivots for SampleSort. If we used those pivots in SampleSort, the partitioning would be identical to the partitioning done by LearnedSort. Obviously, as shown in the previous section, using the model directly is more advantageous as it skips the comparisons as a whole.
This explains why LearnedSort is effective. LearnedSort is an enhanced version of SampleSort, which is already a competitive sorting algorithm. If the learned pivots of LearnedSort are better than the randomly selected pivots of SampleSort, we expect LearnedSort to beat SampleSort. Moreover, LearnedSort skips the comparisons done by SampleSort and relies on the \(O(1)\) predictions of the model, which gives LearnedSort an additional performance boost.
There are some differences between an augmented SampleSort and the implementation of LearnedSort 2.0. These minor details come from the authors iterating to improve LearnedSort empirically.
The major discrepancy is that SampleSort does \(O(\log_{B}N)\) partitioning rounds while LearnedSort does only two. We interpret this as Kristo et al. implementing a very shallow recursion tree with a large base case size. SampleSort implementations generally use \(B=128\) or \(B=256\) buckets and use Insertion Sort when there are 16 elements or less. LearnedSort uses \(B=1000\) buckets, hence assuming two rounds of partitioning with around \(N=10^{9}\) elements of input that leads to around 1000 elements on average to be handled by LearnedSort's base case. We hypothesize that if there were \(N=10^{12}\) or \(N=10^{13}\) elements, LearnedSort's performance would be hurt and a third partitioning round would be helpful. However, that input size requires over a terabyte of RAM, which stops being an in-memory sort problem and starts being an external sort instance. Thus, the implementation by Kristo et al. works well in practice.
Another discrepancy is that SampleSort samples data on every sub-problem while LearnedSort samples data only once. This may be an optimization that comes from practical experience. Instead of sampling a few data points, creating 1000 sub-problems and sampling for each sub-problem again, LearnedSort opts to sample a lot of data in bulk. This works because the recursion tree of LearnedSort is very shallow and because the RMI architecture supports this strategy as the second-level models specialize in parts of the CDF.
Lastly, the RMI used by LearnedSort violates one assumption from our analysis. It does not guarantee that \(x\leq y\implies F(x)\leq F(y)\). In practice, inversions do occur but they are relatively rare. This leads to an almost-sorted sequence, which can be quickly fixed by Insertion Sort.
### Quality of the Pivots
This section analyzes the quality of the learned pivots implicitly used by LearnedSort. For two datasets, the uniform distribution and the Wiki/Edit data, the RMI created by LearnedSort was used with Algorithm 4 to calculate the pivots in the first partitioning step. The RMI pivots were compared with the random pivots used by IPS\({}^{4}\)o.
The sorted data was used to calculate the true CDF, \(P(A\leq p_{i})\), for each pivot \(p_{i}\). The metric used for the quality was the distance between the CDF of the pivots and the CDF of the perfect splitters \(\sum_{i=0}^{B-2}|P(A\leq p_{i})-(i+1)/B|\). For simplicity, we matched the number of pivots used by IPS\({}^{4}\)o with the number of pivots computed by the RMI, although LearnedSort uses more pivots in practice.
The results in Table 2 display that the learned pivots are indeed better than the random pivots.
\begin{table}
\begin{tabular}{|l|l|l|} \hline & Random (255 pivots) & RMI (255 pivots) \\ \hline Uniform & 1.1016 & 0.4388 \\ \hline Wiki/Edit & 0.9991 & 0.5157 \\ \hline \end{tabular}
\end{table}
Table 1. Comparison of Pivot Quality
\begin{table}
\begin{tabular}{|l|l|} \hline \hline
**Algorithm 4**LearnedPivotsForSampleSort(A, l, r)
\begin{table}
\begin{tabular}{|l|l|l|} \hline \hline & Random (255 pivots) & RMI (255 pivots) \\ \hline Uniform & 1.1016 & 0.4388 \\ \hline Wiki/Edit & 0.9991 & 0.5157 \\ \hline \end{tabular}
\end{table}
Table 2. Quality of the pivots for IPS\({}^{4}\)o (Random) and LearnedSort (RMI)
## 4. Parallelization of LearnedSort
One direct consequence from the previous analysis is that the progress in engineering a fast SampleSort transfers to LearnedSort. A relevant limitation of LearnedSort 2.0 is that there is only a sequential version available that cannot use all the cores present in modern CPUs to sort data in parallel. This limits applying LearnedSort to real-world workflows.
To address this limitation, we introduce the Augmented In-place Parallel SampleSort (\(\text{AIPS}^{2}\)o). \(\text{AIPS}^{2}\)o is a hybrid of \(\text{IPS}^{4}\)o with LearnedSort. It is built upon the codebase available from \(\text{IPS}^{4}\)o and augments it with the RMI implementation used in LearnedSort.
```
S \(\leftarrow\) Sample(A, I, r); Sort(S, 0, S.size() - 1); ifInputSizeIsLarge(I, r) and notToManyDuplicates(S)then // we sample more data as the RMI benefits from larger samples R \(\leftarrow\) LargerSample(A, I, r); Sort(R, 0, R.size() - 1); \(\text{rmi}\leftarrow\) BuildRMI(R); return\(rmi\); else \(\text{tree}\leftarrow\) BuildBranchlessDecisionTree(S); return\(tree\);
```
**Algorithm 5**BuildPartitionModel(A, I, r)
How \(\text{AIPS}^{2}\)o selects its partitioning strategy is in Algorithm 5. Essentially, if the input size is sufficiently large and there are not too many duplicates, the routine samples more data and returns a trained RMI Otherwise, it builds and returns the decision tree used in \(\text{IPS}^{4}\)o. For our implementation, we use \(B=1024\) buckets for the RMI. We default to the decision tree with \(B=256\) if the input size is smaller than \(N=10^{5}\) or if there are more than 10% of duplicates in the first sample.
Since \(\text{AIPS}^{2}\)o uses the framework from \(\text{IPS}^{4}\)o, it profits from the parallelization of the latter. Another feature it inherits from \(\text{IPS}^{4}\)o is the handling of duplicates, which avoids the common adversarial case for LearnedSort by using the equality buckets from the decision tree.
There are additional modifications to make \(\text{AIPS}^{2}\)o work as well. The most critical modification is making the RMI monotonic such that \(x\leq y\implies F(x)\leq F(y)\) holds. This is necessary to avoid applying Insertion Sort to guarantee the correctness. To implement a monotonic RMI, we had to constraint the second-level linear models such that \(\max_{x\in R}f_{2}^{(i)}(x)\leq\min_{x\in R}f_{2}^{(i+1)}(x)\). This incurs two additional accesses to an array storing the minimums and maximums when processing an element.
The base case is also modified. Model-based counting sort is not used as the algorithm never forwards the RMI between recursive calls. Instead, SkaSort is used for the base case when there are less than 4096 elements (SkaSort). SkaSort is a fast radix sort that is the base case for \(\text{IPS}^{2}\)Ra.
## 5. Experimental Results
\(\text{AIPS}^{2}\)o is compared against other sorting algorithms on the benchmark presented in the Learned Sort 2.0 paper (Kang et al., 2018). For reproducibility, benchmarks were executed on the **m5zn.metal** instance from AWS. The instance runs an Intel(r) Xeon(r) Platinum 8252C CPU @ 3.80GHz with 48 cores, 768KB of L1 cache, 24MB of L2 cache, 99 MB of L3 cache, and 192 GB of RAM.
The four competitors with \(\text{AIPS}^{2}\)o are the following. \(\text{IPS}^{4}\)o, the state-of-the-art implementation of SampleSort. \(\text{IPS}^{2}\)Ra, the radix sort implementation built on top of the framework for \(\text{IPS}^{4}\)o. Learned Sort, one of the fastest sequential sorting algorithms as discussed earlier. \(\text{std::sort}\) from the C++ STL, as the baseline for the experiment. The implementations were written in C++ and compiled with GCC 11 using the -O3 and -march=native flags.
The benchmark includes sequential and parallel settings. We refer to the sequential versions of the algorithms as \(\text{AIIS}^{2}\)o, \(\text{II}^{5}\)o, and \(\text{II}^{5}\)Ra for consistency as they are not parallel. We provide \(\text{std::execution::par\_unseq}\) as an argument to \(\text{std::sort}\) when executing in parallel. To sort floating point numbers with \(\text{IPS}^{2}\)Ra, we use a key extractor that maps floats to integers. Learned Sort is not in the parallel benchmark because there is only a sequential implementation. The parallel benchmark uses all of the 48 cores available in the machine.
The datasets used in the benchmark consist of synthetic and real-world data. The synthetic portion contains 64-bit double floating-point elements from various probability distributions. The real-world portion contains 64-bit unsigned integer elements mostly from the work of Marcus et al. (Marcus et al., 2018). For \(\mathbf{N}=10^{8}\), data size is 800 MB.
**Synthetic Datasets**
* **Uniform (N = \(10^{8}\))**: Uniform distribution with \(a=0\) and \(b=N\)
* **Normal (N = \(10^{8}\))**: Normal distribution with \(\mu=0\) and \(\sigma=1\)
* **Log-Normal (N = \(10^{8}\))**: Log-normal distribution with \(\mu=0\) and \(\sigma=0.5\)
* **Mix Gauss (N = \(10^{8}\))**: Random additive distribution of five Gaussian distributions
* **Exponential (N = \(10^{8}\))**: Exponential Distribution with \(\lambda=2\)
* **Chi-Squared (N = \(10^{8}\))**: \(\chi^{2}\) distribution with \(k=4\)
* **Root Dups (N = \(10^{8}\))**: Sequence of \(A[i]=i\mod\sqrt{N}\) as proposed in (Kang et al., 2018)
* **Two Dups (N = \(10^{8}\))**: Sequence of \(A[i]=i^{2}+N/2\mod N\) as proposed in (Kang et al., 2018)
* **Zipf (N = \(10^{8}\))**: Zipfian distribution with \(s_{\text{zipf}}=0.75\)
**Real-World Datasets**
* **OSM/Cell_IDs (N = \(2\cdot 10^{8}\))**: Uniformly sampled location IDs from OpenStreetMaps.
* **Wiki/Edit (N = \(2\cdot 10^{8}\))**: The edit timestamps from Wikipedia articles
* **FB/IDs (N = \(2\cdot 10^{8}\))**: The IDs from Facebook users sampled in a random walk of the network graph
* **Books/Sales (N = \(2\cdot 10^{8}\))**: Book popularity data from Amazon
* **NYC/Pickup (N = \(10^{8}\))**: The yellow taxi trip pick-up time stamps
### Sequential Results
The sorting rate of the sequential algorithms is in Figures 1, 2, and 3. The rate is measured by keys per second and indicates the throughput of each algorithm. The numbers are the mean of 10 executions of the algorithms. Higher rates indicate better algorithms.
LearnedSort is the fastest in 9 out of 14 of the datasets, claiming the first spot in the sequential benchmark. I1S2Ra comes second, beating the competitors in 4 datasets. Surprisingly, I1S2Ra outperforms LearnedSort in most of the real-world datasets that were created to benchmark the RMIs that power LearnedSort. I1S4o is the fastest only for one dataset, Root Dups, that it handles gracefully due to its equality buckets.
AI1S2o is outperformed in the sequential benchmark. It is faster than the baseline of std::sort. Nevertheless, the hybrid algorithm is slower than both LearnedSort and I1S4o that provide its inner parts.
We attribute the slower sequential results to the more costly training step of AI1S2o. It is important to recall that the training time is accounted in the sorting time for AI1S2o and LearnedSort. AI1S2o samples more data than I1S4o on each partitioning step, which incurs a penalty as we need to sort those samples. The advantage of having better pivots is offset by the training cost. AI1S2o also spends more time training models than LearnedSort as LearnedSort trains the RMI only once while AI1S2o trains a RMI per recursive call.
As we will see in the next section, AIPS2o is a more competitive parallel algorithm. We found that adjusting the sample size and training time had little to no improvement on the sequential case but improved the parallel performance.
### Parallel Results
The sorting rate of the parallel algorithms is in Figures 4, 5, and 6. The rate is measured by keys per second and indicates the throughput of each algorithm. The rates come from the mean of 10 executions of the algorithms.
Figure 1. Sorting throughput of the sequential algorithms. Higher rates are better.
Figure 3. Sorting throughput of the sequential algorithms. Higher rates are better.
Figure 2. Sorting throughput of the sequential algorithms. Higher rates are better.
AIPS2o is the fastest in 10 out of 14 of the datasets, claiming the first spot in the parallel benchmark. IPS4o comes second finishing as the fastest in 4 out of 14 datasets. std::sort places third. IPS2Ra finishes last, behind the baseline on the majority of cases.
The key to high parallel performance is an algorithm's ability to maximize the use of the hardware. AIPS2o creates the best partition of the data for the majority of cases, which creates many sub-problems of a balanced size. This favours the performance of AIPS2o because it manages to keep every thread of the CPU busy always doing work. It also hurts AIPS2o when the RM does not model the data as accurately. The lowest throughputs of AIPS2o happen on the FB/IDs and Wiki/Edit datasets, which are known to be harder for the RMI than the Books/Sales and OSM/Cell_IDs datasets [20].
By contrast, IPS2Ra does not manage to use all the hardware because its partitions are not balanced. There are no bounds on the number of elements that have the same radix prefix and go in the same bucket. Hence, IPS2Ra may end with threads waiting for work, hurting its sorting rate compared to AIPS2o and IPS4o which always keep threads busy. This is particularly relevant to show that having a fast sequential algorithm does not necessarily imply a fast parallel algorithm and vice-versa.
The benchmarks demonstrate that AIPS2o is a practical algorithm. It is a parallel LearnedSort that achieves excellent sorting rates in many datasets. We expect that continuous work will help AIPS2o become more robust against data distributions like the one from FB/IDs, finally closing the gap between AIPS2o and IPS4o on the cases where the latter wins.
Figure 4. Sorting throughput of the parallel algorithms. Higher rates are better.
Figure 5. Sorting throughput of the parallel algorithms. Higher rates are better.
Figure 6. Sorting throughput of the parallel algorithms. Higher rates are better.
## 6. Conclusion and Future Work
This paper argues that LearnedSort is analogous to a SampleSort with pivots selected by a CDF model. This helps explain the effectiveness of LearnedSort by comparing it to SampleSort. We introduced the Augmented In-place Parallel SampleSort, combining the state-of-the-art implementation of SampleSort with LearnedSort. The benchmarks demonstrated that Augmented In-place Parallel SampleSort is a practical parallel implementation of LearnedSort that can outperform the fastest parallel sorting algorithm in the majority of the tested inputs including both synthetic and real-world data sets.
Future work in this research direction is to explore how machine learning models can be applied to other uses cases in sorting. Some possibilities include:
**GPUSorting**: Can the RMI or other learned indexes be combined with GPU SampleSort (Krishnan et al., 2019)?
**String Sorting**: Can learned indexes targeting strings (Krishnan et al., 2019) be combined with String SampleSort (Blekman et al., 2019)?
**Sampling and Pivot Quality**: Can the quality of the learned pivots improve if combined with better sampling techniques (Krishnan et al., 2019)?
| この研究は、機械学習モデルに基づいて累積分布関数を使用してソートを行う新しいアルゴリズムであるLearnedSortを分析し、並列化する。LearnedSortは、予測に基づいてアルゴリズムを分析し、LearnedSortが学習補助付きサンプルソートであると主張されている。LearnedSortの並列化アルゴリズムは、LearnedSortをサンプルソートの最新実装であるIPS4oと組み合わせることで開発された。合成および現実のデータセットのベンチマークでは、並列処理の改善が、IPS4oおよび他のソートアルゴリズムと比較して、LearnedSortのパフォーマンスに顕著な差が見られた。 |
2308.07381 | Late time HST UV and optical observations of AT~2018cow: extracting a
cow from its background | The bright, blue, rapidly evolving AT2018cow is a well-studied peculiar
extragalactic transient. Despite an abundance of multi-wavelength data, there
still is no consensus on the nature of the event. We present our analysis of
three epochs of Hubble Space Telescope (HST) observations spanning the period
from 713-1474 days post burst, paying particular attention to uncertainties of
the transient photometry introduced by the complex background in which
AT2018cow resides. Photometric measurements show evident fading in the UV and
more subtle but significant fading in the optical. During the last HST
observation, the transient's optical/UV colours were still bluer than those of
the substantial population of compact, young, star-forming regions in the host
of AT2018cow, suggesting some continued transient contribution to the light.
However, a compact source underlying the transient would substantially modify
the resulting spectral energy distribution, depending on its contribution in
the various bands. In particular, in the optical filters, the complex, diffuse
background poses a problem for precise photometry. An underlying cluster is
expected for a supernova occurring within a young stellar environment or a
tidal-disruption event (TDE) within a dense older one. While many recent works
have focused on the supernova interpretation, we note the substantial
similarity in UV light-curve morphology between AT2018cow and several tidal
disruption events around supermassive black holes. Assuming AT2018cow arises
from a TDE-like event, we fit the late-time emission with a disc model and find
$M_{BH} = 10^{3.2{\pm}0.8}$ M$_{\odot}$. Further observations are necessary to
determine the late-time evolution of the transient and its immediate
environment. | Anne Inkenhaag, Peter G. Jonker, Andrew J. Levan, Ashley A. Chrimes, Andrew Mummery, Daniel A. Perley, Nial R. Tanvir | 2023-08-14T18:06:54 | http://arxiv.org/abs/2308.07381v1 | # Late time _Hst_ UV and optical observations of AT 2018cow: extracting a cow from its background
###### Abstract
The bright, blue, rapidly evolving AT2018cow is a well-studied peculiar extragalactic transient. Despite an abundance of multi-wavelength data, there still is no consensus on the nature of the event. We present our analysis of three epochs of _Hubble Space Telescope (HST)_ observations spanning the period from 713-1474 days post burst, paying particular attention to uncertainties of the transient photometry introduced by the complex background in which AT2018cow resides. Photometric measurements show evident fading in the UV and more subtle but significant fading in the optical. During the last _HST_ observation, the transient's optical/UV colours were still bluer than those of the substantial population of compact, young, star-forming regions in the host of AT2018cow, suggesting some continued transient contribution to the light. However, a compact source underlying the transient would substantially modify the resulting spectral energy distribution, depending on its contribution in the various bands. In particular, in the optical filters, the complex, diffuse background poses a problem for precise photometry. An underlying cluster is expected for a supernova occurring within a young stellar environment or a tidal-disruption event (TDE) within a dense older one. While many recent works have focused on the supernova interpretation, we note the substantial similarity in UV light-curve morphology between AT2018cow and several tidal disruption events around supermassive black holes. Assuming AT2018cow arises from a TDE-like event, we fit the late-time emission with a disc model and find \(M_{BH}=10^{3.2\pm 0.8}\,\mathrm{M}_{\odot}\). Further observations are necessary to determine the late-time evolution of the transient and its immediate environment.
keywords: stars: individual: AT2018cow - ultraviolet: stars - supernovae: general - transients: supernovae - transients: tidal disruption events
## 1 Introduction
Multi-wavelength, wide field-of-view surveys at various wavelengths have transformed transient astrophysics. From X-rays with _Swift_(Burrows et al., 2005) and eROSITA (Predehl et al., 2021) through to optical with e.g., the Zwicky Transient Facility (ZTF); Bellm et al. (2019), the All-Sky Automated Survey for Supernovae (ASASSN)1; (Shappee et al., 2014), and the Asteroid Terrestrial-Impact Last Aject System (ATLAS); (Tonry, 2011) and radio surveys (e.g., the VLA sky survey Lacy et al., 2020, the Canadian Hydrogen Intensity Mapping Experiment[CHIME]; CHIME Collaboration et al., 2022, and MeerKAT; Jonas & MeerKAT Team, 2016), we can now identify and follow hundreds to thousands of transients, such as gamma-ray bursts, supernovae and fast radio bursts, per year. These high rates result from the combination of areal coverage, depth and cadence of these surveys, and the intrinsic volumetric rate and luminosity function of the transients under consideration. Due to these large, high cadence, sensitive surveys, events that are intrinsically rare, or that are numerous but faint, are also being detected. At the extremes of parameter space, we detect events whose nature stretches plausible progenitor models. These events are thus extremely valuable for study in their own right.
Footnote 1: [https://www.astronomy.ohio-state.edu/sasssn/](https://www.astronomy.ohio-state.edu/sasssn/)
One class of such peculiar transients are fast blue optical transients (FBOTs; e.g., Drout et al., 2014; Arcavi et al., 2016; Whitesides et al., 2017; Pursiainen et al., 2018; Tampo et al., 2020; Ho et al., 2023). A handful of FBOTs have been discovered over the last decade: CSS161010 (Coppejans et al., 2020), AT2018lug/ZTF18abvkula (Ho et al., 2020), AT2020xnd/ZTF20acigmen (Perley et al., 2021), AT2020mrf (Yao et al., 2022), and the well-known example AT 2018cow (Prentice et al., 2018; Perley et al., 2019). Together, these events form their own class of astrophysical transients, although the FBOT properties are heterogeneous, and the nature of the events is still uncertain. This class of events is characterised by fast rise and decay times, high peak luminosities (absolute peak magnitude \(\lesssim-19\)), and early spectra dominated by a blue
Multiple models were suggested, such as peculiar supernovae (SNe) and magnetars formed in double neutron star mergers (Drout et al., 2014). In SNe the timescale of Ni56 radioactive decay and the diffusion time scale are critical parameters in the light-curve evolution (Arnett, 1982). However, these two time scales are too long to explain the rapid decay and high peak luminosity observed for FBOTs (Drout et al., 2014; Pursiainen et al., 2018).
Footnote 5: [https://archive.stsci.edu/](https://archive.stsci.edu/)
AT 2018cow was the first FBOT discovered in real-time instead of archival searches. The transient rose to peak rapidly (\(>\)5 mags in \(\sim\) 3.5 days), was extremely bright (\(\rm L_{peak}\approx 10^{44}\) erg s\({}^{-1}\); Prentice et al., 2018; Perley et al., 2019) and was detected across the electromagnetic (EM) spectrum. The host galaxy CGCG137\(-\)068 has a luminosity distance of 63.0\(\pm\)4.4 Mpc (redshift z=0.01404\(\pm\)0.00002) (SDSS DR6; Adelman-McCarthy et al., 2008). The combination of high (peak) luminosity and relativey low distance meant that many telescopes and satellites could observe and detect it, and led to an extensive observational campaign.
Observations of AT 2018cow showed that the luminosity decay was too fast to be powered by Ni56 decay (Margutti et al., 2019). In addition, the photospheric radius stayed hot and small for hundreds of days (Perley et al., 2019; Sun et al., 2022). The optical spectra were featureless the first \(\sim\)20 days; after that period, emission lines of hydrogen and helium appeared (Prentice et al., 2018; Margutti et al., 2019; Perley et al., 2019). The spectral evolution has some resemblence to the spectral development of SNe Ibn and IIn (Fox & Smith, 2019; Xiang et al., 2021) although the lines in AT 2018cow appeared later than usual for those supernovae. The X-ray luminosity was high (e.g., Margutti et al., 2019; Kuin et al., 2019) and showed suggestive evidence for the presence of one or more quasi-periodic oscillations (QPOs) (Zhang et al., 2022; Pasham et al., 2021). QPOs are regularly seen in accreting systems, and the combination of a high luminosity and the detection of a QPO, if real, would thus suggest AT 2018cow is caused by an accreting compact object.
Footnote 5: [https://drazzlepac.readthedocs.io/en/latest/astrodrizzle.html](https://drazzlepac.readthedocs.io/en/latest/astrodrizzle.html)
The host galaxy of AT 2018cow appears to be a face on spiral system, and there are several (at least two) star-forming regions that lie close to (within \(\sim\) 170 parsec) the (projected) position of AT 2018cow. Assuming AT 2018cow lies in the plane of the host galaxy and not above or below it, this provides suggestive evidence for a link between massive star evolutionary processes and AT 2018cow (Lyman et al., 2020; Morokuma-Matsui et al., 2019). On the other hand, Sun et al. (2023) suggest that the low extinction in the transient implies that it is more likely on the near side of the disc and is not necessarily embedded in the star-forming regions. It would argue against a link with a massive star progenitor if this is correct.
Combining all the observed properties, the emission of AT 2018cow most likely comes from an engine-driven explosion (e.g., Margutti et al., 2019; Perley et al., 2019). Multiple models have been proposed for AT 2018cow (and FBOTs in general), including magnetars (Prentice et al., 2018; Mohan et al., 2020; Liu et al., 2022), interactions with the circumstellar material (Rivera Sandoval et al., 2018; Pellegrino et al., 2022) and a pre-existing stellar mass BH disrupting or accreting a companion (Metzger, 2022). Among the proposed models, the following two are considered most promising: An engine-powered core-collapse event, where a compact object is formed that accretes progenitor material (Prentice et al., 2018; Perley et al., 2019; Margutti et al., 2019; Mohan et al., 2020), or a tidal disruption event (TDE) of a white dwarf (WD) or main sequence (MS) star by an intermediate mass black hole (IMBH, Kuin et al., 2019; Perley et al., 2019). This class of TDEs may naturally explain the fainter and faster evolution compared to classical TDEs (of stars by a supermassive black hole [SMBH]), as well as provide an explanation for the non-nuclear location of the transient (Maguure et al., 2020). However, the IMBH must reside in a dense stellar environment such that two-body relaxation is efficient enough to scatter a white dwarf (or MS star) into the tidal radius within a Hubble time. Such a dense stellar environment is then a requirement for the TDE interpretation to be viable, although previous research does not provide evidence for such an environment (e.g. Margutti et al., 2019). However, long-lived, luminous emission from AT 2018cow makes detecting any putative (underlying) stellar cluster difficult.
The _Hubble Space Telescope (HST)_ observed AT 2018cow several times over the four-year period since its first detection. Surprisingly, Sun et al. (2022, 2023) detected UV-radiation even more than 4 years after the first detection of AT 2018cow. This emission is consistent with a hot and bright source and Sun et al. (2022) suggest a massive star progenitor is most likely involved.
In this work, we present our analysis of the late-time _HST_ data of AT 2018cow, spanning three epochs between 713 and 1474 days after the initial detection. The filters range from F225W in the UV to F814W in the red part of the optical. We perform photometry in multiple ways and investigate the influence of the background measurement on the photometry outcome. We also investigate whether the detected emission is from AT 2018cow itself or the environment and if there are implications from this distinction for the progenitor scenarios. We investigate if the UV properties can be explained under a TDE scenario and what the implications would be.
All magnitudes are presented in the AB magnitude system unless specified otherwise. Throughout the paper we use \(\rm H_{0}=67.8\,km\,s^{-1}\,Mpc^{-1}\), \(\rm\Omega_{m}=0.308\) and \(\rm\Omega_{\Lambda}=0.692\)(Planck Collaboration et al., 2016).
## 2 Data analysis
For this work we use observations of AT2018cow by _HST_ using the Ultraviolet-Visible (UVIS) channel of the Wide Field Camera 3 (WFC3) at three different late-time epochs. The data we use were taken under program IDs 15974, 16179 and 16925 with PIs A. Levan, A. Filippenko and Y. Chen, respectively. The observations are taken 713 days, 1135 days and 1474 days after the first detection of the transient, which we take to be \(\rm T_{0}=58285.44\)(Perley et al., 2019). We obtain the individual on-the-fly processed images from the Mikulski Archive for Space Telescopes2, these have had flat field and bias corrections applied and have also been corrected for the impact of charge transfer efficiency on the ageing WFC3 CCDs.
Footnote 2: [https://archive.stsci.edu/](https://archive.stsci.edu/)
### Alignment
First we combine the individual images using astrodrizzle from the python package drizzlepack (Hoffmann et al., 2021)3. Here, we set the final pixel scale to final_scale=0.025 to utilize sub-pixel dithering to obtain more finely sampled images and to better sample the _HST_ point spread function (PSF). We use default settings for parameters unless mentioned otherwise. Next, we use the geomap task in iraf(Tody, 1986, 1993) to align the images obtained in the four different filters 713 days after the onset. The sources used for this alignment are the galaxy nucleus [R.A.,Dec] = [16:16:00.582,+22:16:08.286] and a star [R.A.,Dec] = [16:15:59.147,+22:15:58.88] : both are detected in
all four filters. After this, we use xregister to align each filter image obtained at the one (F225W and F336W) or two (F555W and F814W) other epoch(s) to their respective image obtained 713 days after the transient's first detection. We cannot use xregister to align all images across all filters because it uses cross correlation to calculate a shift, which does not work well if there are many sources that are not detected in both images, which is the case here when using observations obtained in different filters. The alignment shifts from geomap and xregister are used to redrizzle the images with an additional shift so the sources align pixel wise in the final images.
### Aperture Photometry
We perform aperture photometry using a circle with a radius of 0.08 arcsec on all the images using dual-image mode in source extractor(Bertin & Arnouts, 1996), except our detection image, F336W at T=713 days, for which we use single image mode. In dual-image mode source detection is done on one image and the measurements are done on the second image. This enforces the use of a fixed position of the aperture across the different filter images. Using dual-image mode prevent us from having to cross match the detected sources between images and forces source extractor to perform photometry at the position of AT 2018cow. The choice of aperture radius (corresponding to a diameter of \(\sim 2\) times the Full Width at Half Maximum (FWHM) ensures we measure most of the emission from AT 2018cow without measuring too much background.
We use the drizzled F336W image at epoch 713 days as our source detection image, because there clearly is still emission at the transient location, and more sources are detected in the F336W than in the F225W image. For the photometry we use default values as mentioned in the source extractor manual4 for parameters not mentioned here and adjust parameters such as the FWHM and pixel scale (0.08 arcsec and 0.025 arcsec/pixel, respectively). We set the detection and analysis thresholds to 3.0 sigma to balance between minimizing contamination from spurious detections of hot pixels and allowing the detection of faint sources in the final output. We subtract the local background from the transient light in the final photometry.
Footnote 4: [https://sextractor.readthedocs.io/em/latest/index.html](https://sextractor.readthedocs.io/em/latest/index.html)
Since the individual images are shifted with respect to each other because of drizzling, certain features such as bad pixels or pixels with cosmic rays removed can influence the quality of the signal in multiple pixels in the final combined image (i.e., the noise in the final pixels is correlated to some degree). This can influence the final photometry, which we take into account by using a weight map (WEIGHT_TYPE = HAP_WEIGHT) in source extractor. This weight map tells source extractor which redrizzled pixels contain bad pixels from the individual images, which improves source detection and error estimation, see the source extractor user manual for full details. We use the weight map that is produced by astrodrizzle during the combination process.
Aperture corrections are done using appropriate values from the table provided on the WFC3 handbook website5 using r=0.08 arcsec values in the UVIS2 table. For comparison to Sun et al. (2022) we report Vega magnitudes based on the zeropoints from the WFC3 instrument handbook6. Photometry is corrected for Galactic foreground reddening following Schlafly and Finkbeiner (2011).
Footnote 5: [https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/photometric-calibration/uvis-encircled-energy](https://www.stsci.edu/hst/instrumentation/wfc3/data-analysis/photometric-calibration/uvis-encircled-energy)
### PSF photometry
We also perform PSF photometry to examine whether the source is point-like or extended. We start by cutting out an image (17 by 17 pixels) away from the host galaxy containing an isolated point source (centred on {RA, Dec}=[16:15:59.254, +22:1621.733] for F555W and F814W, and {RA, Dec} = {16:15:59.148,+22:15:58.379} for F225W and F335W). This point source is used to provide an estimate of the PSF. Although it does not have as high a signal-to-noise ratio as the computed PSFs available, the advantage of this approach is that it measures the PSF directly on the image. Since the star is much brighter than the transient, the impact of photometric noise on the template PSF is minimal.
We now proceed to measure the magnitude of a point source at the location of AT2018cow within our images. We linearly interpolate the template PSF to enable sub-pixel centroiding, confirm this model subtracts out cleanly from the PSF star image, and then perform a fit using the pixels with a central position \(<6.1\) pixels from the best fitting (x,y) position determined before. This best-fit position of AT 2018cow is obtained using a 4-parameter fit on the F225 image at T=1474 d (the highest signal-to-noise value of the four UV images), in which the (x,y) position, the PSF normalisation, and the background are left free to vary. The best-fit (x,y) coordinates are then used as fixed input parameters for the fits on the other images (which is possible because of the pixel-wise alignment described in Section 2.1), leaving a 2-parameter fit (the normalisation and background are the remaining free parameters). We minimize the \(\chi^{2}\) in this area and report the values for the best-fit background and PSF normalisation.
To produce PSF subtracted images, the PSF template multiplied by the best-fit normalisation is subtracted from the data at the best-fit position. To calculate the magnitude of the subtracted point source, we sum the number of electrons/s in the template PSF in a circular area with a 6-pixel radius around the peak of the PSF, and multiply by the best-fit normalisation. We determine the error on the best fitting peak height by performing a two parameter \(\chi^{2}\) fit, leaving the centroid position fixed on the best-fit position allowing only the PSF normalisation and the background to vary. The error on the height is determined using \(\Delta\chi^{2}=2.30\). We calculate the error on the magnitude by multiplying the summed PSF model with the error on the PSF normalisation.
We also perform PSF photometry using dolphot(v2.0, Dolphin, 2000). This software package is specifically designed for PSF photometry in crowded fields. It performs photometry on the individual _fic images and combines the individual measurements into a final (Vega) magnitude for each source. We transform the Vega magnitudes into AB magnitudes using the same difference in zeropoints as mentioned in Section 2.2. We use twackage from drizzlepac to align all _fic images to the drizzled F336W T=713 days image, as this has the sharpest PSF. We then perform PSF photometry for this epoch leaving dolphot free to search for sources and use the output positions of this run as fixed positions for the other filters and epochs using the "warmstart" option in dolphot.
### Aperture photometry on difference images
We compute difference images using hidpants (v5.1.11; Becker, 2015) by subtracting epoch 3 from epoch 1 or 2 to investigate the brightness of any residual emission at the position of AT 2018cow. To perform the subtraction we use default values for the input parameters of hidpants except for bgo, ko, and the rss and rss parameters where we use values of 0.1, 0.05, 5 and 5, respectively. The parameters (bgo, ko, nsx, nsy) are the spatial orders of the background and kernel variations and the number of stamps within a region in x and y direction, respectively. We also change the gain (which is equal to the exposure time for the _HST_ reduced data), and values for the upper and the lower valid data counts for each combination of images we compute a difference image for. We maximize the size of the difference image which is however limited by the need to avoid gaps between the CCDs in the different exposures. We also perform aperture photometry on these difference images in all filters, using the procedure described below.
We measure the flux density of any residual on the difference images by determining the number of electrons/s in a circular aperture of 0.08 arcsec radius centered on the position of AT 2018cow. From this, we subtract the mean background and we convert to magnitudes. To determine the mean and standard deviation of the background flux density in the difference images, we randomly place circular apertures of the same radius as above within 30 pixels of the position of AT 2018cow. In placing these apertures we avoid regions where in the images bright objects are present (see Figure 1 for an example of the placement of these regions in the epoch 1 F555W image). We find a large spread in the value of the background (on average a factor \(\sim 1.5\) for the optical filters and between a factor \(\sim 2\) and \(\sim 33\) for the UV filters), and therefore the magnitude and its uncertainty depend on the flux density in the background. We will come back to this in the Discussion, while in the paper we use the median background to determine the source magnitude in the difference image and the standard deviation on the background as the \(1\sigma\) uncertainty on the magnitude in the difference image.
If the measured number of electrons/s in the aperture at the position of AT 2018cow is lower than the mean background, or of similar value to the standard deviation of the background we determine a \(\sigma\) upper limit. For this, we measure the number of electrons/s in a circular aperture with 0.08 arcsec radius centered on the position of AT 2018cow and we added three times the standard deviation on the background as described above. The signal-to-noise ratio of the detection of a source in the difference images is determined as the flux density in the source divided by the standard deviation in the flux density in the background.
## 3 Results
### Astrometry
We find a frame-to-frame alignment uncertainty of \(0.005-0.024\) arcsec (\(0.19-0.97\) pixels), depending on which combination of frames is looked at. The alignment between images using the same filter is systematically better than alignment between images using different filters.
A relevant question relating to the nature of the late time emission is whether it is dominated by a point-like component that may be due to the transient, or whether it could arise from an underlying compact region. We therefore check if the position of any emission detected in the difference images is consistent with the position of AT 2018cow.
To investigate this we map the early time UV observations (in particular the F336W data) to the later time F555W observations using 10 compact sources which are likely star forming regions within the host galaxy (see Table 2 for the positions of these sources). We then fit a geometric transformation using gdomap, allowing only for a shift in position. The centroid locations for the UV source at 713 days and the compact source in F555W at 1474 days are entirely consistent (\(\delta(x)=0.19\pm 0.25\) pixels and \(\delta(y)=0.01\pm 0.19\) pixels). Furthermore, the location of a faint residual visible in the F555W difference image between epoch 1 and epoch 3 is also consistent with the brightest pixel in all epochs of F555W imaging (\(\delta(x)=0.30\pm 0.36\) pixels and \(\delta(y)=0.06\pm 0.36\) pixels, where the additional uncertainty arises from the centroid error of the residual emission in the F555W image).
### Photometry
#### 3.2.1 Aperture photometry
The results of our aperture photometry can be found in Table 1. In the two UV filters (F225W and F336W) and the F555W filter the source has faded between the first and the third epoch (by \(0.55\pm 0.08\), \(0.39\pm 0.06\) and \(0.23\pm 0.06\) magnitudes, respectively). In the F814W band the magnitudes are consistent with being the same within \(3\sigma\).
#### 3.2.2 Photometry from PSF fitting
In the _right panels_ of Figure 1 we show the residuals after PSF subtraction in high contrast for all epochs and filters. The best-fit position of the centroid of the PSF model (as determined on the F225W T=1474 days image) is marked by red pointers in each image. The _left panels_ show the same images, before subtracting the best-fit PSF model. In general, the emission in the UV filters subtracts out well while the point source subtraction in the optical filters reveals/highlights the presence of residual emission close to and in some cases under the source position. The magnitudes of the subtracted point sources are listed in Table 1 under PSF photometry. We find reduced \(\chi^{2}\) values between 0.5 and 1.1 for the best fits of the PSF normalisation and background value, showing our model describes the data well. Generally, the PSF magnitudes of the subtracted point source are consistent within \(3\sigma\) with those derived through our aperture photometry for all filters, although the PSF magnitudes in the F814W filter are systematically fainter (but still consistent within \(3\sigma\)).
Any small residuals present in the PSF-subtracted images obtained through the UV filters can be explained by the fact that the PSF in the UV varies as a function of source location on the image. Due to various factors (such as the coatings of the optical elements) the UV PSF contains broader wings than the optical PSF and these broad wings have complex features7. Since we try to fit the central part of the PSF to the data, the features in the wings can leave residuals when a template PSF determined at one place of the image is subtracted from a source at another location in the image.
Footnote 7: [https://hst-docs.stsci.edu/wfc3ihb/chapter-6-wis-imaging-with-wfc3/6-6-wvis-optical-performance](https://hst-docs.stsci.edu/wfc3ihb/chapter-6-wis-imaging-with-wfc3/6-6-wvis-optical-performance)
#### 3.2.3 Photometry using dolphot
The results or our PSF photometry using dolphot can be found in Table 1. However, dolphot yields no detection at the position of
AT 2018cow in F814W for any of the observation epochs and in F555W at the epoch at T=1135 days, unless the input source position is fixed, as described in Section 2.3, which is effectively equivalent to forced photometry at the position of AT 2018cow.
#### 3.2.4 Aperture photometry on the difference images
Figure 2 shows the difference images created by subtracting the epoch 3 images from the epoch 1 images for the two optical filters. Here, the position of AT 2018cow is indicated by red markers. In the
Figure 1: _Left panel:_ Three columns of four rows of cutout images close to the location of AT 2018cow for all filters (rows) and epochs (columns). Intensity is given in e\({}^{-}\)/s in a colour scale, with blue being the least intense and yellow most intense. The best fit centroid position of the PSF to the emission at the location of AT 2018cow lies where the two red lines touch. The cross hairs have a length of 0.1 arcsec. _Right panel:_ Three columns of four rows of cutout images showing the residuals of PSF subtraction at the location of AT 2018cow for all filters (rows) and epochs (columns). The exposure times for the last epoch is longer than for the first two epochs, the second epoch having the shortest exposure time of all, which explains the difference in noise properties in the residual images.
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline Filter & Epoch & \# of & Exp. time & Aperture phot. & Aperture phot. & Diff. image \({}^{\dagger}\) & Diff. image \({}^{\dagger}\) & PSF phot. & PSF phot. & dolphot & dolphot \\ & (day) & exp. & (sec) & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) \\ \hline F225W & 713 & 3 & 1116 & \(1.17\pm 0.06\) & 23.73\(\pm\)0.05 & 0.45 \(\pm\) 0.06 & 24.77 \(\pm\) 0.14 & 1.41 \(\pm\) 0.11 & 23.53 \(\pm\) 0.09 & 1.08 \(\pm\) 0.06 & 23.82 \(\pm\) 0.06 \\ F336W & 713 & 3 & 1116 & \(0.87\pm 0.04\) & 24.05\(\pm\)0.04 & 0.28 \(\pm\) 0.04 & 25.28 \(\pm\) 0.15 & 0.82 \(\pm\) 0.07 & 24.11 \(\pm\) 0.09 & 0.75 \(\pm\) 0.03 & 24.21 \(\pm\) 0.05 \\ F555W & 713 & 3 & 1044 & \(0.39\pm 0.02\) & 24.92\(\pm\)0.04 & 0.09 \(\pm\) 0.02 & 26.54 \(\pm\) 0.25 & 0.48 \(\pm\) 0.06 & 24.69 \(\pm\) 0.13 & 0.27 \(\pm\) 0.06 & 25.32 \(\pm\) 0.06 \\ F814W & 713 & 3 & 1044 & \(0.37\pm 0.03\) & 24.97\(\pm\)0.09 & 0.11 \(\pm\) 0.04 & 26.3\({}^{+0.4}_{-0.3}\) & 0.14 \(\pm\) 0.06 & 26.0\({}^{+0.6}_{-0.4}\) & 0.13 \(\pm\) 0.2 & 26.06 \(\pm\) 0.17 \\ \hline F555W & 1135 & 2 & 710 & \(0.33\pm 0.02\) & 25.10\(\pm\)0.06 & \(<0.09\) & \(>26.5\) & 0.35 \(\pm\) 0.07 & 25.07 \(\pm\) 0.22 & 0.18 \(\pm\) 0.02 & 24.79 \(\pm\) 0.10 \\ F814W & 1135 & 2 & 780 & \(0.23\pm 0.03\) & 25.48\(\pm\)0.15 & \(<0.11\) & \(>26.3\) & 0.10 \(\pm\) 0.06 & 26.4\({}^{+1.1}_{-0.5}\) & 0.05 \(\pm\) 0.02 & 27.2\({}^{+0.7}_{-0.4}\) \\ \hline F225W & 1474 & 3 & 1845 & \(0.71\pm 0.04\) & 24.28\(\pm\)0.06 & – & – & 0.76 \(\pm\) 0.08 & 24.20 \(\pm\) 0.11 & 0.54 \(\pm\) 0.04 & 24.57 \(\pm\) 0.07 \\ F336W & 1474 & 3 & 1953 & \(0.61\pm 0.02\) & 24.44\(\pm\)0.04 & – & – & 0.54 \(\pm\) 0.04 & 24.56 \(\pm\) 0.08 & 0.51 \(\pm\) 0.02 & 24.63 \(\pm\) 0.04 \\ F555W & 1474 & 3 & 1149 & \(0.32\pm 0.01\) & 25.15\(\pm\)0.05 & – & – & 0.37 \(\pm\) 0.05 & 24.98 \(\pm\) 0.15 & 0.19 \(\pm\) 0.01 & 25.68 \(\pm\) 0.07 \\ F814W & 1474 & 3 & 2271 & \(0.29\pm 0.02\) & 25.24\(\pm\)0.07 & – & – & 0.08 \(\pm\) 0.04 & 26.6\({}^{+0.6}_{-0.4}\) & 0.08 \(\pm\) 0.01 & 26.53 \(\pm\) 0.17 \\ \hline \end{tabular} \({}^{\dagger}\)This implicitly assumes that any light at the position of the transient at epoch 3 is not due to AT 2018cow.
\end{table}
Table 1: The result of our aperture and difference image photometry for AT2018cow, using a circular aperture of r=0.08 arcsec radius as well as PSF photometry following either our manual procedure (see Section 2.3 for details) or using dolphot. “Diff. image” refers to the image obtained after subtracting the image obtained during the third epoch from the epoch 1 or 2 image under consideration (see Section 2.4 for details). Aperture photometry is performed on the diff. images. Values include aperture correction and a Galactic reddening correction as mentioned in the text. To correct for Galactic extinction we have used \(\rm A_{F225W}=0.524\), \(\rm A_{F236W}=0.334\), \(\rm A_{F555W}=0.214\) and \(\rm A_{F814W}=0.115\). Values without error bars are 3\(\sigma\) upper limits.
F555W difference image (_left panel_) there is residual emission near the position of AT 2018cow. This residual emission is not an artefact due to uncertainties in alignment as there are no such residuals at the positions of other point sources in the difference image. This residual is detected at a signal-to-noise ratio of 4.5 with a magnitude of \(26.54\pm 0.25\), consistent with the difference between the F555W magnitude in epoch 1 and epoch 3 as measured through aperture photometry.
For the observations obtained in the F814W filter, no distinguishable residual emission is present (when looking by eye) in the difference image, as can be seen in the _right panel_ of Figure 2. Following the same procedure as for F555W above we find a signal-to-noise ratio of 3.4 with a magnitude of \(26.3^{+0.4}_{-0.3}\). Subtracting the epoch 3 image and then measuring the flux/magnitude of the residual measures the decaying component in the AT 2018cow light. An alternative way of looking at the difference image is that it assumes all emission at epoch 3 (T=1474 days) is due to an underlying source at the position of AT 2018cow. Under "Diff. image" in Table 1, we list our results for aperture photometry on all difference images created by subtracting the epoch 3 from the epoch 1 and epoch 2 images. For the F555W and F814W epoch 2 minus epoch 3 difference images the measured flux density in the aperture is consistent with that expected due to variations in the background, hence we report 3\(\sigma\) upper limits of \(>26.5\) in F555W and \(>26.3\) in F814W.
### Lightcurve
Out of the three different ways we used to measure the photometry of AT 2018cow the aperture and PSF photometry agree within 3 \(\sigma\). The aperture photometry on the difference images (epoch 1 or epoch 2 minus epoch 3) yield fainter results for the source brightness. This can be explained as follows: through photometry on a difference image we are sensitive only to the component of the light that varied between the epochs under consideration. In the extreme scenario that the third epoch contains _no light_ from AT 2018cow the magnitudes determined through analysis of the difference images are relevant. In the opposite extreme scenario, we assume that _all the light_ detected at the third epoch comes from AT 2018cow. Clearly, whether either or none of these two is a viable assumption may well depend on the filter of the observations under consideration.
We show the brightness evolution of AT 2018cow as a function of time in Figure 3, using the results of our aperture photometry on the images and the difference images, together with early time data from Perley et al. (2019) and Sun et al. (2022) (circles). Even though the effective wavelengths of the filters used in the early UVOT and later _HST_ observations are slightly different, we compare UVOT/UVW1 to _HST_/F235W, UVOT/U to _HST_/F336W, UVOT/V to _HST_/F555W and UVOT/I to _HST_/F814W. Different filters are indicated using different colours and we offset the light-curve in each filter by a constant shown in the figure legend for display purposes. Our aperture photometry measurements are shown with squares and our measurements for AT 2018cow obtained assuming the third epoch contains no transient emission (aperture photometry on the difference images) are indicated with triangles when a residual was detected or downwards pointing arrows when an upper limit to the source magnitude was determined. Comparing the early-time (\(<100\) days after discovery) measured decay in absolute magnitude with absolute magnitude (limits) obtained for the last three _HST_ epochs, it is clear that the detected emission is brighter than expected had this trend continued.
### Comparison of AT 2018cow and compact UV selected sources
Next, we explore whether AT 2018cow is localised in an unusual region of its host galaxy by fitting synthetic spectra of simple stellar populations to 23 compact UV-selected star-forming (SF) regions within the host (plus the location of AT 2018cow). These SF regions were selected by running source extractor in dual image mode in the same way as for AT 2018cow (see Section 2.2) removing sources that are not detected in all four filters at T=713 days. We also removed sources that are detected with a signal-to-noise ratio \(<3\). From these sources we select those that have a constant magnitude (within
Figure 2: The residual flux after subtracting the image obtained at T=1474 d from the T=713 d image for the two optical filters (F555W _left panel_; F814W _right panel_) using ‘notpants’. The location of AT 2018cow is indicated with red thick marks. Residual emission is present at the position of AT 2018cow with a signal-to-noise of 4.5 in the F555W difference image and signal to noise of 3.4 in the F814W difference image (_left panel_; see Section 3.2.4 of the main text for details.)
Figure 4: Greyscale image of the host galaxy of AT 2018cow in the F336W filter at T=713 days., with the ages of compact UV-selected sources that were detected in all four filters indicated by coloured circles. The colours correspond to population ages, indicated by the colour bar and derived from BPASS SED fitting as described in Section 3.4. The location of AT 2018cow is marked by green cross hairs. Number labels for the regions are as in Table 1.
Figure 3: AT 2018cow light-curves in different filters, F225W in blue, F336W in red, F55SW in yellow and F814W in green (with offsets as indicated in the legend). The early time data is from Perley et al. (2019) in transparent circles and Sun et al. (2022) in opaque circles. Our aperture photometry results marked with squares assume all flux measured in the last (third) epoch is due to the transient, whereas for the measurements indicated with triangles and downwards pointing arrows (for upper limits) we assumed that all detected flux in epoch three is unrelated to AT 2018cow. The error bars are at a 1\(\sigma\) confidence level. The horizontal bars through the markers do not indicate uncertainties in the observation date but instead they are the end caps of the error bars on the magnitudes.
\(3\sigma\)) as measured on T=713 days and T=1474 days. Differences in magnitudes between these epochs might be caused by e.g., different orientations of _HST_ during the observations. We ignore epoch 2 in this comparison because the exposure time is shorter and there are only two exposures, resulting in a bad removal of cosmic rays.
Next, we select the sources that behave PSF-like in F336W. We test this by performing aperture photometry using two different values for the radius of the circular aperture and we retained sources only if the difference in their photometry was consistent with the different aperture corrections for a point source given the two different aperture radii. A full list of the positions and magnitudes of the sample can be found in Table 1 in the Appendix C.
To determine the ages of these regions, we make use of BPASS v2.2.1 (Binary Population and Spectral Synthesis, Eldridge et al., 2017; Stanway & Eldridge, 2018) synthetic spectra, assuming a single burst of star formation and a metallicity (by mass fraction) of \(Z=0.01\) (based on the host metallicity found by Lyman et al., 2020). For each region, SED fitting is performed by convolving the BPASS spectra at each age (52 ages spaced logarithmically from 1 Myr to 100 Gyr are used) with the filter response curves for F225W, F336W, F555SW, and F814W (Rodrigo et al., 2012; Rodrigo & Solano, 2020), converting magnitudes to fluxes, and vertically scaling8 the synthetic spectra to minimise \(\chi^{2}\) across both age and different values for the host-intrinsic extinction. The extinction in each filter is calculated by adopting their effective wavelengths and using the python extinction module (Barbary, 2016), with a Fitzpatrick (1999) extinction curve and R\({}_{\rm V}=3.1\). Galactic extinction is already accounted for as described in Section 2.2.
Footnote 8: The scaling is needed as the synthetic spectra are for a \(10^{6}\) M\({}_{\odot}\) population, in Solar luminosity per Angstrom
For each region we determine a best-fit age and extinction A\({}_{\rm V}\). Full results can be found in Appendix C. The extinction values are in the range 0.0-0.6 (in broad agreement with A\({}_{\rm V}=0.2\) as found by Sun et al., 2023, for nearby star forming complexes), and the ages range from 6-25 Myr. These ages are younger than the tens of Myr found by Lyman et al. (2020) for example, but this can be explained by the spaxel size of their MUSE integral field unit data, which averages over larger physical areas than the compact star-forming regions we are probing here.
The reduced \(\chi^{2}\) values (which are the same as the \(\chi^{2}\) values because our fit has one degree of freedom) for the 23 compact star forming regions are typically around \(\sim\)1-10; whereas the fits at the location of AT 2018cow (at both 713 and 1474 days) are notably poorer, with \(\chi^{2}=47\) and 37, respectively. The fits at the location of AT 2018cow favour very little to no extinction, and tellingly, favour the youngest population age available in the BPASS outputs (1 Myr), whilst still failing to reproduce the extremely blue observed SED.
In Figure 4, we show the 23 UV-selected star-forming regions over an F555W image of the host galaxy. Each of the 23 regions is encircled, where the colour of the circle corresponds to the best-fit population age. Young stellar populations are present across the galaxy, with no preference for particularly young star forming regions in the vicinity of AT 2018cow, although there are 2 star forming regions within \(\sim 400\) parsec of the transient (regions 1 and 3, these were unresolved in previous non-_HST_ data sets).
### Spectral Energy Distribution of AT 2018cow
Figure 5 (_left panel_) shows the spectral energy distributions (SEDs) for AT 2018cow as measured at epoch 1 (T=713 d) and at epoch 3 (T=1474 d). The black markers represent measurements from the third epoch, while the grey markers those of the first epoch. The marker symbols are the same as in Figure 3. The coloured bands represent the FWHM of the filter with the corresponding colour in Figure 39. The _right panel_ of Figure 5 shows both possible extremes of the SED of AT 2018cow in red compared to the SEDs of compact UV-selected sources detected in a box of 180x180 pixels centred on the position of AT 2018cow ("neighbours") in green, and "other sources" in the rest of the host galaxy in grey for T=713 d. From this red shaded region it is clear that for either of the two extreme scenarios for the aperture photometry at epoch T=713 d, the F555W\(-\)F225W colour of AT 2018cow is bluer than that of the neighbours. The _left panel_ of Figure 5 shows that the SED for the third epoch lies in between the aperture photometry SED and the difference image SED. Therefore, the T=1474 d SED is also bluer than that of the neighbours.
Footnote 9: [https://hst-docs.stsci.edu/wfc3ihb/chapter-6-uvis-imaging-with-wfc3/6-5-uvis-spectral-elements](https://hst-docs.stsci.edu/wfc3ihb/chapter-6-uvis-imaging-with-wfc3/6-5-uvis-spectral-elements)
We fit a Planck function to the four-filter SEDs at T=713 d, T=1474 d, and to the four-filter SED when assuming none of the third epoch emission is due to AT 2018cow, with the best-fit black body over-plotted in the _left panel_ of Figure 5 in orange, green, and blue, respectively. The best-fit values for the temperature and radius, the calculated luminosity, the number of degrees of freedom, and the reduced \(\chi^{2}\) values are presented in Table 2. The error on the temperature for the fit to the epoch 1 - epoch 3 difference image was calculated by fixing the radius to the best-fit value and finding the value for which \(\Delta\chi^{2}=1\). This was done because the error calculated by the fitting algorithm was larger than the best fitting value for the temperature. Only the reduced \(\chi^{2}\) value for the fit to the epoch 1 SED derived assuming epoch 3 contains no light from AT 2018cow is close to 1 (at a value of 2.2). However, the error on the luminosity is very large due to the large errors on the radius. Due to the sizes of the error bars on the magnitudes obtained with aperture photometry on the difference image, the fit of the Planck function is dominated by the two data points in the UV bands, meaning the fit is almost degenerate for a two-parameter Planck function. This results in a large error on the fit and therefore on the calculated luminosity.
## 4 Discussion
In this paper, we present aperture and PSF photometry of _HST_ data of the FBOAT2018cow. We first compare our results in Table 1 with the results from the epoch 1-3 PSF photometry by Sun et al. (2022) and Sun et al. (2023). We find that our measurements in the UV filters yield a source that is consistent within \(3\sigma\) in the first epoch, while in the last epoch our source is brighter than they report (there are no UV images for the second epoch). In the optical filters our measurements indicate a brighter source in all epochs than found by Sun et al. (2022, 2023). They assumed all the light is emitted by AT 2018cow. Additionally, Sun et al. (2022, 2023) find a steeper decay between epoch 1 and 3 in the UV filters (\(1.02\pm 0.11\) mag and \(0.57\pm 0.07\) mag for F225W and F336W, respectively) than we do (\(0.55\pm 0.08\) mag and \(0.39\pm 0.06\) for F225W and F336W, respectively). Furthermore, they find no evidence for a decay in the source brightness in the optical filters, whereas we do (\(0.23\pm 0.06\) maggs in F555W, and a detection with a signal to noise of 3.4 in the F814W epoch 1 and epoch 3 difference image with a magnitude of \(26.3^{+0.4}_{-0.3}\)). We will investigate possible reasons for these differences below.
Next, we compare with the epoch 1-3 PSF photometry reported in Chen et al. (2023). Our aperture as well as our manual PSF photometry give brighter magnitudes for AT 2018cow than Chen et al. (2023), although the difference is small for the two UV filters it increases for the optical filters. Comparing the magnitudes in the Chen et al. (2023) table 1 with their figure 6 we deduced that their table 1 magnitudes are corrected for extinction. However, if they are not, the differences with our extinction-corrected magnitudes is reduced, especially for the UV filters. However, still, only the measurements in F225W (both epochs) and F555W T=113C days would be consistent withing within the 3\(\sigma\) error. Our dolphot PSF photometry results are consistent within 3\(\sigma\) with the values presented by Chen et al. (2023) in their table 1 if those values are not corrected for Galactic extinction. When leaving the position as a free parameter, dolphot does not find a source in F814W at any epoch and also not in F555W at the epoch T=113C days. Only forced photometry (i.e. keeping the source position fixed) yields a sometimes marginal detection of the source at those epochs and filters.
However, this does not necessarily mean the photometry presented by Sun et al. (2022, 2023), Chen et al. (2023) or our photometry results are wrong. In practice, contributions from other sources besides a point source may influence the measurements, or if no point source is present but if the observed light is dominated by diffuse emission (on the scale of \(\sim\)few times the PSF size) in AT 2018cow's host galaxy galactic disc, PSF photometry provides an upper limit on the magnitude of a point source at the location of AT 2018cow. Instead, aperture photometry may over-estimate the true flux density of the transient if the light from the point source and diffuse emission in the galactic disc are of similar magnitude. In practise, the estimated value of the background flux density under AT 2018cow may influence the determined magnitudes especially in crowded regions like that of AT 2018cow. Next, we investigate the potential influence of the choice of the background region used to estimate the flux density at the position of AT 2018cow.
Using the same 20 background regions we used for the aperture photometry on the difference images (see Figure 14), we measure
Figure 5: _Left panel:_ The spectral energy distribution (SED) of the emission detected at the position of AT 2018cow at T=713 d and T=1474 d. The four vertical coloured bands are centered on the effective wavelength of the filters used for the observations while the width of the vertical bands indicate the passband rectangular width of the filters. Light grey markers are used for the data obtained at T=713 d. Here, the light grey circles indicate the measured flux density assuming all light in the third epoch (T=1474 d) originates from AT 2018cow, whereas light grey triangles are used for measurements obtained assuming none of the third epoch light is due to AT 2018cow. The circles are always at a higher flux density than the triangles. The black symbols represent our measurements of the source flux density obtained at T=1474 d. The lines are Planck functions fitted to the four-point SEDs at T=713 d (orange), T=1474 d (green), and to the grey triangles (blue). The best fitting values for the temperature and the radius, and reduced \(\chi^{2}\) values can be found in Table 2. The fit to the difference image gave unphysical (a negative) values for the temperature when considering the uncertainty on the temperature using both python routines cuvuv_pit and lmfit. To obtain an estimate of the uncertainty on the black body temperature we fixed the radius to the best fitting value and determine for which value for the temperature around the best fitting temperature value \(\Delta\chi^{2}=1\). From the reduced \(\chi^{2}\) values and the Figure we conclude that a single black body function is only a reasonably good description of the SED for the light grey triangles. _Right panel:_ The SEDs of our list of compact UV-detected sources at T=713 d (Table 1 contains selected properties of these sources). The data for AT 2018cow is in red with the marker shapes as mentioned above. We make a distinction between ”neighbours” shown in green and ”other sources” in light grey. See the main text for the definition of ”neighbour” and ”other sources”. Irrespective of the interpretation of the AT 2018cow data at T=1474 d, the F555W–F2225W colour of the source at the position of AT 2018cow is bluer than any of our compact UV-detected sources.
\begin{table}
\begin{tabular}{l l l l l l} \hline Epoch & log(T (K)) & radius (R\({}_{\odot}\)) & luminosity (erg s\({}^{-1}\)) & reduced \(\chi^{2}\) & degrees of freedom \\ \hline
1: T=713 d & \(4.54\pm 0.04\) & \(34\pm 3\) & \((6\pm 2)\times 10^{39}\) & 17.2 & 2 \\
3: T=1474 d & \(4.37\pm 0.02\) & \(43\pm 2\) & \((1.9\pm 0.4)\times 10^{39}\) & 17.9 & 2 \\ \hline Epoch 1 - Epoch 3 & \(5.03\pm 0.04\) & \(9\pm 6\) & \((4_{-5}^{+5})\times 10^{407}\) & 2.2 & 2 \\ \multicolumn{5}{l}{\({}^{\dagger}\) See Section 3.5 for the explanation on how the error bars on the luminosity were determined.} \\ \end{tabular}
\end{table}
Table 2: Results from fitting a Planck black body function to the _HST_ spectral energy distribution for AT 2018cow. These fits are shown in Figure 5.
the median, minimum, and maximum value for the flux density in the background aperture. There is a large spread between these three values. To investigate how the choice of background region influences the flux density measured for AT 2018cow we compare the results based on which of these three values is subtracted from the flux density measured in the aperture centered on the position of AT 2018cow. In Table 3 we show the resulting magnitude measurements for the different background regions. As expected, we find that using a higher background flux density yields a lower flux density for AT 2018cow. Depending on the choice of background in our work and in the work of Chen et al. (2023) both results could be consistent in all filters. We do note that in the F814W filter when using the maximum background flux density, our results are either upper limits when the maximum background flux density was higher than the flux density in the aperture of AT 2018cow, or there are large error bars on our photometry. Clearly, the region used to determine the background flux density greatly influences the value of the magnitude of AT 2018cow.
Next, we investigate if there are filters and epochs where the detected light originates solely from AT2018cow, or if it is possible to determine if the emission is dominated by underlying sources (for instance from diffuse emission in the galactic disk or e.g., a star forming region or cluster) or if it is a combination of both. Understanding the origin of the light is important because it will influence the interpretation of the power source of AT 2018cow.
In the observations obtained through the UV filters the magnitude has decreased between epochs, suggesting that a significant fraction of the detected light is emitted by the fading transient. The SED of the light extracted at the position of AT 2018cow is substantially bluer than that of our compact, UV-selected, star forming regions detected throughout the host of AT2018cow. This is also in line with the notion that the majority, but not necessarily all, of the light detected in the UV arises from the transient. Subtracting a point source from the UV images at the location of AT 2018cow, leaves residuals consistent with noise (see Figure 1). Therefore, we conclude that the emission in the UV filters is dominated by a point source, likely the transient event AT 2018cow. In the optical filter images, comparing the observations at epoch 1 with those at epoch 3 there is evidence that the source faded in addition to light from either AT 2018cow (constant) and/or underlying emission from part of the host galaxy itself.
Overall, a picture emerges where light from the transient is still detected in the UV images, while in optical images we cannot determine if the detected light at epoch 3 is due to AT 2018cow or due to diffuse emission in the galactic disc or, more speculatively, due to a compact source at the same position of AT 2018cow. Note that in the optical images crowding is more important than in the UV images.
The SED of the emission detected at the location of AT 2018cow is consistent with this picture (Figure 5). While the F814W-F555W colour of AT 2018cow is consistent with that of the neighbouring sources, the F336W-F225W colour at the location of AT 2018cow is bluer than that of the sources in the neighbourhood. This and the fact that a single black body does not fit the SED, together with the different variability properties of the UV and optical filters, suggests that the UV and optical parts of the SED are caused by more than one emission type and/or by more than one source. This conclusion does not depend on the assumption for the nature of the light detected at 1474 days (either transient or environment light or a combination thereof). Furthermore, the emission cannot be solely from a stellar population - it is too blue - strongly implying the presence of late-time UV emission from AT 2018cow.
We also searched for BPASS single and binary stellar models, across all possible stellar ages (at Z=0.010), for models satisfying log\({}_{10}\)(T/K)\(>4.7\) and log\({}_{10}\)(L/L\({}_{\odot}\))\(>7.0\). These constraints are derived from fitting a black body to the late-time emission at the location of AT 2018cow (see also Sun et al., 2023). We find no stellar models which are this blue and luminous, and therefore, a dominant contribution from an underlying massive star or binary seems ruled out by the data.
The F555W-F814W colour at the location of AT 2018cow at 1474 days is \(=-0.09\pm 0.08\) and the absolute magnitude is \(-\)\(-\)9. Assuming that the optical bands at epoch 3 are free from light originating from the transient (as we do when taking the magnitudes measured on the difference images), we check what kind of source(s) can explain these values. They are consistent with those expected for an OB association or star-forming region (e.g., Drazinos et al., 2013), and they are broadly consistent with the F555W-F814W colours of the UV-selected, compact star-forming regions shown in Figure 4. The mean F555W-F814W colour (corrected for Galactic but not intrinsic extinction [at the specific location in the host galaxy]) of these regions is 0.02\(\pm\)0.05. Excluding the UV filters, fixing A\({}_{\rm V}=0\) and performing SED fitting as described in Section 3.4, we infer a best-fit population
\begin{table}
\begin{tabular}{l c c c c c c c} \hline Filter & epoch & min background & min background & median background & median background & max background & max background \\ & epoch & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) & F\({}_{\nu}\) (\(\mu\)Jy) & (mag) \\ \hline F225W & 713 & 1.28\(\pm\)0.06 & 23.63 \(\pm\) 0.05 & 1.18\(\pm\)0.06 & 23.71 \(\pm\) 0.05 & 1.07\(\pm\)0.06 & 23.82 \(\pm\) 0.06 \\ F336W & 713 & 0.95\(\pm\)0.04 & 23.95 \(\pm\) 0.05 & 0.88\(\pm\)0.04 & 24.02 \(\pm\) 0.05 & 0.78\(\pm\)0.04 & 24.16 \(\pm\) 0.06 \\ F555W & 713 & 0.49\(\pm\)0.05 & 24.68 \(\pm\) 0.11 & 0.40\(\pm\)0.05 & 24.92 \(\pm\) 0.14 & 0.30\(\pm\)0.05 & 25.22 \(\pm\) 0.19 \\ F814W & 713 & 0.57\(\pm\)0.12 & 24.50 \(\pm\) 0.22 & 0.41\(\pm\)0.12 & 24.9 \(\pm\) 0.3 & 0.18\(\pm\)0.12 & 25.8\({}^{\dagger}_{-0.6}\) \\ \hline F555W & 1135 & 0.42\(\pm\)0.05 & 24.85 \(\pm\) 0.13 & 0.33\(\pm\)0.05 & 25.10 \(\pm\) 0.17 & 0.25\(\pm\)0.05 & 25.41 \(\pm\) 0.22 \\ F814W & 1135 & 0.46\(\pm\)0.12 & 24.8 \(\pm\) 0.4 & 0.26\(\pm\)0.12 & 25.4\({}^{+0.7}_{-0.4}\) & \(<\)0.34\({}^{\dagger}\) & \(>\)25.11\({}^{\dagger}\) \\ \hline F225W & 1474 & 0.76\(\pm\)0.03 & 24.19 \(\pm\) 0.05 & 0.70\(\pm\)0.03 & 24.28 \(\pm\) 0.05 & 0.65\(\pm\)0.3 & 24.37 \(\pm\) 0.05 \\ F336W & 1474 & 0.65\(\pm\)0.03 & 24.37 \(\pm\) 0.05 & 0.61\(\pm\)0.03 & 24.43 \(\pm\) 0.05 & 0.51\(\pm\)0.03 & 24.64 \(\pm\) 0.07 \\ F555W & 1474 & 0.40\(\pm\)0.05 & 24.88 \(\pm\) 0.14 & 0.32\(\pm\)0.05 & 25.13 \(\pm\) 0.17 & 0.22\(\pm\)0.05 & 25.53 \(\pm\) 0.25 \\ F814W & 1474 & 0.47\(\pm\)0.13 & 24.7 \(\pm\) 0.3 & 0.30\(\pm\)0.13 & 25.2\({}^{+0.6}_{-0.4}\) & \(<\)0.43\({}^{\dagger}\) & \(>\)24.8\({}^{\dagger}\) \\ \hline \end{tabular} \({}^{\dagger}\)The flux density value of the background was higher than that in the aperture centred on the position of AT 2018cow, so we report the 3\(\sigma\) upper limit for the maximum background flux density.
\end{table}
Table 3: The result of our aperture photometry for AT2018cow, using a circular aperture of r=0.08 arcsec radius for three different values of the background (see main text for details). The reported magnitudes include the effect of the aperture correction and the Galactic reddening correction. To correct for Galactic extinction we used A\({}_{\rm F225W}=0.524\), A\({}_{\rm F336W}=0.334\), A\({}_{\rm F555W}=0.214\) and A\({}_{\rm F814W}=0.115\). The errors reported are at the 1\(\sigma\) confidence level.
age at the location of AT 2018cow of 20 and 79 Myr at 713 and 1474 days, respectively. Although we cannot determine a precise age with just two optical filters, if we assume no extinction and that the optical light is dominated by the underlying stellar population, the optical spectral slope constrains the age to \(\sim\)100 Myr or less.
Taking the 4-band photometry of AT 2018cow (latest epoch with the median background, see Table 3), and converting it to absolute magnitudes and using BPASS simple stellar populations, we calculate the maximum mass of a cluster that can be present at the location of AT 2018cow before the luminosity in one of the filters exceeds the magnitude plus its \(1\,\sigma\) error. We plot the upper limit on the cluster mass in Figure 6. This upper limit is a strong function of age - as the UV flux reduces with increasing age, the upper limit on the cluster mass increases. An old stellar population at the location of AT 2018cow cannot be ruled out - in particular, we note that a globular cluster can easily be hidden underneath the light of AT 2018cow (based on typical globular cluster ages of several Gyr and masses of \(10^{3}\)-\(10^{6}\)M\({}_{\odot}\), Harris 1996).
### Disc modelling
It has been speculated that AT 2018cow is caused by a tidal disruption event (TDE; e.g., Perley et al. 2019; Kuin et al. 2019). Interestingly, for low mass (\(M_{BH}<10^{6.5}\,M_{\odot}\)) TDEs roughly time-independent UV emission lasting for time scales of years is commonly detected (Van Velzen et al., 2019; Mummery and Balbus, 2020; Wen et al., 2023). Comparing the UV light curve of AT 2018cow (Figure 3) with that of TDEs, for example ASSAN-14li (see e.g., figure 2 in Wen et al. 2023), we note that the UV light curve morphology is similar. Especially the late-time plateau is a distinguishing feature shared by both sources.
To test the hypothesis if the late-time UV emission observed from AT2018cow could come from an evolving accretion flow produced by the tidal disruption of a star by a massive black hole, we follow the procedure set out in Mummery and Balbus (2020), and generate evolving UV light curves by solving the time-dependent general relativistic thin disc equations. In brief, we assume that the tidal disruption of a star results in the formation of a compact ring of material with total mass roughly half that of the disrupted star. This initial ring is assumed to form at the circularisation radius (typically twice the tidal radius) of the incoming star (see also Hayasaki and Jonker 2021). Once this initial condition is specified, by solving the time-dependent relativistic disc equations, the disc density can be propagated out to large times. Once the dynamical evolution of the disc density is solved, the observed disc spectrum can be computed by ray-tracing the emergent flux of each disc annulus. This then allows us to compute late time UV luminosities for a range of black hole and stellar parameters.
The late-time luminosity observed from the location of AT2018cow is, compared to the typical TDE population, at a relatively low level \(\nu L_{\nu}\simeq 10^{39}\) erg/s, at \(\nu\simeq 10^{15}\) Hz. This is much smaller than, for example, the luminosity of the \(\sim 10^{6}M_{\odot}\) black hole mass TDE ASASSN-14li, which had a late time (\(>1\) year) UV luminosity of \(\nu L_{\nu}\simeq 10^{42}\) erg/s. Mummery (2021) showed empirically from fitting the light curves of 9 TDEs that the late time UV plateau luminosity correlates approximately linearly with the black hole mass responsible for the TDE. This empirical result has strong theoretical and numerical support (Mummery and van Velzen et al. in prep.), and suggests that AT2018cow could well be due to a TDE involving an intermediate-mass central black hole.
To test this hypothesis, we numerically sample \(N=10^{5}\) black hole masses uniformly in the range \(10^{1}<M_{BH}/M_{\odot}<10^{7}\). At each black hole mass we sample stellar masses from the Kroupa IMF (Kroupa, 2001), solve the disc equations and "observe" the system at a random inclination (with \(\cos(i)\) sampled uniformly). We sample uniformly the (dimensionless) black hole spin between \(-1<a<1\). As a very conservative constraint on the central black hole mass in AT2018cow, we record all TDE disc systems which produce a UV luminosity at +713 days (the time of the first _HST_ observation) within a factor of 2 of \(3\times 10^{39}\) erg/s at \(\nu=10^{15}\) Hz. The black hole mass distribution of the TDE systems which satisfy this constraint are shown in Figure 7.
A more detailed analysis of the late time AT2018cow light curve and spectrum highlights that an evolving accretion flow provides a plausible explanation of the observed AT2018cow data. It is of course difficult to constrain a best fitting set of parameters from observations in two bands at two epochs, and we do not attempt to measure the precise system parameters of AT2018cow from the late time _HST_ data. Instead, we examine a sub-set (200) of our solutions (Figure 7) which produce particularly "good fits" (as judged by their chi-squared statistic computed from both epochs). For these solutions we generate both optical-UV spectra at \(t=+713\) d and +1474 d, and disc UV light curves from \(t=0\) to \(t=+1500\) d. These disc spectra and light curves are displayed in Figures 8 and 9, respectively. It is clear that an evolving relativistic accretion flow can reproduce the observed late-time properties of AT2018cow.
The central black hole masses inferred from disc modelling (\(M_{BH}\sim 10^{3.2\pm 0.8}M_{\odot}\)) implies that the early-time UV/optical emission observed from AT2018cow is significantly above the Eddington luminosity \(\mathrm{L_{Edd}}\simeq 10^{41}(M_{BH}/10^{3}M_{\odot})\) erg/s. If the early time luminosity is indeed ultimately powered by accretion (which is still uncertain, see e.g., Roth et al. 2020), then it is unlikely that the thin disc models used here would be valid at these very early times (i.e., for the first \(\sim 100\) days). However, by the much later times, which we are interested in, the bolometric luminosities of the disc solutions are typically at the level of a few \(10^{40}\) erg/s (e.g., Fig. 8), suggesting Eddington ratios at the \(10\%\) level, where thin disc models are valid.
Chen et al. (2023) uses a steady state disc model to fit their T=1474 d SED and obtain an estimate for the mass for the BH. However, steady state disc theory predicts an optical/UV disc lumi
Figure 6: The maximum mass of a stellar cluster that can be underlying AT 2018cow as a function of population age. This is determined by the maximum luminosity of a BPASS simple stellar population that can lie at this location without the luminosity in one of the four _HST_ bands exceeding the observed value.
nosity which scales as \((M_{BH}\dot{M})^{2/3}\). This optical/UV luminosity is thus highly degenerate between the (unknown) mass accretion rate \(\dot{M}\), and the black hole mass \(M_{BH}\) (e.g., Frank et al., 2002). However, the late time disc temperature profile in a TDE disc is constrained, as the total initial mass, radial and temporal scales of the disc are known a priori for a given stellar disruption. This initial mass content must then propagate radially according to the standard laws of mass and angular momentum conservation. The resulting late-time optical/UV luminosity of a _TDE disc_ is strongly constrained. We make use of this in our disc model.
If AT2018cow is indeed a TDE, the short time scale and the disc modelling suggests a relatively low-mass BH was responsible for the disruption. Pasham et al. (2021) find a limit of \(M_{BH}<850M_{\odot}\) based on the frequency of the soft QPO. Zhang et al. (2022) find a low frequency QPO, corresponding to a BH mass of \(\sim 10^{4}M_{\odot}\) and they suggest the maximum mass found by Pasham et al. (2021) can be increased to higher mass adding a binary component to the compact object.
A problem for the TDE hypothesis is that the BH responsible for the disruption needs to be embedded in a dense stellar environment for dynamical friction to be efficient enough to bring a star on an orbit towards its tidal radius within a Hubble time (e.g., Stone & Metzger, 2016). Such a dense stellar environment where dynamical interactions occur then enough, may arise in nuclear star clusters, dense young star clusters, or globular clusters. There is evidence of a recent interaction between CGCG 137-068 and a companion galaxy from a ring of high column density gas as well as from a faint tidal tail Lyman et al. (2020); Roychowdhury et al. (2019). If the host galaxy underwent a recent (minor) merger it is conceivable that an IMBH or SMBH, with its nuclear star cluster, is in the process of
Figure 8: Late time (blue = +713 d, red = +1474 d) spectral snapshots of a sample of relativistic accretion disc models for AT 2018cow. These curves show a sub-set of the disc models (Fig. 7) which produced particularly good fits to the data.
Figure 7: The black hole masses consistent with the assumption that AT 2018cow was caused by an tidal disruption event. The distribution of black hole masses has been derived assuming the late time UV emission is due to the accretion disc formed following the disruption (see the main text for details). The mean of the logarithm of black hole mass (M\({}_{\rm BH}\)) is log M\({}_{\rm BH}\) = 3.2\(\pm\)0.8 (with the mass in M\({}_{\odot}\)).
Figure 9: The light curves of the relativistic disc models which produce the spectra displayed in Figure 8. The late time _HST_ data are displayed in blue (F225W) and red (F336W). Early time data in the ultra-violet bands UVW1, UVW2 and UVM2 are displayed in purple. Importantly, a disc model can reproduce the late time AT 2018cow UV emission, without modifying the observed early time AT 2018cow rapid light curve decline. There is no consensus in the TDE community about the origin of the early-time UV (and optical) emission (see, e.g., Roth et al., 2020). The error bars are at a 1\(\sigma\) confidence level, and may be smaller than the marker size.
falling into the center of CGCG 137-068. This, means that a TDE origin of AT 2018cow remains a viable possibility.
However, Michalowski et al. (2019) attributes the presence of the ring of high column density gas reported by Roychowdhury et al. (2019) to internal processes instead of a galaxy merger/interaction. Their observations using H I show no evidence for a concentration of gas near the location of AT 2018cow. They conclude that the host of AT 2018cow differs from the hosts of Gamma-ray bursts (GRBs)/SNs in its properties and therefore the environment of AT 2018cow does not provide evidence for a massive star progenitor for the event, leaving the question on the nature of AT 2018cow wide open.
## 5 Summary and conclusions
Using three epochs of _HST_ observations we investigate the late-time UV and optical emission at the location of AT 2018cow. The main results are that AT 2018cow remains UV-bright, even with evidence for fading in the UV filters (F225W and F336W) between the first and third epoch of _HST_ observations. The magnitude of AT 2018cow in the optical filters (F555W and F814W) can differ by up to a magnitude depending on how the (diffuse galaxy) background at the location of AT 2018cow is determined.
From our analysis we conclude the following: i) The observed UV emission is consistent with being dominated by a fading point source which originates most likely from AT 2018cow. ii) While part of the optical emission could be due to slowly decaying emission from the transient, there is evidence for a contribution of underlying emission, that did not fade between epochs. Some fraction of this could originate in diffuse galactic background light or an underlying point(like) source. iii) The late-time UV emission is reminiscent of late-time UV emission seen for TDEs. The late-time UV luminosity of AT 2018cow is consistent with the disruption of a (low-mass) star by an IMBH. For this scenario to be feasible AT 2018cow needs to reside in a dense (young/old) stellar cluster.
Our research shows that the nature of AT 2018cow is still uncertain. Both model scenarios involving either specific massive star evolution or a tidal disruption of a (white dwarf) star by an intermediate mass black hole have their advantages and disadvantages.
## Acknowledgements
AI thanks Luc Ilysepert for helpful discussions. This work is part of the research programme Athena with project number 184.034.002, which is financed by the Dutch Research Council (NWO). The scientific results reported on in this article are based on data obtained under _HST_ Proposals 15974, 16179 and 16925 with PI A.J. Levan, A. Filippenko and Y. Chen, respectively. This work was supported by a Leverhindre Trust International Professorship grant [number LIP-202-014]. This work makes use of Python packages numpy (Harris et al., 2020), scipy (Virtanen et al., 2020); matplotlib (Hunter, 2007), extinction (Barbary, 2016) and drizzlepac (Hoffmann et al., 2021). This work made use of Astropy:10 a community-developed core Python package and an ecosystem of tools and resources for astronomy (Astropy Collaboration et al., 2013, 2018, 2022). This research made use of Photutils, an Astropy package for detection and photometry of astronomical sources (Bradley et al., 2022). This research has made use of the SVO Filter Profile Service ([http://svo2.cab.inta-csic.es/theory/fps/](http://svo2.cab.inta-csic.es/theory/fps/)) supported from the Spanish MINECO through grant AYA2017-84089 (Rodrigo et al., 2012; Rodrigo and Solano, 2020). This work has made use of v2.2.1 of the Binary Population and Spectral Synthesis (BPASS) models as described in Eldridge et al. (2017) and Stanway and Eldridge (2018).
Footnote 10: [http://www.astropy.org](http://www.astropy.org)
## Data availability
All data used in this paper is publicly available from the _HST_ data archive. A reproduction package for this paper is uploaded to Zenodo ([https://doi.org/10.5281/zenodo.8246571](https://doi.org/10.5281/zenodo.8246571)).
| brightな、青色の、急速に進化するAT2018cowは、多様な波長データの豊富な例として、よく研究されている特殊な宇宙空間の不連続現象です。多様な波長データの豊富さにも関わらず、そのイベントの性質にはまだ合意はありません。私たちは、AT2018cowのバーストから713-1474日後の3つの時代を対象としたHubble宇宙望遠鏡(HST)の観測分析をPresentingしています。特に、複雑な背景の中でAT2018cowが住んでいるため導入されたトランジションの光度測定の不確実性を考慮して。光度測定は、UVで顕著な変化、そして光学ではより微妙な変化が見られます。HSTの最後の観測では、AT2018cowの光学/UV色の変化が、宿主のAT2018cowの、compactな |
2305.19143 | A Tale of Two Laws of Semantic Change: Predicting Synonym Changes with
Distributional Semantic Models | Lexical Semantic Change is the study of how the meaning of words evolves
through time. Another related question is whether and how lexical relations
over pairs of words, such as synonymy, change over time. There are currently
two competing, apparently opposite hypotheses in the historical linguistic
literature regarding how synonymous words evolve: the Law of Differentiation
(LD) argues that synonyms tend to take on different meanings over time, whereas
the Law of Parallel Change (LPC) claims that synonyms tend to undergo the same
semantic change and therefore remain synonyms. So far, there has been little
research using distributional models to assess to what extent these laws apply
on historical corpora. In this work, we take a first step toward detecting
whether LD or LPC operates for given word pairs. After recasting the problem
into a more tractable task, we combine two linguistic resources to propose the
first complete evaluation framework on this problem and provide empirical
evidence in favor of a dominance of LD. We then propose various computational
approaches to the problem using Distributional Semantic Models and grounded in
recent literature on Lexical Semantic Change detection. Our best approaches
achieve a balanced accuracy above 0.6 on our dataset. We discuss challenges
still faced by these approaches, such as polysemy or the potential confusion
between synonymy and hypernymy. | Bastien Liétard, Mikaela Keller, Pascal Denis | 2023-05-30T15:50:29 | http://arxiv.org/abs/2305.19143v1 | # A Tale of Two Laws of Semantic Change:
###### Abstract
Lexical Semantic Change is the study of how the meaning of words evolves through time. Another related question is whether and how lexical relations over pairs of words, such as synonymy, change over time. There are currently two competing, apparently opposite hypotheses in the historical linguistic literature regarding how synonymous words evolve: the Law of Differentiation (LD) argues that synonyms tend to take on different meanings over time, whereas the Law of Parallel Change (LPC) claims that synonyms tend to undergo the same semantic change and therefore remain synonyms. So far, there has been little research using distributional models to assess to what extent these laws apply on historical corpora. In this work, we take a first step toward detecting whether LD or LPC operates for given word pairs. After recasting the problem into a more tractable task, we combine two linguistic resources to propose the first complete evaluation framework on this problem and provide empirical evidence in favor of a dominance of LD. We then propose various computational approaches to the problem using Distributional Semantic Models and grounded in recent literature on Lexical Semantic Change detection. Our best approaches achieve a balanced accuracy above \(0.6\) on our dataset. We discuss challenges still faced by these approaches, such as polysemy or the potential confusion between synonymy and hypernymy.
## 1 Introduction
Recent years have seen a surge to model lexical semantic change (LSC) with computational approaches based on Distributional Semantic Models (DSMs) Tahmasebi et al. (2021). While most research in this area has concentrated on developing approaches for automatically _detecting_ LSC for individual words, as in the dedicated SemEval 2020 shared task Schlechtweg et al. (2020), there has also been some work on validating or even proposing laws of semantic changes through new DSM-based approaches Dubossarsky et al. (2015); Hamilton et al. (2016); Dubossarsky et al. (2017). Ultimately, this line of work is very promising as it can provide direct contributions to the field of historical linguistics.
In this paper, we consider two laws of semantic change that are very prominent in historical linguistics, but that have to date given rise to very little computational modeling studies. Specifically, the Law of Differentiation (LD), originally due to Breal (1897, chapter 2), posits that synonymous words tend to take on different meanings over time; or one of them will simply disappear.1 The same idea is also discussed in more recent work, such as Clark (1993). As an example, the verbs _spread_ and _broadcast_ used to be synonyms (especially in farming), but now the latter is only used in the sense of _transmit_, by means of radio, television or internet. The verbs _plead_ and _beseech_ are synonyms, but _beseech_ is no longer used nowadays compared to _plead_. By contrast, the Law of Parallel Change (LPC),2 inspired from the work of Stern (1921), claims that two synonyms tend to undergo the same semantic change and therefore remain synonyms. As an illustration, Stern (1921, chapter 3 and 4) describes the change of _swiftly_ and its synonyms from the sense of _rapidly_ to the stronger sense of _immediately_. Lehrer (1985) also observes a parallel change affecting animal terms which acquire a metaphorical sense.
Footnote 1: To cite Breal (1897): “_[S]ynonyms do not exist for long: either they differ, or one of the two terms disappears.”_
Footnote 2: Name coined by Xu and Kemp (2015).
These two laws are interesting under several aspects. Firstly, these laws go beyond the problem of detecting semantic change in individual words, as they concern the question of whether a lexical relationship between words, in this case synonymy, is preserved or not through time. Secondly, these laws make very strong, seemingly opposite, predictions
on how synonyms evolve: either their meanings diverge (under LD) or they remain close (under LPC). It is likely that both of these laws might be at work, but they possibly apply to different word classes, correspond to different linguistic or extra-linguistic factors, or operate at different time scales. A large-scale study, fueled by computational methods over large quantities of texts, would be amenable to statistical analyses addressing these questions. In this work, we focus on predicting the persistence (or disappearance) of synonymy through time, as a first step toward more complete analyses.
Prima facie, DSMs appear to provide a natural resource for constructing a computational approach for assessing the importance of these laws, as they inherently -through the distributional hypothesis-capture a notion of semantic proximity, which can be used as a proxy for synonymy. Following this idea, Xu and Kemp (2015) propose the first DSM-based method for predicting how synonymous word pairs of English evolve over time (specifically, from 1890 to 1990). This research decisively concludes that there is "evidence against the Law of Differentiation and in favor of the Law of Parallel Change" for adjectives, nouns and verbs alike (i.e., the three considered POS). However, this pioneering work suffers from some limitations that cast some doubts on this conclusion. First off, the predictions made by their approach are not checked against a ground truth, thus lacks a proper evaluation. Second, the approach is strongly biased against LD, as only pairs in which _both_ words have changed are considered, excluding pairs in which differentiation may occur (e.g. in _spread/broadcast_, only the latter word changed in meaning).
This paper addresses these shortcomings by introducing a more rigorous evaluation framework for testing these two laws and evaluating computational approaches. We build a dataset of English synonyms that was obtained by combining lexical resources for two time stamps (1890 and 1990) that records, for a given list of synonym pairs at time 1890, whether these pairs are still synonymous or not in 1990. The analysis of this dataset reveals that, contra Xu and Kemp (2015) and though using the same initial synonym set, synonymous words show a strong tendency to differentiate in meaning over time. With some variation across POS, we found that between \(55\) and \(80\%\) of synonyms in 1890 are no longer synonyms in 1990.
Moreover, we propose several new computational approaches3, grounded in more recent DSMs, for automatically predicting whether synonymous words diverge or remain close in meaning over time, which we recast as a binary classification problem. Inspired by Xu and Kemp (2015), our first approach is unsupervised and tracks pairwise synchronic distances over time, computed over SGNS-based vector representations. Our second approach is supervised and integrates additional variables into a logistic regression model. This latter model achieves a balanced accuracy above \(0.6\) over the proposed dataset.
Footnote 3: The code used to run experiments in this paper can be found at [https://github.com/blietard/synonyms-sembange](https://github.com/blietard/synonyms-sembange)
## 2 Related Work
Data-driven methods to detect LSC have gained popularity in the recent years (Tahmasebi et al., 2021), using increasingly powerful and expressive word representations, ranging from the simple co-occurrence word vectors (Sagi et al., 2012) to static word embeddings (Schlechtweg et al., 2019) and transformer-based contextualized word representations (Kutuzov et al., 2022; Fourrier and Montariol, 2022). This line of research lead to the development of shared tasks (Zamora-Reina et al., 2022; Schlechtweg et al., 2020; Rodina and Kutuzov, 2020). Most often, these tasks concern the evolution of individual words, in effect focusing on _absolute_ semantic change (of words individually). In this paper, we take a different stand, considering the problem of _relative_ change in meaning among pairs of words, specifically focusing on synonym pairs.
Previous work on word pairs are rare in the current LSC research landscape. A first exception is (Turney and Mohammad, 2019), who also study the evolution of synonyms. They propose a dataset to track how usage frequency of words evolve over time within a sets of synonyms, as well as a new task: namely, to predict whether the dominant (most frequent) word of a synonyms set will change or not. This task is actually complementary to the one we address in this work. While Turney and Mohammad (2019) assume the stability of most synonym pairs between 1800 and 2000, and rather investigate the dynamic inside sets of synonymous words across time, we question this alleged stability and attempt to track whether these words remain synonymous at all in this time period.
Another distinctive motivation of our work is in the empirical, large-scale evaluation of two proposed laws of semantic change, originating from historical linguistics. Previous work investigating laws of semantic change with DSMs include Dubossarsky et al. (2015) and Hamilton et al. (2016), who measured semantic change of words between 1800 and 2000 and attempted to draw statistical laws of semantic change from their observations. Later, Dubossarsky et al. (2017) contrasted these observations and showed that even if these effects may be real, it may be to a lesser extent.
The closest work to the current research is the study of Xu and Kemp (2015), as they already focus on the two laws of Differentiation (LD) and Parallel Change (LPC). Their main motivation was to automatically measure, using DSMs, which of the two laws was predominant between 1890 and 1999. To study which of the two laws actually operates, they focus on word pairs that (i) are synonyms in the 1890s and (ii) where both words changed significantly in meaning between 1890 and the 1990s. First, they represent words as probability distributions of direct contexts, using normalized co-occurrence count vectors. Then, they measure the (synchronic) semantic proximity of words by computing the Jensen-Shannon Divergence between the corresponding distributions. Semantic change in a word is quantified by comparing its semantic space neighborhoods in the 1890s and in the 1990s. Finally, for every selected synonymous pair, they pick a control word pair that has a smaller divergence in the 1890s than the associated synonyms. At a later time in the 1990s, if the divergence for the synonyms is larger than that for the control pair, they decide these synonyms have undergone LD, otherwise they predict LPC. Ultimately, they found that most pairs (around \(60\%\)) have undergone LPC, which would be the dominant law.
The pioneering work of Xu and Kemp (2015) faces a number of shortcomings. First, their restriction to synonymous pairs in which both words changed mechanically excludes certain cases of LD (i.e., where one one word has changed), thus introducing an artificial bias against LD. Moreover, they often select near-synonyms as controls (e.g. _instructive_ and _interesting_) because they constrain control pairs to be _closer_ in divergence in the 1890s than the associated synonym pairs. Furthermore, and more importantly, Xu and Kemp (2015) did not compare their predictions to any ground-truth and there is no evaluation of the reliability of their method. Finally, their choice of word representations is not among the State-of-the-Art for static methods.
In this paper, we consider all synonymous pairs, thus avoiding the bias against LD. We propose different approaches that we compare to Xu and Kemp (2015)'s control pairs, and we provide results obtained with more recent distributional semantic models. Most importantly, we propose a complete evaluation framework to benchmark the different methods, something missing in this prior work.
## 3 Problem Statement
Our overarching goal is to develop new computational approaches that are able to automatically predict which pairs of synonymous words underwent LD or LPC. These predictions could be used as a first step towards providing a more refined and statistically meaningful analysis of the two laws. An important milestone towards developing such an approach is to compare it to some ground truth. Otherwise, there is no way to assess whether statistics obtained for LD or LPC are indeed reliable, a problem faced by Xu and Kemp (2015).
Unfortunately, there is no existing large-scale resource that records instances of LD/LPC, beyond a handful of examples found in research papers and textbooks in historical linguistics. What exists however are historical lists of synonyms, which we can compare to obtain some form of ground truth. This forces us to consider a slightly different methodological framework, focusing on a more constrained prediction task, namely to detect pairs of synonyms at time \(T1\) that have remained synonymous or that are no longer synonymous at time \(T2(>T1)\).
### Formalization
Let us denote \(W^{(T)}\) the set of words (or vocabulary) for a given language (say English) at time \(T\). As language evolves through time, vocabularies at two times \(T1\) and \(T2\) need not have the exact same extensions: e.g., a word \(w\) in \(W^{(T1)}\) might not be in \(W^{(T2)}\) (i.e., \(w\) has disappeared). Making a simplistic, idealized assumption, let \(\mathcal{C}\) be a mostly atemporal and exhaustive discrete set of concepts, and denote \(M_{w}^{(T)}\subset\mathcal{C}\) the meaning of word \(w\) at time \(T\). The definition of \(M_{w}^{(T)}\) as a set allows homonymy and/or polysemy to be accounted for.
Given these notations, we have that \(u\in W^{(T)}\)
and \(v\in W^{(T)}\) are synonyms at a time \(T\) if \(M_{u}^{(T)}\cap M_{v}^{(T)}\neq\emptyset\). We understand that the study of LD / LPC implies to track (i) the change of \(M_{u}^{(T)}\) and \(M_{v}^{(T)}\) over time, (ii) the evolution of \(M_{u}^{(T)}\cap M_{v}^{(T)}\) and (iii) the very persistence of both words in vocabularies \(W^{(T)}\) between \(T1\) and \(T2\). Discussion about formalizing LD and LPC under those conditions can be found in appendix A.1.
### Task Formulation: Tracking Synonyms Change
The presented formulation, though very idealized, should make it clear that the development of a computational system that attempts to directly predict LD and LPC, and even the construction of an evaluation benchmark for evaluating such a system, are very challenging tasks. First, the initial synonym set selection presupposes, not only that one has access to a list of synonyms at \(T1\) and \(T2\), but also that one can reliably predict LSC in one of the two words from \(T1\) to \(T2\); unfortunately, LSC is still an open problem for current NLP models. Second, one typically does not have meaning inventories or automatic systems (e.g. WSD systems) for mapping words to their meanings at different time stamps. Finally, even tracking the disappearance of words through time is not trivial, as it ideally requires full dictionaries at different time stamps.
Given these limitations, we suggest to narrow down our target problem to the task of predicting, for a given pair of synonymous words \((u,v)\) at \(T1\), whether \((u,v)\) are still synonymous or not at \(T2\). Stated a little more formally, we are concerned with the following binary classification problem:
\[f: \mathcal{S}^{(T1)}\rightarrow\{\text{``Syn''},\text{``Diff''}\}\] \[(u,v)\mapsto f((u,v))=\begin{cases}\text{``Syn'' if }\;(u,v)\in \mathcal{S}^{(T2)}\\ \text{``Diff'' otherwise}\end{cases}\]
where \(\mathcal{S}^{(T)}\) is a set of synonymous word pairs at time \(T\), "Syn" indicates that words \((u,v)\) that were synonymous at \(T1\) remain synonymous at \(T2\), while "Diff" signals that they are no longer synonymous at \(T2\). This simpler problem leads to a more operational evaluation procedure, which does not require access to \(M_{u}^{(T^{*})}\) and \(M_{v}^{(T^{*})}\), but only to lists of synonyms \(\mathcal{S}^{(T1)}\) and \(\mathcal{S}^{(T2)}\). See Section 4 for presentation of such procedure. It should be clear that predicting which synonym pairs remain ("Syn") or cease to be synonymous ("Diff"), will provide some information about LPC and LD, although the mapping between the two problems is not one-to-one. Even if "Diff" covers pretty well LD, a pair that is still synonymous at \(T2\) could either be a case of LPC (their shared meaning changed the same way for both words) or a pair of words that simply have not changed in meaning at all (or at least that their shared meaning is unchanged).
Now turning to designing a computational system that detects "Syn" vs. "Diff", a natural question that emerges is whether current DSMs, commonly used for detecting LSC in individual words, are able to capture synonym changes. More specifically, our main hypothesis will be that one can reliably track the evolution of synonymous pairs through their word vector representations at \(T1\) and \(T2\).
This approach will be instantiated into different unsupervised and supervised models in Section 5.
## 4 Evaluation Dataset
This section presents a dataset designed to track the evolution of English synonymous word pairs between two time stamps \(T1\) and \(T2\), with \(T2>T1\). Specifically, the two time periods considered are the 1890's decade (\(T1\)) and the 1990's decade (\(T2\)). For extracting synonymous pairs in the 1890's (noted \(\mathcal{S}^{(T1)}\)), we use Fernald's _English Synonyms and Antonyms_ (Fernald, 1896) as Xu and Kemp (2015) did. Pairs were selected based on a set of specific target words (see appendix A.7). As shown in Table 1, we obtain \(1,507\) adjective pairs, \(2,689\) noun pairs and \(1,489\) verb pairs. To assess whether these word pairs are still synonyms in the 1990's, we use WordNet (Fellbaum and Princeton, 2010), as this lexical database was originally constructed in 1990's. Thus, WordNet provides us with \(\mathcal{S}^{(T2)}\). Specifically, we considered that a pair of words/lemmas \((u,v)\in\mathcal{S}^{(T1)}\) are still synonymous if they point to at least one common _synset_ in WordNet.
The construction of this dataset relies on two crucial hypotheses, which seem reasonable to make. First, both lexical resources rely on the same definition of synonym. Second, \(\mathcal{S}^{(T2)}\) meets some exhaustivity criterion, in the sense that \((u,v)\in\mathcal{S}^{(T1)}\) not appearing in \(\mathcal{S}^{(T2)}\) should indicate that \(u\) and \(v\) are no longer synonymous at \(T2\), and not be due to a lack of coverage of the resource (i.e., a false negative). WordNet is assumed to be exhaustive enough, as we checked that every word involved in at least one synonymous pair has its own entry in
WordNet's database.
Table 1 provides some detailed statistics on the evolution of synonymous pairs between decades 1890's and 1990's, overall and for different parts of speech. A first observation on these datasets is that the proportion of pairs that are still synonyms at \(T2\) ("_Syn_") is globally \(15.1\)%. This implies that most synonymous pairs underwent differentiation. While it does not provide information about how change happened between \(T1\) and \(T2\) for the remaining \(84.9\)%, it's a clue that the Law of Differentiation should be a dominant phenomenon among synonyms.
We exploit the structure of the WordNet database to analyze the different cases of "_Diff_". WordNet includes lexical relations of hyper-/hypo-nymy (e.g., _seatl/bench_) as well as holo-/mero-nymy (e.g., _bike/wheel_) and antonymy (e.g., _small/large_) defined over synsets4. Note that the hyper-/hypo-nymy relation does not exist in WordNet among adjectives. Among nouns and verbs, we observe that around 30% of pairs that were synonyms at \(T1\) are in an hyper-/hypo-nymy relation at \(T2\) and two third of them are direct hypernyms in WordNet (their synsets are direct parent/child) indicating the preservation of a very close semantic link. For a further depiction of the dataset in terms of distance in WordNet's graph, see Figure 3 in appendix A.4.
Footnote 4: As we did for synonyms, we assume that two words \(w_{1}\) and \(w_{2}\) are instances of one of these relations \(R\) if \(R\) holds for one of their corresponding synset pair.
One cannot entirely exclude that \(\mathcal{S}^{(T1)}\) includes some hyper-/hypo-nyms as synonyms. However, even if we extend the notion of synonymy at \(T2\) to include these cases, we would have only around \(45\%\) of all pairs still considered synonyms among nouns and verbs. This indicates that "Diff" largely remains the most common phenomenon with an estimated proportion between \(55\%\) and \(80\%\). This finding contradicts the experimental results reported by Xu and Kemp (2015) with their computational approach (only \(40\%\) of differentiation).
In lack of additional indication that some of these hyper-/hypo-nym cases at \(T2\) are indeed synonyms, or that they may also have been hyper-/hypo-nym at \(T1\), we decided to still consider them as instances of "Diff". Another argument for this decision is precisely that there are well-known reported cases of lexical semantic changes in which the meaning of a particular word in effect "widens" to denote a larger subset (i.e., becomes an hypernym): this is the case of _dog_ in English that used to denote a specific breed of dogs (Traugott and Dasher, 2001).
## 5 Approaches
This section presents two classes of computational approaches, unsupervised and supervised, for predicting whether pairs of synonyms at \(T1\) remain synonyms ("Syn") or cease to be so ("Diff") at a later time \(T2\). Common to all of these approaches is that they are based on two time-aware DSMs, one for each time stamp.
### Time-aware DSMs
Inspired by work on LSC, we rely on separate DSMs for each time stamp \(T1\) and \(T2\), respectively yielding vector spaces \(V^{(T1)}\) and \(V^{(T2)}\) encoding the (possibly changing) word meanings at \(T1\) and \(T2\). Thus, for each synonym pair \((u,v)\), we have two pairs of vectors : \((\mathbf{u}^{(T1)},\mathbf{v}^{(T1)})\in V^{(T1)}\times V^{(T1)}\) and \((\mathbf{u}^{(T2)},\mathbf{v}^{(T2)})\in V^{(T2)}\times V^{(T2)}\).
Specifically, we use pre-computed SGNS (Mikolov et al., 2013) from Hamilton et al. (2016) trained on the _English_ part of the GoogleBooks Ngrams dataset5 for every decade between 1800 and 2000 and extract \(V^{(T1)}\) (1890) and \(V^{(T2)}\) (1990). For any word \(w\in W\) and any time period \(T\), \(\mathbf{w}^{(T)}\in V^{(T)}\) is a single 300 dimensional vector. We ensure synonymy is accurately reflected by checking that synonym pairs have a smaller cosine distance than non-synonymous pairs for both time periods, as in Figure 4 of appendix A.5.
Footnote 5: [https://storage.googleapis.com/books/ngrams/books/datasetsv3.html](https://storage.googleapis.com/books/ngrams/books/datasetsv3.html)
Traditional DSM-based approaches for detecting LSC are based on self-similarities over time for a given word. For instance, for a given time
\begin{table}
\begin{tabular}{r|c c c c} \hline \hline Synonyms pairs & ADJ & NN & VERB & All \\ \hline Synonyms at \(T1\) & 1507 & 2689 & 1489 & 5685 \\ \hline \& synonyms at \(T2\) & 202 & 347 & 311 & 860 \\ \& synonyms at \(T2\)(\%) & 13.4 & 12.9 & 20.9 & 15.1 \\ \hline \& hypernyms at \(T2\) & 0 & 858 & 398 & 1256 \\ \& hypernyms at \(T2\)(\%) & 0.0 & 31.9 & 26.7 & 22.1 \\ \& hypernyms at \(T2\) (1) (\%) & 0.0 & 23.2 & 22.5 & 16.9 \\ \& hyp. at \(T2\) (2) (\%) & 0.0 & 6.9 & 3.5 & 4.1 \\ \& hyp. at \(T2\) (3) (\%) & 0.0 & 1.4 & 0.5 & 0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Numbers of synonymous pairs extracted from Fernald (1896) (\(T1\)) displayed by POS, and numbers of those that are also considered as synonyms or hypernyms/hyponyms in WordNet (\(T2\)) For hypernyms, we detail the proportions of hypernym/hyponym pairs that are separated by 1, 2 or 3 nodes in the WordNet graph.
interval \((T1,T2)\), they compute for each word \(w\) an individual _Diachronic Distance_, noted here \(DD^{(T1,T2)}(w)\). Cosine distance is often used (recall in appendix A.2).
There is no obvious distance for comparing _pairs_ of word vectors, but one can instead rely on comparing the pairwise word vector distance at each time stamp \(T\); we call this _Synchronic Distance_ (denoted SD). The two types of distances for two time stamps \(T1\) and \(T2\) are described in Figure 1. Our unsupervised method, proposed in Sec. 5.2 directly exploit the idea of tracking different types of SD through time, while Sec. 5.3 presents a supervised approach that combines both SD and DD.
### Unsupervised Methods
While we don't have access to \(M_{u}^{(T)}\) and \(M_{v}^{(T)}\), we can represent the meaning of \(u\) and \(v\) using DSM and compare them at a given time to estimate how close they are in meaning. Indeed, if \(M_{u}^{(T)}\cap M_{v}^{(T)}\) changes, this should be reflected in difference of the use contexts of \(u\) and those of \(v\), and so reflected in the distance between \(\mathbf{u}^{(T)}\) and \(\mathbf{v}^{(T)}\). Let
\[SD^{(T)}:W^{(T)}\times W^{(T)}\rightarrow\mathbb{R}^{+}\]
be a measure of **synchronic distance** between vectors representing two words. By construction of \(V^{(T)}\), \(SD^{(T)}(u,v)\) is smaller for words \((u,v)\) that appear in similar contexts than for unrelated words. We assume that there exists a value \(\delta_{T}\) such that
\[\forall(u,v)\in\mathcal{S}^{(T)},\;SD^{(T)}(u,v)\leq\delta_{T}.\]
This entails that for a given pair \((u,v)\):
\[SD^{(T)}(u,v)>\delta_{T}\Rightarrow(u,v)\text{ are not synonyms}.\]
In this setting, one can compare the synchronic distances within \(V^{(T1)}\) and with \(V^{(T2)}\) and decide if the pair differentiated or stayed synonymous.
Let \((u,v)\) be a pair of synonyms at \(T1\), as such we have that \(SD^{(T1)}(u,v)\leq\delta_{T1}\). If \((u,v)\) are not synonyms at time \(T2\) then \(SD^{(T2)}(u,v)>\delta_{T2}\).
Combining these two inequalities, we would say that a pair of synonyms at \(T1\) has differentiated at \(T2\) if:
\[\underbrace{SD^{(T2)}(u,v)-SD^{(T1)}(u,v)}>\delta_{T2}-\delta_{T1}.\]
Ideally one could imagine that the distance threshold \(\delta_{T}\) at which, words cease to be synonyms should be independent of the time period \(T\). Empirically however, because word embeddings are not necessarily build with an enforced scale, there might be a dilation or shrinking in the overall synchronic distances between \(T1\) and \(T2\). Let us assume that
\[\delta_{T2}=\delta_{T1}+\tau,\;\tau\in\mathbb{R}.\]
Our decision rule could then be rewritten as:
\[f(u,v)=\begin{cases}\text{``Diff'' if }\Delta(u,v)\geq\tau\\ \text{``Syns'' otherwise}.\end{cases} \tag{1}\]
This approach is shortly denoted "\(\Delta\)" in section 6. It diverges from the prior work of Xu and Kemp (2015) that chooses to rely on control pairs instead of a threshold. For the sake of comparison, we implemented their method presented as "_XK controls_". It is not the full protocol presented by Xu and Kemp (2015), as (i) the experimental setting is not identical, they filtered out some synonym pairs and we didn't (ii) we use SGNS word representations and cosine distance instead of normalized co-occurrence counts and Jensen-Shannon Divergence. Schlechtweg et al. (2019) provided a longer comparison between word representations.
We propose a statistically-grounded criterion to set the value for the threshold \(\tau\). Since the meaning of most words is expected to remain stable6, we argue that most pairwise distances should remain stable as well. We can then estimate the dilation between the representations in the two time periods by the average gap between the synchronic distances of words.
Footnote 6: Intuitively, someone in 2023 can still understand writings published in the 1890s in their original text, like books from Charles Dickens or Arthur Conan Doyle.
\[\tau=\frac{1}{|W|^{2}}\sum_{(w_{1},w_{2})\in W\times W}\Delta(w_{1},w_{2}) \tag{2}\]
Figure 1: Pairs of word embeddings at 2 time periods and associated diachronic and synchronic distances.
In practice, we experiment with two different types of synchronic distances between words. The first is the cosine distance (see A.2). That is:
\[SD^{(T)}(u,v)=\text{cos-dist}(\mathbf{u}^{(T)},\mathbf{v}^{(T)}).\]
We shortly denote it "SD(cd)". Another measure of semantic proximity is based on the shared word neighborhood between the two vectors \(u\) and \(v\):
\[SD^{(T)}(u,v)=\text{jaccard-dist}(\mathcal{N}_{k}^{(T)}\left(u\right), \mathcal{N}_{k}^{(T)}\left(v\right)),\]
with \(\mathcal{N}_{k}^{(T)}\left(w\right)\) being the set of the \(k\)-nearest neighbors of the point representing \(w\) in the vector space at time \(T\), and _jaccard-dist_ being the Jaccard distance (see appendix A.2). This measure is ranged between 0 and 1, and we denote it "SD(n\(k\))".
### Supervised Methods
Approaches described so far use the labels in the dataset ("Syn" and "Diff") only for evaluation purposes. But one can also use part of the available data to learn a _supervised_ classifier to predicts these labels. Concretely, for most of these models, we trained Logistic Regression (LR) models7
Footnote 7: Implemented with the _scikit-learn_ library for Python8.
Synchronic Distances CombinationIn our unsupervised approach, we compute \(SD^{(T1)}\) and \(SD^{(T2)}\) and their difference, denoted \(\Delta\). This quantity is then compared to a fixed threshold \(\tau\). We propose to investigate two supervised approaches stemming from this: (i) simply tune \(\tau\) and (ii) use a LR model to learn the optimal weighting in the linear combination of the two distances. This latter model is called "LR SD".
Accounting for Individual ChangeMost works about computational approaches to LSC focus on detecting the change of a single word (Tahmasebi et al., 2021), using a diachronic distance, which we noted \(DD^{(T1,T2)}(w)\), across time periods \(T1\) and \(T2\) for individual words \(w\).
In addition to synchronic distances, we input diachronic distances as features for a LR model. The resulting classifier (LR SD+DD) uses the 4 distances represented in Figure 1 as variables: self-similarities across time periods (\(DD\)s), and a distance measure within pairs for each of both time stamps (\(SD\)s). Similarly to synchronic distances defined in Sec. 5.2, we try two definitions of DD. First, we compare sets of neighbors at \(T1\) and \(T2\):
\[\mathrm{DD}(w)=\text{jaccard-dist}(\mathcal{N}_{k}^{(T1)}\left(w\right), \mathcal{N}_{k}^{(T2)}\left(w\right)).\]
We also compute the cosine distance between \(\mathbf{w}^{(T1)}\) and \(\mathbf{w}^{(T2)}\) after aligning the vector space \(V^{(T2)}\) to \(V^{(T1)}\) using Orthogonal Procrustes (Hamilton et al., 2016; Schlechtweg et al., 2019, 2020). Denoting \(\mathbf{w}_{align}^{(T2)}\) the vector \(\mathbf{w}^{(T2)}\) after alignement with Orthogonal Procrustes, we have:
\[\mathrm{DD}(w)=\text{cos-dist}(\mathbf{w}^{(T1)},\mathbf{w}_{align}^{(T2)}).\]
Using Distances and FrequenciesA final step of this process is to add word frequencies for both words at both time periods, as there exist links between usage frequency and semantic change Zipf (1945). We could observe whether adding explicit frequency information helps retrieving discriminatory clues that could be missed by using only distributional representations.
Word frequencies were estimated from the Corpus of Historical American English (COHA) list,9 which has the advantage to be genre-balanced. As variables for both words and both periods to feed our model, we try to add either raw occurrences counts (indicated by "+FR"), either grouped frequency counts ("+FG"). The procedure to create such groups is described in appendix A.6.
Footnote 9: [https://www.ngrams.info/download_coha.asp](https://www.ngrams.info/download_coha.asp)
All FeaturesFor the sake of comparison to previous models, we evaluate LR models that take as input an implementation of each of these features (SD + DD + frequency); and an even larger model (called "LR multi.") that reunites _all_ described implementations of _SD_, _DD_ and frequencies.
Non-linear ModelsAs a further step increasing the model's complexity, we try to combine this full set of available variables in a non-linear fashion. We compare previous models to polynomial features (degree 2) preprocessing10 and a SVM classifier with a Gaussian kernel.
Footnote 10: We also try degrees higher than 2, finding no consistent improvement.
## 6 Experiments
### Experimental Settings
Target Words SelectionWe use a unique vocabulary \(W\) composed of \(6,453\) adjectives, \(16,135\) nouns and \(10,073\) verbs. The process to select words is described in appendix A.7.
Dataset SplitsFor every POS tag, we have a set of word pairs that are synonymous at \(T1\). We call _ALL_ the dataset that comprises all pairs indistinctly of their POS. These datasets (ADJ,NN,VERB or ALL) are individually shuffled and 33% of their samples (pairs) are set aside for testing. For each dataset, a model is trained on the 66% remaining pairs and evaluated on the test part. Presented results are averaged over 20 random train/test splits.
HyperparametersWe train models with combinations of the different definitions of distances and frequency variables. Choice of synchronic distances was between SD(cd) and SD(n\(k\)) with \(k\) in \(\{5,10,15,20,40,100\}\). For DD, we tried neighborhoods with fixed size \(100\), like Xu and Kemp (2015), and Orthogonal Procrustes with cosine distances. For frequency, the choice is between raw counts and groups. The selected models are detailed in Appendix A.9. The ideal value for the SVM's regularization parameter is found using 5-fold cross-validation over the training set.
Evaluation MetricsWe use two standard evaluation metrics: \(F_{1}\) score and _Balanced Accuracy (BA)_. \(F_{1}\) scores were computed for both classes, denoting it "**F\({}_{\textbf{1}}\)(_Syn_)" for _Syn_s and "**F\({}_{\textbf{1}}\)(_Diff_)" for _Diff_. BA is defined as the average of recalls for both classes, and provide a notion of accuracy robust to class imbalance. We also display the percentage of predicted _Diff_ ("%D").
BaselinesThe first two baselines are constant output classifiers, always predicting "_Syn_" or "_Diff_" respectively. They are expected to have a balanced accuracy of \(50\%\), as they would be fully accurate for one class and always wrong for the other. The third baseline (_LR Frequency_) is a Logistic Regression model trained _only_ with frequency variables, without any knowledge on the semantic aspect of the pair (neither _SD_ or _DD_).
### Results
Performances over the test parts of the different datasets are displayed Table 2.
The first observation is that, in line with the dataset's proportions, all models predict a majority of "Diff", even unsupervised ones (including our reimplementation of Xu & Kemp's control pair selection method). While our task does not directly address the question of the opposition between LD and LPC, this is an empirical clue in favor of LD, contradicting Xu and Kemp (2015). However, predicting the right amount of "Diff" does not guarantee the quality of predictions. Indeed, obtained balanced accuracies range between \(0.49\) and \(0.65\).
Considering our unsupervised methods and the \(\Delta\) (tuned \(\tau\)), we find no real improvement over baselines. In particular, they fail to outperform the frequency-based baseline model which performs surprisingly well. On the other hand, Logistic Regression and SVM models substantially improve
\begin{table}
\begin{tabular}{r|c c c c|c c c} \hline \hline
**Dataset** & **ADJ** & **NN** & **VERB** & **ALL** & \multicolumn{3}{c}{**ALL**} \\
**Evaluation metric** & \multicolumn{3}{c}{**Balanced**} & **Accuracy** & \multicolumn{1}{c}{**F\({}_{\textbf{1}}\)(_Syn_)**} & \multicolumn{1}{c}{**F\({}_{\textbf{1}}\)(_Diff_)} & **\%(D)** \\ \hline All (_Syn_) &.50 &.50 &.50 &.50 &.48 & 0 & 0 \\ All (_Diff_) &.50 &.50 &.50 &.50 & 0 &.81 & 100 \\ LR F &.51 &.56 &.59 &.55 &.35 &.74 & 75 \\ \hline XK controls &.52 &.49 &.51 &.50 &.33 &.67 & 65 \\ \(\Delta\) (cd) &.50 &.49 &.51 &.50 &.27 &.73 & 75 \\ \(\Delta\) (n\(k\)) &.48 &.49 &.49 &.50 &.32 &.67 & 66 \\ \hline \(\Delta\) (tuned \(\tau\)) &.51 &.52 &.52 &.51 &.27 &.74 & 79 \\ LR SD &.60 &.62 &.59 &.60 &.48 &.69 & 56 \\ LR SD + DD &.61 &.62 &.60 &.60 &.48 &.69 & 56 \\ LR SD + F &.61 & **.64** &.63 & **.62** &.51 &.71 & 57 \\ LR SD + DD + F & **.62** & **.64** &.63 & **.62** &.50 &.70 & 57 \\ LR multi & **.62** & **.64** & **.65** & **.62** &.51 &.71 & 57 \\ \hline LR multi. poly. degree (2) &.56 &.63 &.62 & **.62** &.50 &.70 & 60 \\ SVM (gaussian) &.60 & **.64** & **.65** & **.62** &.50 &.74 & 63 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performances of the different approaches. Results are averaged over 20 random splits.
over the baselines, Xu & Kemp's control pairs and all \(\Delta\)-based methods. Interestingly, LR SD outperforms \(\Delta\)-based methods despite the fact that they rely on the same components.
The gap between baselines and models is larger for nouns and lesser for verbs. Despite these POS-specific differences, best models are consistently the ones using both SD and frequencies, while DD brings little to no improvement. This can be expected as individual changes of words seem less important on the problem of _Syn/Diff_. However, this factor could be used in future work to distinguish pairs of synonyms (among the _Syn_ class) that did not change and pairs that went under LPC.
We observe that there is a substantial difference in \(F_{1}\) scores between the two classes, \(\mathbf{F_{1}}\)(_Syn_) being lower than \(\mathbf{F_{1}}\)(_Diff_) across all models. Moreover, models with higher \(\mathbf{F_{1}}\)(_Syn_) are often found to be the ones with higher balanced accuracy, even when \(\mathbf{F_{1}}\)(_Diff_) is lower. This is likely linked to the fact that the datasets are highly imbalanced as presented in Table 1: the ground truth proportion of _Syn_ never exceeds 21%. We also remark that Xu and Kemp (2015) decision rule based on control pairs also predicts a majority of _Diff_, contrarily to the results they showed. It may be because the protocol is not fully identical.
### Confounding Factors
Using WordNet, we discuss two aspects that may be sources of errors when detecting a change in synonymy: polysemy and hypernymy. We study predictions of our best performing LR model on the noun dataset.
PolysemyWordNet provides us with different set of synonyms for every entry, corresponding to different senses or usages, and therefore we can measure the polysemy of a word at \(T2\). We found that pairs misclassified as "Syn" tend to be those whose second term has fewer senses (6 senses on average as compared with well classified "Diff" which have 8 senses on average). Indeed, as we use static embeddings and no Word Sense Disambiguation (WSD) method, our model is subject to the complexity brought by polysemy. In a recent shared task about Lexical Semantic Change measures, best performing models are the one using WSD methods Zamora-Reina et al. (2022). This finding highlights the importance of handling polysemy as a potential confounding factor.
Distances in WordNetIn Figure 2 we display the percentage of prediction with respect to shortest distance between the two words of _noun_ pairs in WordNet's graph. The distance \(d\) is the minimum number of nodes separating the two words. We remark that, as expected, the model predicts more and more _Diff_ as \(d\) increases. What is more interesting is that for \(d=1\) (direct hypernymy), there is still an important proportions of predicted _Syn_. This highlights that our model has difficulties to handle hypernymy and confuses it with synonymy.
## 7 Conclusion
In this work, we considered two contradicting laws about the semantic change of synonyms. We discussed the necessary adaptations of the problem statement for this particular type of LSC and elaborated a framework to evaluate models for this new classification problem. The use of linguistic resources from two different time periods allowed us to improve model analysis with respect to prior work on the matter. Then we proposed unsupervised and supervised approaches relying on measures of semantic change extracted or inspired by existing literature on LSC, and also leveraged the usefulness of explicit word usage frequency information. We compared these approaches in our evaluation framework, finding that distances in vector spaces from different time periods should not be considered equally. We also observed that explicit frequency information actually help distributional methods to capture the change of synonymy. Finally we discussed challenges that DSM approaches still face and opened a discussion about the interplay between hypernymy and synonymy.
Figure 2: Proportions of predictions of the models w.r.t. the actual distance \(d\) in WordNet of _noun_ pairs. Pairs with \(d=0\) are synonymous pairs in WordNet.
### Limitations
As mentioned already, the problem _Syn_/_Diff_ does not reflect the initial question of LD/LPC. In particular, the _Syn_ class of pairs that remained synonyms contains pairs that underwent LPC and pairs which shared meaning remained unchanged. The latter does not play a role in the LD/LPC dichotomy and should be discarded for deeper study of the two apparently opposite laws. Also, we restrain the study to some target words that are chosen to occur at both time periods, thus preventing us to fully measure the importance of LD. Indeed, recall that Breal's Law of Differentiation predicts that some synonyms may disappear in the process. Thus, our _Diff_ class could be considered incomplete. However, including such disappeared words would prevent the use of time-aware DSMs.
Section 3 presented synonymy as a symmetrical relation between words. However, a thesaurus like Fermald (1896) displays asymmetrical synonymy: for an entry \(u\) we have a set of synonyms \(v_{1},v_{2},...\) from which we extract pairs \((u,v)\). We observe that \(v\) itself is rarely an entry of the thesaurus, and when it does, \(u\) may not appear in the list of synonyms of \(v\). This is contradictory to WordNet's definition of synonymy that consider this relationship to be symmetrical. However, up to our knowledge, there is no lexical database (like WordNet) being also historical and that could help us ensure the notion of synonymy at both time periods is strictly the same. In the absence of such a resource, we leave potential disagreements in definition between the two linguistic resources to future investigations.
In section 4, we discussed that hyper/hypo-nymy could be misleading. We made the assumption that Fermald (1896) and Wordnet (Fellbaum and Princeton, 2010) used similar-enough notions of synonymy such that our labels _Syn_/_Diff_ are relevant. However, thesaurus like Fermald (1896) are created as a tool for writers and authors to avoid redundancy, thus including wide lists of synonyms that include hypernyms (instead of repeating _the bench_, you could say _the seat_). In section 6.3 we showed that direct hypernymy is misleading for our model. Yet, we still miss guidelines/insights about the possibility to include some cases of hypernymy among synonyms at \(T2\). Another approach would be to remove hypernyms from the source material at \(T1\), which implies to automatically detect them or manually review thousands of pairs.
There are remaining factors that presented approaches do not take in account and that one could think relevant. In particular, further work could investigate the influence of pressure of words on a concept, for instance many words sharing (at least partially) a similar meaning. However, this would require access to list of senses for each word at time \(T1\), which we do not have in Fermald (1896). To this extent, contextualized language models fine-tuned for the different time periods could be helpful.
Finally, because we used pre-computed SGNS embeddings on historical data binned in decade, we have no guarantee that this is the optimal setting for studying Lexical Semantic Change. Maybe different kind of changes could be observed using larger or smaller time periods, and conducting the study over a larger or a smaller time span instead of just a century.
## Acknowledgements
We would like to thank the three anonymous reviewers for their helpful comments on this paper. We would also thank Anne Carlier for the thoughtful discussion about this work. This research was funded by Inria Exploratory Action COMANCHE.
| 意味の変化を研究する学問分野は、単語の意味が時間とともにどのように変化していくかを説明するものであり、単語の関連性について、例えば同義語の関係など、時間の経過とともに変化するかどうか、そしてどのように変化するのかという関連する質問もあります。歴史的言語学文献では、同義語の関係が時間とともにどのように変化していくかの2つの競争関係、つまり、分化の法則(LD)と平行変化の法則(LPC)が存在します。LDは同義語が時間とともに異なる意味を持つ傾向を示し、LPCは同義語が同じ意味変化を遂げ、その結果同義語であるという主張です。これらの法則が歴史的データに基づいて適用される程度を評価するためには、現在、分布的なモデルを用いた研究が限られています。本研究では、LDまたはLPCが特定の単語対にどのように作用するかを検出するための最初のステップを踏み出します。問題をより扱いやすいタ |
2310.14809 | Learning spatio-temporal patterns with Neural Cellular Automata | Neural Cellular Automata (NCA) are a powerful combination of machine learning
and mechanistic modelling. We train NCA to learn complex dynamics from time
series of images and PDE trajectories. Our method is designed to identify
underlying local rules that govern large scale dynamic emergent behaviours.
Previous work on NCA focuses on learning rules that give stationary emergent
structures. We extend NCA to capture both transient and stable structures
within the same system, as well as learning rules that capture the dynamics of
Turing pattern formation in nonlinear Partial Differential Equations (PDEs). We
demonstrate that NCA can generalise very well beyond their PDE training data,
we show how to constrain NCA to respect given symmetries, and we explore the
effects of associated hyperparameters on model performance and stability. Being
able to learn arbitrary dynamics gives NCA great potential as a data driven
modelling framework, especially for modelling biological pattern formation. | Alex D. Richardson, Tibor Antal, Richard A. Blythe, Linus J. Schumacher | 2023-10-23T11:16:32 | http://arxiv.org/abs/2310.14809v2 | # Learning spatio-temporal patterns with Neural Cellular Automata
###### Abstract
Neural Cellular Automata (NCA) are a powerful combination of machine learning and mechanistic modelling. We train NCA to learn complex dynamics from time series of images and PDE trajectories. Our method is designed to identify underlying local rules that govern large scale dynamic emergent behaviours. Previous work on NCA focuses on learning rules that give stationary emergent structures. We extend NCA to capture both transient and stable structures within the same system, as well as learning rules that capture the dynamics of Turing pattern formation in nonlinear Partial Differential Equations (PDEs). We demonstrate that NCA can generalise very well beyond their PDE training data, we show how to constrain NCA to respect given symmetries, and we explore the effects of associated hyperparameters on model performance and stability. Being able to learn arbitrary dynamics gives NCA great potential as a data driven modelling framework, especially for modelling biological pattern formation.
###### Contents
* 1 Introduction
* 2 Model and methods
* 2.1 Model details and parameters
* 2.2 Training techniques
* 2.2.1 Loss functions
* 3 Results
* 3.1 Gray-Scott reaction diffusion equations
* 3.1.1 Effect of training hyperparameters
* 3.2 Image morphing
* 3.2.1 Effect of model hyperparameters
* 3.2.2 Stability analysis
* 4 Discussion
* A Gradient Calculation
* B Videos
## 1 Introduction
Many complex natural phenomena--such as organ growth, the structure of materials or the patterns of neural activity in our brains--are emergent [1]. These are typically characterised by many simple interacting components that collectively exhibit behaviour that is far richer than that of the individual parts, and cannot easily be predicted from them. Emergence is especially prevalent in complex systems of biological nature across a wide range of scales - from gene expression dictating cell fates, interacting cells forming structures during morphogenesis, synaptic connections in the brain, or the interactions of organisms in ecology.
Cellular Automata (CA) provide simple models of spatio-temporal emergent behaviour, where a discrete lattice of 'cells' are equipped with an internal state and a rule that updates each cell state depending on itself and its local neighbours. The classic _Game of Life_[2] is a famous example, where cell states and the update rule utilise simple Boolean logic, but the emergent complexity has fascinated and inspired much research [3, 4]. CA are a natural modelling framework of a wide range of biological processes such as: skin patterning [5, 6], limb polydactyly [7], chimerism [8], cancer [9] and landscape ecology [10]. In these cases the CA rules are constructed with expert knowledge of likely mechanisms, however in general the space of possible CA rules is vast, and there is a non-uniqueness by which several rules can result in qualitatively similar emergent behaviours. As such the inverse problem of inferring mechanistic interactions (CA rules) that might generate a given observed emergent behaviour is much more challenging than the forward problem. Establishing the emergent
consequences of a known set of mechanistic interactions between components is conceptually straightforward -- one sets them up in a computational model or _in vivo_ and then observes the collective behaviour that emerges. In the case of CA, once a rule is defined, any initial condition can trivially be propagated forward to obtain the emergent behaviour.
In this work, we establish and extend the utility of Neural Cellular Automata (NCA, [11])--a special case of CA where each cell state is a real vector, and the update rule is determined by a neural network. The update rule is parameterised by the neural network by minimising a cost function that measures how similar the trajectory generated by iteratively applying the update rule is to training data. Since the gradient of the loss function can be evaluated straightforwardly, gradient-based optimisation allows for efficiently learning (non-unique) local update rules to match desired global behaviour, in the usual manner of machine learning [12]. This allows us to tackle the inverse problem of inferring local mechanistic rules given observed emergent behaviour.
We investigate the potential for NCA to be applied as a data-driven alternative, where the underlying mechanisms are not assumed to be known. This could include biological systems, which are inherently complex and may feature interactions that are not directly measured by a given experimental procedure. We do this by exploring the behaviour of NCAs on two idealised systems. First, we train the NCA on Turing patterns generated by the solutions of certain partial differential equations (PDEs). We show that the underlying dynamics are well represented, for example, by testing on initial conditions that were not part of the training set. Second, to understand the generality of the method, we build on previously observed behaviour of NCA on the artificial problem of morphing from one image to another [11]. Here any underlying dynamics is _a priori_ unknown and likely to be highly complex, as opposed to the PDE learning case where we know and understand the PDE. Nonetheless we show that such dynamics are learnable and are robust to perturbations. In addition, we achieve this with the minimal neural network complexity of NCA [13], in line with the principles of Explainable Artificial Intelligence [14, 15].
In Section 2 we set out the structure of the NCA, and show that it can be viewed as a machine learning approach to approximating the finite difference discretisation of an unknown PDE. There is already a fairly strong link between cellular automata and reaction diffusion equations [5, 6], in that both are used to model similar systems, and a suitably chosen CA rule will correspond to the finite difference approximation of any PDE. In Section 3 we present our results. We first (Section 3.1) benchmark the NCA by assessing its ability to capture the types of Turing patterns [16] that emerge from the Gray-Scott [17] reaction-diffusion equations. These equations describe the population of two chemical species with a nonlinear interaction between them, and is capable of generating a variety of patterns. In Section 3.2, we show that the same basic model and training techniques can also be applied to an image morphing task. Thus we conclude that NCA are capable of constructing microscopic dynamical rules that lead to a wide range of prescribed emergent behaviour. In Section 3.2.2 we further explore constraining NCA to respect basic symmetry requirements placed upon them, and investigate the robustness of trained NCA to initial condition perturbations. We discuss implications for further development of the method
Section 4.
## 2 Model and methods
We now define the NCA model, and discuss the motivations behind design choices and hyperparameters. We further discuss the training methods, as it turns out that most training parameters can be kept constant between tasks. The main exception to this is how frequently to sample in time: PDEs have a clear notion of time, whereas image morphing does not. All the models and software developed here have been implemented within the Tensorflow [18] framework [https://github.com/AlexDR1998/NCA](https://github.com/AlexDR1998/NCA) which permits efficient GPU parallelisation.
### Model details and parameters
Neural Cellular Automata (NCA) are a class of cellular automata defined on a lattice of real vectors, with the local update rule encoded in a neural network. As with all neural network models, there is freedom to choose the network structure and how the input data are preprocessed for training and testing. We refer to the set of choices that are not learned via the training data as _hyperparameters_. Figure 1 shows a schematic of a single NCA update, which maps the state of the system at time step \(n\) to a corresponding state at time \(n+1\), where \(n=0,1,\ldots,\). This update comprises a sequence of stages--depicted counterclockwise starting top left in the figure--which we now describe in turn.
System stateWe take the state of the system to be described through a vector of \(C\) real numbers at each point on an \(M\times M\) lattice. For example, each of the \(C\) values could represent the concentration of a chemical or biological species at a given point in space. These observable channels are shown with coloured shading in Figure 1. These can be augmented with _hidden channels_ (transparent in the figure), the state of which can influence the dynamics within the _observable channels_. In a biological context, these hidden channels could represent concentration profiles of proteins or chemicals that are not measured in a particular experimental setting, but can be inferred by the machine learning algorithm. The number of hidden channels is a hyperparameter of the model. The number of hidden and observable channels sum to \(C\). Mathematically, we denote the state of the system at timestep \(n\) as the vector \(x^{(n)}\in\mathcal{X}=\mathds{I}^{M\times M\times C}\), where \(\mathds{I}\in[a,b]\) is some interval of real numbers. The elements of this vector are \(x_{ijc}^{(n)}\in\mathds{I}\), which \(i\in[1,M]\), \(j\in[1,M]\) and \(c\in[1,C]\) denote the \(x\)-coordinate, \(y\)-coordinate and channel number, respectively. We emphasise that during training (Section 2.2), only the observable channels are compared to data.
Convolution kernelsThe first stage of the update is to apply _convolution kernels_ to the spatial data in each channel. These kernels are chosen to mimic differential operators, such as gradients (Sobel filters) and Laplacians. We denote the set of convolution kernels \(g^{k}\), labelled with the index \(k=1,\ldots,K\). Each kernel \(g^{k}\) is a square matrix, in our case (\(3\times 3\)). This generates the expanded _perception vector_\(z^{(n)}\) whose elements are given by:
\[z^{(n)}_{ijck}=\sum_{\Delta i,\Delta j\in[-1,0,1]}g^{k}_{\Delta i,\Delta j}x^{( n)}_{i+\Delta i,j+\Delta j,c}\equiv g*x^{(n)} \tag{1}\]
Crucially the kernels are applied in parallel on all channels \(C\) independently: that is, all kernels are applied _depthwise_. The idea of decomposing an arbitrary convolution to separate
Figure 1: Schematic of an update step of the NCA. For each \(C\) channel pixel in the \(M\times M\) lattice \(x^{(n)}\) at step \(n\), a perception vector \(z^{(n)}\) is constructed to encode local information via convolution with hard-coded kernels \(K\). This perception vector is fed through a dense neural network \(F_{\theta}\) with trainable weights \(W_{1}\), \(W_{2}\), and biases \(v\). The nonlinear activation function \(u(\cdot)\) is applied on the single hidden layer of the network. The output of this network yields the incremental update to that pixel, which is applied in parallel to all pixels with the stochastic mask \(\sigma\) to determine the lattice state \(x^{(n+1)}\) at step \(n+1\).
depthwise and channel-mixing convolutions [19] was inspired by the deep link between Convolutional Neural Networks (CNNs) and Cellular Automata [11, 20]. In particular, this facilitates representing the NCA in standard Tensorflow code. In principle, kernels can be learnable rather than hard-coded; however this makes the trained models less interpretable and so we do not pursue this approach here. As the \(3\times 3\) convolution kernels only encode information about the Moore neighbourhood (i.e. adjacent and diagonal cells), we would never need any more than 9 kernels, as any more would be linearly dependant on those already present.
The purpose of applying the kernels is to make a clearer correspondence between NCAs and numerical methods for solving PDEs. Essentially, they provide the neural network with a basic set of differential operators to work with. The set of kernels to include is an important hyperparameter. For example, if one anticipates that the update rules should be invariant under a global rotation of the system--i.e., that the dynamics are isotropic--one can justify excluding Sobel kernels and just using identity, Laplacian, and local averages. We already have translational invariance as a direct consequence of the NCA construction, but isotropic symmetry can only be realistically achieved by only using isotropic kernels [21] or data augmentation. We explore this further in Section 3.2.2. The explicit forms of the convolution kernels used in this work are
\[\underbrace{\begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix}}_{\text{identity}},\quad\underbrace{\begin{array}{c} \frac{1}{9}\begin{bmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{bmatrix}}_{\text{average}},\quad\underbrace{\frac{1}{8}\begin{bmatrix} 1&2&1\\ 0&0&0\\ -1&-2&-1\end{bmatrix}}_{\text{Sobel}_{x}},\quad\underbrace{\frac{1}{8}\begin{bmatrix} 1&0&-1\\ 2&0&-2\\ 1&0&-1\end{bmatrix}}_{\text{Sobel}_{y}},\quad\underbrace{\frac{1}{4}\begin{bmatrix} 1&2&1\\ 2&-12&2\\ 1&2&1\end{bmatrix}}_{\text{Laplacian}}. \tag{2}\]
Neural networkThe perception vector \(z^{(n)}\) is then applied to the input layer of a neural network (see lower part of Figure 1). The values on the output layer, \(F(z^{(n)})\), form a vector of increments in the original state space \(\mathcal{X}\). In a deterministic update, one would simply add \(F(z^{(n)})\) to \(x^{(n)}\) to obtain the updated state, \(x^{(n+1)}\). Taking \(F(z^{(n)})\) as an increment, rather than the new state vector \(x^{(n+1)}\), implies that the NCA is a residual, rather than a naive, recurrent neural network (RCNN) [22].
In the present context, residual RCNNs have several benefits. Firstly a residual RCNN is easier to compare to numerical discretisiations of PDEs, aiding interpretability of our models. Secondly the residual RCNN minimises the problem of vanishing or exploding gradients that the naive RCNN would experience. In the naive approach, recurrent iterations of our neural network would lead to exponentially large or small gradient updates, leading to a failure to learn optimal model weights. A consequence of the vanishing gradients problem is that information from previous timesteps is quickly lost, so \(x^{(m)}\) has little bearing on \(x^{(n)}\) for \(m\ll n\). In principle a naive RCNN can learn long term dependencies, but in practice this is very challenging. As such residual RCNNs are better suited to learning long term behaviours, as \(x^{(n+1)}\) depends linearly on \(x^{(n)}\), so information preservation is in some sense the default behaviour of our model. This behaviour is especially clear during training, in that initialising the residual RCNN to perform the identity mapping
is straightforward: one simply arranges for \(F(z^{(n)})=0\) by setting the final layer weights in the neural network to zero. Initialising the weights such that \(F(z^{(n)})=x^{(n)}\), which would be required in the naive case, is much harder. This 'do nothing' update is a better starting point than one that quickly forgets the initial state of the system. In the latter case, the NCA may resort to learning how to construct the desired \(x^{(n)}\) as a global attractor, irrespective of initial conditions, for example 'growing' \(x^{(n)}\) from fixed boundary conditions. Preserving the initial system state allows the model to better learn dynamics particular to those initial conditions, whilst still allowing for boundary driven behaviour to be learned.
It remains to specify the neural network structure, that is, the number and size of any hidden layers, and how they are connected. Here we aim to keep the architecture as simple as possible, as a minimal yet sufficiently functional network architecture has several advantages. Training a small model is computationally cheaper, and smaller models are far more interpretable [15]. Specifically, we use just one hidden layer, as shown in Figure 1. As noted previously, we do not mix spatial data in the neural network, only between channels and kernels. That is, the network shown in Figure 1 is replicated for each pixel \(i,j\) in \(z^{(n)}\), and takes as input the \(K\times C\) elements of \(z^{(n)}\) that correspond to a given pixel, and transforms to \(C\) channel values, consistent with the original state vector.
The hyperparameters associated with this neural network structure are the choice of activation function and size of the hidden layer. We fix the hidden layer size to \(H=4C\) where \(C\) is the number of channels. This way the network size scales with the number of channels, so we just explore them together as one hyperparameter. Denoting the hidden-layer activation function as \(u\), we can specify the mapping \(F\) through the elements of the output vector as
\[f^{(n)}_{ijc}=\sum_{h\in[1,H]}\left(W^{ch}_{1}u\Big{(}\sum_{ \begin{subarray}{c}c^{\prime}\in[1,C]\\ k\in[1,K]\end{subarray}}W^{c^{\prime}kh}_{2}z^{(n)}_{ijc^{\prime}k}\Big{)} \right)+v^{c}\equiv F(z^{(n)}). \tag{3}\]
The weights \(W_{2}\) mix information between the channels and kernels independently of the position \(i,j\) to determine the activation of each hidden node \(h\). The weights \(W_{1}\) then combine the hidden nodes to construct the output value for each channel \(c\), again independently of \(i,j\). We emphasise that the same set of weights and biases is applied at every pixel, consistent with the separation of spatial and channel mixing between the two stages of the process.
Stochastic maskThe final step is to increment a random subset of state vector elements \(x^{(n)}_{ijc}\) by applying a _mask_\(\sigma^{(n)}=(\sigma^{(n)}_{ijc})\), where \(\sigma^{(n)}_{ijc}\) are independent Bernoulli random variables with parameter \(1-p\). That is \(\sigma^{(n)}_{ijc}=0\) with probability \(p\), which is related to the dropout rate in machine learning regularisation, and \(\sigma^{(n)}_{ijc}=1\) otherwise. The purpose of this mask is to break any global synchronisation between cells.
Given the above, the update specified by the NCA is
\[x^{(n+1)}=x^{(n)}+\sigma^{(n)}F(g*x^{(n)}))\equiv\Phi(x^{(n)},\theta) \tag{4}\]
where we have introduced the mapping \(\Phi(\cdot,\theta):\mathcal{X}\rightarrow\mathcal{X}\) from one state vector to the next, where \(\theta\) encodes the network parameters \((W_{1},W_{2},v)\). In terms of individual elements, this corresponds to
\[x_{ijc}^{(n+1)}=x_{ijc}^{(n)}+\sigma_{ijc}^{(n)}f_{ijc}^{(n)} \tag{5}\]
where the elements \(f_{ijc}^{(n)}\) are given by Eq. (3) above. See line 7 of Algorithm 1. Hence, given \(x^{(0)}\), the NCA provides recursively the sequence \((x^{(n)})_{n=0,1,2,\ldots,N}\).
```
1:function\(\Phi(z,\theta)\)
2:\(W_{1},W_{2},v\leftarrow\theta\)
3:\(z\gets g*x\)
4:for all\((i,j)\in[1,M]^{2}\)do
5:\(dx\gets W_{1}u(W_{2}\bullet z[i,j])+v\)
6:if Random() \(\geq p\)then
7:\(x[i,j]\gets x[i,j]+dx\)
8:endif
9:endfor
10:return\(x\)
11:endfunction
```
**Algorithm 1** Pseudocode description of a single NCA update step. Here \(x\) is an \(M\times M\) lattice with \(C\) channels. \(g*x\) represent the convolutions described in Eq.1. \(W_{1}\) and \(W_{2}\) are the neural network weight matrices, with \(v\) being a vector of biases, all of which are encoded in \(\theta\). \(u()\) is the activation function. Note that in practice the For loops in line 4 and convolutions in line 3 are efficiently parallelised in Tensorflow's GPU implementation. Random() samples a uniform \([0,1)\) distribution.
Batch parallelismWhen implementing this model in tensorflow, we make use of _batch parallelism_, where instead of training on one trajectory \((x^{(n)})_{n}\), we train simultaneously on a set (batch) of trajectories \((x^{(n,r)})_{n,r}\), where superscript \(r=1,2,\ldots,R\) denotes the batch number. In effect this just adds an extra batch dimension to \(x^{(n)}\), so \(x_{ijc}^{(n)},z_{ijdk}^{(n)}\) and \(f_{ijc}^{(n)}\) become \(x_{ijc}^{(n,r)},z_{ijdk}^{(n,r)}\) and \(f_{ijc}^{(n,r)}\) respectively. This is normally done to leverage low-level speed-up, as training the network on batches involves matrix-matrix rather than matrix-vector multiplications, which are well optimised on parallel architectures (GPUs). However in the case of NCA, batch parallelism enables far more diverse systems to be learned, for example learning several distinct trajectories by the same rule, or improving stability through data augmentation. It is also crucial for extending the existing NCA training algorithm [11] to learning longer sequences of data, as discussed in section 2.2.
To summarise, the NCA can be described in terms of neural network language as a residual (calculating increments to each pixel) Recurrent Convolutional Neural Network (RCNN) with per-pixel dropout (stochastic mask). The hyperparameters are the number of hidden channels, the set of convolution kernels and activation functions on the hidden layer of the network. The effect of varying the hyperparameters is explored in Section 3.
### Training techniques
The neural network architecture described above is very minimal in comparison to the state-of-the-art in the literature [23], featuring only a single hidden layer applied in parallel. By contrast, the training process is fairly complex. We set out key steps below, with corresponding pseudo-code set out as Algorithm 2. NCA trajectories can be considered as paths in \(\mathcal{X}\) (Figure 2), where the training process constrains the NCA parameters such that the trajectories pass as close (defined by the loss function) as possible to the observed data points in \(\mathcal{X}\). Projecting these paths onto 1D helps visualise the training process, especially when training to multiple batches.
The technique of training NCA is based on backpropagation through time, a typical method for training RNNs [24]. We have established the batch of NCA trajectories \((x^{(n,r)})_{n=1,\ldots,N}\), with \(n\) and \(r\) denoting time and batch respectively. Originally [11], training the NCA consisted of one set of initial states being mapped to one set of final states \(x^{(0,r)}\to x^{(t,r)}\). We extend this to learn the set of transitions \(x^{(n-1,r)}\to x^{(n,r)}\), where the batch parallelism allows us to train the NCA on each transition simultaneously. This allows training NCA to far more diverse and complex dynamics. For clarity we drop the explicit batch index (i.e. set \(r=1\)) for now, the context where it matters is discussed later, but even so batch parallelism is still exploited for learning the different timesteps.
We have a time series of data \((y^{(\delta)})_{\delta=0,1,\ldots,D}\), but only for every \(t\) NCA timesteps, that is at times \(\delta t\) for \(\delta=0,1,\ldots,D\), where \(N=Dt\). Hence we need to compare \(x^{(\delta t)}\) to \(y^{(\delta)}\). We initialise \(x^{(\delta t)}=y^{(\delta)}\) for \(\delta=0,\ldots,D-1\), and propagate each state through \(\Phi^{t}(\cdot,\theta)=\Phi\circ\cdots\circ\Phi(\cdot,\theta)\) (\(t\) nested function compositions). To compute the loss, we compare \(x^{(\delta t)}\) to \(y^{(\delta)}\) for \(\delta=1,\ldots,D\), averaging over different \(\delta\): \(\frac{1}{D}\sum_{\delta=1}^{D}\mathcal{L}(x^{(\delta t)},y^{(\delta)})\), where the loss function \(\mathcal{L}:\mathcal{X}^{2}\rightarrow\mathds{R}\) is a meaningful measure of distance between any pair of \(x^{(i)}\) and \(y^{(j)}\). Training is achieved by minimising the loss function, which requires partial gradients with respect to the trainable parameters \(\theta\) to be evaluated. For the full gradient calculations, see appendix A.
**Algorithm 2** Training an NCA \(\Phi\) to \(R\) data trajectories of length \(D\). Split into \(B\) mini-batches to reduce memory usage. Here \(x^{(n,r)}\) and \(y^{(n,r)}\) denote predicted state and data at step \(n\) of trajectory \(r\) respectively. \(\hat{x}^{(n,r)}\) is a temporary variable to store new intermediate states at each training iteration. Lines 20 and 21 perform gradient normalisation followed by a parameter update handled by the Nadam algorithm (or any other optimiser of choice). The nested For Loops on lines 9,11 and 12 are easily parallelised. Typical choices of mini-batching are such that \(10<(D\times R)//B<100\). \(\Phi^{t}\) denotes \(t\) iterations of \(\Phi\). The choice of \(t\) is an important hyperparameter, and relates to the temporal resolution of the data \(x\). The model parameters \(\theta\) are assumed to either be initialised appropriately, or already partially trained. RandomShuffle\((A,B)\) randomly shuffles \(A\) and splits it into \(B\) equal sized chunks. \(\mathcal{L}(A,B):\mathcal{X}\times\mathcal{X}\rightarrow\mathds{R}\) computes the loss between states \(A\) and \(B\).
```
1:functionTrain(\(\Phi,\theta,y,t\),B,EPOCHS)
2:\(x\gets y\)
3:\(D\gets y.shape[0]\)
4:\(R\gets y.shape[1]\)
5:for\(i\in\) EPOCHS do
6: Grad \(\leftarrow\vec{0}\)
7: DS \(\leftarrow\) RandomShuffle([1,D],B)
8: RS \(\leftarrow\) RandomShuffle([1,R],B)
9:for\(b\in[1,B]\)do
10: Loss \(\leftarrow\) 0
11:for\(\delta\in\) DS\([b]\)do
12:for\(r\in\) RS\([b]\)do
13:\(\hat{x}^{(\delta,r)}\leftarrow\Phi^{t}(x^{(\delta-1,r)},\theta)\)
14: Loss \(\leftarrow\) Loss \(+\mathcal{L}(\hat{x}^{(\delta,r)},y^{(\delta,r)})\)
15:endfor
16:endfor
17: Loss \(\leftarrow\frac{1}{D\times R}\) Loss
18: Grad \(\leftarrow\) Grad \(+\frac{\partial\text{Loss}}{\partial\theta}\)
19:endfor
20: Grad \(\leftarrow\) Norm(Grad)
21: Update(\(\theta\),Grad,\(i\))
22:for\(\delta\in[1,D]\)do
23:for\(r\in[2,R]\)do
24:\(x^{(\delta,r)}\leftarrow\hat{x}^{(\delta,r)}\)
25:endfor
26:endfor
27:endfor
28:return\(\Phi\)
29:endfunction
```
**Algorithm 2** Training an NCA \(\Phi\) to \(R\) data trajectories of length \(D\). Split into \(B\) mini-batches to reduce memory usage. Here \(x^{(n,r)}\) and \(y^{(n,r)}\) denote predicted state and data at step \(n\) of trajectory \(r\) respectively. \(\hat{x}^{(n,r)}\) is a temporary variable to store new intermediate states at each training iteration. Lines 20 and 21 perform gradient normalisation followed by a parameter update handled by the Nadam algorithm (or any other optimiser of choice). The nested For Loops on lines 9,11 and 12 are easily parallelised. Typical choices of mini-batching are such that \(10<(D\times R)//B<100\). \(\Phi^{t}\) denotes \(t\) iterations of \(\Phi\). The choice of \(t\) is an important hyperparameter, and relates to the temporal resolution of the data \(x\). The model parameters \(\theta\) are assumed to either be initialised appropriately, or already partially trained. RandomShuffle\((A,B)\) randomly shuffles \(A\) and splits it into \(B\) equal sized chunks. \(\mathcal{L}(A,B):\mathcal{X}\times\mathcal{X}\rightarrow\mathds{R}\) computes the loss between states \(A\) and \(B\).
```
1:functionTrain(\(\Phi,\theta,y,t\),B,EPOCHS)
2:\(x\gets y\)
3:\(D\gets y.shape[0]\)
4:\(R\gets y.shape[1]\)
5:for\(i\in\) EPOCHS do
6: Grad \(\leftarrow\vec{0}\)
7: DS \(\leftarrow\) RandomShuffle([1,D],B)
8: RS \(\leftarrow\) RandomShuffle([1,R],B)
9:for\(b\in[1,B]\)do
10: Loss \(\leftarrow\) 0
11:for\(\delta\in\) DS\([b]\)do
12:for\(r\in\) RS\([b]\)do
13:\(\hat{x}^{(\delta,r)}\leftarrow\Phi^{t}(x^{(\delta-1,r)},\theta)\)
14: Loss \(\leftarrow\) Loss \(+\mathcal{L}(\hat{x}^{(\delta,r)},y^{(\delta,r)})\)
15:endfor
16:endfor
17: Loss \(\leftarrow\)\(\frac{1}{D\times R}\) Loss
18: Grad \(\leftarrow\) Grad \(+\frac{\partial\text{Loss}}{\partial\theta}\)
19:endfor
20: Grad \(\leftarrow\) Norm(Grad)
21: Update(\(\theta\),Grad,\(i\))
22:for\(\delta\in[1,D]\)do
23:for\(r\in[2,R]\)do
24:\(x^{(\delta,r)}\leftarrow\hat{x}^{(\delta,r)}\)
25:endfor
26:endfor
27:endfor
28:return\(\Phi\)
29:endfunction
```
**Algorithm 3** Training an NCA \(\Phi\) to \(R\) data trajectories of length \(D\).
There are additional practical considerations for optimising this training method. After each iteration, we have the choice of keeping and further propagating the values in \(x^{(\delta t)}\) for \(\delta=1\ldots D\), or re-initialising them: \(x^{(\delta t)}=y^{(\delta)}\). Propagating them allows the NCA to better learn long term dynamics (particularly of the hidden channels) over many training iterations. However, we observe that re-initialising helps speed up the training process in the earlier steps. As both approaches have their advantages, we return to the batch parallelised case and do both. We regard re-initialising the states as a form of data augmentation (Figure 2), so in practice we only re-initialise one batch: \(x^{(\delta t,1)}=y^{(\delta,1)}\). This choice of only re-initialising one batch performs well, but is arbitrary and could be further tuned for specific problems. The NCA is initialised with random hidden layer weights (\(W_{2}\)), and zero final layer weights (\(W_{1}\)) and bias (\(v\)).
When implementing the training procedure, as described in Algorithm 2, the additional subtlety of mini-batching is required [25, 26]. Rather than computing the gradient for transitions \(x^{(\delta-1,r)}\to x^{(\delta,r)}\) for all \(\delta\in[1,D],r\in[1,R]\), we randomly split \([1,D]\times[1,R]\) into \(B\)_mini-batches_, and separately compute the loss gradient for each mini-batch. After iterating through each mini-batch, the gradients are averaged and applied once to the model parameters. The need for mini-batching is due to memory constraints: if a large enough number of batches \(R\) or timesteps \(D\) is used, computing the gradient over the full set of transitions is unfeasible. In Algorithm 2 the memory cost of calculating the gradient (line 18) scales like \(D\times R\times M^{4}\times C^{2}\times t\) (where \(|\mathcal{X}|=M^{2}C\)). By contrast the memory cost of storing and adding to the calculated gradient over each mini-batch is fixed as \(\|\theta\|\) (i.e. does not scale with \(B\)), which is minimal given the small size of the network. \(M\) and
Figure 2: 1D phase space representation of NCA trajectories, predictions \(x^{(\delta t,r)}\) and true states \(y^{(\delta,r)}\). Here \(D=3,R=2\). The first batch (\(x^{(\cdot,1)}\)) is trained with re-initialised intermediate states, whereas the second batch (\(x^{(\cdot,2)}\)) is trained with propagated intermediate states.
are fixed by the spatial (and channel) resolution of the data, but mini-batching reduces the memory burden of \(D\) and \(R\). For the full calculation see appendix A. In the case of \(B=1\), the mini-batching reduces such that the For loops at lines \(9,11\) and \(12\) collapse into simpler loops over \([1,D]\) and \([1,R]\). When training on image morphing [11], the size of data does not require mini-batching as \(D\) and \(R\) are small. To accurately capture PDE dynamics, much larger \(D\) is used, and as such mini-batching is required. We do not explore spatial mini-batching, where random subsets of pixels are tracked for gradients, as this makes efficient parallelisation more challenging, however it could be very useful to further explore as it could enable training of NCA to higher resolution data.
#### 2.2.1 Loss functions
We now turn to the question of how to define the distance between a NCA trajectory and target data. This problem is split naturally into two parts: first, how to find the difference between corresponding (time and batch labelled) points in \(\mathcal{X}\), and then how to combine all these difference measures to the distance between two sets of points in \(\mathcal{X}\). For the latter choice, we adopt an arithmetic mean over the time points and batches considered (as shown in lines \(14\) and \(17\) of Algorithm 2), although these could be weighted (e.g., the states at a specific time or batch being most important).
We compared several loss functions for corresponding points in \(\mathcal{X}\), but we found that the standard euclidean norm \(\mathcal{L}(x,y)=\big{(}\sum_{i,j,c}(x_{ijc}-y_{ijc})^{2}\big{)}^{\frac{1}{2}}\) worked best in all contexts. Probability mass based losses (Hellinger [27] and Bhattacharyya [28]) and distance between spatial Fourier transforms of points in \(\mathcal{X}\) all work well in the PDE modelling case, but perform poorly on image morphing. Various approximations to the Wasserstein distance [29] performed poorly in both contexts but still remain promising given their success in texture synthesis [30, 31, 32]. As such, for the following results we stick to the euclidean distance, although we recommend experimenting with different loss functions depending on the system one is modelling. Any differentiable function \(\mathcal{L}:\mathcal{X}\times\mathcal{X}\to\mathds{R}\) can be used, if minimising it's output constrains the inputs in a desirable way.
Results
We now demonstrate the applicability of the NCA to two contrasting use cases. First, we consider training data comprising numerical solutions of coupled nonlinear reaction-diffusion equations, with parameters that produce Turing patterns. Equations in this class are widely used to model complex biological systems such as in: developmental biology [33, 34]; ecology [35]; and skin pattern morphogenesis [36]. We adopt a representative example that is capable of generating a wide variety of spatial patterns, in the context of chemical reactions [17]. We demonstrate in particular that the NCA can generalise beyond the set of initial conditions in its training data. We then turn to an artificial image morphing problem, inspired by [11], and show that the same NCA is capable of constructing local rules to effect the desired dynamics. In contrast to the reaction-diffusion system, such rules are not known _a priori_, so it is not obvious that they exist. This problem also lends itself to testing the robustness of the rules that result. When training to PDEs, the focus is on exploring training hyperparameters (loss function, time sampling). After determining suitable training hyperparameters, we then show that this generalises to the image morphing task, where we explore model hyperparameters (kernels, activation functions, number of hidden channels).
### Gray-Scott reaction diffusion equations
The Gray-Scott [17] reaction diffusion equations read
\[\partial_{t}A =D_{A}(\partial_{xx}+\partial_{yy})A-AB^{2}+\alpha(1-A)\] \[\partial_{t}B =D_{B}(\partial_{xx}+\partial_{yy})B+AB^{2}-(\gamma+\alpha)B\]
in which \(D_{A},D_{B},\alpha\) and \(\gamma\) are parameters. These describe two species, \(A\) and \(B\), which diffuse in two-dimensional space with diffusion constants \(D_{A}\) and \(D_{B}\), respectively. Species \(A\) grows towards a density of \(A=1\) at a rate \(\alpha\), whilst species \(B\) dies out at rate \(\gamma+\alpha\). The species also undergo the reaction \(A+2B\to 3B\).
With \(D_{A}=0.1,D_{B}=0.05,\alpha=0.06230,\gamma=0.06268\) we obtain maze-like patterning (Figure 3). We solve these PDEs with an Euler finite-difference discretisation scheme, with time-step of 1, for \(N=1024\) steps, and on an \(M\times M\) lattice of size \(M=64\). \(\alpha\) and \(\gamma\) parameterise the patterning type, whereas \(D_{A}\) and \(D_{B}\) re-scale the patterns, and must be chosen in line with the timestep size to achieve numerical stability.
#### 3.1.1 Effect of training hyperparameters
We begin with a basic NCA architecture that employs just \(K=2\) kernels, the identity and Laplacian, \(p=0\) (purely deterministic), \(C=8\) channels in total (so 2 observable and 6 hidden channels) and a rectified linear unit (relu, \(u(z)=\frac{|z|+z}{2}\)) activation function. We found that the Nadam optimiser [37] consistently performed well. Nadam is a modification of the widely used Adam optimiser [38], with the only difference being the use of Nesterov momentum [37]. Optimisers based on Nesterov momentum perform well, both in theoretical convergence and generalisability of trained deep neural networks [37, 39]. Note that we also employ gradient normalisation [40] before passing gradient information to the optimiser - this was found to significantly improve training performance. This just leaves the time sampling (\(t\)) as the main hyperparameter to optimise. Time sampling is subtle in the case of PDEs as numerical integration of the PDE system necessarily involves a discrete integration time step. We can sample the trajectories at coarser intervals, increasing \(t\) in Algorithm 2 such that each NCA update corresponds to a timestep of the PDE solver. In other words, we only compare every \(t^{\text{th}}\) PDE and NCA step.
We found that while training loss increased with greater sampling intervals \(t\), tests on unseen initial conditions achieve comparable loss for most sampling intervals, with modest improvements for greater \(t\) (Figure 4). The training loss is calculated for \(N=1024\) steps (as in Figure 3), whereas the test loss is calculated over \(N=2048\) steps from an unseen initial condition. This demonstrates generalisation both to unseen initial conditions, and longer simulation times. Note that the unseen initial condition used for testing features
Figure 3: Snapshots taken from the training data used for learning PDE dynamics. PDE is run for \(N=1024\) steps with timestep 1 and \(D_{A}=0.1,D_{B}=0.05,\alpha=0.06230,\gamma=0.06268\).
high frequency components not observed at all during training, and that NCA trained with high sampling were sometimes numerically unstable to these high frequency inputs (missing test loss points in Figure 4A, or snapshots at \(t=2\),\(t=6\) in Figure 4B).
Figure 5 shows various snapshots from true (PDE) trajectories alongside the corresponding snapshots from an NCA trained with \(t=32\). This extrapolates an unseen initial condition far beyond the time observed during training (\(n\in[0,1024]\)), demonstrating that the NCA does learn the underlying rules of the dynamics rather than overfitting to the training trajectories. When we considered finer sampling \(t\) (Figure 4), we observe more frequent numerical instabilities, or complete failure to learn dynamics. Coarse time sampling appears to both stabilise these numerical problems, and yield more generalisable models. We posit this is due to coarse time sampling allowing the NCA to be less constrained during training, in that intermediate states may explore more possible states, increasing the chances of finding \(\theta\) that gives the correct dynamics. Fine time sampling perhaps over-constrains the NCA, leading to instabilities or training converging to sub-optimal local minima of the loss
Figure 4: A: loss as a function of time sampling \(t\). Training loss shows the minimum loss during training epochs (averaged over 4 random initialisations, with standard deviation as error bars). Test loss shows how the best trained NCA (minimal training loss) performs on unseen initial conditions. B: snapshots of NCA trajectories (at \(n=2048\)) based on unseen initial conditions, with varying sampling \(t\). Each NCA is trained for 4000 epochs, with a mini-batch size \(B=64\).
landscape. Alternatively, the behaviour at fine time sampling is consistent with overfitting, so coarser time sampling could be considered a regularising technique here.
In summary, we have found that the NCA architecture set out in Section 2 is capable of learning update rules that reproduce the solution of a certain pair of coupled nonlinear PDEs. Our main finding is that good rules can be learnt, but coarse time sampling improves numerical stability, and learns rules that generalise better to unseen initial conditions. In the next section we show that the algorithm still performs well when there is no known underlying integration timestep.
### Image morphing
We now test whether a training method that works well for PDEs also works on the image morphing task. The task comprises an initial image, an intermediate image to morph through, and then a final image that is intended to remain stable (i.e., be an attractor of the dynamics) (Figure 6). This latter requirement is incorporated into the training data by repeating the final image twice. The images shown were downsampled to a resolution of \(60\times 60\) to reduce computational cost. We further impose fixed b
Figure 5: Snapshots of PDE and NCA trajectories from an unseen initial condition. NCA trained with \(C=8\), Identity and Laplacian kernels, relu activation, trained on sampling \(t=32\) for 4000 epochs with euclidean loss.
Figure 6: Image morphing task. Given a space invader initial condition, morph through a microbe and remain stable at a rooster pattern.
is, to insist that the state vector \(x^{(n)}\) vanishes at the boundary points. This is because the system is not periodic, as was the case for the PDE problem.
Reliable training of the NCA requires a careful construction of the training data. A side-effect of the fixed boundary conditions is that the NCA can learn to grow an image from the boundary, rather than from the initial condition. We would like the pattern formation to remain translationally invariant - the image morphing should behave independently of the boundaries, and should only depend on the input images. To enforce this, we embed the training data within a larger lattice, thereby reducing the influence of the boundaries. Translational invariance is further encouraged by training to several copies of the image sequence, each randomly shifted in space, to prevent learning any effective long range boundary interaction. It is possible to train NCA to produce textures as _global_ attractors [13, 30] as textures are translationally invariant. We believe it is impossible to have a fixed pattern (i.e. not a translationaly invariant texture) as a global attractor -- if boundary effects are removed so the whole NCA system is translationaly invariant. Avoiding the desired dynamics being a global attractor ensures that the NCA has learned a rule that maps the input state through the sequence of target states, rather than just generating the target states irrespective of initial conditions. This is further verified by exploring stability under perturbation in Section 3.2.2
We also find that to train the NCA to reach a stable state, in effect mapping an image to itself, augmenting the training data with noise is necessary. Without noise augmentation, the training process crashes as gradients diverge (when training the final stable transition), but adding a very small amount of random noise to the data fixes this. We can understand the effect of this noise as introducing a basin of attraction around the desired final state, and training to noisy images enhances the robustness of the NCA to noisy perturbations.
#### 3.2.1 Effect of model hyperparameters
The training hyperparameter \(t\) that corresponds to the frequency of time sampling cannot be assumed to translate from the PDE case to the image morphing task. As there is no underlying physical mechanism connecting the images, there is nothing to provide a basic unit of time. We are however guided by the fact that the update rules are local, and therefore initially take the number of timesteps to be 64, which is similar to the lattice size and therefore gives sufficient time for information to propagate across it. As we explore this point more below, we find in fact that fewer timesteps can also be sufficient.
While a deterministic update (\(p=0\)) is appropriate in the PDE case, as this was a feature of the training data, for the image morphing problem, we update stochastically with \(p=\frac{1}{2}\), as removing global synchronisation between each cell acts like dropout regularisation, and can be biologically motivated [11]. Note that a choice of \(p=\frac{1}{2}\) effectively halves the number of timesteps between images. We found that varying the update probability \(p\) had very little direct impact on model performance (except for extreme values close to 1 or 0), instead the effective number of timesteps between images was explored.
The system state has 4 observable channels--the red, green, blue and alpha (transparency) components--and 12 hidden channels. Since the images are not rotationally symmetric, and we have no prior underlying mechanisms to constrain symmetries of update rules, we consider adding Sobel kernels to the identity and Laplacian kernels that were used in the PDE case. The Laplacian kernel detects spatial changes (and curvature) in patterns, whereas the Sobel kernels also detect the orientation of any changes. We find that including the symmetry breaking Sobel kernels improves the performance over just using symmetric kernels (Figure 7A,B). This does however break the symmetry of the NCA--such an NCA is unlikely to be stable under rotations or reflections of the initial condition, as discussed in Section 3.2.2. We also find that the best performing activation function is relu (Figure 7C,D). The linear case, \(u(z)=z\), can be thought of as an absence of an activation function. Surprisingly, the overall shape of the final image is reasonably well reproduced, although it clearly lacks the definition achieved with the other activation functions. This justifies the additional complexity of nonlinear activations.
Figure 7: NCA trained on image morphing task with different kernels and activations. 16 channels, 64 steps between images. A,B: training loss and snapshots of NCA with relu activation and various kernels. C,D: training loss and snapshots of NCA with Identity, Sobel and Laplacian kernels, for various activation functions.
Exploring how NCA behaviour scales with number of channels, we find that unsurprisingly more hidden channels performs better (Figure 8A,B), capturing more of the details in the rooster image. The number of channels functions as a clear'model size' parameter, and we find that model performance scales nicely with this measure of model size. We also explore how NCA train for different numbers of timesteps between images (Figure 8C,D). It is surprising that with as few as 8 timesteps, the basic shape and colour of the rooster are correct (although details are better at 16 or 32 steps), which highlights the locality of the update rule. The image resolution is \(60\times 60\), and with 8 timesteps between images only cells less than 16 pixels away can communicate (from initial condition to reaching the stable rooster pattern). However with the stochastic cell updates, the effective communication
Figure 8: NCA trained on image morphing task. Relu activation; Identity, Sobel and Laplacian kernels. A,B: Training loss and snapshots of 16 channel NCAs trained with different time sampling. C.D: Training loss and snapshots of NCAs trained with time sampling of 32, and various numbers of channels.
range from initial to final condition here is halved to just 8 pixels. This emphasises that the update rule is local, in that local structures of the initial condition morph into local structures of the final state at similar locations.
#### 3.2.2 Stability analysis
With a trained NCA, there is an obvious question of stability--if an initial condition is perturbed away from what the NCA is trained on, how does this affect the behaviour? We consider three kinds of perturbations of the initial condition: local perturbations, global perturbations, and symmetry perturbations. Local perturbations change one pixel, and allow us to explore how errors propagate through the spatial part of the NCA. Global perturbations can show how resilient NCA are to noisy inputs. Symmetry perturbations, such as rotations or reflections of initial conditions, allow us to explore how NCA respect desirable symmetries.
Stability under local perturbations depends strongly on how many timesteps between images the NCA is trained on (Figure 9). We find that NCAs with fewer time-steps are more stable to local perturbations, or conversely that allowing more NCA steps between training images gives more time for local perturbations to travel. In both cases the perturbations remain mostly local. Using local perturbations could help calibrate the number of timesteps to use when modelling real systems, in that the NCA should have the same response to perturbations as the underlying data being modelled.
Figure 9: Local stability behaviour of two NCA. A:i 32 channels, 32 steps between images. B: 16 channels, 64 steps between images. Top left heatmap in each case shows how many pixels of the final image change (by more than 0.1 to account for random fluctuations) when that pixel is perturbed in the initial condition. The other images all show snapshots of the final state when the initial condition is perturbed locally, for different perturbation locations.
To address the question of stability with respect to global perturbations, we frame it as an optimisation problem. Let \(\kappa^{n}(x^{(0)},\tilde{x}^{(0)})=\|\tilde{x}^{(0)}\|-\|\Phi^{n}(x^{(0)}+\tilde{ x}^{(0)})-\Phi^{n}(x^{(0)})\|\), where \(\tilde{x}^{(0)}\) is a perturbation of initial condition \(x^{(0)}\). By finding \(\tilde{x}^{(0)}\) that maximises or minimises \(\kappa^{n}\), we can find a maximally perturbed initial condition \(x^{(0)}+\tilde{x}^{(0)}\) that leaves a future state \(x^{(n)}\) unchanged, or a minimal perturbation that destroys \(x^{(n)}\). This allows us to explore the space of initial conditions around which the NCA was trained, and can reveal which features of an initial condition are important. For example, it may be only the edges and corners of an image are learned from. As the whole NCA process is differentiable, we can use gradient based optimisation on \(\kappa^{n}\) to find \(\tilde{x}^{(0)}\). Finding minimal perturbations that destroy \(x^{(n)}\) is similar to adversarial attacks of classifier networks, where small changes to an image completely destroy the behaviour of an image classifier. Figure 10 shows the behaviour of a trained NCA (best performing model shown in figure 8A,B) starting from examples of these adversarial initial conditions. It is possible to find initial conditions that are visually similar to the true initial condition, and yet they destroy the stable rooster pattern (\(x^{(96)}\)). We can also find large perturbations of the initial condition that leave the target state (\(n=96\)) unperturbed, however the long term stability of the rooster pattern is still damaged.
Figure 10: Rightmost column shows extrapolation beyond training time, demonstrating stability of the final state. Top row shows snapshots from unperturbed trajectory. Middle row shows snapshots from minimal initial perturbation that destroys the final state (minimising \(\kappa^{(96)}(x^{(0)},\tilde{x}^{(0)})\)). Bottom row shows snapshots from maximal initial perturbation that preserves the final state (maximising \(\kappa^{(96)}(x^{(0)},\tilde{x}^{(0)})\)). NCA (32 channels; Identity, Sobel and Laplacian kernels; time sampling \(t=32\), relu activation) trained on image morphing task.
Finally, we compare the behaviour of different NCA models on symmetrically perturbed initial conditions (Figure 11). By rotating or flipping the input image we obtain symmetrically perturbed inputs. One NCA is trained on _normal data_, that is, data that has only been translationally perturbed to minimise boundary effects. The other is trained on _augmented data_ that also includes the same training data after applying global rotations about random angles. We also explore the effect of restricting the NCA to include only symmetric kernels, rather than both symmetric and asymmetric kernels. We find that even without any data augmentation, the symmetric kernel NCA already performs very well, although it struggles with the off lattice 45 degree rotations. When trained on rotationally augmented data, the asymmetric kernel NCA improves its performance on rotated inputs, but is still outperformed by the symmetric kernel NCA for on-lattice rotations (with or without data augmentation). Off lattice rotations are the most challenging, and seem to be where data augmentation is necessary even for symmetric kernel NCA. Overall, we find that the symmetric kernel NCA better handles symmetric perturbations, whilst the asymmetric
Figure 11: Behaviour of trained NCA on symmetrically perturbed inputs. Left column shows inputs, middle two shows final state behaviour for NCA with asymmetric kernels (identity, Sobel and Laplacian), rightmost two shows final state behaviour for NCA with symmetric kernels (identity, Laplacian, average). Augmented data examples show NCAs trained to trajectories rotated to random angles.
kernel NCA performs best on the unperturbed data. As one might expect, building symmetry into the model allows it to solve a broader range of symmetrically-related problems, whereas leaving it out promotes specialisation towards a single problem.
## 4 Discussion
We have demonstrated NCA as a framework for modelling emergent spatio-temporal patterning. Many systems in biology are characterised by complex emergent phenomena of locally interacting components, and finding interaction rules or mechanisms that lead to specific emergent behaviours is a challenging inverse problem. By making classic cellular automata differentiable, NCA present a new approach to this class of problems. [11] demonstrated that a trained NCA can generate complex structures (specifically emojis) that remain stable over time and to perturbation. Here we have extended this approach to learn dynamics from snapshots of a pattern at multiple timepoints (rather than just the end-point), i.e., we show the ability to learn dynamic patterns, specifically those arising from PDEs with Turing instabilities that been widely used to study biological pattern formation.
Specifically, we showed in Section 3.1 that NCA can infer update rules equivalent to those obtained by discretising and iterating a set of PDEs. NCA have an inductive bias to learning underlying dynamics, rather than just overfitting to data, due to their minimal parameters and hard coded local kernels. We demonstrate this by presenting the trained NCA with initial conditions that were not part of the training data, and finding that the predicted trajectories are similar to those obtained directly from the PDEs - the trained NCA generalise well. This suggests that an NCA trained on experimental data could be used to predict the behaviour of that system under conditions that have not been directly observed. We have also discussed NCA hyperparameters in more detail than most previous work, which can provide guidance for future exploration. For example, tuning the number of timesteps between images can constrain how far local information is allowed to spread, and the number of channels required to accurately capture a patterning behaviour could function as a heuristic for the complexity of that pattern.
More generally, the findings of Section 3.2 confirm that NCA can be used as a tool to construct local dynamical update rules whose emergent properties satisfy certain constraints. These constraints include the end state, but extend also to the stability of that configuration, invariance of the dynamics under certain symmetry operations and the effect of boundary conditions. Given that for any observed emergent patterning, many possible microscopic update rules could exist, constraining how NCA respect symmetries or behave around boundaries helps reduce the set of possible rules. As the training process for NCA amounts to a differential optimisation procedure, microscopic rules that yield desired emergent behaviour can be efficiently found, even when constraints are imposed, as exemplified by the image morphing task.
Using NCA as an _in silico_ way to study the behaviour of growing systems is a recurring theme in the literature [11, 21], discussed more concretely in [41]. Further applications or extensions of NCA models have been explored in the context of image processing and
synthesis [42, 43]. Here NCA models have been coupled to CLIP text embeddings [44], in line with recent machine learning based text to image techniques. NCA are clearly capable of a diverse range of behaviours, but all the previous literature just focuses on training one set of transitions, rather than full dynamic trajectories. We believe that the extension to learning dynamics of arbitrarily long sequences dramatically increases the already wide range of systems and behaviours NCA can model. We believe that being able to train to sequences of images will better enable NCA to be applicable to modelling real biological systems, and it will probably enable more interesting image synthesis techniques.
Compared to most current machine-learning research, our chosen neural network architecture is economical in terms of the number of trainable parameters. This not only makes training more computationally efficient, but also adheres also to the aesthetic guidance of modelling traditions in physics and mathematics that simpler models are preferred over more complex models when they have comparable descriptive power. Here we have found that a single hidden layer is sufficient to model a variety of systems and reproduce a wide range of behaviours. The reason for this may lie in part due to hidden channels in the state space, as these can encode complex environmental information such as boundary conditions, as well as encoding memory for longer term dynamics. This spatially distributed memory encoding in the hidden channels could be likened to previous work on differentiable neural computers [45]. We note that NCA also link back to older work on amorphous computing [46], providing a connection between modern machine learning and theory of spatially distributed computational frameworks.
There are however a few shortcomings of NCA, mainly the underlying assumption that purely local interactions are sufficient. There will be systems with non-local (or multiscale) interactions that cannot be elegantly explained with purely local rules. We have also assumed that the update rules are constant over time, even though many complex (physical or biological) systems are highly time dependent [47]. Whilst it is possible that the hidden channels in the NCA could encode nonlocality and time-dependence, it might be more natural (and interpretable) to extend the representation of the dynamics in the neural network to incorporate such dependencies explicitly, for example by increasing the size/stride of convolution kernels, or by including an explicit time parameter as a network input. A possible risk with including explicit time dependence is that the NCA could over-fit to the timestep, rather than learning the microscopic interactions that yield the emergent behaviour. To tackle such questions, it might be desirable to augment the loss functions with measures of model complexity as a means to converge on the most parsimonious description, for example by sparsity regularisation. Generalising NCA to better work on multiscale systems could also be worth exploring, for example by coupling NCA-like models on lattices of different resolutions.
Similarities can be drawn between NCA and other data driven equation discovery techniques like SINDy [48] or Extended Dynamic Mode Decomposition (EDMD) [49]. Both SINDy and EDMD have the main purpose of fitting dynamical systems to data, both in the cases of ODEs and PDEs. SINDy enforces parsimonious solutions through parity regularisation, whereas Dynamic Mode Decomposition is analogous to Singular Value De
composition for time series data (and the Extended DMD is a nonlinear generalisation). The key areas where NCA differ is in background motivation, and the sorts of systems they're suited to. NCA act as a bridge between machine learning based image processing, and learning simple models of complex systems; whereas SINDy and EDMD were developed in the context of data driven engineering and fluid dynamics respectively. Cellular automata models are more general than (local) PDEs. Although we restrict ourselves to differential operator kernels, we don't have to - learning arbitrary kernels would provide far more general expressive power, whilst still keeping a minimal (and importantly local) model. The class of models that could be learned with arbitrary kernels includes local PDEs, but is far more general.
A further development of these NCA models is to make them truly distributed during learning. When computing the loss, information of the whole lattice is needed, which places limits on the size of a lattice that can be handled with available computation time and memory. If instead an NCA could be trained with a purely local loss, such that model weights are updated for each pixel based on its neighbours, more advanced training procedures could be exploited. In essence, if the only global communication is the updates to the model weights, rather than the full lattice state, NCA could be trained using online training, or on much larger lattice sizes. An alternative approach to increase the resolution that can be trained on would be to randomly sub-sample elements [50] of the NCA lattice when computing losses.
Although we explored the stability of NCA under perturbations to the trajectory (initial condition), we did not address stability under perturbation of model parameters. This could naturally tie in to interpretation of the trained model, for example, by assessing stability under perturbation of network weights. Performing network pruning [51] could be another powerful approach, yielding even more minimal models. The recently popular field of explainable AI [14, 15] likely offers some other tools that would enable this. A worthwhile further development would be to reverse-engineer a concise analytic expression for the underlying PDE from the trained NCA parameters, for example with symbolic regression techniques like Sparse Identification of Nonlinear Dynamics (SINDy) [48]. Such an approach could be tested with NCAs trained on known PDEs, but it would be interesting to then apply this to systems where we don't know the underlying mechanics, such as image morphing or biological data.
## Acknowledgements
This work has made use of the resources provided by the Edinburgh Compute and Data Facility (ECDF) ([http://www.ecdf.ed.ac.uk/](http://www.ecdf.ed.ac.uk/)). For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. Alex D Richardson was supported by the EPSRC Centre for Doctoral Training in Mathematical Modelling, Analysis and Computation (MAC-MIGS) funded by the UK Engineering and Physical Sciences Research Council (grant EP/S023291/1), Heriot-Watt University and the University of Edinburgh. | ニューラル・細胞型オートマトン（NCA）は、機械学習とメカニズムモデルの強力な組み合わせです。私たちは、画像の時系列とPDEの軌跡から複雑なダイナミクスを学習するようにNCAを訓練しました。私たちの方法は、大規模な動的なEmergent Behavioursを支配する、潜在的な局所ルールを特定するように設計されています。NCAの以前の研究は、静止的なEmergent Structureを学習するのに焦点を当てています。私たちは、NCAを時間変化と安定な構造を同時に捉えるために拡張し、線形非線形偏微分方程式（PDE）のTuringパターン形成のダイナミクスを学習する規則を学習しました。私たちは、NCAはPDEのトレーニングデータを超えて非常に汎用的に学習できることを示し、NCAを与えられた対称性に従って制限する方法を説明し、モデルのパフォーマンスと安定性を評価 |
2304.01425 | A GKP qubit protected by dissipation in a high-impedance superconducting
circuit driven by a microwave frequency comb | We propose a novel approach to generate, protect and control GKP qubits. It
employs a microwave frequency comb parametrically modulating a Josephson
circuit to enforce a dissipative dynamics of a high impedance circuit mode,
autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is
robustly protected against all dominant decoherence channels plaguing
superconducting circuits but quasi-particle poisoning. In particular, noise
from ancillary modes leveraged for dissipation engineering does not propagate
at the logical level. In a state-of-the-art experimental setup, we estimate
that the encoded qubit lifetime could extend two orders of magnitude beyond the
break-even point, with substantial margin for improvement through progress in
fabrication and control electronics. Qubit initialization, readout and control
via Clifford gates can be performed while maintaining the code stabilization,
paving the way toward the assembly of GKP qubits in a fault-tolerant quantum
computing architecture. | Lev-Arcady Sellem, Alain Sarlette, Zaki Leghtas, Mazyar Mirrahimi, Pierre Rouchon, Philippe Campagne-Ibarcq | 2023-04-04T00:20:02 | http://arxiv.org/abs/2304.01425v2 | A GKP qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb
###### Abstract
We propose a novel approach to generate, protect and control GKP qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a high impedance circuit mode, autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasi-particle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a state-of-the-art experimental setup, we estimate that the encoded qubit lifetime could extend two orders of magnitude beyond the break-even point, with substantial margin for improvement through progress in fabrication and control electronics. Qubit initialization, readout and control via Clifford gates can be performed while maintaining the code stabilization, paving the way toward the assembly of GKP qubits in a fault-tolerant quantum computing architecture.
###### Contents
* I Introduction
* II The GKP code
* III Protection of GKP qubits by modular dissipation
* III.1 Stabilization of the code manifold
* III.2 Autonomous quantum error correction
* IV Modular Hamiltonian engineering in a Josephson circuit
* V Modular dissipation engineering in a Josephson circuit
* V.1 Modular dissipators from modular interactions
* V.2 Activating modular interactions in the rotating frame
* VI Implementation with state-of-the-art circuits and control electronics
* VI.1 Limited bandwidth and accuracy of the flux bias signal
* VI.2 Fabrication constraints and disorder
* VI.3 \(1/f\) magnetic flux noise
* VI.4 Quasi-particle poisoning
* VII Fault-tolerant Clifford gates
* VII.1 Clifford gates in the finite-energy GKP code
* VII.2 Clifford gates by slow variation of the engineered dissipation parameters
* VII.3 Example circuit for single and two-qubit Clifford gates
* VIII Conclusion and outlook
## I Introduction
Despite considerable progress realized over the past decades in better isolating quantum systems from their fluctuating environment, noise levels in all explored physical platforms remain far too high to run useful quantum algorithms. Quantum error correction (QEC) would overcome this roadblock by encoding a logical qubit in a high-dimensional physical system and correcting noise-induced evolutions before they accumulate and lead to logical flips. In stabilizer codes, such errors are unambiguously revealed by measuring _stabilizer operators_[1], which commute with the logical Pauli operators and thus do not perturb the encoded qubit. A central assumption behind QEC is that a physical system only interacts with its noisy environment via low-weight operators. For instance, in discrete variable codes such as the toric code [2], the surface code [3; 4] or the color code [5], the logical qubit is encoded in a collection of physical two-level systems devoid of many-body interactions. In bosonic codes such as the GKP code [6; 7], the Schrodinger cat code [8; 9] and the binomial code [10; 11], the qubit is encoded in a quantum oscillator whose interactions, denoted here as _low-weight interactions_, involve a small number of photons [12]. Under these assumptions, noise does not directly induce logical flips between well-chosen code states. Specifically, codes are constructed such that several two-level systems should flip in order to induce a logical flip in the former
case, and that a multi-photonic transition should occur in the latter case. Admittedly, logical flips may occur indirectly as low-weight interactions can generate a high-weight evolution operator, but this evolution takes time and is correctable provided that QEC is performed sufficiently fast.
The aforementioned bosonic codes are appealing for their moderate hardware overhead, but a paradox emerges in their operation: some of their stabilizers are high-weight operators that do not appear naturally in the system interactions. A common strategy to measure these stabilizers is to map their value to an ancilla system via an evolution operator generated from a low-weight interaction. It was successfully employed to stabilize cat codes [13], binomial codes [11] and the GKP code [14], but results in the opening of uncorrectable error channels. As illustrated in Fig. 1a in the case of the GKP code, while the interaction is carefully timed so that the overall evolution operator leaves code states unaffected in the absence of noise, ancilla errors during the interaction propagate as uncontrolled long shifts of the target system, triggering logical flips. Partial QEC of the ancilla [15] or error mitigation [16; 17; 18] was proposed to suppress this advert effect, but the robust implementation of these ideas is a major experimental challenge [19]. An alternative strategy, more robust but experimentally more demanding, consists in engineering high-weight interactions so that the target system only interacts with the ancilla via its stabilizer operators. In this configuration, ancilla noise propagates to the target system as an evolution operator generated by the stabilizers only, which leaves the logical qubit unaffected (see Fig. 1b).
Focusing on the GKP code, the two stabilizers are commuting trigonometric functions of the oscillator position and momentum (high-weight operators), which generate discrete translations along a grid in phase-space. The phase of these so-called _modular operators_[20; 21; 22; 23] reveals spurious small shifts of the oscillator state in phase-space while supporting no information on the encoded qubit state. Most proposals [24; 25; 26; 27; 28; 29] and all experimental demonstrations [30; 14; 31] of GKP state preparation and error-correction are based on variants of phase-estimation [32; 33] of the stabilizers. Phase-estimation falls into the first category of stabilizer measurement strategies described above, and therefore leaves the target system open to uncorrectable error channels. In this paper, we consider the second, more robust strategy and aim at engineering high-weight interactions involving only the two modular stabilizers. The state of the oscillator would then only hop along the GKP code lattice in phase-space (see Fig. 1b for schematic hopping along one phase-space quadrature). But how can we engineer a coupling Hamiltonian involving two modular operators?
An isolated Josephson junction behaves as an inductive element whose dynamics is governed by a modular flux operator. However, in most circuitQED experiments [34], the junction is shunted by a low-impedance circuitry, so that it effectively acts on the circuit modes as a weakly non-linear, low-weight, operator. In contrast, connecting the junction to a circuit whose impedance exceeds the quantum of resistance--a regime recently attained in circuitQED--reveals its truly modular nature [35]. Unfortunately, experimental implementations of the dual _coherent phase-slip element_, whose dynamics is governed by a modular charge operator [36] are not yet coherent enough for practical use [37]. Moreover, the doubly modular Hamiltonian implemented by the association of these two elements would only stabilize
Figure 1: **a) Low-weight interactions.**\(\mathbf{H}=-g\mathbf{p}\mathbf{B}\) is an example of low-weight Hamiltonian employed in recent experiments stabilizing the GKP code. It entails a continuous displacement of a GKP state along the \(q\) quadrature of an oscillator (plain black lines, initial state represented by dashed black lines), conditioned on an ancillary mode observable \(B\). The interaction duration \(\delta t\) is chosen such that the state is displaced by one period of the square GKP lattice after the evolution. However, if noise modifies the value of \(B\) during the interaction (red lightning), the final target state is shifted (red lines) and the GKP qubit may be flipped (see Sec. II). **b) High-weight interactions.**\(\mathbf{H}=-g\mathrm{cos}(2\sqrt{\pi}\mathbf{p})\mathbf{B}\) is a high-weight (modular) Hamiltonian that entails a hopping dynamics along the GKP lattice. If noise modifies the value of \(B\) during the interaction, the relative weights of the final state peaks may be affected but not their positions, so that no logical flip may occur.
a single GKP state and not a two-dimensional code manifold [38]. The \(0-\pi\) qubit [39; 40] is an elementary protected circuit that would circumvent these two pitfalls. In this circuit, an effective coherent phase-slip behavior emerges in the low energy dynamics of an ultra-high impedance _fluxonium_ mode [41; 42]. When appropriately coupled to a _transmon_ mode [43], the quasi-degenerate ground manifold is spanned by a pair of two-mode GKP states [44]. However, fully fledged GKP states are only obtained in an extreme parameter regime currently out of reach [40]. Recently, Rymarz _et al._[45] proposed an alternative approach to offset the lack of a phase-slip element. Building on an idea suggested in the original GKP proposal [6], they realized that two Josephson junctions bridged by a high-impedance gyrator would implement a doubly modular Hamiltonian stabilizing quasi-degenerate GKP states. However, existing gyrators are either far too limited in impedance and bandwidth [46; 47; 48] or rely on strong magnetic fields incompatible with superconducting circuits [49].
In this paper, we propose to engineer a true doubly modular Hamiltonian in the rotating frame of a state-of-the-art Josephson circuit. The method, similar to the twirling-based engineering introduced in Ref. [50], is schematically represented in Fig. 2. A Josephson junction allows the coherent tunneling of Cooper pairs across a high-impedance circuit mode, translating its state by \(\pm 2e\) along the charge axis of phase-space. Modulating the tunneling rate with fast pulses, we ensure that such translations occur every quarter period of the target mode only, and let the state rotate freely in phase-space in-between pulses. As a result, the state evolves in discrete steps on a square grid, which matches the GKP code lattice for the proper choice of target mode impedance. We combine this novel approach with dissipation-engineering techniques successfully employed to stabilize Schrodinger cat states [13; 51], so that the target oscillator autonomously stabilizes in the GKP code manifold. Mathematical analysis and numerical simulations show that this strategy can enhance the logical qubit coherence far beyond that of the underlying circuit. Moreover, we describe how to control encoded qubits with fault-tolerant Clifford gates, paving the way toward a high-fidelity quantum computing architecture based on GKP qubits.
The paper is organized as follows. In Sec. II, we review the properties of idealized GKP states and their realistic, finite-energy counterparts. In Sec. III, we propose a dissipative dynamics based on four modular Lindblad operators stabilizing the finite-energy GKP code, and benchmark its error-correction performances against the dominant decoherence channels plaguing superconducting resonators. In Sec. IV, we show how to engineer a doubly modular Hamiltonian in a high-impedance, parametrically driven Josephson circuit. In Sec. V, we combine this method with reservoir engineering techniques to obtain the target modular dissipation. In Sec.VI, we briefly discuss the impact of various noise processes and that of circuit fabrication constraints and disorder. We refer the reader to the Supplemental materials for a more detailed analysis. Finally, in Sec. VII we sketch how to fault-tolerantly control encoded GKP qubits with Clifford gates.
## II The GKP code
GKP introduced coding _grid states_ as superpositions of periodically spaced position states of a quantum oscillator. For simplicity's sake, we consider throughout this paper square grid states--see Supplemental Materials for generalization to hexagonal grid states--defined as
\[\begin{split}&|+Z_{\infty}\rangle=\sum_{n\in\mathbb{Z}}|n\eta \rangle_{q}=\sum_{n\in\mathbb{Z}}|\frac{2\pi n}{\eta}\rangle_{p}\\ &|-Z_{\infty}\rangle=\sum_{n\in\mathbb{Z}}|(n+\tfrac{1}{2})\eta \rangle_{q}=\sum_{n\in\mathbb{Z}}(-1)^{n}|\frac{2\pi n}{\eta}\rangle_{p}\end{split} \tag{1}\]
where \(\eta=2\sqrt{\pi}\) and \(|r\rangle_{q}\) (respectively \(|r\rangle_{p}\)) denotes an eigenstate with eigenvalue \(r\) of the oscillator normalized position \(\mathbf{q}=(\mathbf{a}+\mathbf{a}^{\dagger})/\sqrt{2}\) (respectively momentum \(\mathbf{p}=(\mathbf{a}-\mathbf{a}^{\dagger})/(i\sqrt{2})\)), \(\mathbf{a}\) being the annihilation operator. One can show that any pair of orthogonal logical states have distant support in phase-space, providing the code robustness against position and momentum shift errors. Since the evolution of an oscillator quasi-probability distribution in phase-space is diffusive under the action of noise coupling via low-weight operators [52], this robustness extends to all dominant error channels in superconducting resonators.
Error-syndromes are extracted by measuring the phase of the code stabilizers \(\mathbf{S}_{q}=e^{i\eta\mathbf{q}}\) and \(\mathbf{S}_{p}=e^{-i\eta\mathbf{p}}\), which is 1 inside the code manifold. Given that the logical qubit can be perfectly decoded as long as the oscillator is not shifted by more than \(\sqrt{\pi}/2\), we define _generalized Pauli_ operators \(\mathbf{Z}=\mathrm{Sgn}\big{(}\mathrm{cos}\big{(}\tfrac{\eta}{2}\mathbf{q}\big{)} \big{)}\), \(\mathbf{X}=\mathrm{Sgn}\big{(}\mathrm{cos}\big{(}\tfrac{\eta}{2}\mathbf{p}\big{)} \big{)}\) and \(\mathbf{Y}=i\mathbf{ZX}\). Here, the superoperator \(\mathrm{Sgn}(\cdot)\) denotes the sign of a real-valued operator and is applied to the logical operators introduced by GKP. With our definition, \(\mathbf{X}\), \(\mathbf{Y}\) and \(\mathbf{Z}\) respect the Pauli algebra composition rules throughout the oscillator Hilbert space and coincide with the logical qubit Pauli operators inside the code manifold. Intuitively, the expectation values of these operators are the qubit Bloch sphere coordinates found when decoding the outcomes of homodyne detections, respectively along \(\mathbf{p}\), \((\mathbf{q}+\mathbf{p})\) and \(\mathbf{q}\). The qubit they define can remain pure whilst the oscillator state is not. Moreover, we verify that they commute with the stabilizers, which can thus be measured without perturbing the encoded qubit. More generally, a noisy environment coupling to the oscillator via the stabilizer operators does not induce logical errors: this is the core
idea guiding our approach.
Even though infinitely squeezed grid states are physically unrealistic, GKP suggested that these desirable features would be retained for the normalized, finitely squeezed states \(|\pm Z_{\Delta}\rangle=\mathbf{E}_{\Delta}|\pm Z_{\infty}\rangle\) where \(\mathbf{E}_{\Delta}=e^{-\Delta\mathbf{a}^{\dagger}\mathbf{a}}\) with \(\Delta\ll 1\). Analogously to the infinitely squeezed case, these two states are \(+1\)-eigenstates of the commuting, normalized, stabilizers \(\mathbf{S}_{q}^{\Delta}=\mathbf{E}_{\Delta}\mathbf{S}_{q}\mathbf{E}_{\Delta}^ {-1}\) and \(\mathbf{S}_{p}^{\Delta}=\mathbf{E}_{\Delta}\mathbf{S}_{p}\mathbf{E}_{\Delta}^ {-1}\). However, they are not orthogonal since their wavefunction peaks are Gaussian with a non-zero standard deviation \(\sigma=(\tanh(\Delta))^{\frac{1}{2}}\). Orthogonal, finite-energy logical states can be rigorously defined as their symmetric and antisymmetric superpositions, and Pauli operators for the finite-energy code can be defined therefrom. Nevertheless, in the following, we retain the encoded qubit as defined by the \(\mathbf{X}\), \(\mathbf{Y}\) and \(\mathbf{Z}\) operators. Even though this definition does not allow the preparation of a pure logical state at finite energy, it is operationally relevant as these observables can be measured experimentally. Moreover, the qubit maximum purity is exponentially close to \(1\) as \(\Delta\) approaches \(0\), so that the encoded qubit is well suited for quantum information processing applications for only modest average photon number in the grid states: we find \(1-(\langle\mathbf{X}\rangle^{2}+\langle\mathbf{Y}\rangle^{2}+\langle\mathbf{Z} \rangle^{2})^{1/2}\simeq 2\times 10^{-8}\) for a pure finite-energy code state containing \(\overline{n}=10\) photons.
## III Protection of GKP qubits by modular dissipation
### Stabilization of the code manifold
In Ref. [53], it was shown that a dissipative dynamics based on four Lindblad operators derived from the two finite-energy code stabilizers and their images by a \(\pi\) rotation in phase space stabilizes the code manifold. More precisely, denoting \(\mathcal{D}[\mathbf{L}]\) the dissipator formed from an arbitrary operator \(\mathbf{L}\) and defined by its action on the density matrix \(\mathcal{D}[\mathbf{L}](\mathbf{\rho})=\mathbf{L}\mathbf{\rho}\mathbf{L}^{\dagger}- \frac{1}{2}(\mathbf{L}^{\dagger}\mathbf{L}\mathbf{\rho}+\mathbf{\rho}\mathbf{L}^{ \dagger}\mathbf{L})\), the finite-energy code states are fixed points of the Lindblad equation
\[\frac{\mathrm{d}\mathbf{\rho}}{\mathrm{d}t}=\Gamma\sum_{k=0}^{3}\mathcal{D}[ \mathbf{M}_{k}](\mathbf{\rho}), \tag{2}\]
where \(\mathbf{M}_{k}=\mathbf{R}_{\frac{k_{B}}{2}}(\mathbf{S}_{q}^{\Delta}-\mathbf{1 })\mathbf{R}_{\frac{k_{B}}{2}}^{\dagger}\), \(\mathbf{R}_{\theta}=e^{i\theta\mathbf{a}^{\dagger}\mathbf{a}}\) performs a rotation by \(\theta\) in phase-space and \(\Gamma\) is the dissipation rate. Indeed, the offsets by \(-\mathbf{1}\) ensure that each Lindblad operator cancels on the code manifold. Moreover, any initial state of the oscillator converges exponentially toward the code manifold at a rate set by \(\Gamma\) and \(\Delta\).
Unfortunately, the \(\mathbf{M}_{k}\) operators are products of trigonometric _and_ hyperbolic functions of \(\mathbf{q}\) and \(\mathbf{p}\), which would prove formidably challenging to engineer in an experimental system. Here, we propose to approximate them to first order in \(\Delta\) by products of trigonometric
Figure 2: **Schematic representation of modular dissipation engineering**. A switch controls the coherent tunneling of Cooper pairs (charge \(2e\)) across a Josephson junction placed in parallel of a two-mode circuit. The target mode (top) has a high impedance \(Z\) such that, in normalized phase-space coordinates, tunneling events translate its state by \(\pm 2\sqrt{\pi}\) along the charge axis. The switch is controlled with a train of sharp pulses (duration \(\delta t\)) activating tunneling every quarter of a period \(T\) of the target oscillator. In between pulses, the oscillator state rotates freely in phase-space at \(\omega=2\pi/T\). Overall, the target mode dynamics is generated by discrete shifts along a square grid matching the GKP lattice (gray grid with period \(2\sqrt{\pi}\) overlaid with Wigner diagrams of the oscillator state). A lower impedance ancillary mode (bottom), also driven by Cooper pair tunneling, dissipates excitations into a cold load (purple wriggled arrow) to ensure that the target mode dynamics is irreversible, autonomously stabilizing the GKP code.
and linear functions of \(\mathbf{q}\) and \(\mathbf{p}\) with the operators
\[\mathbf{L}_{k}=\mathcal{A}\mathbf{R}_{\frac{k_{\pi}}{2}}e^{i\eta\mathbf{q}}( \mathbf{1}-\epsilon\mathbf{p})\mathbf{R}_{\frac{k_{\pi}}{2}}^{\dagger}- \mathbf{1}, \tag{3}\]
where \(\epsilon=\eta\)\(\sinh(\Delta)\) is a small parameter and the scalar factor \(\mathcal{A}=e^{-\eta\epsilon/2}\) originates from the non commutativity of \(\mathbf{q}\) and \(\mathbf{p}\) in the Baker-Campbell-Hausdorff formula.
In order to qualitatively apprehend the dynamics entailed by these modular Lindblad operators, we represent in Fig. 3 the evolution of a displaced code state \(\mathbf{\rho}_{\alpha+i\beta}=e^{-i\alpha\mathbf{p}+i\beta\mathbf{q}}|+Z_{\Delta }\rangle\langle+Z_{\Delta}|e^{+i\alpha\mathbf{p}-i\beta\mathbf{q}}\) over an infinitesimal time step \(\mathrm{d}t\ll 1/\Gamma\). On the top panel, arrows represent the variation of the state center of mass (vector complex coordinates proportional to \(\mathrm{d}\Gamma\mathrm{r}(\mathbf{a}\ \mathbf{\rho}_{\alpha+i\beta})\)). A single attractor at the origin of phase-space pins the grid state normalizing envelope. On the bottom panel, arrows represent the variation of the state position and momentum modulo \(2\pi/\eta\) (vector complex coordinates proportional to \(\mathrm{d}\Gamma\mathrm{r}(\mathrm{Arg}[\mathbf{S}_{\mathbf{q}}\mathbf{\rho}_{\alpha+ i\beta}]+i\mathrm{Arg}[\mathbf{S}_{\mathbf{p}}\mathbf{\rho}_{\alpha+i\beta}])\)). Multiple attractors appear for \(\alpha\), \(\beta=0\) mod \(2\pi/\eta\) pinning the grid peaks onto the GKP code lattice. Note that here, we employ the displaced grid state \(\mathbf{\rho}_{\alpha+i\beta}\) as a sensitive position and momentum shift detector [54], but initializing the oscillator in a less exotic state such as a coherent state centered in \(\alpha,\beta\) yields similar phase portraits, albeit smoothed by the state quadrature fluctuations. These observations hint at a convergent dynamics toward the finite-energy code manifold, irrespective of the oscillator initial state. This contrasts with the Lindblad dynamics based on only two modular dissipators introduced in Ref. [29], for which we observe dynamical instabilities [55].
Quantitatively, we show that, under this four-dissipator dynamics, the expectation values of the infinite-energy code stabilizers converge to their steady state value at a rate \(\Gamma_{c}\gtrsim\mathcal{A}\epsilon\eta\Gamma\) and that the oscillator energy remains bounded [55], proving that the dynamics is indeed stable. Note that, due to the linear approximation of hyperbolic functions we made to obtain the operators (3), the state reached by the oscillator after a few \(1/\Gamma_{c}\) does not strictly belong to the code manifold, but consists in a statistical mixture of shifted code states. In terms of phase-space quasiprobability distribution, this results in broader peaks for the stabilized grid states. Yet, the overlap of a peak with its neighbors remains exponentially small as \(\epsilon\) decreases, so that high-purity encoded states can still be prepared, and population leakage between two orthogonal logical states occurs on a timescale much longer than \(1/\Gamma_{c}\). Quantitatively, we show that when \(\epsilon\ll 1\), the generalized Pauli operators \(\mathbf{X}\) and \(\mathbf{Z}\) decay at a rate \(\Gamma_{L}^{0}=\frac{4}{\pi}\mathcal{A}\epsilon\eta\Gamma e^{-\frac{4}{4 \epsilon\eta}}\), while \(\mathbf{Y}\) decays twice faster, as expected for the square GKP code.
### Autonomous quantum error correction
Given that the confinement strength onto the code manifold \(\Gamma_{c}\) and the residual error rate \(\Gamma_{L}^{0}\) both depend on \(\epsilon\), this value needs to be optimized when correcting for errors induced by intrinsic noise channels. Indeed, \(\epsilon\) should not be too small for the modular dissipation to cancel efficiently the stochastic shifts induced by a low-weight noise process, but not too large for the grid state peaks to be well resolved. In the case of quadrature noise entering the Lindblad dynamics as two spurious dissipators \(\mathcal{D}[\sqrt{\kappa}\mathbf{q}]\) and \(\mathcal{D}[\sqrt{\kappa}\mathbf{p}]\), we show that, in the limit of weak intrinsic dissipation \(\kappa\ll\Gamma_{c}\), the decay rate of the generalized Pauli operators \(\mathbf{X}\) and \(\mathbf{Z}\) reads \(\Gamma_{L}=\frac{4}{\pi}\mathcal{A}\epsilon\eta\Gamma e^{-(\mathcal{A} \epsilon\eta)}\), where \(\tilde{\epsilon}=\epsilon+\frac{\kappa}{2\mathcal{A}^{2}\epsilon\Gamma}\)[55]. The minimum flip rate is obtained for \(\epsilon\simeq(\frac{\kappa}{2\mathcal{A}^{2}\Gamma})^{\frac{1}{2}}\) and reads \(\Gamma_{L}\simeq\frac{4\eta}{\pi}(\frac{\kappa\Gamma}{2})^{\frac{1}{2}}e^{-( \frac{8\Gamma}{q^{2}\pi})^{\frac{1}{2}}}\). This exponential scaling ensures that logical errors can be heavily suppressed for a modest ratio \(\Gamma/\kappa\), as illustrated by Fig. 4a. There, we represent the decay rate of the generalized Pauli operators \(\mathbf{X}\) and \(\mathbf{Z}\) extracted by spectral analysis of the Lindblad superoperator (dashed lines), in quantitative
Figure 3: **Modular dissipation phase portraits**. For a finite-energy code state (\(\sinh(\Delta)=0.2/\eta\)) displaced by \(\alpha+i\beta\) in phase-space, arrows encode the evolution of the state center of mass (top panel) and modular coordinates (bottom panel) entailed by the Lindblad operators (3) over a short time step \(\mathrm{d}t\ll 1/\Gamma\). Arrows length are rescaled to arbitrary units.
agreement with a full Lindblad master equation simulation (dots). The latter is computationally much more costly but proves necessary to investigate more realistic noise models for which no simulation shortcut was found. In particular, we verify numerically that errors entailed by single-photon dissipation, pure dephasing and a Kerr Hamiltonian perturbation all appear to be exponentially suppressed when increasing the modular dissipation rate (see Fig. 4b-d). The logical error rates induced by the two latter processes--entering the Lindblad equation via fourth order polynomials in \(\mathbf{q}\) and \(\mathbf{p}\)--are qualitatively captured by a mean-field approximation which boils down to quadrature noise scaled up by the grid states mean photon number \(\overline{n}=\eta/(2\epsilon)\) (dashed gray lines in Fig. 4c-d). These numerical considerations support the intuition that modular dissipation can suppress errors induced by arbitrary finite-weight noise channels, albeit with degraded performances when considering higher-weight processes. In the limit of infinite-weight noise processes, _i.e._ modular noise channels, errors are not corrected.
## IV Modular Hamiltonian engineering in a Josephson circuit
For the sake of pedagogy, we now describe a control method to engineer a Hamiltonian involving the two modular stabilizers in a simple superconducting circuit. The key ideas of the protocol for modular dissipation engineering described in Sec. V are already present in this toy example. The goal here is to synthesize the GKP Hamiltonian
\[\mathbf{H}_{\mathrm{GKP}}=-E\bigl{(}\cos(\eta\mathbf{q})+\cos(\eta\mathbf{p}) \bigr{)}, \tag{4}\]
in the rotating frame of a superconducting resonator. This Hamiltonian has a degenerate ground state corresponding to the two infinite-energy GKP states \(|\pm Z_{\infty}\rangle\).
We consider the circuit pictured in Fig. 5a. The inductor and capacitor form a quantum oscillator whose conjugate variables are the flux threading the inductor \(\Phi\) and the charge on the capacitor \(Q\). The corresponding operators can be reduced as \(\mathbf{q}_{0}=\frac{1}{\sqrt{\hbar\mathbf{Z}}}\mathbf{\Phi}\) and \(\mathbf{p}_{0}=\sqrt{\frac{Z}{\hbar}}\mathbf{Q}\), where \(Z=\sqrt{L/C}\) is the circuit impedance, so as to verify \([\mathbf{q}_{0},\mathbf{p}_{0}]=i\) and to display equal fluctuations in the vacuum state. The \(LC\) oscillator is placed in parallel of a ring made of two Josephson junctions
Figure 4: **GKP qubit protection by modular dissipation**. The decay rate \(\Gamma_{L}\) of the Pauli operators \(\mathbf{Z}\) and \(\mathbf{X}\) is extracted from numerical simulations (dots) when varying the strength of some intrinsic noise channel relative to the modular dissipation rate \(\Gamma\). For all low-weight noise channels considered, errors appear to be exponentially suppressed in the weak noise limit. **a)** Quadrature noise modeled by two Lindblad operators \(\sqrt{\kappa\mathbf{q}}\) and \(\sqrt{\kappa}\mathbf{p}\). Dashed lines are predictions by spectral analysis of the Lindblad superoperator [55]. **b)** Single-photon dissipation modeled by a Lindblad operator \(\sqrt{\kappa_{1\mathrm{ph}}}\mathbf{a}\). **c)** Pure dephasing modeled by a Lindblad operator \(\sqrt{\kappa_{\phi}}\mathbf{a}^{\dagger}\mathbf{a}\). **d)** Kerr Hamiltonian perturbation of the form \(\frac{K}{2}(\mathbf{a}^{\dagger}\mathbf{a})^{2}\). For (c-d), note the rescaling of the x-axis by \(\eta/\epsilon=2\overline{n}\). For (b-d), dashed gray lines reproduce the dashed colored lines in (a), un-rescaled, for comparison.
with equal energy \(E_{J}\). We apply two magnetic fluxes \(\Phi_{J}^{\rm ext}=\varphi_{0}(\pi-2{\rm Arcsin}(\xi(t)))\) and \(\Phi_{L}^{\rm ext}=-\Phi_{J}^{\rm ext}/2\), where \(\varphi_{0}=\hbar/(2e)\) is the reduced flux quantum and \(\xi\) is an AC bias signal, respectively through the Josephson ring loop and the loop formed with the inductor. In presence of these flux biases, the Josephson ring behaves as a single junction with time-varying energy and null tunneling phase [51], acting on the \(LC\) resonator via the Hamiltonian
\[{\bf H}_{J}(t)=-2E_{J}\xi(t){\rm cos}({\bf\Phi}/\varphi_{0}). \tag{5}\]
Designing the circuit to have an impedance \(Z=2R_{Q}\), where \(R_{Q}=\frac{h}{4e^{2}}\simeq 6.5\) k\(\Omega\) is the resistance quantum, the circuit Hamiltonian in reduced coordinates reads
\[{\bf H}_{0}(t)=\frac{\hbar\omega}{2}({\bf q}_{0}^{2}+{\bf p}_{0}^{2})-2E_{J} \xi(t){\rm cos}(\eta{\bf q}_{0}), \tag{6}\]
where \(\omega=1/\sqrt{LC}\). We now place ourselves in the interaction picture to cancel out the dynamics of the linear part of the circuit. In the \((q,p)\) frame rotating at \(\omega\), the sole remaining dynamics is governed by the Josephson term, a modular function of the now rotating quadrature operator \({\bf q}_{0}(t)={\rm cos}(\omega t){\bf q}+{\rm sin}(\omega t){\bf p}\). This operator aligns with \({\bf q}\) or \({\bf p}\) every quarter period of the oscillator (see Fig. 5b). The idea is to bias the Josephson ring with a train of short flux pulses in order to activate Josephson tunneling at these precise instants only (see Fig. 5c). Letting \(\xi(t)\simeq\xi_{1}{\rm III}_{\frac{\pi}{2\omega}}(t)\) where \(\xi_{1}\) is the integrated amplitude of each pulse and \({\rm III}_{T}\) denotes a Dirac comb of period \(T\), in the Rotating Wave Approximation (RWA), we obtain the effective Hamiltonian
\[\begin{split}{\bf H}_{\rm RWA}&=-2E_{J}\overline{ \xi(t){\rm cos}(\eta{\bf q}_{0}(t))}\\ &=-E\big{(}{\rm cos}(\eta{\bf q})+{\rm cos}(\eta{\bf p})\big{)} \\ &={\bf H}_{\rm GKP}\end{split} \tag{7}\]
with \(E=\frac{2E_{J}\omega}{\pi}\xi_{1}\). It is straightforward to combine this doubly modular Hamiltonian with a small quadratic potential \(\frac{\hbar\delta}{2}({\bf q}^{2}+{\bf p}^{2})\) with \(\hbar\delta\ll E\) in order to get finite-energy GKP states as quasi-degenerate ground states [45]. Indeed, such a weakly confining potential is simply obtained by increasing the duration between the pulses of the bias train to \(\frac{\pi}{2(\omega-\delta)}\).
Here, we stress that we described this method as an example of modular dynamics engineering only. It does not provide a protected qubit _per se_ as would a circuit implementing the same Hamiltonian in the laboratory frame [45; 6]. Indeed, the GKP code states are not stable upon loss of a photon. For a system directly governed by the static Hamiltonian \({\bf H}_{\rm GKP}\) and prepared in the ground manifold, photon emission into a cold bath would violate energy conservation and photon loss thus does not occur. This argument does not hold when \({\bf H}_{\rm GKP}\) is engineered in the rotating frame from a time-dependent Hamiltonian. In that case, photon emission into the environment can occur even at zero temperature, pulling the oscillator state out of the ground manifold of \({\bf H}_{\rm GKP}\). Stabilization of the GKP code manifold could still be achieved by coupling the circuit to a _colored_ bath engineered to enforce energy relaxation in the rotating frame [56].
## V Modular Dissipation Engineering in a Josephson Circuit
### Modular dissipators from modular interactions
Armed with the previous example, we now turn to engineering the modular dissipative dynamics described in Sec. III. First, we note that the Lindblad operators (3) can be substituted with the following linear combinations
\[\begin{split}{\bf L}_{q,s}&=({\bf L}_{0}+{\bf L}_{ 2})/\sqrt{2}\\ {\bf L}_{q,d}&=({\bf L}_{0}-{\bf L}_{2})/(\sqrt{2}i) \\ {\bf L}_{p,s}&=({\bf L}_{1}+{\bf L}_{3})/\sqrt{2}\\ {\bf L}_{p,d}&=({\bf L}_{1}-{\bf L}_{3})/(\sqrt{2}i) \end{split} \tag{8}\]
Second, following a standard procedure [55], each Lindblad operator \({\bf L}_{r,l}\) with \(r=q\) or \(p\), \(l=s\) or \(d\) is obtained by coupling the target mode \(a\) to an ancillary mode \(b\), damped at rate \(\kappa_{b}\), via an interaction Hamiltonian
\[{\bf H}_{k,l}^{\rm int}=\hbar g{\bf L}_{r,l}{\bf b}^{\dagger}+h.c. \tag{9}\]
Indeed, adiabatically eliminating the mode \(b\) in the limit \(g\ll\kappa_{b}\), the two-mode dynamics reduces to a single-mode dissipative dynamics with the desired Lindblad operator \({\bf L}_{r,l}\), at a rate \(\Gamma=4g^{2}/\kappa_{b}\). Third, we define rotated quadrature operators of the target and ancillary modes \({\bf r}_{a}^{\pm}=e^{\pm i\frac{\pi}{4}\pi^{\rm a}{\bf a}^{\dagger}{\bf a}}\)\({\bf r}_{a}^{0}\)\(e^{\mp i\frac{\pi}{4}\pi^{\rm a}{\bf a}^{\dagger}{\bf a}^{\dagger}{\bf a}}\) and \({\bf p}_{b}^{\pm}=e^{\pm i\frac{\pi}{4}{\bf b}^{\dagger}{\bf b}}\)\({\bf p}_{b}\)\(e^{\mp i\frac{\pi}{4}\pi^{\rm b}{\bf b}^{\dagger}{\bf b}}\), where \({\bf r}_{a}^{0}={\bf q}_{a}\) for \(r=q\) (respectively \({\bf r}_{a}^{0}={\bf p}_{a}\) for \(r=p\)), and we remark that the Hamiltonian (9) is approximated by
\[\begin{split}{\bf H}_{r,l}^{\rm int}\simeq& 2\hbar g\ \big{(}&\quad{\cal A}\,\,{\rm cos}(\eta{\bf r}_{a}^{0}-\delta_{l} \frac{\pi}{2}){\bf q}_{b}\\ &+{\cal A}\,\,{\rm cos}(\eta{\bf r}_{a}^{+}-\delta_{l}\frac{\pi}{ 2}){\bf p}_{b}^{+}\\ &-{\cal A}\,\,{\rm cos}(\eta{\bf r}_{a}^{-}-\delta_{l}\frac{\pi}{ 2}){\bf p}_{b}^{-}\\ &-\,\,(1-\delta_{l}){\bf q}_{b}\,\,\big{)}\end{split} \tag{10}\]
at first order in \(\epsilon\)[57], with the convention \(\delta_{s}=0\), \(\delta_{d}=1\). The first three terms in this Hamiltonian have the same form and can be activated in the rotating frame of a two-mode Josephson circuit as described in the next section. The fourth term is trivially implemented by driving the ancillary mode resonantly.
Note that activating simultaneously four Lindblad operators necessitates four distinct ancillary modes. A hardware-efficient alternative consists in activating them sequentially, leveraging a single ancillary mode and switching from one operator to the next at a
rate slower than \(\kappa_{b}\)--giving the ancillary mode sufficient time to reach its steady state and justifying its adiabatic elimination--but faster than \(\Gamma\)--accurately reproducing the target four-dissipator dynamics by Trotter decomposition. This strategy drastically reduces the experimental software complexity, at the cost of a fourfold reduction of the modular dissipation rate \(\Gamma\). With these considerations in mind, we now focus on the activation of a single Lindblad operator and assume that the full target dynamics is easily derived thereof.
### Activating modular interactions in the rotating frame
The method and circuit to activate arbitrary modular interactions is analogous to the GKP Hamiltonian engineering technique described in Sec. IV. Here, we consider the multimode circuit pictured in Fig. 6a. The Josephson ring is shunted by the target resonator with impedance \(Z_{a}=2R_{Q}\) placed in series with a low-impedance dissipative ancillary mode \(b\) (\(Z_{b}\ll R_{Q}\), \(\kappa_{b}\sim\omega_{b}Z_{b}/R_{b}\)). Note that this circuit should not necessarily represent a physical device: suffices it to represent the Foster decomposition [58; 59; 60] of a linear environment connected to the two ports of the Josephson ring.
Compared to Sec. IV, the DC flux bias point is modified following
\[\begin{split}\Phi_{J}^{\mathrm{ext}}&=\varphi_{0} \big{(}\pi+2\mathrm{Arcsin}(\xi(t))\big{)}\\ \Phi_{L}^{\mathrm{ext}}&=-\frac{\Phi_{J}^{\mathrm{ ext}}}{2}+\varphi_{0}\frac{\pi}{4}\end{split} \tag{11}\]
in order to give a non-trivial phase to the Josephson tunneling [51]. The circuit Hamiltonian then reads
\[\mathbf{H}_{0}(t)=\hbar\omega_{a}\mathbf{a}^{\dagger}\mathbf{a}+\hbar\omega_{ b}\mathbf{b}^{\dagger}\mathbf{b}+2E_{J}\xi(t)\,\cos\big{(}\frac{\mathbf{\Phi}}{ \varphi_{0}}-\frac{\pi}{4}\big{)} \tag{12}\]
where the generalized phase operator across the series of resonators reads \(\mathbf{\Phi}=\varphi_{0}(\eta_{a}\mathbf{q}_{a}^{0}+\eta_{b}\mathbf{q}_{b}^{ 0})\), and the vacuum phase fluctuations of each mode across the Josephson ring are given by \(\eta_{a}=\sqrt{2\pi Z_{a}/R_{Q}}=2\sqrt{\pi}\) and \(\eta_{b}=\sqrt{2\pi Z_{b}/R_{Q}}\ll 1\). Importantly, these values do not need to be fine-tuned in circuit fabrication as one can adapt the system controls to accommodate a value of \(\eta_{a}\) exceeding \(2\sqrt{\pi}\) (see Sec. VI and Supplemental materials).
Placing ourselves in the rotating frame of both \(a\) and \(b\), the Hamiltonian becomes
\[\mathbf{H}(t)=2E_{J}\xi(t)\,\cos\big{(}\eta_{a}\mathbf{q}_{a}^{0}(t)+\eta_{b} \mathbf{q}_{b}^{0}(t)-\pi/4\big{)} \tag{13}\]
where the quadrature operators \(\mathbf{q}_{a}^{0}\) and \(\mathbf{q}_{b}^{0}\) rotate in phase-space. We now consider the AC bias signal
\[\begin{split}\xi(t)&=\sum_{j=0,+,-}\xi^{j}(t)\\ &\simeq\sum_{j=0,+,-}s_{j}\xi_{1}\big{(}\mathrm{III}_{\frac{2\pi} {\omega_{a}}}(t-t_{j,r})\ +\ \delta_{l}\mathrm{III}_{\frac{2\pi}{\omega_{a}}}(t-t_{j,r}-\frac{\pi}{\omega_{a} })\\ &\qquad\qquad\qquad\qquad\times\cos(\omega_{b}t\ -\ \theta_{j})\end{split} \tag{14}\]
consisting in three trains of short pulses with integrated amplitude \(\xi_{1}\) and sign \(s_{j}\) (\(s_{j}=+1\) for \(j=0\) or \(j=+\), \(s_{j}=-1\) for \(j=-\)) modulating carriers at \(\omega_{b}\), the pulses within each train being separated by half a period of the target resonator and having either constant (\(\delta_{l}=1\)) or alternating signs (\(\delta_{l}=-1\)). Each train is offset by \(t_{j,r}\), defined as \(t_{j,q}=\frac{j\epsilon}{2\eta_{\omega_{a}}}\) and \(t_{j,p}=t_{j,q}+\frac{\pi}{2\omega_{a}}\), and is responsible for the activation of one of the modular interactions in the target Hamiltonian (10) by triggering Josephson tunneling when the rotating operator \(\mathbf{q}_{a}^{0}(t)\) aligns or anti-aligns with the corresponding target mode quadrature \(\mathbf{r}_{a}^{j}\). If the pulses are sufficiently narrow (see Sec. VI for details) and assuming that \(\omega_{a}\) and \(\omega_{b}\) are not commensurable, the RWA yields only terms of the form \(\cos(\pm\eta_{a}\mathbf{r}_{a}^{j})(e^{-i\theta_{j}}\mathbf{b}+e^{i\theta_{j}} \mathbf{b}^{\dagger})\) and
Figure 6: **Engineering modular interactions.****a)** A Josephson ring is placed in parallel of a high-impedance target resonator (green) and a low-impedance, dissipative, ancillary resonator (black). The Josephson tunneling amplitude \(2E_{J}\xi(t)\) and phase are adjusted with the control fluxes \(\Phi_{J,L}^{\mathrm{ext}}\) biasing the circuit. **b)** Each modular Lindblad operator (8) is activated with an AC bias signal consisting in the sum of three pulse trains \(\xi=\xi^{-}+\xi^{0}+\xi^{+}\) (trains respectively colored in purple, black, and orange), pulses within each train being separated by half a period of the target resonator. Each train modulates a carrier at \(\omega_{b}\), the three carriers being phase-shifted by \(\sim\pm\frac{\pi}{2}\) from one another (dashed lines with same colors as the pulse trains). **c)** In frequency domain, the bias signal \(\mathcal{F}[\xi](\omega)=\sum_{k\in\mathbb{Z}}\tilde{\xi}(k)(\delta(\omega- \omega_{b}-k\omega_{a})+\delta(\omega+\omega_{b}+k\omega_{a}))\) is a real-valued frequency comb centered at \(\pm\omega_{b}\) and whose amplitude \(\tilde{\xi}(k)\) oscillates with a period \(\frac{4\pi\pi}{\varepsilon}\). The signal represented in **b-c** corresponds to the activation of \(\mathbf{L}_{q,s}\) and contains only even harmonics \(k\in 2\mathbb{Z}\). For readability, we set \(\epsilon=1\) and \(\omega_{b}=2.3\ \omega_{a}\) in **b** (respectively \(\omega_{b}/\omega_{a}\rightarrow\infty\) in **c** so that the comb does not overlap with its mirror image centered at \(-\omega_{b}\)), which is not the regime in Table 1.
\(\sin(\pm\eta_{a}\mathbf{r}_{a}^{j})(e^{-i\theta_{j}}\mathbf{b}+e^{+i\theta_{j}} \mathbf{b}^{\dagger})\)[61]. Finally, choosing pulses with constant (\(\delta_{l}=1\)) or alternating (\(\delta_{l}=-1\)) signs ensures that only cosine _or_ sine operators survive the RWA--depending on which Lindblad operator is targeted--and the carrier phases \(\theta_{j}=j(j\frac{\pi}{2}+\frac{\pi n}{4})\) are chosen to match the phases of each ancillary mode operator in the target Hamiltonian. In Fig. 6b-c, we represent the total bias signal when activating \(\mathbf{L}_{q,s}\). In time domain, it consists in a train of pulse triplets, while in frequency domain, it is a frequency comb centered at \(\pm\omega_{b}\) (mirror image around \(-\omega_{b}\) not shown) and whose amplitude oscillates with a period \(\frac{4\pi n}{e}\omega_{a}\). The signals activating other Lindblad operators are obtained by alternating the pulses sign in time domain and/or alternating the harmonics sign in frequency domain.
Overall, the target Hamiltonian (10) is activated at a rate \(g=E_{J}\eta_{b}\omega_{a}\xi_{1}/(2\sqrt{2}\pi\hbar\mathcal{A})\). Note that to reach this effective Hamiltonian, we performed an adiabatic elimination of the ancillary mode--requiring \(g\ll\kappa_{b}\)--and a RWA--requiring \(\kappa_{b}\ll\omega_{a}\)[55]. Moreover, we choose \(\omega_{a}\ll\omega_{b}\) to avoid frequency collisions that would enable high-order processes involving multiple photons of the ancilla in the RWA. Given that protection of the logical qubit requires the modular dissipation rate \(\Gamma\sim\frac{g^{2}}{\kappa_{b}}\) to be larger than the target resonator photon loss rate \(\kappa_{a}\), the system parameters should respect
\[\kappa_{a}\ll g^{2}/\kappa_{b}\ll\kappa_{b}\ll\omega_{a}\ll\omega_{b}. \tag{15}\]
This regime is attainable in a state-of-the-art circuit (see Tab. 1) comprising a high-impedance mode resonating in the 100 MHz range. This unusually low resonance frequency is needed to respect the above hierarchy, and to ensure that flux bias pulses are sufficiently short with respect to the target oscillator period, as detailed in the next section.
## VI Implementation with state-of-the-art circuits and control electronics
The goal of this section is to propose realistic experimental parameters for the stabilization of GKP qubits and to estimate the impact of various experimental imperfections. We first remind the reader that the impact of intrinsic, low-weight noise processes affecting the target resonator was analyzed in Sec. III and shown to be robustly suppressed by the modular dissipation. Here, we consider the noise sources induced by the dissipation engineering itself.
### Limited bandwidth and accuracy of the flux bias signal
A central hypothesis to the dissipation engineering technique detailed in Sec. III is that the width of the flux pulses that bias the circuit is negligible with respect to the target oscillator period. In frequency domain, this figure of merit directly relates to the number of harmonics \(N\) in the frequency comb forming the bias signal \(\xi\) (see Fig. 6c). This number should be quantitatively optimized: on the one hand, it should not be too small for the aforementioned hypothesis to hold, but picking an unnecessarily large \(N\) would place prohibitive constraints on the circuit design--for a fixed control signal bandwidth, one can only increase \(N\) by decreasing the target mode resonance frequency--and limit the modular dissipation rate for a given maximum value of the bias signal \(\xi_{\text{max}}\)[62].
To this end, we perform numerical simulations, in the RWA [55], considering Lindblad operators activated by a bias signal \(\tilde{\xi}_{N}(k)\) obtained by truncating the Fourier series \(\tilde{\xi}(k)\) (setting \(\tilde{\xi}(k)=0\) for \(|k|>N\), see Fig. 6c for a representation of \(\tilde{\xi}(k)\)). The evolution of the target oscillator state is computed for the corresponding imperfect modular dissipation in absence of any other decoherence channel. The decay rate of the generalized Pauli operators \(\mathbf{X}\) and \(\mathbf{Z}\) is extracted for each value of \(N\), and represented in Fig. 7. Truncation of the bias comb leads to spurious logical flips at a rate independent of \(\epsilon\) and exponentially decreasing with \(N\). In the long term, this scaling is encouraging as one does not need to increase the control signal bandwidth indefinitely to robustly protect the encoded information. In the short term, combs containing \(N\sim 100\) harmonics are needed
Figure 7: **Truncated frequency comb.** Decay rate of the generalized Pauli operators \(\mathbf{X}\) and \(\mathbf{Z}\) under modular dissipation engineered with a frequency comb containing a finite number of harmonics \(N\) (dots), in absence of intrinsic noise of the target resonator. A finite bandwidth signal yields spurious errors at a rate decreasing exponentially with \(N\). The decay rate found for \(N\rightarrow\infty\) does not quantitatively match the rate found for the ideal dissipators (3) (dashed lines), which is yet to be understood.
to suppress the logical error rate significantly beyond the break-even point (see Tab. 1). Limiting microwave drives to the 0-20 GHz range, which corresponds to the bandwidth of standard laboratory equipment and is below the typical plasma frequency of Josephson junctions [63], this places the target mode resonance frequency in the sub-GHz range (see Tab. 1).
Delivering a precise, wideband, microwave signal to a superconducting circuit cooled down in the quantum regime is a major experimental challenge. If this signal is generated at room temperature, one needs to account for _a priori_ unknown dispersion of the feedlines. Therefore, the complex amplitudes of \(2N+1\) phase-locked, monochromatic microwave signals need to be individually calibrated (see [55] for quantitative estimates of the impact of miscalibration). Recent advances in digital synthesis of microwave signals allows for the automation of these calibrations. An alternative strategy consists in generating the frequency comb directly on-chip with a dedicated Josephson circuit [64; 65] in order to deliver a precise, wideband comb with no need for complex calibrations.
### Fabrication constraints and disorder
Inaccuracy on the energy of Josephson junctions is the main source of disorder in superconducting circuits, with a typical mismatch of the order of a few percents from the targeted value to the one obtained in fabrication. In the circuit depicted in Fig. 6a, this leads to uncertainty on the value of the _superinductance_\(L_{a}\), typically implemented by a chain of Josephson junctions [66], and to a small energy mismatch between the two ring junctions. Fortunately, these parameters do not need to be fine-tuned in our approach.
Indeed, an inductance \(L_{a}\) differing from its nominal value only results in a modified target mode impedance \(Z_{a}\), and therefrom in modified phase fluctuations across the Josephson ring \(\eta_{a}=\sqrt{2\pi Z_{a}/R_{Q}}\). Here we remind the reader that the target value \(\eta_{a}=2\sqrt{\pi}\) was chosen to match the length of the square GKP lattice unit cell. However, as detailed in Sec. VII, there exists a continuous family of GKP codes whose diamond-shaped unit cells still have an area of \(4\pi\), but longer edges. As long as \(\eta_{a}>2\sqrt{\pi}\), one simply adjusts the timing of flux bias pulses to stabilize such a non-square code. We verify in simulation that the accuracy with which this adjustment needs to be performed is well within reach of current experimental setups [55].
We now consider the effect of a small asymmetry of the circuit Josephson ring. We remind the reader that in our dissipation engineering scheme, the effective Josephson energy is cancelled by threading the ring with half a quantum of magnetic flux--corresponding to the DC contribution in \(\Phi_{J}^{\text{ext}}\)--except at precise instants when it is activated with sharp flux pulses--corresponding to the AC contribution in \(\Phi_{J}^{\text{ext}}\). Mismatch between the two junction energies lead to imperfect cancellation in-between pulses, potentially generating shifts of the target oscillator state by \(\eta_{a}\) along a random axis in phase-space (see Fig. 2). As detailed in Supplemental materials, this advert affect can be mitigated by slightly adjusting the circuit DC bias point so that the imperfectly cancelled Josephson Hamiltonian becomes non-resonant and drops out in the RWA. This RWA is only valid if the energy mismatch between junctions is much smaller than the target mode frequency \(\omega_{a}\), placing a new constraint on the circuit parameters. In Tab. 1, we choose a Josephson energy as low as \(E_{J}=h\times 500\) MHz--which we still consider experimentally realistic while keeping the junctions plasma frequency above 20 GHz [55]--such that a 2% mismatch should be tolerable. We leave quantitative analysis of the robustness of this strategy for future work and note that it may be combined with the method sketched in the next section for a more robust suppression of the impact of imperfectly cancelled Josephson energy.
### \(1/f\) magnetic flux noise
While its microscopic origin is still debated, low-frequency magnetic flux noise (referred to as \(1/f\) noise) is ubiquitous in superconducting circuits [67]. In practice, such noise will induce slow drifts in the DC bias point of our proposed circuit, which cannot be detected and compensated on short (\(\sim 1\) ms) timescales. A small offset to the magnetic flux \(\Phi_{L}^{\text{ext}}\) threading the rightmost loop of the circuit (see Fig. 6 and Eq. 11) is not expected to affect significantly the performances of our protocol. Indeed, it only impacts the phase of the Josephson term in (13), slightly unbalancing the rates of the engineered modular dissipators (8). On the other hand, an offset to the magnetic flux threading the Josephson ring \(\Phi_{J}^{\text{ext}}\) results in an imperfectly cancelled Josephson energy in between fast bias pulses, similar to that induced by a mismatch of the two junctions energy. Unfortunately, here, adapting the circuit bias to make the spurious Josephson Hamiltonian non-resonant is not an option as the magnetic flux offset is unknown. Quantifying the impact of imperfect cancellation of the Josephson energy on the lifetime of GKP qubits will be the subject of future work.
A possible strategy--not investigated in this work--to mitigate the impact of such imperfect cancellation is to dynamically vary the target mode phase fluctuations across the ring \(\eta_{a}\) with a periodic window signal such that \(\eta_{a}=2\sqrt{\pi}\) only an narrow windows covering the triplets of pulses forming the bias signal \(\xi(t)\) (pulses represented in Fig. 6b). If \(\eta_{a}\ll 2\sqrt{\pi}\) outside these windows, the imperfectly cancelled Josephson Hamiltonian only generates short displacements of the oscillator, which are cor
rected by the modular dissipation. Note that, although requiring a more complex circuit, controlling the value of \(\eta_{a}\) is anyhow needed in order to perform protected gates on GKP qubits (see Fig. 9 for an example circuit allowing such control).
### Quasi-particle poisoning
Quasi-particles are excitations of the circuit electron fluid above the superconducting gap [68]. The probability for such excitations should be negligible at the working temperature of circuit QED experiments (10 mK), but normalized densities of quasi-particles in the range \(x_{qp}\sim 10^{-5}-10^{-7}\) are typically observed. A quasi-particle with charge \(e\) tunneling through the Josephson ring is expected to translate the target mode by \(\pm\sqrt{\pi}\) in normalized units, which can directly lead to a logical flip. In term, this uncorrected error channel could limit the coherence time of the logical qubit. Quantitative estimates of the logical error rate induced by a given density of quasi-particles will be sought in a future work. Note that quasi-particle poisoning is detrimental to all circuitQED architectures, and is thus actively investigated. Recent progress in identifying and suppressing sources of out-of-equilibrium quasi-particles [69; 70; 71; 72], as well as in trapping and annihilating them [73; 74; 75; 76; 77; 78; 79] could conceivably lead to efficient suppression strategies in the near future.
## VII Fault-tolerant Clifford gates
Following the definition given by GKP [6], we define as _fault-tolerant_ an operation on logical qubits that does not amplify shift errors of the embedding oscillators. Therefore, a fault-tolerant operation does not significantly increase the logical error rate of the qubits being controlled compared to idling qubits. Moreover, fault-tolerance requires that no decoherence channel generating long shifts of the oscillator state be opened during the operation, and that the gate be performed exactly. In the infinite-energy code limit, GKP proposed to perform fault-tolerant Clifford gates with simple low-weight drive Hamiltonians applied to the embedding oscillators [6]. In this section, we extend this result and derive target evolutions implementing Clifford gates in the finite-energy code. In contrast with the infinite-energy code case, these evolutions are not unitary and thus not trivially driven: a practical driving scheme remains to be found. Fortunately, one can circumvent the problem by slowly varying the parameters of the dissipation described in the previous sections such that its fixed points follow the desired code states trajectory in phase-space throughout the gate. In the limit where the gate duration \(T_{\text{gate}}\) is much longer than \(1/\Gamma_{c}\) (\(\Gamma_{c}\) is the confinement rate onto the code manifold, see Sec. III), we expect dissipation to coral the target state with no additional drive, as was proposed for the control of cat qubits [80]. Quantifying the spurious logical error probability for finite-time gates time will be the subject of a future work.
### Clifford gates in the finite-energy GKP code
Remarkably, the target evolutions proposed by GKP to implement Clifford gates in the infinite-energy code correspond to continuous symplectic mappings of the target oscillator phase-space coordinates. In detail, for a control parameter \(u\) varying continuously from 0 to 1 during the gate, these transformations read:
\[\begin{split}\text{\it Hadamard gate}\\ \mathcal{S}_{u}^{H}:&\mathbf{q}\rightarrow\cos(u \frac{\pi}{2})\mathbf{q}+\sin(u\frac{\pi}{2})\mathbf{p}\\ &\mathbf{p}\rightarrow-\sin(u\frac{\pi}{2})\mathbf{q}+\cos(u \frac{\pi}{2})\mathbf{p}\end{split} \tag{16}\]
The corresponding evolution is a quarter turn rotation of the target state \(\mathbf{U}_{u}^{H}=e^{iu\frac{\pi}{2}\mathbf{a}^{\dagger}\mathbf{a}}\) (see Fig. 8a).
\begin{table}
\begin{tabular}{|c|c|c|} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline \hline Target mode inductance & \(L_{a}\) & 14 \(\mu\)H \\ (inductive energy) & & (\(h\times\)12 MHz) \\ \hline Josephson junction energy & \(E_{J}\) & \(h\times\)500 MHz \\ \hline Target mode capacitance & \(C_{a}\) & 80 fF \\ (charging energy) & & (\(h\times\)240 MHz) \\ \hline Target mode frequency & \(\omega_{a}\) & 2\(\pi\times\)150 MHz \\ \hline Target mode photon loss rate & \(\kappa_{a}\) & 2\(\pi\times\)300 Hz \\ \hline Ancillary mode frequency & \(\omega_{b}\) & 2\(\pi\times\)5 GHz \\ \hline Ancillary mode phase & \(\eta_{b}\) & 0.3 \\ fluctuations across the ring & & \\ \hline Ancillary mode photon loss rate & \(\kappa_{b}\) & 2\(\pi\times\)0.5 MHz \\ \hline Number of harmonics in bias comb & \(N\) & 100 \\ \hline Maximum modulation signal & \(\xi_{\text{max}}\) & 0.2 \\ \hline Modular interaction rate & \(g\) & 2\(\pi\times\)100 kHz \\ \hline Modular dissipation rate & \(\Gamma\) & 2\(\pi\times\)20 kHz \\ \hline Decay rate of \(\mathbf{X}\) and \(\mathbf{Z}\) & \(\Gamma_{L}\) & 2\(\pi\times\)4 Hz \\ Pauli operators & & \\ \hline \end{tabular}
\end{table}
Table 1: **Proposed circuit parameters.** The target mode has an impedance \(Z_{a}=2R_{Q}\) and resonates in the radio-frequency range to allow biasing of the circuit with a frequency comb containing \(N=100\) harmonics within a 20 GHz bandwidth. This requires to load the circuit with an ultra-high inductance, which can be implemented at the cost of only a small number of parasitic modes appearing in the operating band [55] with state-of-the-art techniques [42]. The two Josephson junctions energy is chosen low enough that a 2% energy mismatch may be compensated _in situ_ (see Sec. VI.2), and other parameters are chosen to respect the hierarchy (15). The estimated decay rate of the generalized Pauli operators only accounts for errors induced by photon loss of the target mode and by truncation of the bias frequency comb. The latter error channel significantly dominates over the former, so that increasing further the number of harmonics in the bias comb would yield a much more robust GKP qubit.
_Phase gate_
\[\begin{split}\mathcal{S}_{u}^{P}:&\mathbf{q}\to\mathbf{q} \\ &\mathbf{p}\to\mathbf{p}-u\mathbf{q}\end{split} \tag{17}\]
The corresponding evolution consists in squeezing and rotating the target state \(\mathbf{U}_{u}^{P}=e^{iu\mathbf{q}^{2}/2}\) (see Fig. 8b).
_CNOT gate_
\[\begin{split}\mathcal{S}_{u}^{C}:&\mathbf{q}^{ \alpha}\to\mathbf{q}^{\alpha}\\ &\tilde{\mathbf{p}}^{\alpha}\to\mathbf{p}^{\alpha}-u\mathbf{p}^{ \beta}\\ &\mathbf{q}^{\beta}\to\mathbf{q}^{\beta}+u\mathbf{q}^{\alpha}\\ &\tilde{\mathbf{p}}^{\beta}\to\mathbf{p}^{\beta}\end{split} \tag{18}\]
Here the joint evolution of the control and target oscillators labeled \(\alpha\) and \(\beta\) reads \(\mathbf{U}_{u}^{P}=e^{iu\mathbf{q}^{\alpha}\mathbf{p}^{\beta}}\) (see Fig. 8c) and is the combination of two-mode squeezing
Figure 8: **Clifford gates by slow variation of the modular dissipation parameters.** The Hadamard **(a)**, Phase **(b)** and CNOT **(c)** gates are each applied by continuously distorting the stabilized GKP lattice structure in phase-space. In boxes, the oscillator field states are represented by their standard deviation contours (red circles), along with the GKP lattice axes (blue and yellow) in the single mode (a-b) or bipartite (c) phase space, before and after a gate. At the end of each gate, the distorted lattice aligns with the initial one. In our proposed architecture, each oscillator is connected to at least two Josephson rings, each ring being connected to a dissipative ancillary mode (trash can icon) and responsible for the activation of a pair of Lindblad operators (\(\mathbf{L}_{r,s},\mathbf{L}_{r,d}\)) (see Eq. 8), where \(r=q\) or \(p\) before and after the gate. Lattice distortion is induced by rotating the target quadrature \(r\) by an angle \(\phi\) (controlled by the timing of the pulses biasing the ring) and simultaneously adjusting the phase fluctuations of the oscillator across the ring \(\eta\) (tunable coupling symbolized by cylinder pierced by a diagonal arrow). The parameter \(u:0\to 1\) is slowly varied during the gate so that the oscillator states remain at a fixed point of the dissipation at all time.
and photon exchange (beam-splitter Hamiltonian).
We now note that the infinite-energy square code is entirely defined by its two stabilizers \(\mathbf{S}_{q}=e^{i\eta\mathbf{q}}\) and \(\mathbf{S}_{p}=e^{-i\eta\mathbf{P}}\). The code properties--namely the stabilizers and generalized Pauli operators commutation rules, the code states definition--are all inferred from the canonical commutation relation of the quadrature operators \([\mathbf{q},\mathbf{p}]=i\). Since symplectic transformations preserve commutation relations, the same modular functions of symplectically transformed variables \(e^{i\eta\mathcal{S}_{u}(\mathbf{q})}\) and \(e^{-i\eta\mathcal{S}_{u}(\mathbf{p})}\), where \(\mathcal{S}_{u}\) is one of the three aforementioned transformations, are the stabilizers of another GKP code. In other words, Clifford gates are applied by continuously distorting the GKP lattice in phase-space so that the final lattice structure overlaps with the initial one, and that an exact gate has been applied to the encoded qubit (see Fig. 8). The same scheme is directly applicable to the finite-energy code, after normalizing all operators with \(\mathbf{E}_{\Delta}=e^{-\Delta\mathbf{q}^{\ast}\mathbf{a}}\). The target evolutions now read \(\mathbf{V}_{u}^{\Delta}=\mathbf{E}_{\Delta}\mathbf{U}_{u}\mathbf{E}_{\Delta}^ {-1}\), and are in general non-unitary. As for the stabilizers of the distorted code, they read \(\mathbf{E}_{\Delta}e^{i\eta\mathcal{S}_{u}(\mathbf{q})}\mathbf{E}_{\Delta}^ {-1}\) and \(\mathbf{E}_{\Delta}e^{-i\eta\mathcal{S}_{u}(\mathbf{p})}\mathbf{E}_{\Delta}^ {-1}\). Note that with this definition, the lattice structure is distorted, but the code states normalizing envelope remains Gaussian-symmetric.
### Clifford gates by slow variation of the engineered dissipation parameters
We now detail how to adapt the dissipation engineering technique described in Sec. V to stabilize a finite-energy code distorted by \(\mathcal{S}_{u}\). We consider the architecture depicted in Fig. 8, in which a target mode is connected to two rings, each one coupled to at least one dissipative ancillary mode (trash can icon). Each ring activates one pair of Lindblad operators \((\mathbf{L}_{r,s},\mathbf{L}_{r,d})\) as defined in Eq. 8. For an idling logical qubit, these operators are modular functions of one of the oscillator quadratures \(r=q\) or \(p\). When a gate is applied, the quadrature needs to be substituted with the symplectically transformed quadrature \(\mathcal{S}_{u}(r)\). First focusing on single-qubit gates, this transformed quadrature is parametrized by its angle \(\phi\) and its length in phase-space. Adjusting the value of \(\phi\) only requires to time-shift the control pulses biasing the corresponding ring. Indeed, these pulses are organized following a train of triplets patterns (see Fig. 6b), triplets being separated by half a period of the oscillator, and the timing of the central pulse \(t\) within each half-period setting the angle \(\phi\) following \(\phi=2\omega_{a}t\) mod \(2\pi\) (see Sec. V for details). The Hadamard gate (see Fig. 8a) is based on such controls only, slowly varied to respect the adiabaticity condition \(\Gamma_{c}T_{\text{gate}}\gg 1\). On the other hand, the phase gate necessitates to vary both the angle _and_ length of the generalized quadrature \(\mathcal{S}_{u}^{P}(p)\). Varying the latter is equivalent to adjusting the phase fluctuations \(\eta\) of the target mode across the corresponding ring. In Fig. 8b, we symbolize this control by a tunable coupler (cylinder pierced by an arrow) connecting the target mode with the ring (see Fig. 9 for a more detailed circuit). Altogether, when applying a phase gate, the coupling of ring 2 to the target mode is slowly increased while the ring bias pulses are slowly time shifted in order to ramp \(\phi\) and \(\eta\) simultaneously.
Similar controls are employed to apply a two-mode CNOT gate. Here, two of the four transformed quadratures \(\mathcal{S}_{u}^{\mathcal{C}}(r)\)--with \(r=q^{\alpha},\ p^{\alpha},\ q^{\beta}\) or \(p^{\beta}\)--combine a fixed contribution from one mode and a varying contribution from the other one. This requires to couple with adjustable strength one of the two rings stabilizing the control oscillator to the target oscillator, and _vice versa_. Thus, in Fig. 8c, the coupling of ring 2--responsible for the activation of \((\mathbf{L}_{q^{\alpha},s},\mathbf{L}_{q^{\alpha},d})\) when the logical qubits are idle--to the mode \(\beta\) is slowly ramped up during the gate. As a consequence, the ring witnesses increasing phase fluctuations \(\eta_{2}^{\beta}\) from the oscillator \(\beta\), while the phase fluctuations \(\eta_{2}^{\alpha}\) from the mode \(\alpha\) remain constant. Moreover, the ring bias signal, which consists in a train of pulse triplets separated by half a period of mode \(\alpha\) when the qubits are idle, is enriched with a second train of pulses during the gate, separated by half a period of mode \(\beta\). They both modulate the same carriers at the frequency of the ancillary mode attached to the ring 2. Similar controls are applied to the ring 4, responsible for the other pair of varied Lindblad operators.
A few comments are in order about these gates and the proposed architecture. First, we remind the reader that once the evolution implementing a gate is complete, the GKP lattice of each oscillator retrieves its initial square structure. As a consequence, the control parameters \(\phi\) and \(\eta\), which have been varied throughout the gate, can be returned to their initial values. While the parameters variation needs to be slow during the gate, this last adjustment can be made on a much shorter timescale. The flux pulse trains biasing the ring being controlled should be interrupted during this stage in order not to inadvertently generate a modular dissipation misaligned with the oscillator GKP lattice. Second, when not applying a CNOT gate, the phase fluctuations \(\eta_{2}^{\beta}\) and \(\eta_{4}^{\alpha}\) need not be perfectly nullified. Indeed, spurious phase fluctuations of a mode across a ring have negligible impact as long as they are much smaller than \(2\sqrt{\pi}\), since the ring can then only generate short, correctable displacements of the mode. As Supplemental materials, we propose, based on similar arguments, a protected readout strategy for the logical qubits, which does not introduce spurious dephasing out of measurement times.
### Example circuit for single and two-qubit Clifford gates
In Fig. 9, we give a more concrete example circuit implementing the architecture of Fig. 8. The phase variables of the two modes supporting logical qubits, labelled \(\alpha\) and \(\beta\), are defined across two capacitors (in green). Each capacitor is placed in series with two Josephson rings, each ring being coupled to an ancillary dissipative mode (in brown) via a small shared inductance \(L_{c}\), neglected in the following. Furthermore, the rings shunt inductances are made tunable--for instance by implementing them with chains of Josephson rings controlled with an external magnetic field (not shown)--and a fraction of the shunt inductance of rings 2 and 4 is shared between the two target modes owing to the horizontal cross-connection. Altogether, adjusting the value of all inductances allows one to control the phase fluctuations of each target mode across one or two rings, as in the architecture of Fig. 8c. In detail, for the mode \(\alpha\), these read \(\eta_{i}^{\alpha}=p_{i}(\frac{\pi_{\alpha}}{R_{Q}})^{1/2}(\frac{L^{\alpha}}{C} )^{1/4}\) where \(L^{\alpha}=L_{1}^{\alpha}+L_{2}^{\alpha}+L_{4}^{\alpha}\) is the total inductance shunting the capacitance \(C^{\alpha}\), and \(p_{i}=\frac{L_{i}^{\alpha}}{L^{\alpha}}\) for \(i=1,2,4\) is the participation ratio of the inductance \(i\). Similar formulas are derived for the mode \(\beta\). Note that it is straightforward to extend this circuit to support a larger number of logical qubits, with a connectivity solely limited by the number of rings in which a mode can participate with large phase fluctuations.
## VIII Conclusion and outlook
In this paper, we have proposed a novel scheme to generate, error-correct and control GKP qubits encoded in a high-impedance Josephson circuit by dissipation engineering. Numerical simulations indicate that logical errors of the encoded qubits stemming from most error channels are exponentially suppressed when the engineered dissipation rate increases. In a state-of-the-art circuit, the logical qubit lifetime could extend orders of magnitude beyond the single photon dwell time in the embedding resonator, a feat never realized so far. Arguably, at this level of error suppression, quasi-particle poisoning, which opens an uncorrected error channel, could limit the device performances. Steady progress in understanding and controlling sources of quasi-particles in superconducting devices [69, 70, 71] could conceivably overcome this roadblock in the near future.
The circuit we propose to embed multiple GKP qubits controllable with protected Clifford gates is remarkably simple (see Fig. 9) and is fabricated in a parameter regime which, though demanding (see Table 1), should prove easier to achieve than alternative proposals to encode GKP qubits at the hardware level [40, 39, 45]. Moreover, circuit parameters do not necessitate fine-tuning so that our protocol is robust against fabrication disorder. Schematically, such robustness and ease of fabrication is made possible by transferring the complexity of quantum error-correction from the hardware to the microwave control domain. Indeed, our system needs to be driven with a precise microwave frequency comb spanning a 20 GHz range. Recent progress in digital synthesis of microwaves should prove instrumental in generating and delivering such a broadband signal with sufficient accuracy. Alternatively, direct on-chip synthesis of microwave frequency combs appears compatible with the circuits we consider [64, 65], and would drastically reduce control complexity.
On the long term, the relative simplicity of Clifford gates and the robustness of our multi-GKP qubit architecture to spurious microwave cross-talks paves the way for the concatenation of these bosonic qubits into a discrete variable code such as the surface code [81, 82, 83, 84, 85]. Given that the coherence time of GKP qubits stabilized by modular dissipation should extend far beyond single and two-qubit gate time--which is set by the confinement rate onto the code manifold in our approach--the hope is that such a _surface-GKP code_ would operate well below threshold, implementing a fault-tolerant, universal quantum computer with minimum hardware overhead.
Figure 9: **Example circuit stabilizing two controllable qubits.** The logical qubits are encoded in two target oscillators labeled \(\alpha\) and \(\beta\), each supported by a capacitor (green) shunted by Josephson rings. Each ring couples to a dissipative ancillary mode (brown) through a small shared inductance. All rings have tunable shunt inductances (piered by diagonal arrows), and the horizontal cross-connection ensures that the rings 2 and 4 participate in both modes. Varying all inductances allows adjusting independently the phase fluctuations of each mode across the rings 2 and 4, and the phase fluctuations of mode \(\alpha\) (respectively \(\beta\)) in the ring 1 (respectively 2). By slowly varying these control parameters along with the timing of the pulse trains biasing each ring (not shown), one applies an arbitrary, fault-tolerant, Clifford gate to the encoded qubits.
## Acknowledgments
We thank W. C. Smith and R. Lescanne for fruitful discussions on inductively shunted circuits, M. Burgelman for fruitful discussions about higher-order averaging methods, and M. H. Devoret, A. Eickbusch and S. Touzard for stimulating discussions on the GKP code. We thank the maintainers of the CLEPS computing infrastructure from the Inria of Paris for providing the computing means necessary to speed up the parameter sweeps presented in the figures. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreements No. 884762, No. 101042304 and No. 851740). A.S. and P.C.-I. acknowledge support from the Agence Nationale de la Recherche (ANR) under grants HAMROQS and SYNCAMIL. The authors acknowledge funding from the Plan France 2030 through the project ANR-22-PETQ-0006.
During the final stages of preparation of this manuscript, we became aware of the recent preprint [86]. The authors propose to engineer a Hamiltonian close to the infinite-energy GKP Hamiltonian, with techniques similar to that exposed in Section IV, and provide a detailed study of a potential implementation in circuitQED and of its application for the preparation of GKP states. We emphasize that in our study, Hamiltonian engineering is introduced mainly as a preliminary pedagogical tool to present our control scheme in a simpler context; while our true focus lies in engineering a dissipative dynamics stabilizing finite-energy GKP states, and designing fault-tolerant logical gates compatible with this exotic dissipation.
| 新しいアプローチを用いて、GKP qubitsの生成、保護、制御を行います。このアプローチは、 Josephson回路をマイクロ波周波数 comb パラメータ的に変調させて、高インピーダンス回路モードの耗散的なダイナミクスを強制することで、高インピーダンス回路モードの安定性を自動的に保証し、有限エネルギーのGKPコードを安定化させます。エンコードされたGKP qubitsは、超伝導回路に多く存在する主要な脱調経路に対して、 quasi-particle poisoning 以外に非常に安定しています。特に、散乱エンジニアリングのために利用される補助モードからのノイズは論理レベルに伝播しません。最新の実験装置を用いて、エンコードされたqubitsの寿命は、実現可能な点を超える2桁の増加を達成でき、製造と制御電子機器の進歩を通じてさらに大きな可能性を秘めています。Clifford gatesを用いた量子ビットの初期化、 |
2303.12398 | Multiscale Attention via Wavelet Neural Operators for Vision
Transformers | Transformers have achieved widespread success in computer vision. At their
heart, there is a Self-Attention (SA) mechanism, an inductive bias that
associates each token in the input with every other token through a weighted
basis. The standard SA mechanism has quadratic complexity with the sequence
length, which impedes its utility to long sequences appearing in high
resolution vision. Recently, inspired by operator learning for PDEs, Adaptive
Fourier Neural Operators (AFNO) were introduced for high resolution attention
based on global convolution that is efficiently implemented via FFT. However,
the AFNO global filtering cannot well represent small and moderate scale
structures that commonly appear in natural images. To leverage the
coarse-to-fine scale structures we introduce a Multiscale Wavelet Attention
(MWA) by leveraging wavelet neural operators which incurs linear complexity in
the sequence size. We replace the attention in ViT with MWA and our experiments
with CIFAR and Tiny-ImageNet classification demonstrate significant improvement
over alternative Fourier-based attentions such as AFNO and Global Filter
Network (GFN). | Anahita Nekoozadeh, Mohammad Reza Ahmadzadeh, Zahra Mardani | 2023-03-22T09:06:07 | http://arxiv.org/abs/2303.12398v4 | # Multiscale Attention via Wavelet Neural Operators for Vision Transformers
###### Abstract
Transformers have achieved widespread success in computer vision. At their heart, there is a Self-Attention (SA) mechanism, an inductive bias that associates each token in the input with every other token through a weighted basis. The standard SA mechanism has quadratic complexity with the sequence length, which impedes its utility to long sequences appearing in high resolution vision. Recently, inspired by operator learning for PDEs, Adaptive Fourier Neural Operators (AFNO) were introduced for high resolution attention based on global convolution that is efficiently implemented via FFT. However, the AFNO global filtering cannot well represent small and moderate scale structures that commonly appear in natural images. To leverage the coarse-to-fine scale structures we introduce a Multiscale Wavelet Attention (MWA) by leveraging wavelet neural operators which incurs linear complexity in the sequence size. We replace the attention in ViT with MWA and our experiments with CIFAR and ImageNet classification demonstrate significant improvement over alternative Fourier-based attentions such as AFNO and Global Filter Network (GFN).
## 1 Introduction
The success of transformer networks in Natural Language Processing (NLP) tasks has motivated their application to computer vision. Among the prominent advantages of transformers is the possibility of modeling long-range dependencies among the input sequence and supporting parallel processing compared to Recurrent Neural Networks (RNN). In addition, unlike Convolutional Neural Networks (CNN), they require minimal inductive biases for their design. The simple design of transformers also enables processing of multi-modality contents (such as images, video, text, and speech) by using the same processing blocks. It exhibits excellent scalability for large-size networks trained with huge datasets. These strengths have led to many improvements in vision benchmarks using transformer networks [25, 9, 6].
A key component for the effectiveness of transformers is the proper mixing of tokens. Finding a good mixer is challenging because it needs to scale with the sequence size. The Self-Attention (SA) block in the original transformer suffers from quadratic complexity. In order to make mixing efficient, several ideas have been introduced. Recently, Adaptive Fourier Neural Operator (AFNO) have been proposed to replace SA by leveraging the geometric structure of images via learning a global convolution operator in the Fourier space. As a major shortcoming of AFNO, it is a global operator, and thus can miss the fine and moderate scale structures that are quite present in natural images [5, 9].
To overcome the shortcomings of AFNO, one needs to effectively mix tokens at different scales [5]. To this end, we propose the use of wavelet transform, which is known as an effective multiscale representation for natural images in image processing. In order to learn a multiscale mixer, we adapt a variation of Wavelet Neural Operator (WNO) that has been studied for solving PDEs in fluid mechanics [24]. We modify the design to account for high-resolution natural images with discontinuities due to objects and edge structures. After the architectural modifications, the MWA attention layer is shown in Fig. 1.
The input image is first transformed into the wavelet domain using two-dimensional Discrete Wavelet Transform (2D-DWT). Then, all coefficients from the last decomposition level are convolved with the learnable weights, and subsequently undergo a nonlinear GeLU activation. Then, an inverse 2D-DWT reconstructs the pixel level tokens. For 2D-DWT (and its inverse), we choose Haar wavelet whit decomposition level m=1 for faster speed. We conducted experiments for classification, and the experiments show that our MWA has a better performance and accuracy than SA block [3] and Fourier based attentions including AFNO [5] and the Global Filter Network (GFN) [18].
## 1 Introduction
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The comparison of MWA with AFNO, GFN and SA block is mentioned in Tab. 1 in terms of the number of parameters and complexity.
## 2 Related Works
Several works have been introduced to improve the efficiency of the attention mechanism in transformers. We divide them into three main categories.
**Graph based attention.** graph-based SA methods include: (1) sparse attention with fixed patterns; which reduces the attention by limiting the field-of-view to predetermined patterns such as local windows in sparse transformers [2]; (2) sparse attention with learnable patterns; where a fixed pattern is learned from data; e.g., axial transformer [7] and reformers [10]; (3) memory; another prominent method is to use a peripheral memory module that can access multiple tokens at the same time. A common form is global memory that can access the entire sequence e.g., set transformers [12]; (4) low-rank methods approximate the SA via a low-rank matrix; e.g., linformers [26]; (5) kernel methods; use kernel trick to approximate linear transformers [8]; (6) recurrence is another method to improve the efficiency of the transformer e.g., compressive transformer [17].
**MLP based attention.** Several works have recently been proposed that use MLP to replace self-interesting layers in feature transformation and fusion, such as ResMLP [23], which replaces layer normalization with affine transformation. A recently proposed gMLP [15] also uses a spatial gating unit to reweight features in the spatial dimension. However, all models including MLP that are used to combine tokens spatially have two basic drawbacks: (1) Similar to SA, MLPs still require quadratic complexity with the sequence size; (2) MLP mixers have static weights, and they are not dynamic with respect to the input.
**Fourier based attention.** Recently, FNet, GFN, and AFNO models have been presented, which incur linear complexity. FNet [13] is an efficient transformer where each layer consists of a Fourier transform sublayer followed by a feedforward sublayer. Basically, the SA layers are replaced by Fourier transform with no learnable weights, and two- dimensional Discrete Fourier Transform (2D-DFT) is applied to embed the sequence length and hidden dimension.
Another efficient transformer is the Global Filter Network (GFN), which aims to replace SA with a static global convolution filter. GFN however lacks adaptivity [18].
AFNO, was introduced from the operator learning perspective to solve the shortcomings of GFN, by introducing weight sharing and block diagonal structure on the learnable weights that makes it scalable [5]. AFNO however, suffers from global biases and does not represent multiscale appearing commonly in natural images. The novelty of our proposed method is to account for multiscale structures using MWA attention.
## 3 Preliminaries and problem statement
Consider a two-dimensional \(3\times m\times n\) RGB image \(x\) that is divided into small and non-overlapping patches. After patching with patch size \(p\), the image can be seen as a two-dimensional grid \(3\times h\times w\), where \(h=m/p\), and \(w=n/p\). Each RGB patch then undergoes a linear projection that creates tokens with a \(d\)-dimensional embedding, namely \(d\times h\times w\). In order to preserve the position information, a \(d\)-dimensional position embedding is also added to each token. Since then, the transformer network processes the two-dimensional sequence of tokens by mixing them over the layers using the attention module that creates the final representation for end tasks [5, 9, 21].
SA learns similarity among tokens [6]. However, quadratic complexity with the sequence size hinders the training of high-resolution images [22]. Our goal is to replace attention with a compute and memory efficient module that is aware of the multiscale structures in the natural images for downstream tasks. Before delving into the details of multiscale attention, let us overview global-scale attention based on AFNO.
### Adaptive Fourier Neural Operator
Neural operators learn the mapping between two functions in continuous space [1]. They are trained only once and once after training, they can do prediction for any input function. Neural operators are typically used for solving PDEs. They can also be extended to computer vision, since images can be treated as RGB-valued functions [14]. This generalization allows us to take advantage of operator learning in computer vision.
In order to leverage the geometric structure of images, the AFNO, relies on convolution with a global filter that is as big as the input tokenized image. AFNO efficiently implements global convolution via FFT, which is inspired by the Fourier Neural Operator (FNO). However, FNO has a \(d\times d\) weight matrix for each token (\(d=n/p\) is the token grid dimension), so the number of parameters becomes very large for high resolution inputs. To reduce the number of parameters, AFNO imposes a block-diagonal structure on the weights. It then shares the weights among the features and truncates certain frequency components using soft-thresholding and shrinkage operations [5]. However, FFT is not suitable to represent images with non-periodic patterns. Natural images usually exhibit multiscale structures, and AFNO can miss non-periodic and small-to-medium scale structures. To model multiscale structures, our idea is to leverage wavelet transform and consequently the wavelet neural operators, that take advantage of wavelets and have
been very successful for solving PDEs with sudden changes as discussed in the next part.
## 4 Wavelet transform and Wavelet Neural Operator
### Wavelet transform for signal representation
Let \(\psi(x)\in L^{2}\ (R)\) be a canonical mother wavelet that is local in both time and frequency domains. Let also \(W(\Gamma)\) and \(W^{-1}(I_{w})\) be the forward wavelet transform and the inverse wavelet transform of an arbitrary function \(\Gamma:\ D\to R^{d}\). Then, the wavelet transform and the inverse are the transforms of the function \(\Gamma\) with scaling and displacement parameters \(\alpha\in R\) and \(\beta\in R\). They are obtained as follows using the following integral pairs [24],
\[(W^{-1}\Gamma)(x)=\frac{1}{c_{\psi}}\iint_{0}^{+\infty}\Gamma_{w}(\alpha,\beta )\frac{1}{\sqrt{|\alpha|}}\bar{\psi}\left(\frac{x-\beta}{\alpha}\right)d\beta \frac{d\alpha}{a^{2}} \tag{1}\]
\[(W\Gamma)(\alpha\,\beta)=\int_{D}\Gamma(x)\frac{1}{\sqrt{|\alpha|}}\psi\left( \frac{x-\beta}{\alpha}\right)dx \tag{2}\]
Where \((\Gamma_{w})(\alpha,\beta)=(W\Gamma)(\alpha,\beta)\psi\big{(}(x-\beta)\alpha \big{)}\in\)
\(L^{2}(R)\) is scaled and transferred to the mother wavelet. By scaling and shifting, the desired wavelets can be obtained from the mother wavelet. Each set of wavelet functions forms an orthogonal set of basis functions.
Note; that term \(C_{\psi}\) is the admissible constant which ranges in \(0\leq\ C_{\psi}\leq\infty\). The expression for \(C_{\psi}\) is given as follows,
\[C_{\psi}=2\pi\int_{D}\frac{|\psi(\alpha)|^{2}}{|\omega|}d\omega \tag{3}\]
In signal representation theory, wavelet decomposition has proven successful in compressible representation with a few basis functions compared with Fourier transform. This comes from the nature of wavelet bases that can well represent trends, breakpoints, and discontinuities in higher derivatives and similarities [28]. We aim to rely on the spatial and frequency localization power of wavelets to learn the relationship between tokens and thus learn the multiscale patterns at the internal layers of transformers. Considering these features, we adapt the WNO, which we discuss in the next section [24, 4, 20].
### Wavelet Neural Operator
The class of shift-equivariant kernels has the favorable property that they can be analyzed as linear combinations of eigen functions [19]. A powerful class of eigen function are the wavelet transform bases, where according to the convolution theorem, multiscale convolution in the spatial domain is equal to multi plication in the wavelet transform domain. Accordingly, the WNO can be defined next [24].
**Definition (Kernel integral operator).** The kernel integral operator \(K\) is defined as follows:
\[K(x)(s)=\int_{D}k(s;t)x(t)dt\ ;\ \ \ seD \tag{4}\]
With a continuous kernel function \(k\): \(D\ \times\ D\ \to\ R^{d\times d}\). For the special case of Green's kernel, \(k(s;t)\) can be expressed as \(k(s;t)=\ k(s-t)\), and the integral of Equ. (4) leads to multiscale convolution defined below.
**Definition (Multiscale convolution kernel operator).** Assuming that as \(k(s;t)=\ k(s-t)\), The kernel integral of Equ. (4) is rewritten as follows:
\[K(x)(s)=\int_{D}k(s-t)x(t)dt;\ \ \ s\epsilon D \tag{5}\]
The Green's kernel has a useful regularization effect that can capture multi-scale interactions. In addition, it can be used to effectively implement multiscale convolution using Discrete Wavelet Transform (DWT).
**Definition (Wavelet neural operator).** For the continuous input \(x\ \epsilon D\) and kernel \(k\) and the kernel integral at token \(s\), the wavelet neural operator is defined as follows:
\[K(x)(s)=W^{-1}\big{(}W(x)\ \cdot\ W(k)\big{)}(s)\ ;\ \ seD \tag{6}\]
Here, \(\cdot\) denotes matrix multiplication, and \(W\) and \(W^{-1}\) represent the forward DWT and the inverse DWT.
## 5 Multiscale Wavelet Attention
Inspired by WNO, for RGB images, our idea is to combine the tokens using DWT. We make fundamental modifications to adapt the WNO operator to images to account for high-resolution natural images with object-induced discontinuities and edge structures. In the proposed MWA, for faster and better performance, we use 2D-DWT to combine tokens. 2D-DWT enjoys fast implementations with GPU support [11].
In MWA, images are converted into high-frequency and low-frequency components using 2D-DWT. In essence, high-frequency components represent edges in the image, while low-frequency components represent smooth regions. According to Fig. 1, the first branch in the two-dimensional array calculates four components as follows: the approximation component (LL) that represent low frequency components; the detail components that account for high frequencies such as horizontal (HL), vertical (LH) and diagonal (HH). In this work, we use all the coefficients of the last decomposition level. In DWT, we transform the mother wavelet to calculate the wavelet coefficients on scales with powers of two. In this case, the wavelet \(\varphi(x)\) is
defined as follows [24]:
\[\psi_{m,\,t}(x)=\frac{1}{\sqrt{2^{m}}}\psi\left(\frac{x-t2^{m}}{2^{m}}\right) \tag{7}\]
Where the parameters \(m\) and \(2\) are the scaling and shifting parameters and the forward DWT wavelet transform is shown below [24]:
\[(WT)(m,t)=\frac{1}{\sqrt{2^{m}}}\int_{D}\Gamma(x)\Psi\left(\frac{x-t2^{m}}{2^{ m}}\right)dx \tag{8}\]
By fixing the scale parameter \(m\) in a certain integer and shifting \(t\), the DWT coefficients at the level of \(m\) can be obtained. Since the use of a one-way filter bank is very effective for decomposing a signal into multiple frequency subbands, 2D-DWT is implemented as a filter bank, which works as a sequence of low-pass and high-pass filters. In the implementation of the filter bank, by passing the image through a low-pass filter and a high-pass filter, the image is decomposed into details and approximate coefficients. If \(r(n)\) and \(s(n)\) represent low-pass and high-pass filters, respectively, then two convolutions in the form \(\mathrm{z}_{high}(n)=(x*s)(n)\) and \(\mathrm{z}_{low}(n)=(x*r)(n)\) are executed, where \(n\) is the number of discretization points. While the detail coefficients \(\mathrm{z}_{high}(n)\) are preserved, the approximate coefficients \(\mathrm{z}_{low}(n)\) are recursively filtered by passing it through low-pass and high-pass filters until the total number of decomposition levels is exhausted [24, 16, 29]. At each level, the length of the image is halved due to conjugate symmetry. The general architecture of the MWA model is shown in Fig. 1.
Our model takes non-overlapping \(h\times w\) grid of patches as input and project each patch into \(d\) dimensional space. We define the input token tensor \(x\in R^{h\times d\times w}\) and the weight tensor \(w\in R^{(h\times w/2^{m})\times d\times d}\) for parameterization of the kernel. MWA performs a sequence of operations for Each token \((m,n)\in[w]\times[h]\), which we will discuss below.
**First step:** Unlike AFNO, which combines tokens with Discrete Fourier Transform (DFT), MWA combines tokens representing different spatial locations using DWT as
\[\mathrm{z}_{m,n}=[DWT(x)]_{m,n} \tag{9}\]
Only from the wavelet coefficients with the highest scale, a parametrization space with limited dimension is obtained.
In general, the length of the wavelet coefficients are also affected by the number of vanishing moments of the orthogonal mother wavelet. Thus, we use the coefficients \(Z_{m,n}\) at the highest level of analysis.
**Second step:** While AFNO uses the multiplication between the learnable weight tensor and the coefficients obtained from the DFT, we use the convolution between a learnable weight tensor and coefficients of the last level of decomposition as follows:
\[\mathrm{\hat{z}}_{m,n}=\mathrm{z}_{m,n}*\ w_{m,n} \tag{10}\]
**Third step:** Unlike AFNO, which uses Inverse Discrete Fourier Transform (IDFT) to recover tokens after mixing, we use Inverse Discrete Wavelet Transform (IDWT) to update and separate tokens by using:
\[\mathrm{\gamma}_{m,n}=[IDWT(\mathrm{\hat{z}})]_{m,n} \tag{11}\]
Using 2D-DWT, we can generate fine image details as well as the rough approximation of the image. Note, DWT and IDWT are well supported by CPU and GPU, so the proposed model has good performance on hardware.
**Fourth step:** weighted skip connections are added using two convolution layers with different kernel sizes (second and third branches of Fig. 1). These convolution layers facilitate learning the identity mapping and have been proven useful for learning high frequency details.
In general, the architectural highlights are as follows:
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & **Complexity (FLOPs)** & **Parameter Count** & **Interpretation** \\ \hline SA & \(N^{2}d+3Nd^{2}\) & \(3d^{2}\) & Graph Global Conv \\ GFN & \(Nd+N\,d\,\,log\,N\) & \(Nd\) & Dephtwise Global Conv \\ AFNO & \(Nd^{2}\) / \(k+N\,d\,\,log\,N\) & \((1+4\) / \(k)d^{2}+4d\) & Adaptive Global Conv \\ MWA & \(1\cdot 5k_{1}Nd^{2}\) / \(g_{1}+1\cdot 5k_{2}Nd^{2}\) / \(g_{2}\) & \((k_{1}\) / \(g_{1}+k_{2}\) / \(g_{2})d^{2}\) & Multi Scale Conv \\ \hline \hline \end{tabular}
\end{table}
Table 1: Complexity, parameter count, and interpretation for MWA, AFNO, GFN, and SA. \(N=hw\), \(d\) and \(K\) refer to the sequence size, channel size, and block count in AFNO. Also, \(k_{1},\ k_{2}\) are kernel size for MWA, and \(g_{1},g_{2}\) is the number of groups, respectively.
Network parameters are learned in the wavelet space, which are localized both in frequency and spatial domains, and thus they can learn multiscale patterns in images effectively.
* WNO is adopted from continuous PDEs and modified for discrete images by adding more nonlinearity and adding convolutional skip connections. Also, both the approximation and detail coefficients of the wavelet transform are used to model the attention.
* Our model is more flexible than SA because both DWT and IDWT have no learnable parameters and can process sequences with arbitrary length.
## 6 Complexity
In this section we quantify the operation count for the proposed MWA attention. For DWT, the input is simultaneously decomposed using a low-pass filter \(r(n)\) and a high-pass filter \(s(n)\). In case of Haar Wavelet, the high-pass and low-pass filters have a fixed length, each of which form \(z_{high}(n)=(r*\chi)(n)\) and \(z_{low}(n)=(s*\chi)(n)\), which \(O(N)\) complexity for the sequence size \(N\). DWT also uses these two filters for decomposition. Thus, the implementation of DWT filter bank has complexity of \(O(N)\)[28]. Decomposing the input using a wavelet with level \(m\) results in an image of length \(n/2m\).
The convolution of the analyzed coefficients of the last level and the weights has a complexity of \(O(KNd^{2}/2mg)\) (in our proposed architecture, the level of analysis is \(m\)=1). The decomposition level and number of groups plays an important role in increasing the speed of our proposed architecture. The input convolution and weights with the kernel size \(k\) and the number of groups \(g\) also have complexity \(O(kNd^{2}/g)\)[27]. The overall complexity of the architecture is shown in Tab. 1.
## 7 Experiments
We conduct experiments to confirm the effectiveness of MWA and compare the results with different Fourier based transformers. We perform our experiment on CIFAR and Tiny-ImageNet datasets as widely used small and medium-scale benchmarks for image classification.
**Datasets**. As mentioned, we adopt CIFAR and Tiny-ImageNet datasets. CIFAR-10 contains 60,000 images from 10 class categories, while CIFAR-100 contains 60,000 from 100 class categories. Tiny-ImageNet also contains 100,000 images with 200 classes. We report the accuracy on test data.
**Comparisons**. We compare our method with the attention block in the main transformer and the AFNO and GFN Fourier transform methods, which have similar FLOPs and number of parameters, and we see that our method can clearly perform well in small and medium sized data such as CIFAR and Tiny-ImageNet (see Tab. 2, Tab. 3 and Tab. 4). One of the problems with transformers is that they require a lot of data for training, and they perform poorly on medium and low data, but our method can perform better than previous transformers on small datasets.
### Architecture and training
The proposed MWA block consists of three major components. The first component converts the input image taken from the previous layer into a wavelet domain using 2D-DWT (horizontal, vertical and diagonal approximation coefficients and details). Then convolution is performed on all the approximate coefficients and details of the last level of decomposition and learnable weights, which then undergo GeLU nonlinear activation. Then, an inverse 2D-DWT reconstructs the pixel-level tokens. For 2D-DWT and its inverse, we choose Haar Wavelet with decomposition level \(m=1\). For skip connections we use two-dimensional convolution with a different kernel sizes 1\(\times\)1 and 3\(\times\)3, followed by nonlinear GeLU activation. Finally, all three branches are gathered and passed through a non-linear GeLU activation.
We use the ViT-S/4 configuration for experimenting on CIFAR10-100 and Tiny-ImageNet datasets. The ViT-S/4 configuration has 12 layers and a hidden size of 384, and a token size of 4\(\times\)4 is used to model the sequence size. We use global average pooling at the last layer to produce output softmax probabilities for classification.
We trained all models for 300 epochs with Adam optimizer and cross-entropy loss using a learning rate of 5 \(\times\)10\({}^{4}\). We also use five epochs of linear learning-rate warm-up. We use a cosine decay schedule with a minimum value of 10\({}^{5}\), along with a smooth gradient cut-off to stabilize the training that does not exceed a value of 1, and the weight-decay regularization is set to 0.05.
In particular, we use 12 transformer layers and adjust the hyperparameters of interest in AFNO and MWA to achieve a close and comparable number of parameters. More details about each model are provided below.
* SA uses 8 attention heads and a hidden size of 384 [3].
* GFN uses a hidden size of 384 [18].
* AFNO uses a hidden size 384, and a sparsity threshold of 0.1 (with 3-4 blocks to reach 16M-17M parameters) [5].
* MWA uses a hidden size 384, 2D ensemble convolution with kernel size 3 and 1 as learnable weights (together with a different number of groups to arrive at a parameter count of 16-17M).
Note that the ViT backbone used in our experiments has a patch size of 4 compared to the patch size of 16 used in the original ViT architecture for image classification. As a result, we observe that self-attention performs poorly compared with MWA and Fourier based methods.
### CIFAR Classification
We perform image classification experiments with the MWA mixer module and using the backbone ViT- S/4 on the CIFAR-10 and CIFAR-100 dataset containing 10,000 test sets with 10 and 100 classes, respectively, with resolution \(32\times 32\). We measure performance using top-1 and top-5 accuracy along with flops for different model parameters.
**CIFAR classification:** Classification results for different mixers are shown Tab. 2 and Tab. 3. It can be seen that the proposed MWA using DWT, can learn multiscale as well as non-periodic patterns in the images better than the Fourier transform, which leads to higher than 1% accuracy improvements over existing Fourier-based mixers such as AFNO and GFN.
### Tiny-ImageNet classification
We perform image classification experiments with the MWA mixer module and using the backbone ViT-S/4 on the Tiny-ImageNet dataset that contains 100,000 images of 200 classes downsized to 64\(\times\)64 colored images. Each class has 500 training images, 50 validation images and 50 test images. We measure performance through top-1 and top-5 validation accuracy along with flop and model parameters.
**Tiny-ImageNet classification:** Classification results for different mixers are shown in Tab. 4. It is observed that our proposed MWA, thanks to multiscale wavelet features that exist in natural images, outperforms global Fourier based methods including AFNO and GFN by more than 1% in top-1 accuracy. It also significantly outperforms SA when the patch size is chosen to be 4.
We introduced Multiscale Wavelet Attention (MWA) for transformers to effectively learn small-to-large range dependencies among the image pixels for representation learning. MWA adapts wavelet neural operators from PDEs and fluid mechanics after making basic corrections to WNO for natural images. MWA incurs linear complexity in the sequence size and enjoys fast algorithms for wavelet transform.
Our experiments for image classification on CIFAR and ImageNet data show the superior accuracy of our proposed MWA block compared with alternative Fourier based attentions. There are still important directions to pursue. One of those pertains to more extensive evaluations with larger datasets and complex images involving multiscale features. Also, studying the performance of MWA for larger networks and data is an important next step that demands sufficient computational resources.
| Transformersによるコンピュータビジョン分野での成功は広範である。その核心には、自注意（SA）機構がある。これは、入力の各トークンを他のすべてのトークンとウェイト付きで関連付ける、誘導的バイアスである。標準的なSA機構は、シーケンスの長さに対して、二次関数的に複雑であるため、高解像度ビジョンで出現する長いシーケンスに対する利便性に欠ける。近々、PDEのオペレータ学習をインスピレーションに、高解像度への注意を導くために、AdaptiveFourier Neural Operators（AFNO）が導入された。AFNOは、FFTを用いて効率的に実装されるグローバル畳み込みに基づいており、高解像度での注意を導く。しかし、AFNOのグローバルフィルターは、自然画像で見られる小規模・中規模の構造をうまく表現することができない。小規模・中規模の構造を捉 |
2305.16270 | Strange Random Topology of the Circle | We characterise high-dimensional topology that arises from a random Cech
complex constructed on the circle. Expected Euler characteristic curve is
computed, where we observe limiting spikes. The spikes correspond to expected
Betti numbers growing arbitrarily large over shrinking intervals of filtration
radii. Using the fact that the homotopy type of the random Cech complex is
either an odd-dimensional sphere or a bouquet of even-dimensional spheres, we
give probabilistic bounds of the homotopy types. By departing from the
conventional practice of scaling down filtration radii as the sample size grows
large, our findings indicate that the full breadth of filtration radii leads to
interesting systematic behaviour that cannot be regarded as "topological
noise". | Uzu Lim | 2023-05-25T17:26:48 | http://arxiv.org/abs/2305.16270v2 | # Strange Random Topology of the Circle
###### Abstract.
We characterise high-dimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers growing arbitrarily large over shrinking intervals of filtration radii. Using the fact that the homotopy type of the random Cech complex is either an odd-dimensional sphere or a bouquet of even-dimensional spheres, we give probabilistic bounds of the homotopy types. By departing from the conventional practice of scaling down filtration radii as the sample size grows large, our findings indicate that the full breadth of filtration radii leads to interesting systematic behaviour that cannot be regarded as "topological noise".
Key words and phrases:Mathematical Institute, University of Oxford, Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG
_E-mail address_: [email protected]
## 1. Introduction
A conventional wisdom in topological data analysis says the following: if we construct a simplicial complex from a random sample drawn from a manifold, then the topology of the simplicial complex approximates the topology of the manifold. Indeed this is true if we scale down the connectivity radius smaller as the sample size grows larger, but what happens when the connectivity radius stays the same?
We study the strange random topology of the circle, where we find high-dimensional topology arise in a systematic way. We find intervals of filtration radii in which the random Cech complex constructed from the circle is homotopy equivalent to bouquets of spheres, with positive probabilities. Here, a bouquet of spheres is the wedge sum \(\vee^{a}\mathbb{S}^{k}\). It was known that only \(a=1\) is allowed if \(k\) is odd, and all \(a\geq 1\) are allowed if \(k\) is even [4]. We show that the single odd sphere \(\mathbb{S}^{2k+1}\) appears with high probability over long intervals of filtration radii. The bouquet of even sphere \(\vee^{a}\mathbb{S}^{2k}\) appears with a smaller but positive probability over shrinking intervals of filtration radii. In particular, we show that \(a\) can get arbitrarily
large for \(\vee^{a}\mathbb{S}^{2k}\). To get to our conclusions, we use the expected Euler characteristic and the stability theorem of persistence diagram. Let's describe the setup more precisely.
**Setup.** We define the circle \(\mathbb{S}^{1}\) as the quotient space \(\mathbb{S}^{1}=[0,1]/\sim\), as the interval of length 1, glued along endpoints: \(0\sim 1\). A bouquet of spheres \(\vee^{a}\mathbb{S}^{k}\) is defined as the wedge sum of \(a\) copies of \(\mathbb{S}^{k}\).1 For a positive integer \(n\), let \(\mathbf{X}_{n}\) be the i.i.d.2 sample of size \(n\), drawn uniformly from \(\mathbb{S}^{1}\). The Cech complex of filtration radius \(\leq r\) is denoted by \(\check{\mathrm{C}}(\mathbf{X}_{n},r)\). In doing this construction, we always use the intrinsic topology of the circle, i.e. the Cech complex is a nerve complex constructed from arcs. We use the following notation for the expected Euler characteristic and expected Betti number, for Theorems A1 and A2:
Footnote 1: We take the convention that for a topological space \(K\), we define the 0-th wedge sum \(\vee^{0}K=*\), the singleton point set, and the 1st wedge sum \(\vee^{1}K=K\) itself.
Footnote 2: Independently and identically distributed
Theorem A1 (Expected Euler Characteristic). Let \(n>0\) and \(r\in(0,1)\). The following equality holds:
\[\bar{\chi}\left(n,\frac{1-r}{2}\right)=\sum_{k=1}^{\lfloor 1/r\rfloor}{n \choose k}(1-kr)^{k-1}(kr)^{n-k}\]
In particular, \(\bar{\chi}(n,r)\) is a continuous piecewise-polynomial function in \(r\).
Theorem A2 (Expected Betti Number). Let \(k\geq 1\). Given \(\epsilon>0\), the following uniform bounds hold for all \(r\in\left[\frac{k}{2k+2},\frac{k+1}{2k+4}\right)\) when \(n\) is sufficiently large:
\[\frac{\bar{\chi}(n,r)}{n}-\epsilon\leq\frac{\bar{b}_{2k}(n,r)}{n}\leq\frac{ \bar{\chi}(n,r)}{n}\]
By directly plotting the expected Euler characteristic, we observe interesting limiting behaviours. Figure 1 shows graphs of \(f_{n}(r)=n^{-1}\cdot\bar{\chi}(n,r)\), which are _normalised_ versions of \(\bar{\chi}\). As \(n\) becomes larger, \(f_{n}(r)\) shows peaks that converge to a sequence of narrow spikes. By definition \(\bar{\chi}(n,r)=n\cdot f_{n}(r)\), and therefore we see that as \(n\) becomes larger, \(\bar{\chi}(n,r)\) also becomes _arbitrarily large_ at those spikes. Theorem A2 then says that the expected Betti
number of a certain homological dimension is precisely responsible for each spike. We also compute later (Proposition 3.4) that height of the \(k\)-th limiting spike is precisely given by:
\[\omega_{k}=\frac{(k-1)^{k-1}}{k!e^{k-1}}\]
Using the Euler characteristic formula, we can now obtain probabilistic bounds on homotopy types. As stated before, all homotopy types arising from nerve complexes of circular arcs were completely classified in [4]: they are either \(\mathbb{S}^{2k+1}\) for some \(k\geq 0\), or \(\vee^{a}\mathbb{S}^{2k}\) for some \(a,k\geq 0\). We observe that \(\chi(\mathbb{S}^{2k+1})=0\), whereas \(\chi(\vee^{a}\mathbb{S}^{2k})=a+1\). Therefore in Figure 1, limiting spikes indicate contribution from the even-dimensional sphere bouquets \(\vee^{a}\mathbb{S}^{2k}\) with large \(a\), and the plateaus indicate contribution from the odd-dimensional spheres. Recalling that \(\mathbf{X}_{n}\) is an i.i.d. sample of size \(n\) drawn from \(\mathbb{S}^{1}\), we introduce Theorems B and C:
Figure 1. The graphs of _normalised_ expected Euler characteristics, \(f(r)=N^{-1}\cdot\bar{\chi}(N,r)\), for \(N\in\{5,10,\ldots 100\}\) and \(r\in[0,1]\). Yellow curves correspond to larger \(N\).
Theorem B (Odd Spheres). Let \(k\geq 0\) be an integer, and also let \(\epsilon,\delta>0\). Suppose that \(|r-\nu_{k}|\leq\tau_{k}-\epsilon\). Then for sufficiently large \(n\), the following homotopy equivalence holds with probability at least \(1-\delta\):
\[\tilde{\mathrm{C}}(\mathbf{X}_{n},r)\simeq\mathbb{S}^{2k+1}\]
where
\[\nu_{k}=\frac{2k^{2}+4k+1}{4(k+1)(k+2)},\quad\tau_{k}=\frac{1}{4(k+1)(k+2)}\]
We note that \(\nu_{k}=\frac{1}{2}(\frac{k+1}{2k+4}+\frac{k}{2k+2})\) and \(\tau_{k}=\frac{1}{2}(\frac{k+1}{2k+4}-\frac{k}{2k+2})\), so that Theorem B covers most of each interval \(r\in[\frac{k}{2k+2},\frac{k+1}{2k+4}]\).
Theorem C (Even Spheres). Let \(k\geq 2\), \(\eta\in(0,1)\). Suppose that \(|r-\rho_{k,n}|\leq\sigma_{k,\eta}/n\). Then for sufficiently large \(n\), the following homotopy equivalence holds with probability at least \(\eta\cdot k\omega_{k}\):
\[\tilde{\mathrm{C}}(\mathbf{X}_{n},r)\simeq\vee^{a}\mathbb{S}^{2k-2},\quad \text{for some }\frac{(1-\eta)\omega_{k}\cdot n}{2}\leq a+1\leq\frac{n}{k}\]
where
\[\rho_{k,n}=\frac{n(k+1)}{2k(n-1)},\quad\sigma_{k,\eta}=\frac{(1-\eta)^{3}(k \omega_{k})^{3}}{320\sqrt{k+2}},\quad\omega_{k}=\frac{(k-1)^{k-1}}{k!e^{k-1}}\]
To see Theorem C in action, one may simply set \(\eta=1/2\) to obtain results.
**Structure of the paper.3** In Section 2 we prove Theorem A1, i.e. compute the expected Euler characteristic precisely. In Section 3 we analyse the limiting spikes of the expected Euler characteristics. In Section 4 we use the classification of homotopy types arising from a nerve complex of circular arcs, to give constraints on homotopy types and compute probabilistic bounds. Theorem A2 and Theorem C are proven in Section 4. In Section 5 we use the classical method of stability of persistence diagram to prove Theorem B; this section works separately and doesn't use the Euler characteristic method.
Footnote 3: We remark that the theorems A1, A2, B, C aren’t proven in sequential order; they are arranged in that order for a clean exposition of the main results.
Theorem C takes the most work to prove. It is a simplified version of Theorem 4.8, which has a few more parameters that can be tweaked to obtain similar variants of Theorem C. Theorem 4.8 is obtained by combining three ingredients: Propositions 3.4, 4.5, and 4.7.
**Related works.**
The classical result of Hausmann shows that the Vietoris-Rips complex constructed from the manifold with a small scale parameter recovers the homotopy type of the manifold [12]. Another classical result of Niyogi, Smale, Weinberger shows that if a Cech complex of small filtration radius is constructed from a finite random sample of a Euclidean submanifold, then the homotopy type of the manifold is recovered with high confidence. [13]
Much work has been done for recovering topology of a manifold from its finite sample, when connectivity radius is scaled down with the sample size at a specific rate [9][11][13][10]. A central theme of this body of work is the existence of phase transitions when parameters controlling the scaling of connectivity radius are changed. For a comprehensive survey, see [16] and [8].
In comparison, the setting when connectivity radius is not scaled down with sample size is studied much less. Results on convergence of the topological quantities have been studied [15][18], but not much attention has been devoted to analysing specific manifolds.
This paper builds on two important works that characterised the Vietoris-Rips and Cech complexes of subsets of the circle: [4] and [1]. Several variants of these ideas were studied, for ellipse [6], regular polygon [7], and hypercube graph [2]. Randomness in these systems were studied using dynamical systems in [5]. One key tool to further study the topology of Vietoris-Rips and Cech complexes arising from a manifold is metric thickening [3]. Using this tool, the Vietoris-Rips complex of the higher-dimensional sphere has been characterised up to small filtration radii [14].
## Acknowledgements.
The author is grateful to Henry Adams and Tadas Temcinas for valuable discussions that led up to this paper. The author would also like to thank Vidit Nanda and Harald Oberhauser for their contributions during initial stages of this research.
Uzu Lim is supported by the Korea Foundation for Advanced Studies.
## 2 Expected Euler characteristic
In this section we compute the expected Euler characteristic precisely. We start with a simple calculation that also briefly considers the Vietoris-Rips complex, but soon after we only work with the Cech complex. Let \(\mathrm{VR}(\mathbf{X}_{n},r)\) denote the Vietoris-Rips complex of threshold \(r\). The following proposition reduces computation of expected values to the quantities \(T_{k}\) and \(Q_{k}\), defined below:
**Proposition 2.1**: _For each \(n>0\), let \(\mathbf{X}_{n}\) be the iid sample drawn uniformly from \(\mathbb{S}^{1}\). Then we have that:_
\[\mathbb{E}[\chi(\mathrm{VR}(\mathbf{X}_{n},r))]= \sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}T_{k}(r)\] \[\mathbb{E}[\chi(\breve{C}(\mathbf{X}_{n},r))]= 1+\sum_{k=1}^{n}(-1)^{k}\binom{n}{k}Q_{k}(2^{-1}-r)\]
_where \(T_{k}(r)\) is the probability that every pair of points in \(\mathbf{X}_{k}\) are within distance \(r\), and \(Q_{k}(r)\) is the probability that open arcs of radius \(r\) centered at points of \(\mathbf{X}_{k}\) cover \(\mathbb{S}^{1}\). Expectation is taken over the iid sample \(\mathbf{X}_{n}\)._
Denoting by \(s_{k}(K)\) the number of \(k\)-simplices in a simplicial complex \(K\), we have that:
\[\mathbb{E}[s_{k}(\mathrm{VR}(\mathbf{X}_{n},r))]=\binom{n}{k}T_{k}(r)\]
and thus
\[\mathbb{E}[\chi(\mathrm{VR}(\mathbf{X}_{n},r))]=\sum_{k=0}^{n-1}(-1)^{k} \mathbb{E}[s_{k}(\mathrm{VR}(\mathbf{X}_{n},r))]=\sum_{k=1}^{n}(-1)^{k-1} \binom{n}{k}T_{k}(r)\]
The relation for the Cech complex is derived in the same way, except we note the following: the probability that arcs of radius \(r\) centered at points of \(\mathbf{X}_{k}\) intersects nontrivially is equal to \(1-Q_{k}(2^{-1}-r)\). This is by De Morgan's Law: for any collection of sets \(\{U_{j}\subseteq\mathbb{S}^{1}\}_{j\in J}\), we have \(\cap_{j\in J}U_{j}=\emptyset\) iff \(\cup_{j\in J}U_{j}^{\mathrm{c}}=\mathbb{S}^{1}\). In the case of circle (of circumference 1), complement of a closed arc of radius \(r\) is an open arc of radius \(2^{-1}-r\). Applying this logic, we obtain:
\[\mathbb{E}[\chi(\breve{C}(\mathbf{X}_{n},r))]=\sum_{k=1}^{n}(-1)^{k-1}\binom{ n}{k}(1-Q_{k}(2^{-1}-r))\]
which is easily seen to be the same as the asserted expression (note that \(\sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}=1\).)
The \(Q_{k}\) were computed by Stevens in 1939 [17]. We reproduce the proof for completeness.
**Theorem 2.2** (Stevens).: _If \(k\) arcs of fixed length \(a\) are independently, identically and uniformly sampled from the circle of circumference 1, then the probability that these arcs cover the circle is equal to the following:_
\[Q_{k}(a/2)=\sum_{l=0}^{\lfloor 1/a\rfloor}(-1)^{l}\binom{k}{l}(1-la)^{k-1}\]
Proof.: The proof is an application of inclusion-exclusion principle. Consider the set \(E=\{(x_{1},\ldots x_{k})|0\leq x_{1}<\cdots<x_{k}<1\}\). For each collection of indices \(J\subseteq\{1,\ldots k\}\), define \(\bar{E}_{J}\) and \(E_{J}\) as the following subsets of \(E\):
\[E_{J}= \{(x_{1},\ldots x_{k})\in E|j\in J\iff x_{j+1}-x_{j}>a\}\] \[\bar{E}_{J}= \{(x_{1},\ldots x_{k})\in E|j\in J\implies x_{j+1}-x_{j}>a\}= \bigsqcup_{J^{\prime}\supseteq J}E_{J^{\prime}}\]
By definition, we have \(\operatorname{Vol}(E_{\emptyset})=Q_{k}(a/2)\). To compute it, we apply the inclusion-exclusion principle for the membership of each \(E_{J}\) over \(\bar{E}_{J^{\prime}}\) whenever \(J^{\prime}\supseteq J\). Noting the relation \(\sum_{l=1}^{k}(-1)^{l+1}\binom{k}{l}=1\), we see that:
\[1=\sum_{J\subseteq\{1,\ldots k\}}\operatorname{Vol}(E_{J})=\operatorname{Vol }(E_{\emptyset})-\sum_{\emptyset\neq J\subseteq\{1,\ldots k\}}(-1)^{\#J} \operatorname{Vol}(\bar{E}_{J})\]
Finally, if \(l=\#J\) and \(l\leq\lfloor 1/a\rfloor\), then \(\operatorname{Vol}(\bar{E}_{J})=(1-la)^{n-1}\). This is because demanding gap conditions \(x_{i+1}-x_{i}>a\) at \(l\) places is equivalent to sampling \(n-1\) points from an interval of length \(1-la\)4. Meanwhile if \(l>\lfloor 1/a\rfloor\), then we always have \(\operatorname{Vol}(\bar{E}_{J})=0\). Plugging these into the above equation, we get:
Footnote 4: This can be seen more precisely by considering the collection \(E^{\prime}\) of \((y_{1},\ldots y_{k-1})\) defined by \(y_{i}=x_{i+1}-x_{i}>0\) and \(\sum y_{i}\leq 1\), and then considering the subset \(E^{\prime}_{J}\) defined by \(y_{i}>a\) for \(i\in J\). The quantity of interest is \(\operatorname{Vol}(E^{\prime}_{J})/\operatorname{Vol}(E^{\prime})\). Furthermore, the map \((y_{1},\ldots y_{k-1})\mapsto(y_{1}-\mathbf{1}_{1\in J},\ldots y_{k-1}- \mathbf{1}_{k-1\in J})\) isometrically maps \(E^{\prime}_{J}\) to \((1-la)\cdot E^{\prime}\), so that \(\operatorname{Vol}(E^{\prime}_{J})=(1-la)^{k-1}\operatorname{Vol}(E^{\prime})\) due to the \((k-1)\)-dimensional volume scaling. This is exactly the original claim.
\[\operatorname{Vol}(E_{\emptyset})=1+\sum_{l=1}^{\lfloor 1/a\rfloor}(-1)^{l} \binom{k}{l}(1-la)^{n-1}\]
as desired. \(\square\)
We then get the following:
**Theorem 2.3** (Theorem A1).: _Expected Euler characteristic of random Cech complex on a circle of unit circumference obtained from \(n\) points and filtration radius \((1-r)/2\) is:_
\[\bar{\chi}\left(n,\frac{1-r}{2}\right)=\sum_{k=1}^{\lfloor 1/r\rfloor}{n\choose k }(1-kr)^{k-1}(kr)^{n-k}\]
_In particular, \(\bar{\chi}(n,r)\) is a continuous piecewise-polynomial function in \(r\)._
Proof.: Substituting the \(Q_{k}\) expression in, we get:
\[\bar{\chi}\left(n,\frac{1-r}{2}\right)= 1+\sum_{k=1}^{n}(-1)^{k}{n\choose k}Q_{k}(r)\] \[= 1+\sum_{l=0}^{\lfloor 1/r\rfloor}\sum_{k=1}^{n}(-1)^{k+l}{n \choose k}{k\choose l}(1-rl)^{k-1}\] \[= \sum_{l=1}^{\lfloor 1/r\rfloor}\sum_{k=1}^{n}(-1)^{k+l}{n \choose k}{k\choose l}(1-rl)^{k-1}\]
where we switched the order of summation in the second equality, and isolating the \(l=0\) part cancels out the \(1\) in the third equality. Noting that \({n\choose k}{k\choose l}={n\choose l}{n-l\choose k-l}\), we further get:
\[\bar{\chi}\left(n,\frac{1-r}{2}\right) =\sum_{l=1}^{\lfloor 1/r\rfloor}(-1)^{l}{n\choose l}(1-rl)^{-1} \sum_{k=l}^{n}{n-l\choose k-l}(rl-1)^{k}\] \[=\sum_{l=1}^{\lfloor 1/r\rfloor}{n\choose l}(1-rl)^{l-1}\sum_{k=0 }^{n-l}{n-l\choose k}(rl-1)^{k}\] \[=\sum_{l=1}^{\lfloor 1/r\rfloor}{n\choose l}(1-rl)^{l-1}(rl)^{n-l}\]
## 3. Limit behaviour of Euler characteristic
We prove a sequence of lemmas in this section to characterise the limiting spikes in Figure 1. The main idea is that only one summand in the expected Euler characteristic contributes mainly to the spike, and this is a polynomial term that can be studied with calculus. The
main results of this section are Propositions 3.3 and 3.4. The two lemmas leading up to it are exercises in calculus that explain the specific situation of our expected Euler characteristic.
**Lemma 3.1**.: _For \(a,b\geq 1\), the function \(f(t)=t^{a}(1-t)^{b}\) satisfies the following:_
_(a) In the range \(0\leq t\leq 1\), \(f(t)\) achieves the unique maximum value at \(t=a/(a+b)\):_
\[\max_{0\leq t\leq 1}f(t)=f\left(\frac{a}{a+b}\right)=\frac{a^{a}b^{b}}{(a+b)^{a+b}}\]
_Also, \(f(t)\) is increasing on \(t\in(0,a/(a+b))\) and decreasing on \(t\in(a/(a+b),1)\)._
_(b) The following linear lower bounds hold:_
\[f(t)\geq u\bigg{(}(a+b)vt-av+1\bigg{)}\text{, when }0<t<\frac{a}{a+b}\] \[f(t)\geq u\bigg{(}-(a+b)vt+av+1\bigg{)}\text{, when }\frac{a}{a+b}<t<1\]
_where_
\[u=\frac{a^{a}b^{b}}{(a+b)^{a+b}},\quad v=\sqrt{\frac{a+b}{ab}}\]
_(c) For each \(\lambda\in[0,1]\), we have that:_
\[\left|t-\frac{a}{a+b}\right|<\frac{(1-\lambda)\sqrt{ab}}{(a+b)^{3/2}}\implies t ^{a}(1-t)^{b}>\lambda u\]
Proof.: The first two derivatives are:
\[f^{\prime}(t)= \bigg{(}a-(a+b)t\bigg{)}t^{a-1}(1-t)^{b-1}\] \[f^{\prime\prime}(t)= \bigg{(}(a+b)(a+b-1)t^{2}+2a(1-a-b)t+a(a-1)\bigg{)}t^{a-2}(1-t)^{ b-2}\]
The first derivative vanishes at \(t\in\{a/(a+b),0,1\}\) and the second derivative vanishes at \(t\in\{t_{0}\pm\eta_{0},0,1\}\) where
\[t_{0}=\frac{a}{a+b},\quad\eta_{0}=\frac{1}{a+b}\sqrt{\frac{ab}{a+b-1}}>\frac{ \sqrt{ab}}{(a+b)^{3/2}}=\eta_{1}\]
The first derivative is positive at \((0,a/(a+b))\) and negative at \((a/(a+b),1)\). Thus the maximum at \(t\in[0,1]\) is given by:
\[f(t_{0})=\frac{a^{a}b^{b}}{(a+b)^{a+b}}\]
Thus
\[f(t)\geq\frac{f(t_{0})}{\eta_{1}}(t-t_{0})+f(t_{0}),\,\text{when}\,\,0<t<t_{0}\]
\[f(t)\geq\frac{-f(t_{0})}{\eta_{1}}(t-t_{0})+f(t_{0}),\,\text{when}\,\,t_{0}<t<1\]
and
\[\pm\frac{f(t_{0})}{\eta_{1}}(t-t_{0})+f(t_{0})= \frac{a^{a}b^{b}}{(a+b)^{a+b}}\left(\pm\frac{(a+b)^{3/2}}{\sqrt{ab }}\left(t-\frac{a}{a+b}\right)+1\right)\]
(c) follows from the linear bound of (b).
**Lemma 3.2**.: _Let \(m,n\geq 1\) be integers and define:_
\[f_{m,n}(t)=\binom{n}{m}(mt)^{m-1}(1-mt)^{n-m}\]
_Then \(f_{m,n}\) satisfies the following:_
_(a) \(f_{m,n}(t)\) is increasing when \(0<t<t_{0}\) and decreasing when \(t_{0}<t<1/m\) where \(t_{0}=\frac{1}{n-1}(1-\frac{1}{m})\)._
_(b) The maximum over \(0<t<1/m\) is given by:_
\[\max_{0<mt<1}f_{m,n}(t)=f_{m,n}(t_{0})=\binom{n}{m}\frac{(m-1)^{m-1}(n-m)^{n-m }}{(n-1)^{n-1}}\]
_(c) For each \(\lambda\in[0,1]\), we have that:_
\[|t-t_{0}|<\frac{(1-\lambda)\sqrt{(m-1)(n-m)}}{m(n-1)^{3/2}}\implies f_{m,n}(t)> \lambda f_{m,n}(t_{0})\]
_(d) The normalised limit of maximum as \(n\to\infty\) is given by:_
\[\lim_{n\to\infty}\frac{\max_{0<t<1/m}f_{m,n}(t)}{n}=\frac{(m-1)^{m-1}}{m!e^{m -1}}\]
Proof.: (a)-(c) follow from the previous lemma. For (d), we compute:
\[\lim_{n\to\infty}\frac{\max_{0<t<1/m}f_{m,n}(t)}{n}= \frac{(m-1)^{m-1}}{m!}\lim_{n\to\infty}(n-1)(n-2)\cdots(n-m+1) \frac{(n-m)^{n-m}}{(n-1)^{n-1}}\] \[= \frac{(m-1)^{m-1}}{m!}\lim_{n\to\infty}\frac{(n-m)^{n-m}}{(n-1)^ {n-m}}\]
and also
\[\lim_{n\to\infty}\frac{(n-m)^{n-m}}{(n-1)^{n-m}}=\lim_{n\to\infty}\left(1- \frac{m-1}{n-1}\right)^{n-m}=\lim_{n\to\infty}\left(1-\frac{m-1}{n-1}\right)^ {n-1}=\frac{1}{e^{m-1}}\]
which gives the desired expression.
Proposition 3.3.: _Suppose that \(m,n\) are integers with \(2\leq m<\sqrt{n}\). The following holds for \(\bar{\chi}(n,r)\)._
_(a) The following bounds hold:_
\[a_{m,n}\leq\frac{\bar{\chi}(n,s_{m,n})}{n}\leq M\leq a_{m,n}+b_{m,n}\]
_where_
\[M= \max\left\{\frac{1}{n}\bar{\chi}\left(n,\frac{1-r}{2}\right)\; \bigg{|}\;r\in\left(\frac{1}{m+1},\frac{1}{m}\right)\right\}\] \[s_{m,n}= \frac{(m-1)n}{2(n-1)m}\] \[a_{m,n}= \binom{n}{m}\frac{(m-1)^{m-1}(n-m)^{n-m}}{n(n-1)^{n-1}}\] \[b_{m,n}= en^{m-1}\left(1-\frac{1}{m+1}\right)^{n-1}\]
_(b) We have the following limits:_
\[\lim_{n\to\infty}a_{m,n}=\frac{(m-1)^{m-1}}{m!e^{m-1}},\quad\lim_{n\to\infty}b _{m,n}=0\]
_(c) Suppose additionally that \(n>2m^{2}\). Then for each \(\lambda\in[0,1]\), we have that:_
\[\left|r-\frac{n-m}{(n-1)m}\right|<\frac{(1-\lambda)\sqrt{(m-1)(n-m)}}{m(n-1)^ {3/2}}\implies\frac{1}{n}\bar{\chi}\left(n,\frac{1-r}{2}\right)>\lambda a_{m,n}\]
_This condition for \(r\) in particular satisfies \(r\in\left(\frac{1}{m+1},\frac{1}{m}\right]\)._
Proof.: Let \(r\in\left(\frac{1}{m+1},\frac{1}{m}\right]\) and also write \(r=\frac{1}{m}-t\), with \(t\in\left[0,\frac{1}{m(m+1)}\right]\). Then we may rewrite the normalised expected Euler characteristic as follows:
\[\bar{\chi}\left(n,\frac{1-r}{2}\right)= \sum_{k=1}^{m}\binom{n}{k}(1-kr)^{k-1}(kr)^{n-k}\] \[= \sum_{k=1}^{m}\binom{n}{k}\left(1-\frac{k}{m}+kt\right)^{k-1} \left(\frac{k}{m}-kt\right)^{n-k}\]
We now claim that the \(k=m\) term is the dominant one among the above summands. As such, we split the above sum as:
\[\bar{\chi}\left(n,\frac{1-r}{2}\right)=f_{m,n}(t)+E\]
where
\[f_{m,n}(t)= \binom{n}{m}(mt)^{m-1}(1-mt)^{n-m},\] \[E= \sum_{k=1}^{m-1}\binom{n}{k}\left(1-\frac{k}{m}+kt\right)^{k-1} \left(\frac{k}{m}-kt\right)^{n-k}\]
Since \(m<\sqrt{n}\), we have \(s_{m,n}=\frac{1}{n-1}(1-\frac{1}{m})<\frac{1}{m(m+1)}\). Therefore, the previous Lemma tells us that \(f_{m,n}(t)\) achieves (global) maximum at \(\tilde{s}\in\left(0,\frac{1}{m(m+1)}\right]\), with the maximum value given by:
\[f_{m,n}(\tilde{s})=n\cdot a_{m,n}\text{, where }a_{m,n}=\binom{n}{m}\frac{(m-1)^ {m-1}(n-m)^{n-m}}{n(n-1)^{n-1}}\]
We also bound \(E\) as follows, using the inequality \(\frac{m}{m+1}<1-mt\leq 1\):
\[E= \sum_{k=1}^{m-1}\binom{n}{k}\left(1-\frac{k}{m}(1-mt)\right)^{k-1 }\left(\frac{k}{m}(1-mt)\right)^{n-k}\] \[\leq \sum_{k=1}^{m-1}\binom{n}{k}\left(1-\frac{1}{m+1}\right)^{k-1} \left(1-\frac{1}{m}\right)^{n-k}\] \[\leq \sum_{k=1}^{m-1}\frac{n^{k}}{k!}\left(1-\frac{1}{m+1}\right)^{n-1}\] \[\leq en^{m-1}\left(1-\frac{1}{m+1}\right)^{n-1}\]
This shows (a). Now (b) follows from the previous Lemma and the fact that \((1-\frac{1}{m+1})^{n}\) term causes exponential decay for \(b_{m,n}\).
(c) follows from (c) of the previous Lemma. We additionally impose the condition \(n>2m^{2}\), so that the endpoints of \(t\) satisfying the condition fall in the interval \(t\in\left[0,\frac{1}{m(m+1)}\right)\).
**Proposition 3.4**.: _Let \(m\geq 2\), \(\epsilon>0\). The following holds for sufficiently large \(n\):_
\[r\in\left[\alpha^{-},\alpha^{+}\right]\implies\frac{1}{n}\bar{\chi}\left(n, \frac{1-r}{2}\right)\in\left[(1-\epsilon)\omega_{m},(1+\epsilon)\omega_{m}\right]\]
_where_
\[\alpha^{\pm}= \frac{n-m}{(n-1)m}\left(1\pm\frac{\epsilon\sqrt{m-1}}{n}\right), \quad\omega_{m}=\frac{(m-1)^{m-1}}{m!e^{m-1}}\]
Proof.: This follows directly from the previous Proposition. \(\alpha^{\pm}\) are slight relaxations of the interval in (c), where we set \(\lambda=1-\epsilon\):
\[\left[\frac{n-m}{(n-1)m}-\epsilon R_{1},\frac{n-m}{(n-1)m}+\epsilon R _{1}\right]\supseteq\left[\frac{n-m}{(n-1)m}(1-\epsilon R_{2}),\frac{n-m}{(n-1 )m}(1+\epsilon R_{2})\right]\] \[\text{where }R_{1}=\frac{\epsilon\sqrt{(m-1)(n-m)}}{m(n-1)^{3/2}}, \quad R_{2}=\frac{\sqrt{m-1}}{n}\]
## 4. Random homotopy types
### Constraints on homotopy types
Let \(\mathbf{U}_{n}=\{i/n\mid i=0,1,\ldots n-1\}\subset\mathbb{S}^{1}\) be the set of \(n\) equally spaced points. Let \(\mathcal{N}(n,k)\) be the nerve complex on \(\mathbf{U}_{n}\) defined by the open cover consisting of closed intervals \([i/n,(i+k)/n]\).
**Lemma 4.1**.: _We have that:_
\[\breve{C}(\mathbf{U}_{n},r)=\breve{C}\!\!\left(\mathbf{U}_{n},\frac{\lfloor 2 rn\rfloor}{2n}\right)=\mathcal{N}(n,\lfloor 2rn\rfloor)\]
The following result is from [4]:
**Proposition 4.2**.: \[\mathcal{N}(n,k)\simeq\begin{cases}\vee^{n-k-1}\mathbb{S}^{2l}&\text{if } \frac{k}{n}=\frac{l}{l+1}\\ \mathbb{S}^{2l+1}&\text{if }\frac{k}{n}\in\left(\frac{l}{l+1},\frac{l+1}{l+2} \right)\end{cases}\]
_Note that if \((k,n)=(jl,j(l+1))\), then \(n-k-1=j-1\), so that \(\vee^{n-k-1}\mathbb{S}^{2l}=\vee^{j-1}\mathbb{S}^{2l}\)._
Using the above, we easily show that:
**Proposition 4.3**.: _Given \(r\in(0,1/2)\), the following two subsets of \(\mathbb{Z}^{3}\) are equal:_
\[\left\{(n,a,b)\;\middle|\;\breve{C}(\mathbf{U}_{n},r)\simeq\vee^{a}\mathbb{S} ^{2b}\right\}=\left\{((a+1)(b+1),a,b)\;\middle|\;b+1\leq\tilde{r}^{-1},\;a+1 \leq\frac{1}{1-(b+1)\tilde{r}}\right\}\]
_where \(\tilde{r}=1-2r\). In particular, if \(\tilde{r}^{-1}\in[k,k+1)\), then \(b\in\{0,1,2,\ldots k-1\}\) and we have \(a\leq k-1\) when \(b\leq k-2\)._
Proof.: To have \(\mathcal{N}(n,\lfloor 2rn\rfloor)=\tilde{\mathrm{C}}(\mathbf{U}_{n},r)\simeq\lor^{a} \mathbb{S}^{2b}\), we see from the previous Proposition that the condition is given by \((\lfloor 2rn\rfloor,n)=((a+1)b,(a+1)(b+1))\). This determines \(n\) from \((a,b)\). The condition on \(\lfloor 2rn\rfloor\) is then:
\[(a+1)b \leq 2r(a+1)(b+1)<(a+1)b+1\] \[\Longleftrightarrow\tilde{r}(b+1)\leq 1,\;a<\tilde{r}(a+1)(b+1)\] \[\Longleftrightarrow(b+1)\leq\tilde{r}^{-1},\;(a+1)<(1-\tilde{r} (b+1))^{-1}\]
as desired.
**Remark.** At fixed \(l\), let \(k=b+k_{0}\). Then \(\frac{1}{1-(b+1)/k}=1+\frac{b+1}{k_{0}-1}\) and changing \(k_{0}\) by a single value can have a heavy effect on the upper bound.
**Proposition 4.4**.: _Let \(r\in(0,1/2)\) and \(n\) be given; define \(\tilde{r}=1-2r\) and let \(k=\lfloor\tilde{r}^{-1}\rfloor\). We have the following relations between subsets of \(\mathbb{Z}^{2}\):_
\[\left\{(a,b)\;\middle|\;\tilde{\mathrm{C}}(\mathbf{Y},r)\simeq \lor^{a}\mathbb{S}^{2b},\;\mathbf{Y}\subset\mathbb{S}^{1},\;\#\mathbf{Y}=n\right\}\] \[= \bigg{\{}(a,b)\;\middle|\;\tilde{\mathrm{C}}(\mathbf{U}_{m},r) \simeq\lor^{a}\mathbb{S}^{2b},\,m\leq n\right\}\] \[\subseteq \bigg{\{}(a,b)\;\middle|\;b+1\leq k,\;a+1\leq\min\left(\frac{n}{b +1},\frac{1}{1-(b+1)\tilde{r}}\right)\bigg{\}}\] \[\subseteq \bigg{\{}(a,b)\;\middle|\;b+1\leq k-1,\;a+1\leq\frac{k}{k-b-1} \bigg{\}}\cup\bigg{\{}(a,k-1)\;\middle|\;a+1\leq\frac{n}{k}\bigg{\}}\]
_where in the final expression, \(k/0=\infty\) by convention._
Proof.: The first equality holds because for every \(\mathbf{Y}\subset\mathbb{S}^{1}\), there exists \(\mathbf{Y}^{\prime}\subset\mathbf{Y}\) such that \(\tilde{\mathrm{C}}(\mathbf{Y},r)\simeq\tilde{\mathrm{C}}(\mathbf{Y}^{\prime},r )\simeq\tilde{\mathrm{C}}(\mathbf{U}_{m},r)\), where \(m=\#\mathbf{Y}^{\prime}\)[4]. The first inclusion follows from the previous Proposition. The second inclusion follows from separating the two cases \(b+1<k\) and \(b+1=k\).
### Probabilistic bounds
For a topological space \(K\), we define the following notation for probability:
\[p(K,n,r)=\mathbb{P}[\tilde{\mathrm{C}}(\mathbf{X}_{n},r)\simeq K]\]
We generally have the following:
\[\bar{\chi}(n,r)=\mathbb{E}[\chi(\check{\mathrm{C}}(\mathbf{X}_{n},r))]=\sum_{K} \chi(K)\cdot p(K,n,r)\]
where the sum is well-defined because there are only finitely many combinatorial structures that \(\check{\mathrm{C}}(\mathbf{X}_{n},r)\) can take. Furthermore if we let \(k=\lfloor(1-2r)^{-1}\rfloor\), then Proposition 4.4 tells us that:
\[\{K\,|\,p(K,n,r)>0\}\subseteq\bigg{\{}\vee^{a}\mathbb{S}^{2b}\,\bigg{|}\,b+1 \leq k-1,a+1\leq\frac{k}{k-b-1}\bigg{\}}\cup\bigg{\{}\vee^{a}\mathbb{S}^{2k-2} \,\bigg{|}\,a+1\leq\frac{n}{k}\bigg{\}}\]
From this we infer that5:
Footnote 5: By convention, in the summation we only consider \(a\leq 0\) when \(b=0\) and instead consider \(a>0\) when \(b>0\). This is so that the singleton set \(\vee^{a}\mathbb{S}^{2b}=*\) is counted only once.
\[\bar{\chi}(n,r)= A_{<k}+A_{k}\] \[\text{where }A_{<k} =\sum_{\begin{subarray}{c}0\leq b\leq k-2\\ (a+1)(k-b-1)\leq k\end{subarray}}(a+1)\cdot p(\vee^{a}\mathbb{S}^{2b},n,r)\] \[A_{k} =\sum_{1<a+1\leq n/k}(a+1)\cdot p(\vee^{a}\mathbb{S}^{2k-2},n,r)\]
where we used \(\chi(\vee^{a}\mathbb{S}^{2b})=a+1\). Since sum of probabilities is \(1\), applying the constraint \((a+1)(k-b-1)\leq k\) implies that \(A_{<k}\leq k\). This implies the following:
Proposition 4.5.: _The following holds:_
\[A_{k}\leq\bar{\chi}(n,r)\leq k+A_{k}\]
_where_
\[A_{k}=\sum_{1<a+1\leq n/k}(a+1)\cdot p(\vee^{a}\mathbb{S}^{2k-2},n,r)\]
Corollary 4.6 (Theorem A2).: _Let \(k\geq 2\). Given \(\epsilon>0\), the following hold for sufficiently large \(n\):_
\[1-2r\in\left(\frac{1}{k+1},\frac{1}{k}\right]\implies\frac{\bar{\chi}(n,r)}{ n}-\epsilon\leq\frac{\bar{b}_{2k-2}(n,r)}{n}\leq\frac{\bar{\chi}(n,r)}{n}\]
Now we're interested in controlling probabilities that \(\vee^{a}\mathbb{S}^{2k-2}\) appear, with large \(n\). For this, we further define following:
\[p_{a}= p(\vee^{a}\mathbb{S}^{2k-2},n,r)\] \[l= \lfloor n/k\rfloor-1\] \[\tilde{\delta}= \lceil\delta n/k\rceil-1\] \[A_{k,\delta}:= \sum_{\tilde{\delta}\leq a\leq l}(a+1)\cdot p_{a}=\sum_{\delta n/ k\leq a+1\leq n/k}(a+1)\cdot p_{a}\] \[B_{k,\delta}:= \sum_{\tilde{\delta}\leq a\leq l}p_{a}\]
To produce bounds for \(B_{k,\delta}\), we split \(A_{k}\) into two parts:
\[A_{k}=\left(2p_{1}+3p_{2}+\cdots+\tilde{\delta}p_{\tilde{\delta}-1}\right)+ \left((\tilde{\delta}+1)p_{\tilde{\delta}}+\cdots+(l+1)p_{l}\right)\]
from which it directly follows that:
\[(\tilde{\delta}+1)B_{k,\delta}\leq A_{k}\leq\tilde{\delta}(1-B_{k,\delta})+(l +1)B_{k,\delta}\]
and therefore
\[\implies (\tilde{\delta}+1)B_{k,\delta}\leq A_{k}\leq\tilde{\delta}+(l+1- \tilde{\delta})B_{k,\delta}\] \[\implies \frac{A_{k}-\tilde{\delta}}{l+1-\tilde{\delta}}\leq B_{k,\delta} \leq\frac{A_{k}}{\tilde{\delta}+1}\] \[\implies \frac{A_{k}-\lceil\delta n/k\rceil+1}{\lfloor n/k\rfloor-\lceil \delta n/k\rceil+1}\leq B_{k,\delta}\leq\frac{A_{k}}{\lceil\delta n/k\rceil}\] \[\implies \frac{A_{k}-\delta n/k}{(1-\delta)(n/k)+1}\leq B_{k,\delta}\leq \frac{A_{k}}{\delta n/k}\]
In summary, we have the following:
**Proposition 4.7**.: _Let \(n\in\mathbb{Z}^{+}\), \(\delta\in(0,1)\), \(r\in(0,1/2)\) be given, and let \(k=\lfloor(1-2r)^{-1}\rfloor\). The following holds:_
\[\frac{kA_{k}-\delta n}{(1-\delta)n+k}\leq B_{k,\delta}\leq\frac{kA_{k}}{ \delta n}\]
_where_
\[A_{k}=\sum_{1<a+1\leq n/k}(a+1)p_{a},\quad B_{k,\delta}:=\sum_{\delta n/k\leq a +1\leq n/k}p_{a},\quad p_{a}:=p(\vee^{a}\mathbb{S}^{2k-2},n,r)\]
Now Propositions 3.4, 4.5, 4.7 imply the following, which is a more general version of Theorem C:
**Theorem 4.8**.: _Let \(r\in[\frac{1}{4},\frac{1}{2})\) and let \(k=\lfloor(1-2r)^{-1}\rfloor\). Given \(\epsilon,\delta\in(0,1)\), the following implication holds for large enough \(n\):_
\[1-2r\in[\alpha^{-},\alpha^{+}]\implies B_{k,\delta}\in[\beta^{-}-\epsilon, \beta^{+}+\epsilon]\]
_where_
\[\alpha^{\pm}= \frac{1}{k}\frac{n-k}{n-1}\bigg{(}1\pm\frac{\sqrt{k-1}}{n}\cdot \frac{\delta(1-\delta)}{5}\cdot\epsilon\bigg{)},\] \[\beta^{-}= \frac{k\omega_{k}-\delta}{1-\delta},\quad\beta^{+}=\frac{k\omega _{k}}{\delta}\] \[\omega_{k}= \frac{(k-1)^{k-1}}{k!e^{k-1}}\] \[B_{k,\delta}:= \sum_{\delta n/k\leq a+1\leq n/k}p(\vee^{a}\mathbb{S}^{2k-2},n,r)\]
_The bounds \(\beta^{\pm}\) satisfy \(\beta^{-}\leq k\omega_{k}\leq\beta^{+}\). Also \(\beta^{-}>0\) iff \(\delta<k\omega_{k}\) and \(\beta^{+}<1\) iff \(\delta>k\omega_{k}\)._
Proof.: We first describe the heuristic reasoning for the bounds, which is rather simple. Proposition 4.7 gives us:
\[\frac{kA_{k}-\delta n}{(1-\delta)n+k}\leq B\leq\frac{kA_{k}}{\delta n}\]
By Proposition 3.4 and 4.5, the upper bound has the following approximations:
\[\frac{kA_{k}}{\delta n}\approx\frac{k\bar{\chi}}{\delta n}\approx\frac{k\omega _{k}}{\delta}\]
and similarly the lower bound has the following approximations:
\[\frac{kA_{k}-\delta n}{(1-\delta)n+k}\approx\frac{kA_{k}-\delta n}{(1-\delta) n}\approx\frac{k\bar{\chi}-\delta}{1-\delta}\approx\frac{k\omega_{k}-\delta}{1-\delta}\]
The actual proof becomes more complicated due to using a different choice of \(\epsilon\) in applying Proposition 3.4.
Let \(\epsilon^{\prime}=\delta(1-\delta)\cdot\epsilon/5\). We apply Proposition 3.4 with \(\epsilon^{\prime}\) taking the role of \(\epsilon\), and this gives the choice of \(\alpha^{\pm}\) in the theorem. Therefore \(r\in[\alpha^{-},\alpha^{+}]\) implies the following:
\[(1-\epsilon^{\prime})\omega_{k}\leq\frac{\bar{\chi}}{n}\leq(1+\epsilon^{ \prime})\omega_{k} \tag{4.1}\]
Before going further, we note the following inequalities for \(\epsilon^{\prime}\), which we will use later:
\[\epsilon^{\prime}=\frac{\delta(1-\delta)\epsilon}{4+1}\leq\frac{ \delta(1-\delta)\epsilon}{4+\delta(1-\delta)\epsilon}\] \[\implies\frac{\epsilon^{\prime}}{1-\epsilon^{\prime}}\leq\frac{ \delta(1-\delta)\epsilon}{4} \tag{4.2}\] \[\implies\frac{\epsilon^{\prime}}{1-\epsilon^{\prime}}\leq\min \left(4\delta,\delta^{-1}-1,1\right)\cdot\frac{\epsilon}{4}\]
**Upper bound.**
By Equation (4.1) and Proposition 4.5, we have:
\[\frac{k\omega_{k}}{\delta}\geq\frac{1}{1+\epsilon^{\prime}}\frac{k\bar{\chi}} {\delta n}\geq\frac{1}{1+\epsilon^{\prime}}\frac{kA_{k}}{\delta n}\]
By Equation (4.2), we have that:
\[\frac{1}{1+\epsilon^{\prime}}\frac{kA_{k}}{\delta n}\geq\frac{kA_{k}}{\delta n }-\epsilon\]
Then Proposition 4.7 applies and we have the upper bound.
**Lower bound.**
By Equation (4.1) and Proposition 4.5, we have:
\[\frac{k\omega_{k}-\delta}{1-\delta}\leq\frac{1}{1-\delta}\bigg{(}\frac{1}{1- \epsilon^{\prime}}\frac{k\bar{\chi}}{n}-\delta\bigg{)}\leq\frac{1}{1-\delta} \bigg{(}\frac{1}{1-\epsilon^{\prime}}\frac{k^{2}+kA_{k}}{n}-\delta\bigg{)}\]
Let \(L_{0}\) be the right hand side. We rewrite it as follows:
\[L_{0}=L_{1}+E_{1}=L_{2}+E_{1}+E_{2}\]
where
\[L_{1}= \frac{kA_{k}-\delta n}{(1-\delta)(1-\epsilon^{\prime})n},\,E_{1}= \frac{\delta\epsilon^{\prime}+k^{2}/n}{(1-\delta)(1-\epsilon^{\prime})}\] \[L_{2}= \frac{kA_{k}-\delta n}{(1-\delta)n+k},\,E_{2}=\frac{kA_{k}- \delta n}{(1-\delta)(1-\epsilon^{\prime})n}\cdot\frac{k+(1-\delta)n\epsilon^{ \prime}}{(1-\delta)n+k}\]
By Equation (4.2), the relation \(kA_{k}\leq n\) and by taking \(n\) large enough, we see that
\[E_{1},E_{2}\leq\epsilon/2\]
This implies that:
\[\frac{k\omega_{k}-\delta}{1-\delta}-\epsilon\leq L_{0}-\epsilon=L_{2}+E_{1}+E _{2}-\epsilon\leq L_{2}\]
Then again Proposition 4.7 applies and we have the lower bound.
We remark that Theorem C is obtained by setting \(\epsilon=\delta=(1-\alpha)k\omega_{k}/2\). The gap \(\alpha^{+}-\alpha^{-}\) is replaced by a smaller but simpler quantity.
## 5 Odd spheres
We prove Theorem B using the stability of persistence diagram. In this case, we will be using the Cech complex constructed from the full set of the circle, and then bound the Gromov-Hausdorff distance between the full circle and a finite sample of it. We use the following result from [1]:
**Theorem 5.1**.: _The homotopy types of the Rips and Cech complexes on the circle of unit circumference are as follows:_
\[\operatorname{VR}(\mathbb{S}^{1},r) \simeq\begin{cases}\mathbb{S}^{2l+1}&\text{,if }\frac{l}{2l+1}<r<\frac{l+1}{2l+3}\\ \bigvee^{\mathfrak{c}}\mathbb{S}^{2l}&\text{,if }r=\frac{l}{2l+1}\end{cases}\] \[\tilde{C}(\mathbb{S}^{1},r) \simeq\begin{cases}\mathbb{S}^{2l+1}&\text{,if }\frac{l}{2l+2}<r<\frac{l+1}{2l+4}\\ \bigvee^{\mathfrak{c}}\mathbb{S}^{2l}&\text{,if }r=\frac{l}{2l+2}\end{cases}\]
_where \(\mathfrak{c}\) is the cardinality of the continuum._
We also note the stability of persistence:
**Theorem 5.2** (Stability of Persistence).: _If \(X,Y\) are metric spaces and \(\mathcal{D}_{k}M\) is the \(k\)-dimensional persistence diagram of persistence module \(M\), then_
\[\operatorname{d}_{B}(\mathcal{D}_{k}\mathbf{VR}(X),\mathcal{D}_{k }\mathbf{VR}(Y)) \leq\operatorname{d}_{GH}(X,Y)\] \[\operatorname{d}_{B}(\mathcal{D}_{k}\tilde{C}(X),\mathcal{D}_{k }\tilde{C}(Y)) \leq\operatorname{d}_{GH}(X,Y)\]
_where \(\operatorname{d}_{GH}\) denotes the Gromov-Hausdorff distance._
The following proposition is a more precise version of Theorem B, which specifies an explicit lower bound for the probabilities of homotopy equivalence:
Proposition 5.3.: _For each \(l\geq 0\) and \(t\in(\frac{l}{2l+2},\frac{l+1}{2l+4})\), the following holds with probability at least \(Q_{n}(r^{\prime}/2)\):_
\[\tilde{C}(\mathbf{X}_{n},t)\simeq\mathbb{S}^{2l+1}\]
_where \(r^{\prime}\) is:_
\[r^{\prime}=\frac{1}{4(l+1)(l+2)}-\left|t-\frac{2l^{2}+4l+1}{4(l+1)(l+2)}\right|\]
Proof.: Consider a random sample \(\mathbf{X}_{n}=(X_{1},\ldots X_{n})\). Then with probability \(Q_{n}(r/2)\), arcs of radius \(r\) centered at \(\mathbf{X}_{n}\) covers \(\mathbb{S}^{1}\), so that \(\mathrm{d}_{GH}(\mathbf{X}_{n},\mathbb{S}^{1})\leq\mathrm{d}_{H}(\mathbf{X}_{ n},\mathbb{S}^{1})\leq r\). This implies:
\[\mathrm{d}_{B}(\mathcal{D}_{k}\tilde{C}(\mathbf{X}_{n}),\mathcal{D}_{k}\tilde {C}(\mathbb{S}^{1}))\leq\mathrm{d}_{GH}(\mathbf{X}_{n},\mathbb{S}^{1})\leq r\]
For each \(l\geq 0\), we have that:
\[\mathcal{D}_{2l+1}\tilde{C}(\mathbb{S}^{1})=\left\{\left(\frac{l}{2l+2},\frac{ l+1}{2l+4}\right)\right\}\]
so that the definition of the bottleneck distance implies that
\[\exists(u,v)\in\mathcal{D}_{2l+1}\tilde{C}(\mathbf{X}_{n})\]
\[\text{with }\frac{l}{2l+2}-r\leq u\leq\frac{l}{2l+2}+r\] \[\frac{l+1}{2l+4}-r\leq v\leq\frac{l+1}{2l+4}+r\]
This implies that whenever \(\frac{l}{2l+2}+r\leq t\leq\frac{l+1}{2l+4}-r\), we have:
\[1\leq\dim H_{2l+1}\tilde{C}(\mathbf{X}_{n},t)\]
and due to the enumeration of possible homotopy types, we have that:
\[\tilde{C}(\mathbf{X}_{n},t)\simeq\mathbb{S}^{2l+1}\]
The condition translates to \(\left|t-\frac{1}{2}\left(\frac{l}{2l+2}+\frac{l+1}{2l+4}\right)\right|<\frac{ 1}{2}\left(\frac{l+1}{2l+4}-\frac{l}{2l+2}\right)-r\), or equivalently
\[\left|t-\frac{2l^{2}+4l+1}{4(l+1)(l+2)}\right|<\frac{1}{4(l+1)(l+2)}-r\]
and thus we obtain the proof. | We high-dimensional topologiesをランダムなCech複素体で構築した円上のCech複素体から発生するものを特徴付けます。期待されるエウレカ特性曲線は計算され、限られたスパイクが観察されます。スパイクは、フィルタリング半径が縮小する間に期待されるベティ数が増えていくことを意味します。ランダムなCech複素体の Homotopy 型が奇数次元球体または偶数次元球体の束であることを証明することで、 Homotopy 型の確率的上限を提示します。サンプルサイズが増大するにつれてフィルタリング半径を縮小する従来の方法とは異なり、私たちの発見は、フィルタリング半径の全幅が「トポロジーノイズ」として見られる可能性のある興味深い系的な挙動を示しています。 |
2301.04383 | quasiconformal mappings and a Bernstein type theorem over exterior
domains in $\mathbb{R}^2$ | We establish the H\"{o}lder estimate and the asymptotic behavior at infinity
for $K$-quasiconformal mappings over exterior domains in $\mathbb{R}^2$. As a
consequence, we prove an exterior Bernstein type theorem for fully nonlinear
uniformly elliptic equations of second order in $\mathbb{R}^2$. | Dongsheng Li, Rulin Liu | 2023-01-11T10:05:13 | http://arxiv.org/abs/2301.04383v1 | # Quasiconformal mappings and a Bernstein type theorem over exterior domains in \(\mathbb{R}^{2}\)
###### Abstract.
We establish the Holder estimate and the asymptotic behavior at infinity for \(K\)-quasiconformal mappings over exterior domains in \(\mathbb{R}^{2}\). As a consequence, we prove an exterior Bernstein type theorem for fully nonlinear uniformly elliptic equations of second order in \(\mathbb{R}^{2}\).
Key words and phrases:Quasiconformal Mappings, Exterior Bernstein Type Theorem, Fully Nonlinear Elliptic Equations, Asymptotic Behavior This research is supported by NSFC 12071365. Dongsheng Li [email protected] School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, P.R.China 710049. Rulin Liu [email protected] School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, P.R.China 710049.
in \(\mathbb{R}^{n}\) is a quadratic polynomial if we assume the concavity of \(F\) and the boundedness of the Hessian \(D^{2}u\). For \(n=2\), the same conclusion follows from the Nirenberg estimate [10] and the boundedness of \(D^{2}u\) without the concavity of \(F\).
In 2020, Li, Li and Yuan [8] established a higher dimensional exterior Bernstein type theorem for the fully nonlinear elliptic equation (1.2), namely, for \(n\geq 3\), the solution of (1.2) in \(\mathbb{R}^{n}\setminus\bar{B}_{1}(0)\) tends to a quadratic polynomial as \(|x|\to\infty\) if \(F\) is convex (or concave or the level set of \(F\) is convex) and \(D^{2}u\) is bounded. As applications of this theorem, the authors obtained the exterior Bernstein type theorems of Monge-Ampere equations, special Lagrangian equations, quadratic Hessian equations and inverse harmonic Hessian equations for \(n\geq 3\). As for \(n=2\), the authors studied these three specific equations one by one to obtain the corresponding exterior Bernstein type theorem instead of establishing the general theorem to equation (1.2). Indeed, the method in [8] does not work for two dimensional problems. Roughly speaking, there are two steps in [8] to establish the exterior Bernstein type theorem. First, by the concavity of \(F\) and the boundedness of \(D^{2}u\), the authors made use of the Evans-Krylov estimate and the weak Harnack inequality to show the existence of the limit \(A\) of \(D^{2}u\) at infinity, which actually holds for all \(n\geq 2\). Second, it is crucial to get the decay rate of \(|D^{2}u-A|\) as \(|x|\to\infty\). This can be done by using barrier functions as \(n\geq 3\) while unfortunately, such barrier does not exist as \(n=2\).
In this paper, we establish the exterior Bernstein type theorem for fully nonlinear elliptic equation (1.2) in \(\mathbb{R}^{2}\) by using \(K\)-quasiconformal mappings. The main result goes as the following.
**Theorem 1.1**.: _Let \(u\) be a viscosity solution of (1.2) in the exterior domain \(\mathbb{R}^{2}\setminus\bar{\Omega}\), where \(F\in C^{1,1}\) is a fully nonlinear uniformly elliptic operator with ellipticity constants \(\lambda\) and \(\Lambda\), and \(\Omega\) is a bounded domain of \(\mathbb{R}^{2}\). If \(\left\|D^{2}u\right\|_{L^{\infty}(\mathbb{R}^{2}\setminus\bar{\Omega})}\leq M <+\infty\), then there exists a unique symmetric matrix \(A\in\mathbb{R}^{2\times 2}\), \(b,e\in\mathbb{R}^{2},c,d\in\mathbb{R}\) such that for any \(0<\alpha<1\),_
\[u(x)=\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x+d\log|x|+c+e\frac{x}{|x|^{2}}+O \left(|x|^{-1-\alpha}\right)\text{ as }|x|\to\infty,\]
_where_
\[d=\frac{1}{2\pi}\left(\int\limits_{\partial\Omega}u_{\nu}\mathrm{d}s+\iint \limits_{\mathbb{R}^{2}\setminus\bar{\Omega}}(\Delta u(x)-\mathrm{tr}A) \mathrm{d}x_{1}\mathrm{d}x_{2}-\mathrm{tr}A|\Omega|\right), \tag{1.3}\]
\(\nu\) _is the unit outward normal of the boundary \(\partial\Omega\). Furthermore, if \(F\) is smooth, then we have_
\[\left|D^{k}\left(u(x)-\frac{1}{2}x^{\mathrm{T}}Ax-b\cdot x-d\log|x|-c-e\frac{ x}{|x|^{2}}\right)\right|=O\left(|x|^{-1-\alpha-k}\right)\text{ as }|x|\to\infty\]
_for all \(k\in\mathbb{N}\)._
**Remark 1.2**.: _In Theorem 1.1, the concavity (or convexity or convexity of the level set \(\{N|F(N)=0\}\)) of \(F\) is not needed that is however an essential assumption in [8]._
As aforementioned, we will use \(K\)-quasiconformal mappings to study equation (1.2) over exterior domains. \(K\)-quasiconformal mappings play a special role in studying the Holder continuity of solutions of two dimensional second order partial differential equations, which was developed by Morrey [9], Nirenberg [10] and Finn and Serrin [4]. In this paper, we will demonstrate the asymptotic behavior of \(K\)-quasiconformal mappings at infinity over exterior domains (Cf. Theorem 2.2 in Section 2). By using this result to (1.2) over exterior domains, we shall not only show \(D^{2}u\) has a limit \(A\) at infinity, but get the decay rate of \(|D^{2}u-A|\) as \(|x|\to\infty\) as well. After this, Theorem 1.1 will be proved by standard arguments.
The organization of this paper goes as follows. In section 2, we study the Holder continuity and asymptotic behavior at infinity of \(K\)-quasiconformal mappings over exterior domains, which implies the gradient Holder estimate and the gradient asymptotic behavior at infinity of solutions of linear elliptic equations over exterior domains. In section 3, we give the proof of Theorem 1.1.
## 2. Exterior \(K\)-quasiconformal mappings
Let's begin with the definition of exterior \(K\)-quasiconformal mappings in \(\mathbb{R}^{2}\setminus\bar{\Omega}\). We refer to [5] for the original definition of \(K\)-quasiconformal mappings.
**Definition 2.1**.: _A mapping \(w(x)=(p(x),q(x))\) from \(\mathbb{R}^{2}\setminus\bar{\Omega}\)\((\Omega\subset\mathbb{R}^{2}\) is bounded \()\) in \(x=(x_{1},x_{2})\) plane to \(w=(p,q)\) plane is exterior \(K\)-quasiconformal in \(\mathbb{R}^{2}\setminus\bar{\Omega}\) if \(p,q\in C^{1}\left(\mathbb{R}^{2}\setminus\bar{\Omega}\right)\) and_
\[p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}\leq 2K\left(p_{1}q_{2}-p_{2}q_{1}\right) \tag{2.1}\]
_holds for all \(x\in\mathbb{R}^{2}\setminus\bar{\Omega}\) with some constant \(K>0\), where \(p_{i}=\frac{\partial p(x)}{\partial x_{i}},q_{i}=\frac{\partial q(x)}{\partial x _{i}},i=1,2\)._
For \(K\)-quasiconformal mappings, the apriori interior Holder estimate is well known (Cf. [10, Lemma 2] and [4, Theorem 1]).
For exterior \(K\)-quasiconformal mappings, we have the following Holder estimate over exterior domain and the asymptotic behavior at infinity.
**Theorem 2.2**.: _Let \(w=(p,q)\) be exterior \(K\)-quasiconformal in \(\mathbb{R}^{2}\setminus\bar{\Omega}\)\((\Omega\subset\mathbb{R}^{2}\) is bounded \()\) with \(K\geq 1\), and suppose \(|w|\leq M\). Then, for any \(\Omega^{\prime}\supset\supset\Omega\) with \(d=\mathrm{dist}(\Omega,\partial\Omega^{\prime})\),_
\[|w(x)-w(y)|\leq C\left|x-y\right|^{\alpha},x,y\in\mathbb{R}^{2}\setminus \overline{\Omega^{\prime}}.\]
_and \(w(x)\) tends to a limit \(w(\infty)\) at infinity such that_
\[|w(x)-w(\infty)|\leq C|x|^{-\alpha}\text{ for any }x\in\mathbb{R}^{2}\setminus \overline{\Omega^{\prime}}, \tag{2.2}\]
_where \(\alpha=K-(K^{2}-1)^{\frac{1}{2}}\), \(C\) depends only on \(K,d\) and \(M\)._
**Remark 2.3**.: _The results in Theorem 2.2 are also valid for \(p,q\in W^{1,2}_{\rm loc}(\mathbb{R}^{2}\setminus\bar{\Omega})\cap L^{\infty}( \mathbb{R}^{2}\setminus\bar{\Omega})\)._
To prove Theorem 2.2, we first state the following Holder continuity of \(K\)-quasiconformal mappings with singularities.
**Lemma 2.4** ([4, Theorem 3]).: _Let \(w=(p,q)\) be \(K\)-quasiconformal in a domain \(\Omega\) of \(x=(x_{1},x_{2})\) plane, except at a set \(T\) of isolated points in \(\Omega\). Assume \(|w|\leq M\). Then \(w\) can be defined, or redefined, at the points of \(T\) so that the resulting function is continuous in \(\Omega\), and in any compact subregion \(\Omega^{\prime}\) of \(\Omega\) with \(d={\rm dist}(\Omega^{\prime},\partial\Omega)\), \(w(x)\) satisfies a uniform Holder inequality_
\[|w(x)-w(y)|\leq C|x-y|^{\alpha},x,y\in\Omega^{\prime}, \tag{2.3}\]
_where \(\alpha=K-(K^{2}-1)^{\frac{1}{2}}\), \(C\) depends only on \(K,d\) and \(M\)._
We prove Theorem 2.2 by making use of the Kelvin transform. For this purpose, we establish the following lemma, which states that the Kelvin transform of an exterior \(K\)-quasiconformal mapping is \(K\)-quasiconformal with an isolated singularity.
**Lemma 2.5**.: _Let \(w=(p,q)\) be exterior \(K\)-quasiconformal in \(\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\). Let \(\tilde{p}\) and \(\tilde{q}\) be the Kelvin transform of \(p\) and \(q\) respectively, namely_
\[\tilde{p}(x)=p\left(\frac{x}{|x|^{2}}\right),\tilde{q}(x)=q\left(\frac{x}{|x| ^{2}}\right),x\in B_{1}(0)\setminus\{0\}.\]
_Then, \(\tilde{w}=(\tilde{q},\tilde{p})\) is \(K\)-quasiconformal in \(B_{1}(0)\setminus\{0\}\)._
Proof.: Calculating directly, we have
\[\tilde{p}_{1} =\left(|x|^{-2}-2x_{1}^{2}|x|^{-4}\right)p_{1}+\left(-2x_{1}x_{2} |x|^{-4}\right)p_{2},\] \[\tilde{p}_{2} =\left(-2x_{1}x_{2}|x|^{-4}\right)p_{1}+\left(|x|^{-2}-2x_{2}^{2} |x|^{-4}\right)p_{2},\] \[\tilde{q}_{1} =\left(|x|^{-2}-2x_{1}^{2}|x|^{-4}\right)q_{1}+\left(-2x_{1}x_{2} |x|^{-4}\right)q_{2},\]
and
\[\tilde{q}_{2}=\left(-2x_{1}x_{2}|x|^{-4}\right)q_{1}+\left(|x|^{-2}-2x_{2}^{2 }|x|^{-4}\right)q_{2}.\]
It's easy to see that
\[\tilde{p}_{1}^{2}+\tilde{p}_{2}^{2}+\tilde{q}_{1}^{2}+\tilde{q}_{2}^{2}=|x|^{ -4}\left(p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}\right),\]
and
\[\tilde{p}_{1}\tilde{q}_{2}-\tilde{p}_{2}\tilde{q}_{1}=-|x|^{-4}\left(p_{1}q_{ 2}-p_{2}q_{1}\right).\]
Since \(w=(p,q)\) is exterior \(K\)-quasiconformal over \(\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\), we deduce by Definition 2.1 that \(p\) and \(q\) satisfy (2.1) in \(\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\) for some \(K\geq 1\). So, we obtain that in \(B_{1}(0)\setminus\{0\}\),
\[\tilde{p}_{1}^{2}+\tilde{p}_{2}^{2}+\tilde{q}_{1}^{2}+\tilde{q}_{2}^{2}\leq 2K \left(\tilde{p}_{2}\tilde{q}_{1}-\tilde{p}_{1}\tilde{q}_{2}\right),\]
which implies \(\tilde{w}=(\tilde{q},\tilde{p})\) is \(K\)-quasiconformal in \(B_{1}(0)\setminus\{0\}\).
Proof of Theorem 2.2.: Assume without loss of generality that \(B_{1}(0)\subset\Omega\). Let \(\tilde{p}\) and \(\tilde{q}\) be the Kelvin transform of \(p\) and \(q\) respectively given by Lemma 2.5. Let
\(\hat{\Omega}=\left\{\frac{x}{|x|^{2}}\Big{|}x\in\mathbb{R}^{2}\setminus\bar{\Omega}\right\}\) and for any \(\Omega^{\prime}\supset\supset\Omega\), \(\tilde{\Omega}=\left\{\frac{x}{|x|^{2}}\Big{|}x\in\mathbb{R}^{2}\setminus\overline {\Omega^{\prime}}\right\}\). Then by Lemma 2.5, \(\tilde{w}=(\tilde{q},\tilde{p})\) is \(K\)-quasiconformal in \(\hat{\Omega}\setminus\{0\}\) with \(K\geq 1\). Since \(|w|\leq M\) implies \(|\tilde{w}|\leq M\), applying Lemma 2.4 to \(\tilde{w}\) with \(T=\{0\}\), we know that
\[|\tilde{w}(x)-\tilde{w}(y)|\leq C|x-y|^{\alpha},x,y\in\tilde{\Omega},\]
which implies that \(\tilde{w}(x)\) has a limit \(\tilde{w}(0)\) at \(0\) and for all \(x\in\tilde{\Omega}\),
\[|\tilde{w}(x)-\tilde{w}(0)|\leq C|x|^{\alpha},\alpha=K-\left(K^{2}-1\right)^{ \frac{1}{2}}.\]
Transforming back to exterior domain, we have that
\[|w(x)-w(y)|\leq C|x-y|^{\alpha},x,y\in\mathbb{R}^{2}\setminus\overline{\Omega ^{\prime}}\]
and \(w(x)\) has a limit \(w(\infty)=\tilde{w}(0)\) at infinity with
\[|w(x)-w(\infty)|\leq C|x|^{-\alpha},x\in\mathbb{R}^{2}\setminus\overline{ \Omega^{\prime}},\]
where \(\alpha=K-(K^{2}-1)^{\frac{1}{2}}\), \(C\) depends only on \(K,d\) and \(M\).
The theorem is therefore proved.
Next we consider linear elliptic equation
\[L(u)=a_{11}(x)u_{11}(x)+2a_{12}(x)u_{12}(x)+a_{22}(x)u_{22}(x)=0, \tag{2.4}\]
where \(L\) is uniformly elliptic, that is, there exist \(0<\lambda\leq\Lambda\) such that
\[\lambda(\xi_{1}^{2}+\xi_{2}^{2})\leq a_{11}\xi_{1}^{2}+2a_{12}\xi_{1}\xi_{2}+ a_{22}\xi_{2}^{2}\leq\Lambda(\xi_{1}^{2}+\xi_{2}^{2}),\forall\xi=(\xi_{1},\xi_{2}) \in\mathbb{R}^{2} \tag{2.5}\]
and
\[\frac{\Lambda}{\lambda}\leq\gamma \tag{2.6}\]
for some constant \(\gamma\geq 1\).
For uniformly elliptic equation (2.4) in a domain \(\Omega\) of \(\mathbb{R}^{2}\), it follows from the interior Holder estimate of \(K\)-quasiconformal mappings that its bounded solutions have interior \(C^{1,\alpha}\) estimate [5, Theorem 12.4].
For uniformly elliptic equation (2.4) over exterior domain in \(\mathbb{R}^{2}\), we can establish the gradient Holder estimate and the gradient asymptotic behavior of solutions at infinity by the virtue of Theorem 2.2.
**Theorem 2.6**.: _Let \(\Omega\) be a bounded domain of \(\mathbb{R}^{2}\) and \(u\in C^{2}(\mathbb{R}^{2}\setminus\bar{\Omega})\) be a solution of equation (2.4) in \(\mathbb{R}^{2}\setminus\bar{\Omega}\). Suppose \(|Du(x)|\leq M\). Then for any \(\Omega^{\prime}\supset\supset\Omega\) with \(d=\operatorname{dist}(\Omega,\partial\Omega^{\prime})\),_
\[|Du(x)-Du(y)|\leq C\left|x-y\right|^{\alpha},x,y\in\mathbb{R}^{2}\setminus \overline{\Omega^{\prime}}\]
_and \(Du(x)\) has a limit \(Du(\infty)\) at infinity with_
\[|Du(x)-Du(\infty)|\leq C|x|^{-\alpha},x\in\mathbb{R}^{2}\setminus\overline{ \Omega^{\prime}}, \tag{2.7}\]
_where \(\alpha\) depends only on \(\gamma\), \(C\) depends only on \(\gamma,d\) and \(M\)._
**Remark 2.7**.: _The results in Theorem 2.6 are also valid for \(u\in W^{2,2}(\mathbb{R}^{2}\setminus\bar{\Omega})\)._
Proof of Theorem 2.6.: Assume without loss of generality that \(\lambda=1\). Let \(p=u_{1},q=u_{2}\). By equation (2.4), (2.5) and (2.6), we have (see details in [5])
\[p_{1}^{2}+p_{2}^{2}\leq a_{11}p_{1}^{2}+2a_{12}p_{1}p_{2}+a_{22}p_{2}^{2}=a_{22} J,J=p_{2}q_{1}-p_{1}q_{2},x\in\mathbb{R}^{2}\setminus\bar{\Omega}\]
and
\[q_{1}^{2}+q_{2}^{2}\leq a_{11}J,x\in\mathbb{R}^{2}\setminus\bar{\Omega}.\]
Noticing that \(2\leq a_{11}+a_{22}=1+\Lambda\leq 1+\gamma\), we arrive at
\[p_{1}^{2}+p_{2}^{2}+q_{1}^{2}+q_{2}^{2}\leq\left(a_{11}+a_{22}\right)J\leq \left(1+\gamma\right)J,x\in\mathbb{R}^{2}\setminus\bar{\Omega},\]
which implies that \(w=(q,p)\) is exterior \(K\)-quasiconformal over \(\mathbb{R}^{2}\setminus\bar{\Omega}\) with \(K=\frac{1+\gamma}{2}\). Since \(|Du|\leq M\) in \(\mathbb{R}^{2}\setminus\bar{\Omega}\), Theorem 2.2 therefore asserts that for any \(\Omega^{\prime}\supset\supset\Omega\),
\[|Du(x)-Du(y)|\leq C|x|^{-\alpha},x,y\in\mathbb{R}^{2}\setminus\overline{ \Omega^{\prime}}\]
and \(Du(x)\) tends to a limit \(Du(\infty)=(p(\infty),q(\infty))\) at infinity with
\[|Du(x)-Du(\infty)|\leq C|x|^{-\alpha},x\in\mathbb{R}^{2}\setminus\overline{ \Omega^{\prime}},\]
where \(\alpha\) depends only on \(\gamma\), \(C\) depends only on \(\gamma,d\) and \(M\).
## 3. Exterior Bernstein type theorem
In this section, we give the proof of the exterior Bernstein type theorem, i.e., Theorem 1.1. As we remarked before, we don't need the concavity or convexity of \(F\).
We find the limit \(A\) of the Hessian \(D^{2}u\) at infinity and estimate the decay rate of \(|D^{2}u-A|\) first.
**Theorem 3.1**.: _Let \(u\) be as in Theorem 1.1. Then there exists a symmetric matrix \(A\in\mathbb{R}^{2\times 2}\) such that_
\[D^{2}u(x)\to A\text{ as }|x|\to\infty\]
_and_
\[|D^{2}u(x)-A|\leq C|x|^{-\alpha}\text{ as }|x|\to\infty,\]
_which implies_
\[\left|u(x)-\frac{1}{2}x^{\mathrm{T}}Ax\right|\leq C|x|^{2-\alpha}\text{ as }|x|\to\infty, \tag{3.1}\]
_where \(\alpha\in(0,1)\) is a constant depending only on \(\lambda\) and \(\Lambda\), \(C\) is a positive constant depending only on \(\lambda\), \(\Lambda\), and \(M\)._
**Remark 3.2**.: _If \(u\in C^{2}\), then we don't need \(F\in C^{1,1}\) in Theorem 3.1._
Proof of Theorem 3.1.: By the virtue of the Nirenberg estimate, we can see that viscosity solutions to the equation (1.2) in \(\mathbb{R}^{2}\) are always \(C^{2,\alpha}\) for some \(\alpha\in(0,1)\) depending only on the ellipticity constants of \(F\). It follows from \(F\in C^{1,1}\) and the Schauder estimate that \(u\in C^{3,\gamma}(\mathbb{R}^{2}\setminus\bar{\Omega})\) for any \(\gamma\in(0,1)\). Then we take derivative
with respect to \(x_{k}\) (\(k=1,2\)) on both sides of equation (1.2) to obtain
\[a_{ij}(x)v_{ij}(x)=0,x\in\mathbb{R}^{2}\setminus\bar{\Omega}, \tag{3.2}\]
where \(a_{ij}(x)=F_{M_{ij}}\left(D^{2}u(x)\right)\) and \(v(x)=u_{k}(x)\).
Since \(\left\|D^{2}u\right\|_{L^{\infty}\left(\mathbb{R}^{2}\setminus\bar{\Omega} \right)}\leq M\), we know \(|Dv(x)|\leq M\). Applying Theorem 2.6 to equation (3.2) in \(\mathbb{R}^{2}\setminus\bar{\Omega}\), we have that \(Dv(x)\) tends to a limit \(Dv(\infty)\) at infinity and for any \(\Omega^{\prime}\supset\supset\Omega\),
\[|Dv(x)-Dv(\infty)|\leq C|x|^{-\alpha},x\in\mathbb{R}^{2}\setminus\overline{ \Omega^{\prime}}.\]
Then by the arbitrarity of \(k\), we conclude that there exists a symmetric matrix \(A\in\mathbb{R}^{2\times 2}\) such that \(D^{2}u(x)\to A\) as \(|x|\to\infty\) and
\[\left|D^{2}u(x)-A\right|\leq C|x|^{-\alpha}\text{ as }|x|\to\infty.\]
It follows that
\[\left|u(x)-\frac{1}{2}x^{\mathrm{T}}Ax\right|\leq C|x|^{2-\alpha}\text{ as }|x|\to\infty,\]
where \(\alpha\in(0,1)\) depends only on \(\lambda\) and \(\Lambda\), \(C>0\) depends only on \(\lambda\), \(\Lambda\) and \(M\).
Based on Theorem 3.1, we will find the finer asymptotic behavior of \(u\) by standard arguments. To do this, we need the following three lemmas which are well known. For readers' convenience, we show the proofs of them. Lemma 3.3 gives the higher order estimates. Lemma 3.4 and Lemma 3.5 are used to determine the linear term, logarithm term and constant term of the asymptotics of \(u\).
**Lemma 3.3**.: _Let \(\phi\) be a viscosity solution of the equation_
\[F\left(D^{2}\phi(x)+A\right)=0,x\in\mathbb{R}^{2}\setminus\bar{B}_{1}(0),\]
_where \(F\in C^{1,1}\) is a fully nonlinear uniformly elliptic operator with ellipticity constants \(\lambda\) and \(\Lambda\), and \(A\in\mathbb{R}^{2\times 2}\) is symmetric matrix, satisfying \(F(A)=0\). Suppose that for some constants \(\beta>0\) and \(\rho<2\),_
\[|\phi(x)|\leq\beta|x|^{\rho},x\in\mathbb{R}^{2}\setminus\bar{B}_{1}(0).\]
_Then there exists some constant \(r=r(\beta,\rho)\geq 1\) such that for \(k=0,1,2,3\),_
\[\left|D^{k}\phi(x)\right|\leq C|x|^{\rho-k},x\in\mathbb{R}^{2}\setminus\bar{B }_{r}(0),\]
_where \(C\) depends only on \(\lambda\), \(\Lambda\), \(\beta\) and \(\rho\)._
Proof.: By \(F\in C^{1,1}\), the Nirenberg estimate and the Schauder estimate, \(\phi(x)\in C^{3,\gamma}\) for any \(\gamma\in(0,1)\).
Fix \(x\in\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\) with \(|x|>6\) and let
\[\bar{\phi}(y)=\left(\frac{2}{|x|}\right)^{2}\phi\left(x+\frac{|x|}{2}y\right), y\in B_{1}(0).\]
Since
\[F(A)=0\]
\[F(D^{2}\bar{\phi}(y)+A)=0,y\in B_{1}(0),\]
we see that
\[\bar{a}_{ij}(y)\bar{\phi}_{ij}(y)=0,y\in B_{1}(0),\]
where \(\bar{a}_{ij}(y)=\int_{0}^{1}F_{M_{ij}}\left(tD^{2}\bar{\phi}(y)+A\right)\mathrm{ d}t\). By the Schauder estimate, we have that for \(k=0,1,2,3\),
\[\left|D^{k}\bar{\phi}(0)\right|\leq\|\bar{\phi}\|_{L^{\infty}(\bar{B}_{1}(0))} \leq C|x|^{\rho-2},\]
which implies
\[\left|D^{k}\phi(x)\right|\leq C|x|^{\rho-k},\]
where \(C\) depends only on \(\lambda\), \(\Lambda\), \(\beta\) and \(\rho\).
**Lemma 3.4**.: _Suppose \(f(x)=O(|x|^{-\beta})\) as \(|x|\to\infty\) with \(\beta>1\). Then for any \(\varepsilon>0\), the equation_
\[\Delta u(x)=f(x)\text{ in }\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\]
_has a solution \(u(x)=O(|x|^{2-\beta+\varepsilon})\) as \(|x|\to\infty\)._
Proof.: Let
\[u(x)=-\frac{1}{2\pi}\int\limits_{\mathbb{R}^{2}\setminus\bar{B}_{1}(0)}(\log| x-y|-\log|y|)f(y)\mathrm{d}y.\]
Then
\[\Delta u(x)=f(x),x\in\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\]
and for any \(\varepsilon>0\),
\[|u(x)|\leq C(\varepsilon)|x|^{2-\beta+\varepsilon},x\in\mathbb{R}^{2}\setminus \bar{B}_{1}(0).\]
**Lemma 3.5**.: _Let \(u(x)=O(|x|^{\beta})\) be a smooth solution of_
\[\Delta u(x)=0,x\in\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\]
_for some \(0<\beta<2\). Then_
\[u=b\cdot x+d\log|x|+c+O\left(|x|^{-1}\right)\text{ as }|x|\to\infty, \tag{3.3}\]
_where \(b\in\mathbb{R}^{2}\), \(c,d\in\mathbb{R}\). Particularly, for \(0<\beta<1\), (3.3) holds with \(b=0\)._
Proof.: Let \(\xi(z)=u_{1}(x)-iu_{2}(x),z=x_{1}+ix_{2}\). Then \(\xi(z)\) is an analytic function in \(\mathbb{R}^{2}\setminus\bar{B}_{1}(0)\) and the growth of \(\xi(z)\) is at most of order \(|z|^{\beta-1}\). Since \(0<\beta<2\), the Laurent expansion of \(\xi(z)\) has the form
\[\xi(z)=a_{0}+a_{-1}z^{-1}+a_{-2}z^{-2}+\cdots,z\in\mathbb{R}^{2}\setminus\bar{ B}_{1}(0), \tag{3.4}\]
where \(a_{0},a_{-1},a_{-2},\cdots\) are all complex numbers. Thus we have
\[Du(x)=D(b\cdot x+c_{1})+D(a_{-1}\log|x|+c_{2})+O(|x|^{-2})\text{ as }|x|\to\infty,\]
where \(b=(\text{Re }a_{0},-\text{Im }a_{0})^{\mathrm{T}},c_{1},c_{2}\in\mathbb{R}\). Since \(\text{Re}\int a_{-1}z^{-1}=\text{Re}(a_{-1}\log z)\) as a part of expansion of a real function \(u\), \(a_{-1}\) must be a real number. Integrating the above,
we see that
\[u=b\cdot x+d\log|x|+c+O(|x|^{-1})\text{ as }|x|\to\infty,\]
where \(c\in\mathbb{R},d=a_{-1}\in\mathbb{R}\).
Particularly, for \(0<\beta<1\), (3.4) holds with \(a_{0}=0\). Therefore, the above equality also holds with \(b=0\).
Proof of Theorem 1.1.: We divide the proof into six steps.
_Step 1. Improving estimate (3.1)._
Let
\[\varphi(x)=u(x)-\frac{1}{2}x^{\mathrm{T}}Ax.\]
Then by Theorem 3.1,
\[\varphi(x)=O(|x|^{2-\alpha})\]
and \(\varphi(x)\) satisfies
\[F(D^{2}\varphi(x)+A)=0,x\in\mathbb{R}^{2}\setminus\bar{\Omega}. \tag{3.5}\]
Suppose \(R_{0}\geq 1\) such that \(\Omega\subset B_{R_{0}}(0)\). It follows from Lemma 3.3 that for all \(x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0)\),
\[|D\varphi(x)|\leq C|x|^{1-\alpha},\big{|}D^{2}\varphi(x)\big{|}\leq C|x|^{- \alpha},\big{|}D^{3}\varphi(x)\big{|}\leq C|x|^{-1-\alpha}. \tag{3.6}\]
Taking derivative to both sides of equation (3.5) with respect to \(x_{k}\) (\(k=1,2\)), we know that \(\varphi_{k}\) satisfies equation
\[a_{ij}(x)\left(\varphi_{k}(x)\right)_{ij}=0,x\in\mathbb{R}^{2}\setminus\bar{B }_{R_{0}}(0), \tag{3.7}\]
where \(a_{ij}(x)=F_{M_{ij}}\left(D^{2}\varphi(x)+A\right)\). Since it follows from Theorem 3.1 that \(D^{2}\varphi(x)\to 0\) as \(|x|\to\infty\), we know
\[a_{ij}(x)\to F_{M_{ij}}(A)\text{ as }|x|\to\infty. \tag{3.8}\]
Assuming without loss of generality that \(F_{M_{ij}}(A)=\delta_{ij}\), then by \(F\in C^{1,1}\),
\[|\delta_{ij}-a_{ij}|\leq C|x|^{-\alpha} \tag{3.9}\]
for some \(C>0\). We obtain that for all \(x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0)\),
\[\varphi_{k}(x)=O(|x|^{1-\alpha})\]
and
\[\Delta(\varphi_{k})(x)=\left(\delta_{ij}-a_{ij}(x)\right)(\varphi_{k})_{ij}(x )=O(\big{|}x|^{-\alpha}|x|^{-1-\alpha}\big{)}=O\left(|x|^{-1-2\alpha}\right). \tag{3.10}\]
By Lemma 3.4, for any \(0<\varepsilon<\alpha\), there exists
\[v(x)=O(|x|^{1-2\alpha+\varepsilon})\]
satisfying the equation (3.10). Then
\[\Delta(\varphi_{k}-v)(x)=0,x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0) \tag{3.11}\]
and
\[\varphi_{k}(x)-v(x)=O(|x|^{1-\alpha}).\]
Therefore Lemma 3.5 states
\[\varphi_{k}(x)-v(x)=d\log|x|+c+O\left(|x|^{-1}\right)\text{ as }|x|\to\infty\]
for some \(b\in\mathbb{R}^{2},c\in\mathbb{R}\). Hence, for \(k=1,2\),
\[\varphi_{k}(x)=O(|x|^{1-2\alpha+\varepsilon}).\]
By the arbitrarity of \(k\), we see
\[\varphi(x)=O(|x|^{2-2\alpha+\varepsilon}).\]
Since \(0<\varepsilon<\alpha\), we have improved the estimate (3.1) a little.
We repeat the arguments above \(n\) times, where \(n\) is determined by the following way. Fix \(0<\varepsilon<\alpha\) and let \(n\) be an integer such that \(0<1-2^{n}\alpha+(2^{n}-1)\varepsilon<\frac{1}{8}\), i.e. \(n=\left[\log_{2}\frac{\frac{7}{8}-\varepsilon}{\alpha-\varepsilon}\right]+1\). Then we get an appropriate improved estimate
\[\varphi(x)=O(|x|^{2-2^{n}\alpha+(2^{n}-1)\varepsilon})=O(|x|^{1+\delta}),x \in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0)\]
with \(\delta=1-2^{n}\alpha+(2^{n}-1)\varepsilon<\frac{1}{8}\).
_Step 2. Determining the linear term._
We obtain by Lemma 3.3 that for \(\delta\in(0,\frac{1}{8})\) and all \(x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0)\),
\[|D\varphi(x)|\leq C|x|^{\delta},|D^{2}\varphi(x)|\leq C|x|^{-1+\delta}.\]
Since \(\varphi(x)\) satisfies equation
\[\bar{a}_{ij}(x)\varphi_{ij}(x)=0,x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0), \tag{3.12}\]
where \(\bar{a}_{ij}(x)=\int_{0}^{1}F_{M_{ij}}\left(tD^{2}\varphi(x)+A\right)\mathrm{d}t\), it follows from \(F\in C^{1,1}\) that for some \(C>0\),
\[|\bar{a}_{ij}(x)-\delta_{ij}|\leq C|x|^{-1+\delta}.\]
Thus
\[\Delta\varphi(x)=\left(\delta_{ij}-\bar{a}_{ij}(x)\right)\varphi_{ij}(x)=O \left(|x|^{-2+2\delta}\right),x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0).\]
Then Lemma 3.4 implies that for any \(\varepsilon\in(0,\frac{1}{8})\), there exists
\[v(x)=O(|x|^{2\delta+\varepsilon}),\]
satisfying
\[\Delta(\varphi-v)(x)=0,x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0).\]
Since
\[\varphi(x)-v(x)=O(|x|^{1+\delta}),\]
it follows from Lemma 3.5 that there exists \(b\in\mathbb{R}^{2}\) such that
\[\varphi(x)-v(x)=b\cdot x+O(\log|x|).\]
Hence
\[\varphi(x)=b\cdot x+O(|x|^{2\delta+\varepsilon}).\]
_Step 3. Determining the logarithm term and constant term._
Let
\[\bar{\varphi}(x)=u-\left(\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x\right).\]
Then
\[\bar{\varphi}(x)=O(|x|^{2\delta+\varepsilon})\]
and \(\bar{\varphi}(x)\) satisfies equation (3.12).
By Lemma 3.3, we see that for all \(x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0)\),
\[|D\bar{\varphi}(x)|\leq C|x|^{-1+2\delta+\varepsilon},|D^{2}\bar{\varphi}(x)| \leq C|x|^{-2+2\delta+\varepsilon}.\]
Consequently, for some \(C>0\),
\[|\bar{a}_{ij}-\delta_{ij}|\leq C|x|^{-2+2\delta+\varepsilon}\]
and
\[\Delta\bar{\varphi}(x)=(\delta_{ij}-\bar{a}_{ij})\bar{\varphi}_{ij}=O(|x|^{-4 +4\delta+2\varepsilon}).\]
Since \(\delta,\varepsilon\in(0,\frac{1}{8})\), then by Lemma 3.4, there exists
\[v(x)=O(|x|^{-2+\varepsilon^{\prime}})\]
with \(\varepsilon^{\prime}\in(0,1)\), satisfying
\[\Delta(\bar{\varphi}-v)(x)=0\]
and
\[\bar{\varphi}(x)-v(x)=O(|x|^{2\delta+\varepsilon}).\]
Thus, Lemma 3.5 leads to
\[\bar{\varphi}(x)=d\log|x|+c+O\left(|x|^{-1}\right)\text{ as }|x|\to\infty\]
for some \(c,d\in\mathbb{R}\), namely,
\[u(x)=\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x+d\log|x|+c+O(|x|^{-1}). \tag{3.13}\]
_Step 4. Determining the \(\frac{x}{|x|^{2}}\) term._
Let
\[\hat{\varphi}(x)=u(x)-\left(\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x+d\log|x|+c \right).\]
Then
\[D^{2}\hat{\varphi}=D^{2}u-A+O(|x|^{-2}).\]
By (3.13), \(D^{2}u=A+O(|x|^{-2})\), which implies
\[\left|D^{2}\hat{\varphi}\right|=O(|x|^{-2}).\]
Since \(\hat{\varphi}(x)\) satisfies equation (3.12) with \(\bar{a}_{ij}(x)=\int_{0}^{1}F_{M_{ij}}\left(t\left(D^{2}\hat{\varphi}(x)+D^{2} (d\log|x|)\right)+A\right)\mathrm{d}t\), we have that for some \(R_{0}\geq 1\) such that \(\Omega\subset B_{R_{0}}(0)\),
\[\Delta\hat{\varphi}(x)=(\bar{a}_{ij}(x)-\delta_{ij})\hat{\varphi}_{ij}(x)=:f( x)=O(|x|^{-2}|x|^{-2})=O(|x|^{-4}),x\in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0).\]
Let \(\psi(x)=\hat{\varphi}(\frac{x}{|x|^{2}})\) and \(\tilde{f}(x)=f(\frac{x}{|x|^{2}})\) be the Kelvin transform of \(\hat{\varphi}(x)\) and \(f(x)\) respectively. Then we see
\[\psi(x)=O(|x|)\]
and
\[\Delta\psi(x)=|x|^{-4}\tilde{f}(x)=:g(x)=O(1),x\in B_{\frac{1}{R_{0}}}(0).\]
From \(g\in L^{p}(B_{1/R_{0}}(0))\) for any \(p>2\), it follows that \(\psi(x)\in W^{2,p}(B_{1/R_{0}}(0))\) and hence \(\psi(x)\in C^{1,\alpha}(B_{1/R_{0}}(0))\) for \(\alpha=1-\frac{2}{p}\in(0,1)\). Then there exists \(e\in\mathbb{R}^{2}\) and \(\tilde{c}\in\mathbb{R}\) such that for some \(C>0\),
\[|\psi(x)-(e\cdot x+\tilde{c})|\leq C|x|^{1+\alpha},x\in B_{\frac{1}{R_{0}}}(0).\]
Since \(\psi(0)=0\) implies \(\tilde{c}=0\), we go back to exterior domain to get
\[\left|\hat{\varphi}(x)-e\cdot\frac{x}{|x|^{2}}\right|\leq C|x|^{-1-\alpha},x \in\mathbb{R}^{2}\setminus\bar{B}_{R_{0}}(0),\]
which leads to
\[u=\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x+d\log|x|+c+e\frac{x}{|x|^{2}}+O(|x|^{-1 -\alpha}).\]
_Step 5. Calculating the value of \(d\)._
Let \(Q(x)=\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x+c\). Then
\[u(x)=Q(x)+d\log|x|+O(|x|^{-1})\]
and
\[\Delta(u-Q)(x)=O(|x|^{-3})\]
is integrable. Let \(\nu\) be the unit outward normal of boundaries \(\partial\Omega\) and \(C_{R}=\partial B_{R}(0)\). Then by the divergence theorem, we have that for some \(R>0\) large enough,
\[\iint\limits_{B_{R}(0)\setminus\bar{\Omega}}\Delta(u-Q)(x)\mathrm{ d}x_{1}\mathrm{d}x_{2} =\int\limits_{\partial(B_{R}(0)\setminus\bar{\Omega})}(u-Q)_{\nu }\mathrm{d}s\] \[=\int\limits_{C_{R}}(d\log|x|+O(|x|^{-1}))_{\nu}(x)\mathrm{d}s- \int\limits_{\partial\Omega}(u-Q)_{\nu}\mathrm{d}s\] \[=d\int\limits_{C_{R}}\frac{x}{|x|^{2}}\cdot\nu\mathrm{d}s+O\left( \frac{1}{R}\right)-\int\limits_{\partial\Omega}u_{\nu}\mathrm{d}s+\int_{ \partial\Omega}Q_{\nu}\mathrm{d}s\] \[=2\pi d+O\left(\frac{1}{R}\right)-\int\limits_{\partial\Omega}u_ {\nu}\mathrm{d}s+\int_{\Omega}\Delta Q\mathrm{d}x\] \[=2\pi d+O\left(\frac{1}{R}\right)-\int\limits_{\partial\Omega}u_ {\nu}\mathrm{d}s+\mathrm{tr}A|\Omega|.\]
Letting \(R\to\infty\), we get (1.3).
_Step 6. Improving smoothness of the error._
Furthermore, suppose \(F\) is smooth. Let
\[\tilde{\varphi}(x)=u-\left(\frac{1}{2}x^{\mathrm{T}}Ax+b\cdot x+d\log|x|+c+e\frac {x}{|x|^{2}}\right).\]
Then, the Schauder estimate asserts that for all \(k\in\mathbb{N}\),
\[\big{|}D^{k}\tilde{\varphi}(x)\big{|}\leq C(k)|x|^{-1-\alpha-k}.\]
We complete the proof of Theorem 1.1.
**Remark 3.6**.: _(i). If the equation has some divergence structure, then we can obtain another representation for the constant \(d\), for example, the Monge-Ampere equations, the special Lagrangian equations and the inverse harmonic Hessian equations. We refer to [1] and [8] to see details._
_(ii). By the virtue of Theorem 1.1, we have expansion for the solutions to the Monge-Ampere equations, the special Lagrangian equations and the inverse harmonic Hessian equations at infinity in \(\mathbb{R}^{2}\setminus\bar{\Omega}\), namely, any solution tends to a quadratic polynomial plus a logarithm term and \(e\frac{x}{|x|^{2}}\) with the error at least \(|x|^{-1-\alpha}\), which is finer than the results in [1] and [8]._
| ```
H\"{o}lder estimatesと、∞への漸近的挙動を、$\mathbb{R}^2$の外部領域上の $K$-quasiconformal mapping に対して確立します。その結果として、完全非線形、一様性Elliptic方程式の2次の$\mathbb{R}^2$における外部Bernstein型定理を証明します。
``` |
2305.18249 | Evolution of QPOs in GX 339-4 and EXO 1846-031 with Insight-HXMT and
NICER | We conduct a spectral and timing analysis of GX 339-4 and EXO 1846-031 with
the aim of studying the evolution of Type-C QPOs with spectral parameters. The
high cadence data from Insight-HXMT and NICER allow us to track them. Type-C
QPOs appear at the end of low-hard state and/or hard-intermediate state. The
results reveal that the QPO frequency is closely related to the inner disk
radius and mass accretion rate in the two sources. Such a correlation is nicely
consistent with the dynamic frequency model. | Zuobin Zhang, Honghui Liu, Divya Rawat, Cosimo Bambi, Ranjeev Misra, Pengju Wang, Long Ji, Shu Zhang, Shuangnan Zhang | 2023-05-29T17:15:50 | http://arxiv.org/abs/2305.18249v2 | # Evolution of QPOs in Gx 339\(-\)4 and Exo 1846\(-\)031 with _Insight_-Hxmt and _Nicer_
###### Abstract
We conduct a spectral and timing analysis of GX 339\(-\)4 and EXO 1846\(-\)031 with the aim of studying the evolution of Type-C QPOs with spectral parameters. The high cadence data from _Insight_-HXMT and _NICER_ allow us to track them. Type-C QPOs appear at the end of low-hard state and/or hard-intermediate state. The results reveal that the QPO frequency is closely related to the inner disk radius and mass accretion rate in the two sources. Such a correlation is nicely consistent with the dynamic frequency model.
Subject headings:High energy astrophysics; X-ray astronomy; Low mass X-ray Binary; Stellar mass black holes
## 1. Introduction
Quasi-periodic oscillations (QPOs) refers to narrow peaks structure in the power density spectrum (PDS) that are commonly observed in X-ray binaries (XRBs) (van der Klis, 2005). In black hole systems, QPOs are mainly split into low-frequency QPOs (LFQPOs, centroid frequency \(0.1-30\) Hz), and high frequency QPOs (HFQPOs, centroid frequency \(\geq 60\) Hz) (Belloni, 2010). Samimi et al. (1979) reported the'sporadic quasi-periodic behaviour' in the light curve of GX 339\(-\)4, and Motch et al. (1983) reported the first rigorous detection of QPOs for the same source. It was immediately recognized that the QPOs would have been a powerful tool to study the accretion process around black holes.
Over the last forty years, especially after the launch of _RXTE_, we have accumulated a lot of knowledge about QPOs. Using a truncated power-law to fit the broadband noise in PDS, and a Lorentz function with the center frequency of \(\nu_{\rm QPO}\) to fit the low-frequency QPOs, Wijnands et al. (1999) found that there is an obvious positive correlation between the truncation frequency \(\nu_{\rm b}\) and the frequency of the low-frequency QPOs \(\nu_{\rm LF}\). Psaltis et al. (1999) reported that there is also a good positive correlation between the frequency of low-frequency QPOs and the frequency of the broadband noise (or high-frequency QPOs) in low mass XRBs, including black holes and neutron stars.
We have observed QPOs in most black hole XRBs, and realized that low frequency QPOs can be divided into three types: Type-A, -B, and -C QPOs, based on quality factor, noise type, fractional rms, and phase delay (e.g., Wijnands et al., 1999; Sobczak et al., 2000; Casella et al., 2005; Motta et al., 2011). Different types QPOs occupy different regions on the hardness-intensity diagram, as well as obviously distribute in different areas on the center frequency and rms plots (e.g., Motta et al., 2011). The phenomenon of rapid transition between different types of QPOs has been found in some sources, and the time scale of this phenomenon can be very short (10 s) (e.g. Homan et al., 2020). In this work, we only focus on the Type-C QPOs.
Type-C QPOs appear in the early stage of the outburst, particularly in the hard-intermediate state and the end of low-hard state. The centroid frequency varies from a few mHz to \(\sim 10\) Hz, and is tightly correlated with the spectral state. Vignarca et al. (2003) reported a positive correlation between the centroid frequency and the photon index \(\Gamma\). Motta et al. (2011) found Type-C QPOs trace out a well-defined track, and the centroid frequency obviously correlate with the corona flux and disk flux. The dependence of the QPO frequency and photon energy was illustrated by Qu et al. (2010).
In addition to the phenomenological study of QPOs, many studies has been done on the theoretical explanation behind it. Most theoretical models explain QPO phenomenon through the following two different mechanisms: instabilities of the corona-disk system (e.g., Titarchuk & Fiorito, 2004; Mastichiadis et al., 2022; Varniere et al., 2012) or the geometrical effects of general relativity (e.g., Stella & Vietri, 1998; Ingram et al., 2009). Titarchuk & Fiorito (2004) introduced a transition layer in corona-disk system that can explain the QPO phenomenon in XRBs. The disk-corona natural frequency model was proposed by Mastichiadis et al. (2022), and they argued that type-C QPOs arise from the interaction of the hot corona with the cold accretion disk. Varniere et al. (2012) suggested that LFQPOs could result from
the relativistic accretion-ejection instability (AEI). The geometrical effects model mainly refers to the precession of the corona region. This model interprets the QPOs as a Lense-Thirring precession of the innermost region of the accretion disk (e.g., Stella and Vietri, 1998; Ingram et al., 2009). In recent years, more and more observations have been analyzed to test these models. However, a unified model that can explain all QPO behaviors has not been found yet.
Recently, Misra et al. (2020) identified the QPO frequency of GRS 1915+105 as the relativistic dynamic frequency of a truncated accretion disk with _AstroSat_ observations of that source. The authors found a strong correlation between the QPO frequency divided by the accretion rate and the inner disk radius. The correlation is actually consistent with the prediction of dynamic frequency under the assumption of a standard relativistic accretion model (Novikov and Thorne, 1973). Liu et al. (2021) extended the relation to cover a wider range of variations, and confirmed the high spin nature of the black hole in GRS 1915+105 with the data of _Insight_-HXMT (dubbed HXMT; Zhang et al., 2014). We note that GRS 1915+105 is a persistent source with particular properties (Belloni et al., 2000). We would like to test if this relation holds for other sources different from GRS 1915+105, and we notice that there are two appropriate sources, GX 339\(-\)4 and EXO 1846\(-\)031, in the archive.
The XRB transient GX 339\(-\)4 is a typical low mass X-ray binary (LMXB) discovered in 1973 (Markert et al., 1973). It goes into bright outburst every a few years and all four X-ray states typically seen in XRBs have been detected in this system (e.g., Miyamoto et al., 1995; Homan and Belloni, 2005; Plant et al., 2014). GX 339\(-\)4 is located at 8-12 kpc with a black hole mass of 4-11 M\({}_{\bigodot}\)(Zdziarski et al., 2019). Strong relativistic reflection signatures have been found in this source in the hard and soft states (e.g., Garcia et al., 2015; Miller et al., 2004; Liu et al., 2022). Previous studies have found that the black hole in GX 339\(-\)4 has a very high spin (\(a_{*}\sim 0.95\), Garcia et al., 2015; Parker et al., 2016). The inclination angle of the accretion disk should have an intermediate value (Furst et al., 2015; Parker et al., 2016). Motta et al. (2011) systematically studied the properties and the behaviour of QPOs, as a function of the integrated broad-band variability and the spectral parameters. The authors suggested that the frequencies of all QPOs (including Type-C QPOs) correlate with the disk flux.
EXO 1846\(-\)031 was discovered by the European X-ray Observatory Satellite (_EXOSAT_) when it went into outburst in April 1985 and is considered a LMXB (Parmar et al., 1993; Draghis et al., 2020). _CGRO_/BATSE detected a second outburst in 1994 (Zhang et al., 1994). Since then, the source was in a quiescent state for 25 years. EXO 1846\(-\)031 had a new outburst in 2019, which was monitored by X-ray missions (e.g., _MAXI_/GSC; HXMT; _NuSTAR_) and radio missions (e.g., _MeerKAT_; _AMI-LA_). _NuSTAR_ conducted a high-quality observation on August 3, 2019 with a 22.2 ks exposure time. Draghis et al. (2020) reported strong relativistic reflection features with the extremely sensitive _NuSTAR_ spectra, and argued that the source is a black hole with a nearly maximal spin parameter (\(a_{*}=0.997\)) at disk inclination of \(\theta=73^{\circ}\). EXO 1846\(-\)031 is located at 2.4-7.5 kpc according to the previous studies on X-ray and radio data (Parmar et al., 1993; Williams et al., 2022), with a black hole mass of \(\sim 9\) M\({}_{\bigodot}\)(Draghis et al., 2020; Williams et al., 2022). Liu et al. (2021) reported the observational results from a detailed timing analysis of EXO 1846\(-\)031 2019 outburst with the observations of HXMT and _NICER_.
In this work, we focus on the latest HXMT and _NICER_ observations of GX 339\(-\)4 and EXO 1846\(-\)031, and present a detailed temporal and spectral analysis. The paper is organized as follows. Sec. 2 presents the observational data reduction. The spectral timing analysis is reported in Sec. 3. We discuss the results and report our conclusions in Sec. 4 and Sec. 5, respectively.
## 2. Observations and Data Reduction
### Data selection
Starting from February 2021, GX 339\(-\)4 went into a new outburst that lasted for a few months. Fig. 1 shows the long-term light curve in the 2-20 keV band and the corresponding hardness ratio observed with _MAXI_ GSC. The hardness is defined as the ratio between the count rates at 4-10 keV and 2-4 keV. HXMT and _NICER_ extensively observed the 2021 outburst of the source. We went through all available HXMT and _NICER_ data and picked out those observations that show Type-C QPO signatures. The selected observations analyzed in this work are marked in the light curve of GX 339\(-\)4 in Fig. 1. Information about these observations is listed in Tab. A1.
The 2019 outburst of EXO 1846\(-\)031 was first detected by _MAXI_/GSC on 2019 July 23 (Negoro et al., 2019), and it lasted about 3 months. The _MAXI_ X-ray Hardness-Intensity diagram (HID) of the outburst shows a characteristic q-shaped hysteresis of X-ray binaries in outburst (Williams et al., 2022; Liu et al., 2021). The long-term _MAXI_ light curve and the corresponding hardness ratio are shown in Fig. 2. HXMT and _NICER_ conducted high-cadence pointing observations of EXO 1846\(-\)031. Type-C QPOs appear during the transition from hard to soft state (Liu et al., 2021). We selected observations showing Type-C QPO signatures. The selected observations are marked in the light curve in Fig. 2 and listed in Tab. A2.
### Data reduction
HXMT covers the broadband energy range of 1-250 keV with low-energy, medium-energy, and high-energy detectors (Zhang et al., 2020; Chen et al., 2020; Cao et al., 2020; Liu et al., 2020). The light curves and spectra are extracted with the HXMT data analysis software (HXMTDAS) version 2.05 and CALDB version 2.06, following the official user guide. The background is estimated by the standalone scripts hebkgmap, mebkgmap, and lebkgmap(Guo et al., 2020; Liao et al., 2020). The data are screened following the recommended criteria, i.e., an elevation angle \(>\)10\({}^{\circ}\), a geomagnetic cutoff rigidity \(>\)10 GeV, a pointing offset angle \(<\)0.1, and at least 300 s away from the South Atlantic Anomaly.
The _NICER_ data are processed with the _NICER_ data analysis software (NICERDAS) version 2021-04-01_V008 and CALDB version 20210707. We use the standard fil
tering criteria: the pointing offset is less than 54'', and the pointing direction is more than 40deg away from the bright Earth limb, more than 30deg away from the dark Earth limb, and outside the South Atlantic Anomaly. In addition, we remove the data of detectors # 14 and # 34 which are affected by episodes of increased electronic noise, and we select events that are not flagged as "overshoot" or "undershoot" resets (EVENT_FLAGS = bxxxx00) or forced triggers (EVENT_FLAGS=bx1x000). The standard _NICER_ reduction routine nicerl2 is used to process the data. The cleaned events are barycenter-corrected using the FTOOL barycorr. We extract the energy spectra of the background in each observation using the nibackgen3C50 tool (Remillard et al., 2022). The Redistribution Matrix File (RMF) and Ancillary Response File (ARF) are created by using the tasks nicerrmf and nicerarf, respectively.
## 3. Data Analysis
### Timing analysis
We extract HXMT LE and _NICER_ XTI light curves with a time resolution of 1 ms from the full energy band
Figure 1.— _MAXI_ GSC light curve and corresponding hardness of GX 339\(-\)4 starting from 2021 February 5 (MJD = 59250). The hardness is defined as the ratio between the count rates at 4–10 keV and 2–4 keV bands. The vertical black lines mark the HXMT observations analyzed in this work and the vertical red lines mark the _NICER_ observations.
Figure 2.— _MAXI_ GSC light curve and corresponding hardness of EXO 1846\(-\)031 starting from 2019 June 16 (MJD = 58650). The hardness is defined as the ratio between the count rates at 4–10 keV and 2–4 keV bands. The vertical black lines mark the HXMT observations analyzed in this work and the vertical red lines mark the _NICER_ observations.
(1-10 keV in HXMT; 0.5-10 keV in _NICER_) for each HXMT observation and _NICER_ observation. In order to calculate hardness ratios, we also produce LE light curves from the 1-5 keV and 5-10 keV bands, and produce XTI light curves from the 0.5-4 keV and 4-10 keV bands.
We carefully check the extracted light curves from all observations of GX 339\(-\)4 and find there are two _NICER_ observations (ObsID: 4133010103, 4133010104) that show a relatively strong variability of count rate and hardness. Fig. 3 shows the light curves of these two _NICER_ observations. The gaps in the light curves are due to the low Earth orbit of the telescope or the SAA. We can clearly see that the source went through a period of luminosity increase and hardness decrease. Comparing with the location of these two observations in Fig. 1 (the last red dotted line; in fact, since the two lines are quite close, they look like one line), we conclude that the two observations are during the hard-to-soft transition so the hardness keeps getting lower. Then we divide the observations according to the light curve, counting each snapshot as a sub-observation. We check over all sub-observation data and pick out those with exposure \(>200\) s. The selected sub-observations are numbered 4133010103-1 through 4133010103-9, and 4133010104-1 through 4133010104-13, as shown in Fig. 3. The other light curves do not show strong variability in the count rate i.e., no distinctive evidence of flares, dips, or state transitions, making it safe for timing and spectral analysis to characterize the source properties.
For EXO 1846\(-\)031, the count rate of the source remain fairly stable during each HXMT and _NICER_ interval, and the hardness does not change dramatically. Therefore, we conclude that we can carry out timing and spectral analysis in the unit of one observation.
To measure the QPO frequency of GX 339\(-\)4 and EXO 1846\(-\)031, we employ the Python package Stingray(Huppenkothen et al., 2019) to create PDS for each observation. The light curve is splited into 64 s segments, and then the final PDS is generated by averaging all 64 s segments. The PDS is normalized according to the "rms" method (Belloni & Hasinger, 1990), and logarithmically rebinned so that each bin size is 1.02 times larger than the previous bin. Note that we focus on the HXMT LE 1-10 keV light curve and the _NICER_ XTI 0.5-10 keV light curve to extract PDS and search for QPO signal. The 8-30 keV HXMT ME light curves have been analyzed in the same way and return consistent measurements of the QPO frequencies. So we report only the results from LE data in this work.
We use XSPEC v12.12.1 (Arnaud, 1996) to analyze the
Figure 3.— Light curves of GX 339\(-\)4 by _NICER_ in the 0.5–10 keV band (top panel) and corresponding hardness (bottom panel) (ObsID: 4133010103, 4133010104). The hardness is defined as the ratio between the count rates in the 4–10 keV and 0.5–4 keV bands. The source undergoes a process of flux increase and hardness decrease during this period. The intervals with exposure \(>200\) s are marked with yellow shadow. The selected sub-observations are numbered 4133010103-1 through 4133010103-9, and 4133010104-1 through 4133010104-13.
PDS. The typical PDS of XRBs manifests broad components and one or two narrow peaks at different frequencies, corresponding to broad band noise, the possible QPO fundamental and (sub) harmonics, respectively. We need at least one narrow Lorentzian for the QPO to fit the Poisson-extracted PDS (Belloni et al., 2002). More narrow Lorentzians are sometimes included to model harmonic peaks. All QPOs we detect have a quality factor (Q) greater than 4 and detection significance greater than 3\(\sigma\). Fig. A1 and Fig. A2 show a typical PDS and the fit results with several Lorentzian models for GX 339\(-\)4. Fig. A3 and Fig. A4 show the counterpart for EXO 1846\(-\)031. The QPO frequencies for each observation are listed in Tab. A3 for GX 339\(-\)4, and Tab. A4 for EXO 1846\(-\)031.
### Spectral analysis
For spectral analysis of the HXMT data, we consider the LE data in the 2-10 keV band and ME data in the 8-20 keV band. ME data above 20 keV and HE data are ignored because of the very high background. Note that we ignore the data below 2 keV of the LE instrument in spectral analysis (instead of the 1 keV for timing analysis) because of calibration uncertainties in the low energy band. For _NICER_ data, we consider the 1-10 keV band in this section, ignoring the data below 1 keV because of calibration issues.
The HXMT and _NICER_ spectra are fitted with the XSPEC (v12.12.1) package, using the recommended photoelectric cross sections of Verner et al. (1996) and element abundances of Wilms et al. (2000). The \(\chi^{2}\) statistics is employed and all parameter uncertainties are estimated at 90% confidence level, corresponding to \(\Delta\chi^{2}=2.71\). All spectra are grouped to ensure a minimum counts of 20 per bin. A systematic error of 1% is added in the _NICER_ spectra.
The HXMT and _NICER_ spectra of GX 339\(-\)4 are fitted with the model combination Tobbs \(\times\) (simpl \(\times\) kerrd + relxill). Tobabs is included to account for absorption by the interstellar medium. We set its column density (\(n_{\rm H}\)) to be a free parameter for _NICER_ spectra. While with HXMT spectra, we can not constrain its column density (\(n_{\rm H}\)), so we fix it at best-fit value, \(0.55\times 10^{22}\) cm\({}^{-2}\), which is consistent with the result of the _NICER_ data and the value in literature (e.g., Wang et al., 2020; Liu et al., 2022). kerrd accounts for the thermal emission from the geometrically thin and optically thick accretion disk (Ebisawa et al., 2003), in which the black hole distance, mass, and inclination angle of the accretion disk are set to 8.4 kpc, 9.0 M\({}_{\bigodot}\), and 30\({}^{\circ}\)(Parker et al., 2016), respectively. The spectral hardening factor of kerrd is set to 1.7 (Shimura and Takahara, 1995). simpl(Steiner et al., 2009) is used to take into account for the Comptonization of disk photons by the corona. The source has been found to have strong reflection features (Liu et al., 2022), and we use the full reflection model relxill(Garcia et al., 2014) to fit them. The spin parameter (\(a_{\star}\)) is fixed at 0.95 (Parker et al., 2016), and the index of the emissivity profile is fixed at 3 because it cannot be constrained by the fit. The best-fit values and uncertainties of GX 339\(-\)4 are shown in Tab. A3. Fig. A1 and Fig. A2 show typical spectra and fit results of HXMT data and _NICER_ data, respectively.
In the case of EXO 1846\(-\)031, the best-fit model combination is Tobabs \(\times\) (simpl \(\times\) kerrd + relxill) for HXMT spectra. The black hole distance, mass, and inclination angle of the accretion disk are set to 4.5 kpc (Williams et al., 2022), 10.0 M\({}_{\bigodot}\)(Williams et al., 2022, Draghis et al., 2020) and 73\({}^{\circ}\)(Draghis et al., 2020), respectively. The spin parameter (\(a_{\star}\)) in relxill is fixed at 0.998 (Draghis et al., 2020). We use a simple power-law to model the emissivity profile (\(q_{\rm in}=q_{\rm out}\) free). The other parameters are set exactly as in the case of GX 339\(-\)4. For _NICER_ spectra, we notice that there are still some large residuals in the soft X-ray band with the same model, including a Gaussian-like emission near 1.1 keV and edge-like shapes near 1.8 keV. These energies correspond to features in the effective area of _NICER_ versus energy (e.g., Wang et al., 2020), where 1.1 and 1.8 keV are attributed to sodium and silicon, respectively. Therefore, we adopt the following model for the _NICER_ spectra: Tobabs \(\times\) (simpl \(\times\) kerrd + relxill + gaussian) \(\times\) edge. This calibration issue arises in EXO 1846\(-\)031 because the source has a high interstellar absorption, which makes the photon count rate in the lower energy band relatively low, making the calibration issue prominent. Typical spectra and fit results of HXMT and _NICER_ are shown in Fig. A3 and Fig. A4. In Tab. A4, we summarize the best-fit values and errors of EXO 1846\(-\)031.
## 4. Results and Discussion
Fig. 4 and Fig. 5 show the evolution of inner radius (\(R_{in}\)) and QPO frequency (\(f_{\rm QPO}\)) with time for GX 339\(-\)4 and EXO 1846\(-\)031, respectively. Generally speaking, we clearly see that the value of \(f_{\rm QPO}\) monotonically increases with time. The behaviour is consistent with that reported in Motta et al. (2011) and Liu et al. (2021). It has also been observed in other XRBs, for example, XTE J1859+226 (Casella et al., 2004). In addition, a notable feature for both sources is the decrease of \(R_{in}\). For GX 339\(-\)4, the inner disk moves toward the ISCO (Innermost Stable Circular Orbit), from \(>50R_{\rm g}\) to \(\sim 7R_{\rm g}\) (\(R_{\rm g}\), gravitational radius), which coincide with the result in the previous study (e.g. Wang-Ji et al., 2018; Wang et al., 2020). Although there is some variable feature, EXO 1846\(-\)031 shows a similar trend.
Correlation between the parameters involved in temporal and spectral analysis are shown in Fig. 6 and Fig. 7. An interesting result is the relationship between the photon index (\(\Gamma\)) and the QPO frequency (\(f_{\rm QPO}\)). The results we get from both sources share the same tendency, as shown in the bottom panels of Fig. 6 and Fig. 7. There is a strong positive correlation between \(f_{\rm QPO}\) and \(\Gamma\) of the power-law in the beginning which flattens or starts reversing at the highest values of the \(f_{\rm QPO}\). The turnoff in the correlation is not apparent in GX 339\(-\)4, while it is evident in EXO 1846\(-\)031 (around \(\Gamma\sim 2.7\)). A similar kind of correlation have been reported in a number of other LMXBs (e.g., Vignarca et al., 2003; Titarchuk and Fiorito, 2004; Titarchuk and Seifina, 2009; Furst et al., 2016). Titarchuk and Fiorito (2004) introduced the transition layer (TL) model to explain the observed correlations. The TL model depicts how the QPOs related to the corona properties (e.g., the size, optical depth, temperature and spectral index), and predicts the correlation between photon index and QPO frequency. The results
we get are in good agreement with the model's predictions, except for the observations of EXO 1846\(-\)031 with \(f_{\rm QPO}>5.18\), where a negative correlation between \(f_{\rm QPO}\) and \(\Gamma\) appears. A universal explanation of this correlation between \(\Gamma\) and \(f_{\rm QPO}\) is still missing.
The upper left panel of Fig. 6 shows a broad anti-correlation between the QPO frequency (\(f_{\rm QPO}\)) and inner radius (\(R_{in}\)) in GX 339\(-\)4. This anti-correlation is not particularly significant in EXO 1846\(-\)031, and we can only see a general tendency of larger \(R_{in}\) corresponding to a smaller frequency. The same correlation between the QPO frequency and the disk inner radius was reported in other sources (e.g., GRS 1915\(+\)105; Rodriguez et al., 2002). The Lense-Thirring precession model would predict anti-correlation between \(f_{\rm QPO}\) and \(R_{in}\), and a direct dependence of the QPO frequency on the inner radius (Ingram et al., 2009; Ingram and Done, 2010). To check the possibility of modeling the results with the relativistic precession model, we use equation 2 in Ingram et al. (2009) to fit data points. The model cannot explain the results we obtained, both in the case of GX 339\(-\)4 and EXO 1846\(-\)031, as shown in the plot.
The variation of the frequency with the accretion rate is shown in the upper right panels of Fig. 6 and Fig. 7. Liu et al. (2021) reported a strongest correlation between the QPO frequency (\(f_{\rm QPO}\)) and mass accretion rate (\(\dot{M}\)) in GRS 1915\(+\)105 with HXMT data. We do not find any significant correlation between them in GX 339\(-\)4, while there is a weak anti-correlation in EXO 1846\(-\)031. In fact, a positive correlation between \(f_{\rm QPO}\) and \(\dot{M}\) is proposed in the TL model by Titarchuk and Fiorito (2004), and in the disk-corona natural frequency model by Mastichiadis et al. (2022). Fig. 3 of Titarchuk and Fiorito (2004) depicts the positive correlation between \(f_{\rm QPO}\) and the \(\gamma\)-parameter (which is proportional to mass accretion rate), which is opposite to what we find. Besides, Mastichiadis et al. (2022) argue that type-C QPOs could arise from the interaction of the hot corona with the cold accretion disk, and predict a formula \(f_{0}\propto\dot{M}^{1/2}\) below a certain mass accretion rate (Fig. 5 of Mastichiadis et al. (2022)). The results we get do not fit well with the predictions of that model. These discrepancies may suggest that the "transition layer model" and the disk-corona natural frequency model are not favored in our case.
Misra et al. (2020) identified QPOs as the dynamic frequency of a truncated relativistic accretion disk in the case of GRS 1915\(+\)105. The dynamic frequency is defined as the ratio of the sound propagation velocity of the inner disk to the truncation radius, i.e., the inverse
Figure 4.— Evolution of disk inner radius \(R_{\rm in}\) (left panel) and QPO frequency (\(f_{\rm QPO}\)) along with MJD in GX 339\(-\)4. The black points indicate HXMT data, and red dots indicate _NICER_ data.
Figure 5.— Evolution of disk inner radius \(R_{\rm in}\) (left panel) and QPO frequency (\(f_{\rm QPO}\)) along with MJD in EXO 1846\(-\)031. The black points and red points represent the results of HXMT data and _NICER_ data, respectively.
of the sound crossing time (Misra et al., 2020). Based on the assumption that the accretion disk is a standard relativistic accretion disk (Novikov and Thorne, 1973), the dynamic frequency is a function of the inner radius (\(R_{\rm in}\)), black hole spin (\(a_{*}\)), mass accretion rate (\(\dot{M}\)) and a normalization factor (\(N\)). Liu et al. (2021) extended the results in Misra et al. (2020) to a larger range of accretion rates with HXMT data of GRS 1915+105, and confirmed the high spin nature of the source. Following the work of Misra et al. (2020) and Liu et al. (2021), we illustrate the relation between QPO frequency (\(f_{\rm QPO}\)) divided by accretion rate (\(\dot{M}\)) and disk inner radius (\(R_{\rm in}\)) in Fig. 8 and Fig. 9. Both sources show negative correlation between \(f_{\rm QPO}/\dot{M}\) and \(R_{\rm in}\), Moreover, the correlation follows the prediction of the dynamic frequency model.
We fit the relation between \(f_{\rm QPO}/\dot{M}\) and \(R_{\rm in}\) using Equation (3) in Misra et al. (2020). The fit returns \(a_{*}=0.9978\pm 0.0009\) and \(N=0.281\pm 0.025\) for EXO 1846\(-\)031, indicating a rapidly spinning black hole. This result is consistent with what has been reported by analyzing the blurred reflection spectra (e.g., Dragish et al., 2020; Abdikamalov et al., 2021). The best-fit curve is shown in Fig. 9. In the case of GX 339\(-\)4, the fit returns \(a_{*}=0.603\pm 0.026\) and \(N=1.02\pm 0.05\). Such a low spin result is somewhat different from the result obtained by analyzing the blurred reflection spectra or
Figure 6.— Correlation between the parameters involved in the temporal and spectral analysis in the case of GX 339\(-\)4. Correlation of the QPO frequency vs. inner disk radius and the QPO frequency vs. accretion rate are shown in the upper left and upper right panels. The two central panels illustrate the accretion rate vs. inner disk radius and inner disk radius vs. photon index. The photon index vs. QPO frequency is depicted in the bottom panel. The black and red crosses denote the results of HXMT data and _NICER_ data, respectively. In the left top panel, the dashed gray lines represent the correlation of the frequency and inner radius predicted by Lense–Thirring precession model. The lines from left to right depict \(a_{*}=0.3\), 0.5, 0.7, 0.9 and 0.998, respectively.
thermal spectra (e.g. Reis et al., 2008; Ludlam et al., 2015; Garcia et al., 2015; Parker et al., 2016; Wang et al., 2020). We note that for this source we do not have data below 6 \(R_{\rm g}\). The relativistic effects are more evident at lower \(R_{\rm in}\) (3 \(\sim\) 5 \(R_{\rm g}\)). Hence, data points at lower \(R_{\rm g}\) plays a crucial role in the estimation of spin parameter value. In Fig. 8, we simultaneously show two curves, \(a_{\star}\) = 0.603 and \(a_{\star}\) = 0.900. It is worth noting that the most important difference between the two curves is reflected in the region with low \(R_{\rm in}\). This also proves our view that a reasonable fitting value cannot be obtained because of the lack of data with relatively small \(R_{\rm in}\).
The middle right panels of Fig. 6 and Fig. 7 show that the inner disk radius tends to decrease when the photon index (\(\Gamma\)) increases. The behaviour is consistent with that expected during a hard-to-soft transition. A noteworthy positive correlation between the mass accretion rate (\(\dot{M}\)) and the inner radius (\(R_{in}\)) in EXO 1846\(-\)031 is described in the middle left panel of Fig. 7. A similar relationship was reported in GRS 1915+105 (Misra et al., 2020; Liu et al., 2021; Rawat et al., 2022) and MAXI J1535-571 (Garg et al., 2022). The correlation is beyond the expectation of truncated disk model (Done et al., 2007). But Dullemond & Spruit (2005) predicted a positive correlation between \(\dot{M}\) and \(R_{in}\) (see their Fig. 8), calculating the evaporation of the cool accretion disk on account of the ion-bombardment. An alternative explanation is discussed in Abramowicz et al. (1978), where the authors suggested a larger inner edge is required when the mass accretion rate increases to dissipate the angular momentum of accretion material.
## 5. Conclusion
Figure 7.— Correlation between the parameters involved in temporal and spectral analysis in the case of EXO 1846\(-\)031. It is organized as in Fig. 6.
\begin{table}
\begin{tabular}{c c c c} \hline \hline Mission & Obs. ID & Start data & Exposure (s) \\ \hline & P0304024026 & 2021-03-12 & 2274 \\ & P0304024028 & 2021-03-14 & 1401 \\ HXMT & P0304024032 & 2021-03-18 & 1597 \\ & P0304024035 & 2021-03-22 & 1669 \\ & P0304024036 & 2021-03-24 & 1193 \\ & P0304024038 & 2021-03-26 & 2088 \\ \hline & 3558011402 & 2021-03-17 & 1595 \\ & 3558011501 & 2021-03-19 & 7560 \\ _NICER_ & 4133010101 & 2021-03-19 & 2030 \\ & 4133010102 & 2021-03-20 & 1860 \\ & 4133010103 & 2021-03-26 & 6111 \\ & 4133010104 & 2021-03-27 & 8709 \\ \hline \hline \end{tabular}
\end{table}
Table 1HXMT and _NICER_ observations of GX 339\(-\)4 analyzed in this work. For HXMT, the listed exposure time is for the LE instrument.
Figure 8.— Variation of QPO frequency divided by the accretion rate vs. inner disk radius in the case of GX 339\(-\)4. The results of HXMT data and _NICER_ data are denoted with black and red crosses, respectively. The orange curve represents the best fit ( \(a_{*}=0.603\), \(N=1.02\)), and the blue curve corresponds to \(a_{*}=0.900\), \(N=0.52\).
Figure 9.— Variation of QPO frequency divided by the accretion rate vs. inner disk radius in the case of EXO 1846\(-\)031. The black and red crosses denote the results of HXMT data and _NICER_ data, respectively. The blue curve represents the best fit ( \(a_{*}=0.9978\), \(N=0.281\)). The orange curve represents the best fit ( \(a_{*}=0.754\), \(N=0.853\)) when we only include the data with \(R_{\rm in}>5~{}R_{\rm g}\). It proves that the data with small \(R_{\rm in}\) are important to fit the spin parameter.
\begin{table}
\begin{tabular}{c c c c} \hline \hline Mission & Obs. ID & Start data & Exposure (s) \\ \hline & P021405000101 & 2019-08-02 & 718 \\ & P021405000102 & 2019-08-02 & 1436 \\ & P021405000103 & 2019-08-02 & 762 \\ & P021405000104 & 2019-08-02 & 718 \\ & P021405000105 & 2019-08-02 & 1715 \\ & P021405000106 & 2019-08-03 & 563 \\ HXMT & P021405000107 & 2019-08-03 & 656 \\ & P021405000301 & 2019-08-05 & 700 \\ & P021405000302 & 2019-08-05 & 1102 \\ & P021405000303 & 2019-08-05 & 678 \\ & P021405000401 & 2019-08-06 & 718 \\ & P021405000502 & 2019-08-07 & 691 \\ & P021405000503 & 2019-08-07 & 539 \\ & P021405000601 & 2019-08-08 & 1130 \\ & P021405000701 & 2019-08-08 & 1163 \\ & P021405000702 & 2019-08-09 & 1795 \\ \hline & 2200760101 & 2019-07-31 & 5658 \\ & 2200760102 & 2019-08-01 & 1165 \\ & 2200760103 & 2019-08-02 & 2562 \\ & 2200760104 & 2019-08-03 & 1488 \\ & 2200760105 & 2019-08-04 & 1130 \\ & 2200760106 & 2019-08-05 & 3564 \\ & 2200760107 & 2019-08-06 & 912 \\ _NICER_ & 2200760108 & 2019-08-07 & 927 \\ & 2200760109 & 2019-08-08 & 3293 \\ & 2200760110 & 2019-08-09 & 4629 \\ & 2200760112 & 2019-08-11 & 2749 \\ & 2200760113 & 2019-08-12 & 3341 \\ & 2200760114 & 2019-08-13 & 7154 \\ & 2200760115 & 2019-08-13 & 8181 \\ & 2200760116 & 2019-08-15 & 4703 \\ & 2200760117 & 2019-08-16 & 8739 \\ & 2200760118 & 2019-08-17 & 4875 \\ & 2200760119 & 2019-08-17 & 3341 \\ & 2200760120 & 2019-08-19 & 3894 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Left: a typical PDS of GX 339\(-\)4 from HXMT data (Obs ID: P0304024028). Right: the HXMT spectrum and residuals to the best-fit model for the same observation. Data from the LE and ME detector are denoted in black and red, respectively.
Figure 17: Left: a typical PDS of EXO 1846\(-\)031 from HXMT data (Obs ID: 2200760106). Right: the _NICER_ spectrum and residuals to the best-fit model for the same observation.
Figure 18: Left: a typical PDS of EXO 1846\(-\)031 from HXMT data (Obs ID: P021405000105). Right: the HXMT spectrum and residuals to the best-fit model for the same observation. Same as Fig. 17, data from the LE and ME detector are denoted in black and red, respectively.
Figure 17: Left: a typical PDS of EXO 1846\(-\)031 from _NICER_ data (Obs ID: 2200760106). Right: the _NICER_ spectrum and residuals to the best-fit model for the same observation. | GX 339-4とEXO 1846-031の光学スペクトルとタイミング解析を実施し、C型QPOの進化を研究するために、そのスペクトルパラメータに基づいて分析します。Insight-HXMTとNICERからの高 cadence データを用いて、これらのQPOを追跡します。C型QPOは低硬状態の終わりと/またはハード中間状態の終わりに現れます。結果から、QPOの周波数と、2つのソースの内部ディスク半径と質量 accretion 速度は密接に関連していることがわかります。このような相関は、動的周波数モデルと nicely consistent です。
Please note:
* I'm looking for a translation that is accurate, concise, and natural-sounding.
* It is important to capture the nuance of the original sentence. |
2306.01248 | How Ready are Pre-trained Abstractive Models and LLMs for Legal Case
Judgement Summarization? | Automatic summarization of legal case judgements has traditionally been
attempted by using extractive summarization methods. However, in recent years,
abstractive summarization models are gaining popularity since they can generate
more natural and coherent summaries. Legal domain-specific pre-trained
abstractive summarization models are now available. Moreover, general-domain
pre-trained Large Language Models (LLMs), such as ChatGPT, are known to
generate high-quality text and have the capacity for text summarization. Hence
it is natural to ask if these models are ready for off-the-shelf application to
automatically generate abstractive summaries for case judgements. To explore
this question, we apply several state-of-the-art domain-specific abstractive
summarization models and general-domain LLMs on Indian court case judgements,
and check the quality of the generated summaries. In addition to standard
metrics for summary quality, we check for inconsistencies and hallucinations in
the summaries. We see that abstractive summarization models generally achieve
slightly higher scores than extractive models in terms of standard summary
evaluation metrics such as ROUGE and BLEU. However, we often find inconsistent
or hallucinated information in the generated abstractive summaries. Overall,
our investigation indicates that the pre-trained abstractive summarization
models and LLMs are not yet ready for fully automatic deployment for case
judgement summarization; rather a human-in-the-loop approach including manual
checks for inconsistencies is more suitable at present. | Aniket Deroy, Kripabandhu Ghosh, Saptarshi Ghosh | 2023-06-02T03:16:19 | http://arxiv.org/abs/2306.01248v2 | # How Ready are Pre-trained Abstractive Models and LLMs for Legal Case Judgement Summarization?
# How Ready are Pre-trained Abstractive Models and LLMs for Legal Case Judgement Summarization?
Aniket Deroy
IIT Kharagpur
West Bengal 721302, India
[email protected]
&Kripabandhu Ghosh
IISER Kolkata
West Bengal 741246, India
[email protected]
&Saptarshi Ghosh
IIT Kharagpur
West Bengal 721302, India
[email protected]
###### Abstract
Automatic summarization of legal case judgements has traditionally been attempted by using extractive summarization methods. However, in recent years, abstractive summarization models are gaining popularity since they can generate more natural and coherent summaries. Legal domain-specific pre-trained abstractive summarization models are now available. Moreover, general-domain pre-trained Large Language Models (LLMs), such as Chat-GPT, are known to generate high-quality text and have the capacity for text summarization. Hence it is natural to ask if these models are ready for off-the-shelf application to automatically generate abstractive summaries for case judgements. To explore this question, we apply several state-of-the-art domain-specific abstractive summarization models and general-domain LLMs on Indian court case judgements, and check the quality of the generated summaries. In addition to standard metrics for summary quality, we check for inconsistencies and hallucinations in the summaries. We see that abstractive summarization models generally achieve slightly higher scores than extractive models in terms of standard summary evaluation metrics such as ROUGE and BLEU. However, we often find inconsistent or hallucinated information in the generated abstractive summaries. Overall, our investigation indicates that the pre-trained abstractive summarization models and LLMs are not yet ready for fully automatic deployment for case judgement summarization; rather a human-in-the-loop approach including manual checks for inconsistencies is more suitable at present.
## 1 Introduction
Summarization of legal case judgements is a practical and important problem in the legal domain, given that the extreme length and complexity of such documents make it difficult even for Law practitioners to read them fully. Traditionally, case judgements are summarized by humans, i.e., Law practitioners. For instance, most Legal information systems provide case summaries/headnotes written by Law practitioners. To reduce the human effort in summarization, there have been many efforts over the years to automate the summarization of case judgements (Bhattacharya et al., 2021; Deroy et al., 2023).
There are two broad approaches for summarization - Extractive (where some important sentences are selected from the input document to form the summary) and Abstractive (where the model attempts to understand the document and generate a summary on its own). The reader is referred to the comprehensive surveys by Nenkova et al. Nenkova and McKeown (2012) and Wafaa et al. El-Kassas et al. (2021) for more details on various types of summarisation algorithms.
For summarization of legal case judgements, extractive summarization models have mostly been applied over the years (Bhattacharya et al., 2021; Polsley et al., 2016; Liu and Chen, 2019; Zhong et al., 2019). But in recent times, the research community is preferring the use of _abstractive_ summarization models, primarily because abstractive methods are said to generate more 'natural' and 'coherent' summaries. As a result, a few recent works have started training abstractive models for legal document summarization (Shukla et al., 2022; Feijo and Moreira, 2023). Domain-specific pre-trained versions of popular abstractive summarization models, such as Google's Pegasus (Zhang et al., 2020), have been released specifically for legal summarization (e.g., Legal Pegasus - [https://huggingface.co/nsi319/legal-pegasus](https://huggingface.co/nsi319/legal-pegasus)). Moreover, recent times have seen the advent of general-purpose Large Language Models (LLMs) such as ChatGPT and DaVinci that have the ability to generate high-quality text as well as the ability to summarize text without additional training. A big advantage of these _pre-trained_ abstractive summarization mod
els and LLMs is that they can be applied without further training. In fact, LLMs are already being used for summarization in other domains, e.g., news summarization (Zhang et al., 2023). But, to our knowledge, these LLMs have not been much used for legal case judgement summarization to date.
In such a scenario, it is natural to ask - _how ready are the pre-trained abstractive summarization models and the LLMs that are available today, for off-the-shelf application for legal case judgment summarization?_ In this paper, we attempt to answer this question.
We apply state-of-the-art abstractive summarization models specifically meant for the legal domain - such as Legal-Pegasus ([https://huggingface.co/nsi319/legal-pegasus](https://huggingface.co/nsi319/legal-pegasus)) and Legal-LED ([https://huggingface.co/nsi319/legal-led-base-16384](https://huggingface.co/nsi319/legal-led-base-16384)) - as well as recently developed Large Language Models such as DaVinci and ChatGPT, on a dataset of Indian Supreme Court case judgements (containing gold standard summaries written by Law practitioners). We also apply some extractive summarization models on the same dataset for comparison. We report a large number of summary quality metrics for all the models, including traditional metrics such as ROUGE, METEOR and BLEU (that match model-generated summaries with gold standard summaries) and metrics for quantifying the consistency of summaries with respect to the original document.
We observe that the summaries generated by abstractive models achieve slightly higher ROUGE, METEOR, BLEU scores than those generated by the extractive models. However, the abstractive summaries have various problems, including incomplete sentences/words, multiple sentences being merged meaninglessly, as well as more serious errors such as inconsistent and hallucinated information. For instance, we observe that the abstractive summarization models and LLMs sometimes generate wrong dates and wrong person names in the summaries, and also confuse different persons associated with a case. Thus our contributions in this work are as follows:
(1) We apply pre-trained abstractive summarization models and LLMs (and a few extractive summarization models for comparison) on a set of Indian court case judgements, and report several metrics that include not only traditional summarization evaluation metrics, but also metrics for the consistency of the generated summaries.
(2) To our knowledge, this paper is the first analysis of the consistency of abstractive summaries in the legal domain. We show that, though abstractive models often achieve higher ROUGE, BLEU, METEOR scores than extractive models, abstractive summaries often contain hallucinated or inconsistent information.
(3) We present several examples of errors, including presence of hallucinated or inconsistent information, in case judgement summaries generated by state-of-the-art LLMs and pre-trained abstractive summarization models. To our knowledge, this is the first study to demonstrate such examples.
Our analyses show that the pre-trained abstractive summarization models and LLMs need to be further improved before they can be readily used for case judgement summarization by legal experts.
## 2 Related work
**Summarization of legal case judgements:** Traditionally, _extractive_ summarization models have been used to summarize legal case judgements. A variety of methods have been tried including optimization techniques (Bhattacharya et al., 2021), multi-task learning (Agarwal et al., 2022), Machine Learning-based classification (Liu and Chen, 2019), and so on. The extractive models that have been tried include both unsupervised (Bhattacharya et al., 2021) and supervised (Agarwal et al., 2022; Liu and Chen, 2019) models.
In recent times, there have been a few works on _abstractive_ summarization of legal case judgements. Our recent prior work (Shukla et al., 2022) applied various abstractive models such as BART, Legal-LED and Legal-Pegasus on Indian and UK court judgements. There are prior works on semantic segmentation of long legal documents in low resource settings, which discuss how to handle long legal documents (which are generally larger than the input length of encoder-decoder based models) to perform abstractive legal document summarization (Moro and Ragazzi, 2022). There are works which try to improve abstractive summarization of legal case judgements using textual entailment (Feijo and Moreira, 2023).
**Hallucinations in large language models:** In the context of natural language processing (NLP), hallucination refers to a phenomenon where a language model generates text that is not true or accurate based on the input it has been given. This
can happen for a variety of reasons, such as a lack of training data, bias in the training data, or limitations in the language model architecture (see Ji et al. (2023) for a survey).
There have been studies on hallucination specifically in abstractive summaries. Since hallucinations are undesirable in summaries, various works have tried to reduce hallucinations in the summaries generated by the abstractive summarization models Filippova (2020); Zhao et al. (2020).
The advent of Large Language Models (LLMs) like ChatGPT, and their increased use in academic writing is raising further concerns about the integrity and accuracy of the generated text Alkaissi and McFarlane (2023). While such models are trained on vast amounts of data and can produce high-quality content, there is always a risk that the generated text may contain inaccuracies, biases, or even outright fabrications. For example, language models trained on Wikipedia and other online sources have been found to generate more sexist and racist content Stanczak and Augenstein (2021). Additionally, LLMs can also generate text that is inconsistent with established scientific facts or that presents misleading information.
**Novelty of this work:** There has been little attempt to analyse how various _abstractive_ summarization methods and LLMs (such as ChatGPT) perform in summarizing legal case judgements. Also, to our knowledge, hallucination has not been studied earlier in the context of legal summarization. This work takes the first step towards understanding how prepared the abstractive summarization models / LLMs are today for the task of automatic case judgement summarization.
## 3 Dataset
We reuse a dataset of Indian Supreme Court judgements from our prior work Shukla et al. (2022). The dataset, called IN-Abs, contains a total of 7,130 legal judgements from the website of the Legal Information Institute of India1, along with a single abstractive summary for every judgement. The summaries (also known as 'headnotes') have been written by Law experts appointed by Legal Information Institute of India.
Footnote 1: [http://www.liiofindia.org/in/cases/cen/INSC/](http://www.liiofindia.org/in/cases/cen/INSC/)
Out of the total set of 7,130 judgement-summary pairs in the dataset, 7,030 judgement-summary pairs are considered as the training set and the other 100 judgements are considered as the test set. Some of the supervised abstractive/extractive models considered in this work have been trained or fine-tuned over the IN-Abs train set. All summarization models are evaluated over the IN-Abs test set (100 documents).
Table 1 represents the number of documents in the training and test sets, along with the average number of words present in a legal judgement and a gold standard summary. Further details about the IN-Abs dataset are available in Shukla et al. (2022).
## 4 Methods for summarizing legal case judgements
We have tried a variety of summarization models in this work. There are 3 main categories of summarization methods applied in this work: (1) General-domain Large Language models, (2) Legal domain-specific abstractive summarization models, and (3) Extractive Summarization models.
### General-domain Large Language Models
We try out two popular Large language Models (LLMs), namely, Text-Davinci-003 and Turbo-Gpt-3.5, both developed by OpenAI.2
Footnote 2: Details of the two LLMs are available at [https://platform.openai.com/docs/models/](https://platform.openai.com/docs/models/).
**Text-Davinci-003** (which we refer to as **Davinci** in short) is a transformer-based language model with 175 billion parameters, making it one of the largest and most advanced language models to date. The language model has been trained on a diverse range of text data, including web pages, books, scientific articles, and other sources of human-written text. OpenAI has not provided detailed information on the exact sources of the training data, but it is known that the model has been trained on a
\begin{table}
\begin{tabular}{|p{42.7pt}|p{42.7pt}|p{42.7pt}|p{42.7pt}|} \hline & Nos. of Documents & Avg nos. of words in document & Avg nos. of words in gold-standard summary \\ \hline Train set & 7,030 & 4,368.49 & 839.75 \\ \hline Test set & 100 & 4,782.71 & 932.01 \\ \hline \end{tabular}
\end{table}
Table 1: Statistics of the IN-Abs train set and test set, containing (case judgement, summary) pairs from the Indian Supreme Court. The train set is used to train extractive models and fine-tune pre-trained abstractive models. All summarization models in this work are applied and evaluated over the test set.
massive scale text dataset using a combination of supervised and unsupervised learning methods.
**Turbo-GPT-3.5** (popularly known as **Chat-GPT**) is a language model which is based on the GPT-3 architecture developed by OpenAI. The model is said to have approximately 154 billion parameters. Turbo-GPT-3.5 was trained on a diverse range of text data, including web pages, books, scientific articles, and other sources of human-written text including _chats_, using a combination of supervised and reinforcement learning methods. The model has been optimized for speed and performance, with efficient use of memory and computation resources.
Davinci is said to be the largest and most powerful model till date, which performs the best on many complex NLP tasks. ChatGPT is a cheaper model with slightly fewer parameters; though it is said to be 'optimized for chat', ChatGPT also performs very well in many types of NLP tasks.
Both these LLMs take as input a 'prompt' and generate text in response. Specifically for the summarization task, the prompt consists of (i) the text to be summarized, which we refer to as <text to summarize> and (ii) an 'instruction' that tells the model that the input text has to be summarized. For both the LLMs - Text-Davinci-003 and Turbo-GPT-3.5 - we consider two variations giving two different prompts for summarization, as explained below.
**Variations of Text-Davinci-003:** We try these two variations of the model:-
(i) **davinci-tldr**: for this model, the prompt is "<text to summarize> Tl;Dr". In other words, the text to be summarized is passed first followed by "Tl;Dr" which is an inbuilt identifier for summarization.3
Footnote 3: [https://platform.openai.com/examples/default-tldr-summary](https://platform.openai.com/examples/default-tldr-summary)
(ii) **davinci-summ**: for this model, the prompt is "<text to summarize> Summarize the document in <XX> words" where XX is a number representing the target length of the output summary in number of words, i.e., the maximum number of words in the summary to be generated. How the target length XX is decided will be explained below.
**Variations of Turbo-GPT-3.5 (ChatGPT):** Similar to what we did for the Davinci model, we try the following two variations:-
(i) **chatgpt-tldr**: here the prompt is "Tl;Dr <text to summarize>". In other words, the inbuilt identifier for summarization "Tl;Dr" is sent first, followed by the text to summarize.
(ii) **chatgpt-summ**: for this model, the prompt is "Summarize the document in <XX> words <text to summarize>" where XX is a number representing the target length of the output summary (in words). The choice of the target length is discussed below.
**Chunking of long legal documents:** LLMs such as ChatGPT and DaVinci impose restrictions over the length of input that can be given at once. In particular, Text-Davinci-003 and Turbo-GPT-3.5 have a _limit of 4,096 tokens for (Prompt + generated text)_, where every 'token' represents approx. 4 characters. On average, one token corresponds to \(\frac{3}{4}\) of an English word, or 100 tokens approximately corresponds to 75 words.4
Footnote 4: Tokens are explained in detail at [https://help.openai.com/en/articles/4936856-what-are-tokens-and-how-to-count-them](https://help.openai.com/en/articles/4936856-what-are-tokens-and-how-to-count-them).
Since most legal case judgements are longer than this limit (having more than 4,300 words on average), we have to follow a divide and conquer strategy to summarize long legal documents using these LLMs. Given the limit of 4,096 tokens for (Prompt + generated text), we choose to send at most 1,024 words as the text to be summarized (as part of the prompt, as described above) at a time to these LLMs. Thus, we chunk the legal documents of length higher than 1,024 words and then pass the chunks (one at a time) into Turbo-Gpt-3.5 / Text-Davinci-003 to obtain the output summaries for the chunks. The summary for every chunk (of size 1,024 or less) is obtained from these models and then the summaries of all chunks are appended together (in the same order as of the chunks) to form the final output summary for the case judgement document. For legal documents with length less than 1,024 words, the entire document is passed into the model at once, to obtain the summary.
Note that the performance of summarization models may depend on the size of chunks. We conducted experiments with a subset of the documents considering two chunk sizes - 1,024 words and 2,048 words. We observed ChatGPT to perform slightly better with 1,024-word chunks, as per all the summarization evaluation metrics (the metrics will be detailed in the next section). Whereas, Davinci gave slightly better values for a few of the
metrics with 1,024-word chunks, and better values for the other metrics with 2,048-word chunks. For simplicity and consistency, in this work, we report all results considering chunks of size at most 1,024 words for all models. Further exploration of the dependence of summarization performance on the chunk size is left as future work.
**Deciding the target summary length for a chunk:** When some text is sent to a LLM for summarization, we need to specify the target summary length in the'max tokens' hyperparameter, i.e., the maximum number of words in the summary to be generated.
Suppose a chunk of text of length 1024 words from a document \(D\) is sent to a LLM for summarization. Let the length of document \(D\) be \(|D|\) words, and the length of the gold standard summary of \(D\) be \(|S|\) words. Then the target summary length for the chunk is specified as \(\frac{|S|}{|D|}\times 1024\) words. In other words, we ask the LLM to summarize each chunk considering the same compression ratio as for the whole document and the gold standard summary.
There is an inherent limitation in this method, which is as follows. In reality, all parts of the document are _not_ equally important, hence different chunks should possibly be allocated different lengths in the final summary. In contrast, this method allocates the same length in the summary for all chunks. However, there is no simple way of knowing the relative importance of different chunks in a legal case judgement.
**Implementation details:** The LLMs stated above have been run using the OpenAI API5. The hyperparameters of Text-Davinci-003 and Turbo-GPT-3.5 are indicated in Table 2. We use the default values for the hyperparameters 'presence penalty', 'frequency penalty' and 'temperature'. The'max tokens' hyperparameter indicates the maximum number of words in the summary to be generated for an input chunk of text; it is computed as described above.
Footnote 5: [https://platform.openai.com/docs/api-reference/completions](https://platform.openai.com/docs/api-reference/completions)
### Legal domain-specific abstractive summarization models
While the LLMs described in the previous section are general-domain (not trained for any particular domain or task), we now consider some abstractive summarization models that are specifically designed for summarization in the legal domain.
One such model is **Legal-Pegasus** (which we abbreviate to **LegPegasus**). This model is based on the google/pegasus-cnn_dailymail model developed by Google, which is designed to perform abstractive summarization task. LegPegasus has been specifically designed for the legal domain by finetuning it on the'sec-litigation-releases' dataset consisting of more than 2,700 litigation releases and complaints concerning civil lawsuits in various courts in the USA (and their summaries) brought by the US Securities and Exchange Commission. The LegPegasus model is available at [https://huggingface.co/nsi319/legal-pegasus](https://huggingface.co/nsi319/legal-pegasus) and has a maximum input sequence length of 1024 tokens.
Another abstractive summarization model specifically designed for the legal domain is **Legal-LED** (Legal Longformer Encoder Decoder) which we abbreviate as **LegLED**. The LegLED model is based on the Longformer architecture, a transformer-based neural network architecture that has been specifically designed for processing long sequences of text. The LegLED, available at [https://huggingface.co/nsi319/legal-led-base-16384](https://huggingface.co/nsi319/legal-led-base-16384), has been finetuned on the same'sec-litigation-releases' dataset as described above, to make it suitable for summarization in the legal domain.
As stated above, both LegPegasus and LegLED have been finetuned over legal documents and their summaries from the US Courts of Law. To make the models more suitable for summarizing Indian legal documents, our prior work Shukla et al. (2022) further finetuned the models over the IN-Abs training set (containing 7,030 Indian case judgements and their summaries, as stated in Section 3). We call these models **LegPegasus-IN** and **LegLED-IN** since they have been specifically finetuned for summarizing Indian legal documents.
**Chunking of long legal documents:** Since the domain-specific abstractive models also have restrictions of the number of input tokens, we follow a similar chunking-based strategy to handle long legal documents, as was described in Section 4.1. We chunk the legal documents (of length higher than 1,024 words) into chunks of at most 1,024 words and then pass one chunk at a time into the summarization models. The summary for every chunk is obtained from these models and then ap
pended together (in the same order as the chunks in the source document) to form the final output summary. The target summary length of each chunk is decided as described in Section 4.1. For documents shorter than 1,024 words, the entire summary of the document is obtained at once.
### Extractive summarization models
We consider some extractive summarization models for comparison with the abstractive models and LLMs. In our prior works (Deroy et al., 2023; Shukla et al., 2022), we applied several extractive summarization methods on the IN-Abs dataset. We observed that the three methods (i) CasteSummarizer, (ii) BertSum, and (iii) SummaRunner/RNN_RNN performed perform well over the IN-Abs dataset across most metrics. So we include the following three extractive methods in the comparison.
**(1) Case Summarizer**(Polsley et al., 2016) is an unsupervised method that identifies the most relevant sentences or phrases of a legal case document based on a metric like TF-IDF. CaseSummarizer adjusts sentence scores using occurrences of known entities, dates, and proximity to section headings.
**(2) BertSum**(Liu, 2019) is a supervised summarization model that uses the Bidirectional Encoder Representations from Transformers (BERT) architecture. This model treats summarization as a binary classification problem where every sentence (in the document) is labeled as 1 if the sentence is suitable for inclusion in the summary, and 0 otherwise. The model is trained (over a training set containing documents and gold standard summaries) to identify sentences that are suitable for inclusion in the summary.
**(3) SummaRunner/RNN_RNN**(Nallapati et al., 2017) is a supervised model that attempts to identify the most important sentences in a text and generate a concise summary. Similar to BertSum, this model considers summarization as a classification problem, and also analyzes the relationships between sentences in a document to select those that contain the most relevant information.
For all the three extractive models stated earlier, we use the implementations made available in our prior work (Shukla et al., 2022). The supervised models BertSum and SummaRunner/RNN_RNN models have been trained on the 7,030 (legal document, summary) pairs in the IN-Abs train dataset. More details about the training procedure are available in (Shukla et al., 2022).
## 5 Comparing performances of summarization models
In the previous section, we described several summarization models, including LLMs, domain-specific abstractive models, and extractive models. We now compare the quality of summaries generated by the different methods along two aspects - (1) their match with the gold standard summaries, and (2) their consistency with the input documents.
### Match with gold standard summaries
We first discuss the metrics used for measuring the match with gold standard summary, and then compare the performances of the different summarization models according to those metrics.
#### 5.1.1 Metrics
We use the following well-known metrics that compare a model-generated summary with the gold-standard summary (written by domain experts) and give a score, where higher scores imply higher
\begin{table}
\begin{tabular}{|l|l|} \hline
**Model** & **Hyperparameters** \\ \hline chatgpt-ldr & temperature=0.7, max tokens = gold-std summary length * 1024/Document length. \\ \hline chatgpt-summ & temperature=0.7, max tokens = gold-std summary length * 1024/Document length. \\ \hline davinci-ldr & Presence penalty=1.0, frequency penalty=0.0, temperature=0.7, \\ & max tokens = gold-std summary length * 1024/Document length. \\ \hline davinci-summ & Presence penalty=1.0, frequency penalty = 0.0, temperature=0.7, \\ & max tokens = gold-std summary length * 1024/Document length. \\ \hline LegPegasus & max tokens = gold-std summary length * 1024/Document length. \\ \hline LegPegasus-IN & max tokens = gold-std summary length * 1024/Document length. \\ \hline LegLED & max tokens = gold-std summary length * 1024/Document length. \\ \hline LegLED-IN & max tokens = gold-std summary length * 1024/Document length. \\ \hline \end{tabular}
\end{table}
Table 2: Hyperparameters of the legal domain-specific abstractive models and LLMs used in the work. ‘max tokens’ indicates the maximum number of words in the summary to be generated for an input chunk of text of length 1,024 words. Here ‘gold-std summary length’ is the actual length (number of words) of the gold standard summary for the given document.
match with the gold-standard (and hence a better quality summary).
**(1) ROUGE**[11] (Recall Oriented Understudy of Gisting Evaluation) is possibly the most popular metric used for measuring the quality of a summary generated by a summarization model. In particular, we calculate _Rouge-2_ precision, recall and F1 scores that measure the bigram match between gold standard summaries and model-generated summaries, and _Rouge-L_ precision, recall and F1 scores which measures Longest Common Subsequence-based match between generated summaries and the gold standard summaries.
**(2) METEOR**[1] calculates the harmonic mean of unigram precision and recall and is generally used for evaluating machine translation output. Prior works have also used this metric to evaluate summaries [1]. Here we use this metric to calculate the unigram overlap between a model-generated summary and the gold standard summary.
**(3) BLEU**[12] (Bilingual Evaluation Understudy) is a metric generally used for evaluating machine translation output, but it can also be used for measuring how well a model-generated summary matches with a gold standard summary.
For all the above metrics, we use the implementations from the SummEval package ([https://github.com/Yale-LILY/SummEval](https://github.com/Yale-LILY/SummEval)) which is a well-known package for evaluation of summarization.
#### 5.1.2 Comparative results
Table 3 shows the performance of all the summarization models (across the three families) that we have applied in this work, over the IN-Abs dataset. The best value for every metric in every family of summarization models is shown in blue-colored and boldfaced font.
We observe that out of the three families of summarization models, the legal domain-specific abstractive models achieve the best metric scores (better than both LLMs and extractive models). Extractive models achieve better scores than the general-domain LLMs for most of the metrics (ROUGE-2 scores, METEOR, BLEU), though the general-domain LLMs achieve slightly higher ROUGE-L scores. We perform Student T-test at 95% confidence interval to check if the best-performing abstractive model / LLM is performing statistically significantly better than the best-performing extractive model (individually for each metric). We see the improvements over the best extractive model are statistically significant only for the ROUGE-L metrics. The entries marked with an asterisk in Table 3 indicate the values that are statistically significantly higher than the best value achieved by an extractive model for the same metric.
Out of the domain-specific abstractive models, LegPegasus-IN and LegLED-IN performed the best. The improvements in their performance over LegPegasus and LegLED show the benefits of in-domain finetuning (as stated in Section 4, LegPegasus and LegLED are finetuned over US legal documents, but LegPegasus-IN and LegLED-IN are additionally finetuned on Indian legal documents similar to the IN-Abs test set).
Though the LLMs (chatgpt and davinci) achieve lower metric values than the best-performing abstractive and extractive models, their performance is creditable - even though the LLMs have not been specifically trained over any legal dataset, they perform better than some of the extractive and abstractive models that are trained over legal data, at least according to certain metrics. For instance, davinci-summ achieves higher ROUGE-L F1 score than LegPegasus, LegLED and all the extractive models. Among the two variations of the LLMs, the'summ' variations achieve a little better scores than the 'ldr' variations as per most metrics.
### Consistency of summaries
We now check how consistent model-generated summaries are with the original documents. This check is important particularly for abstractive summarization models and LLMs which are known to hallucinate in text generation. We first describe the metrics, and then discuss comparative results.
#### 5.2.1 Metrics
The following metrics compare the model-generated summary with the original document and estimate how consistent the summary is with the document. All these metrics give a score in the range \([0,1]\); the higher the score, the more consistent is the summary.
**(1) SummaC** - This metric [1] is based on Natural Language Inferencing (NLI) which is a task in Natural Language Processing that involves determining the relationship between two sentences. One of the sentences is considered as a 'hypothesis' and the other sentence is considered as
a 'premise'. NLI is the task of determining whether the given hypothesis logically follows from the premise. Typically, a NLI model will give a score representing how likely the hypothesis sentence is to logically follow from the premise sentence.
Given a (document, summary) pair, SummaC segments both the document and the summary into sentence units, and then leverages NLI models to effectively detect inconsistencies in the summary with respect to the document. In simple terms, NLI scores are computed for each sentence in the (model-generated) summary, to estimate the likelihood that this sentence logically follows from some sentence in the original document. Lower NLI scores for a particular sentence \(s\) in the summary implies a higher mismatch between this sentence and the sentences in the original document, thus indicating a higher likelihood that this sentence \(s\) contains hallucinated information. The NLI scores obtained by different sentences in the summary are then combined to give a single SummaC score for the given (document, summary) pair. Thus, a higher SummaC score for a summary indicates that the summary is more consistent with respect to the original legal document (more details can be found in Laban et al. (2022)).
**(2) NumPrec** - Numbers are an important part of a legal case judgement, because there are important numbers like dates, statute identifiers (e.g., Act and Section numbers), monetary values, terms of punishment, etc. It is important that these numbers are faithfully represented in the summary. The NumPrec metric measures what fraction of the numbers present in the model-generated summary is also present in the source document. The numbers are identified using the standard Python library.
**(3) NEPrec** - Named Entities (NEs) are also very important in a legal case judgement. If entities like persons, organizations, etc. get changed in the summary, then not only will significant information be lost, but also the summary may become misleading. To detect the amount of inconsistency in a summary in terms of named entities, we calculate the metric called NEPrec that measures what fraction of the Named Entities present in the model-generated summary is also present in the source document. In this work, we detect Named Entities (from both the original document and the summaries) using the standard Spacy Toolkit available at [https://spacy.io/api/entityrecognizer](https://spacy.io/api/entityrecognizer).
Note that the NumPrec and NEPrec metrics are dependent on the ability to detect numbers and named entities accurately. In particular, it is quite challenging to identify all types of named entities from Indian legal documents Kalamkar et al. (2022). Hence the metric values are dependent on the accuracy of the Spacy toolkit used for this purpose.
#### 5.2.2 Comparative results
Table 4 shows the performance of the LLM and abstractive summarization that we have applied in
\begin{table}
\begin{tabular}{|l|c c c|c c c|c c|} \hline
**Model** & **R2-P** & **R2-R** & **R2-F1** & **RL-P** & **RL-R** & **RL-F1** & **ME** & **BLEU (\%)** \\ \hline \hline \multicolumn{10}{|c|}{**General-domain Large Language models**} \\ \hline chatgpt-ldr & **0.3391** & 0.1428 & 0.1729 & **0.2956*** & 0.1785 & 0.2149 & 0.1634 & 7.39 \\ \hline chatgpt-summ & 0.1964 & 0.1731 & 0.1818 & 0.2361 & **0.2087** & 0.2188 & **0.1962** & 10.82 \\ \hline davinci-ldr & 0.2338 & 0.1255 & 0.1563 & 0.2846 & 0.1529 & 0.1901 & 0.1412 & 6.82 \\ \hline davinci-summ & 0.2202 & **0.1795** & **0.1954** & 0.2513 & 0.2058 & **0.2234** & 0.1917 & **11.41** \\ \hline \multicolumn{10}{|c|}{**Legal domain-specific abstractive models**} \\ \hline LegPegasus & 0.1964 & 0.1203 & 0.1335 & 0.2639 & 0.1544 & 0.1724 & 0.1943 & 13.14 \\ \hline LegPegasus-IN & **0.2644** & 0.2430 & 0.2516 & **0.2818*** & 0.2620 & 0.2698 & 0.1967 & 18.66 \\ \hline LegLED & 0.1115 & 0.1072 & 0.1085 & 0.1509 & 0.1468 & 0.1477 & 0.1424 & 8.43 \\ \hline LegLED-IN & 0.2608 & **0.2531** & **0.2550** & 0.2769 & **0.2691*** & **0.2711*** & **0.2261** & **19.81** \\ \hline \multicolumn{10}{|c|}{**Extractive models**} \\ \hline CaseSummarizer & **0.2512** & **0.2269** & **0.2381** & **0.2316** & **0.2085** & **0.2191** & 0.1941 & 15.46 \\ \hline SummaRunner/RNN\_RNN & 0.2276 & 0.2103 & 0.2180 & 0.1983 & 0.1825 & 0.1893 & **0.2038** & 17.58 \\ \hline BertSum & 0.2474 & 0.2177 & 0.2311 & 0.2243 & 0.1953 & 0.2082 & 0.2037 & **18.16** \\ \hline \end{tabular}
\end{table}
Table 3: Performance of summarization models from three families, that we have applied in this work. All metric values are averaged over the 100 documents in the IN-Abs test set. The metrics respectively are Rouge-2 precision, Rouge-2 recall, Rouge-2 F1 score, Rouge-L precision, Rouge-L recall, Rouge-L F1 score, METEOR and BLEU scores. The best value for every metric, for every family of summarization models, is shown in blue-bold. Entries with an asterisk (*) indicate a value that is statistically significantly higher (by the Student T-test at 95% confidence interval) than the best value achieved by an extractive summarisation model (the value shown in blue-bold) for the same metric.
this work, over the IN-Abs dataset. All metric values are averaged over 100 documents. Note that it is meaningless to compute the metrics for extractive methods, since all the three metrics will be 1.0 by definition for any extractive method.
We now see some potential consistency issues with the LLMs and abstractive models. The SummaC scores for the LLMs are in the range \([0.5,0.65]\) which show relatively lower consistency compared to the domain-specific abstractive models. The NEPrec and NumPrec scores are higher, often higher than \(0.9\); still these values indicate presence of some inconsistent / hallucinated named entities and numbers in the abstractive summaries.
Among the domain-specific abstractive models, LegPegasus and LegLED have got relatively low scores (especially LegLED) which indicates substantial presence of hallucinated content in their summaries. LegPegasus-IN and LegLED-IN have consistently got higher scores (across all metrics) than the LegPegasus and LegLED models, which again shows the benefits of domain-specific fine-tuning.
### Takeaways from this section
The analyses in this section allows us to compare between extractive and abstractive summarization models, both trained over Indian legal documents. We see the abstractive models perform better than the extractive models according to standard metrics such as ROUGE, METEOR and BLEU (Table 3). Also the supervised models perform better than LLMs such as Davinci and ChatGPT.
However, abstractive models seem to have problems with consistency (Table 4). Some of the named entities / parts of the summary may be inconsistent with the original document. We look for the presence of such inconsistencies in the next section.
## 6 Inconsistencies in abstractive summaries
The analysis in Section 5.2 indicates that some parts of the summaries generated by abstractive models and LLMs may _not_ be consistent with the original documents. To understand what kind of inconsistencies are present in the summaries, we manually observed a large number of (document, summary) pairs from our dataset. In particular, we observed those sentences that obtained relatively low SummaC scores, and those sentences that contained numbers and named entities that could not be matched with the original documents (while computing NERPrec and NumPrec). We also observed the relevant parts in the main document to understand the errors/inconsistencies.
We found several different types of errors and inconsistency in the abstractive summaries. Table 5, Table 6, Table 7 show some example errors/inconsistencies in the summaries generated by the abstractive models and LLMs for three specific Indian Supreme Court documents (which are mentioned in the table captions). The tables show the name of the model, an extract from the summary showing the error, and an explanation of the error.
We observed some common types of errors in most summaries generated by almost all abstractive models and LLMs, such as **two sentences being merged** (leaving the first sentence incomplete) - for examples, see Table 5 error-3, Table 6, error-1 and Table 7 error-4. These errors mostly happen at the boundary of chunks.
We also observed more serious errors such as **wrong numbers being generated in the summary**, which are not present in the original document. For instance, Table 6 error-5 shows a wrong year being mentioned in the summary - this table refers to a case heard in 1961; hence the year '2019' in the LegLED summary is clearly hallucinated.
We noticed one strange type of error particularly in summaries generated by LegLED - even when the models are summarizing Indian case judgements, names of U.S. Courts and names of U.S. statutes come up in the summaries, which are not at all related to the input document. Examples of such hallucinations are shown in Table 5, error-4 and error-5, and Table 7 error-2. Such hallucina
\begin{table}
\begin{tabular}{|l l l l|} \hline
**Model** & **SummaC** & **NEPrec** & **NumPrec** \\ \hline \hline \multicolumn{4}{|c|}{**General-domain Large Language models**} \\ \hline chatgpt-ldr & 0.5719 & 0.8612 & 0.9498 \\ \hline chatgpt-summ & 0.5762 & **0.9172** & **0.9612** \\ \hline davinci-sum & **0.6356** & 0.8599 & 0.9323 \\ \hline davinci-ldr & 0.6080 & 0.8331 & 0.9123 \\ \hline \multicolumn{4}{|c|}{**Leg domain-specific abstractive models**} \\ \hline LegPegasus & 0.6333 & 0.8429 & 0.9483 \\ \hline LegPegasus-IN & 0.7368 & **0.8542** & **0.9952** \\ \hline LegLED & 0.6563 & 0.7199 & 0.8192 \\ \hline LegLED-IN & **0.8552** & 0.8276 & 0.9769 \\ \hline \end{tabular}
\end{table}
Table 4: Consistency metrics of all abstractive methods and LLMs that we have applied in this work. All metric values are averaged over 100 documents in the IN-Abs dataset. The best value for every metric for each family of summarization models is highlighted.
tions are probably due to the fact that LegLED has been trained on US legal document-summary pairs, and the model has a tendency of generating US court / statute names that it has seen during training. Importantly, we did _not_ observe this type of error in the LegLED-IN summaries, which shows that domain-specific fine-tuning can help to reduce hallucinations. Also we did _not_ observe this particular type of error in the summaries generated by the LLMs (ChatGPT or DaVinci).
There are also examples of **errors in named entities**, e.g., a case where LegLED confused the name of a judge with the name of a lawyer (Table 7 error-1) and a case where chatgpt-summ mistakenly thought the lawyers representing the appellants to be the appellants themselves (Table 5 error-2). Such errors are very difficult to detect by automatic methods, and can lead the summaries to be misleading.
## 7 Concluding discussion
We have tried a wide range of Large Language Models (e.g., Text-Davinci-003 and Turbo-Gpt-3.5) and domain-specific abstractive summarization models (e.g., Legal-LED, Legal-Pegasus) on a dataset of Indian Supreme Court case judgements, and calculated a wide range of metrics. Apart from the standard metrics of evaluation like ROUGE, METEOR, BLEU, we also calculate non-traditional metrics for evaluation of summary consistency like Numprec, NERprec and SummaC.
We observe that the domain-specific fine-tuning improves the performance of abstractive models (LegPegasus-IN and LegLED-IN) in terms of both match with gold standard summary and consistency. LLMs such as Turbo-GPT-3.5 (ChatGPT) and Text-Davinci-003 also perform well in a zero-shot setting, considering they have not been trained specifically on legal documents. However, these LLMs also sometimes generate inconsistent text in summaries.
In general, we see that the abstractive models often outperform the extractive models in terms of metrics such as ROUGE, METEOR and BLEU (Table 3). However, the abstractive models are fraught with issues like inconsistencies and hallucinations in the generated summaries. Some of the problems can be mitigated by _domain-specific fine
\begin{table}
\begin{tabular}{|p{34.1pt}|p{28.5pt}|p{142.3pt}|p{142.3pt}|} \hline
**id** & **Model** & **Extract from summary showing error** & **Explanation of error** \\ \hline
1 & davinci-summ & The language used, Deoria, praying that the proceedings before the Nyaya Panchayat and its order dated December 25, 1963, be quashed... & As per the source document, ‘Deoria’ is the name of a place, not the name of a language. So the sentence in the summary is meaningless. \\ \hline
2 & chatgpt-summ & The appellants, consisting of R Chari, M K Ramamurthi, Vineet Kumar, and Shyamala & The names mentioned are actually that of the lawyers who represented the appellants, not the appellants themselves. The source document states “A. S. R. Chari, M. K Ramamurthi, Vineet Kumar and Shyamala Pappu, **for the** appellants”. The summarization model has mistakenly thought these names to be of the appellants themselves. \\ \hline
3 & chatgpt-ldr & Mahabir filed an application under sections 4 and 5 of theThe case involves allegations of content of court & Incomplete sentence, where the name of the statute (Act) has been omitted in the summary. The most similar sentence in the main document is “On May 21, 1964, Mahabir filed an application under ss. 4 and 5 of the Comtempt of Courts Act, 1952,...” \\ \hline
4 & LegLED &... violating the antifraud provisions of Section 17(a) of the Securities Act of 1933, Section 10(b) of the Securities Exchange Act of 1934 and Rule 10b-5 thereafter,... & There is a lot of hallucination in this part of the summary. The phrases “Section 17(a) of the Securities Act of 1933” and “Section 10(b) of the Securities Exchange Act of 1934 and Rule 10b-5” are all hallucinated. In particular, the Securities Act and Securities Exchange Act are Acts of the USA and are totally unrelated to the source document (which is a case in India). \\ \hline
5 & LegLED & On December 20, 1963, the U.S. District Court for the Southern District of New York entered a final judgment finding a judicial officer guilty of contempt of court for disobeying the order of the U.S. District Court for the Southern District of New York. \\ \hline \end{tabular}
\end{table}
Table 5: Examples of errors in abstractive summaries generated by different models for the Indian Supreme Court judgement available at indiankanoon.org/doc/1234444/. The errors in the summaries have been marked in red. The last column explains the error.
tuning_; for instance, while LegLED often generates names of US courts/statutes while summarizing Indian documents, these errors are considerably lesser in LegLED-IN which is further fine-tuned on Indian legal data. Some of the errors can also be potentially detected and addressed by careful post-processing of the generated summaries. However, some of the errors committed by abstractive models are subtle and much more difficult to detect automatically, e.g., confusing the names of appellants and the names of the lawyers representing the appellants (see the third example in Table 5). To our knowledge, this is the first work to demonstrate examples of such complex errors in abstractive summaries of legal case judgments.
So, as expressed by the experiments reported in this paper, we conclude (1) pre-trained abstractive summarization models and LLMs are not yet ready for fully automatic summarization in a complex domain such as Law; possibly a human-in-the-loop approach is more suitable where a legal expert can monitor the quality of the summaries generated by these methods, and (2) better methods need to be designed to detect complex types of errors in abstractive summaries. In future, we plan to pursue these directions towards improving abstractive summarization in the legal domain.
\begin{table}
\begin{tabular}{|p{42.7pt}|p{113.8pt}|p{113.8pt}|} \hline
**id** & **Model** & **Extract from summary showing error** & **Explanation of error** \\ \hline \hline
1 & chatgpt-tldr & The article examines three circumstances to determine whether the property in goods passedThe document discusses two separate legal cases related to the taxation... & The first sentence is left incomplete and two sentences are merged. \\ \hline
2 & LegPegasus & On September 27, 1960, the Supreme Court of India dismissed an appeal by Daulatram Rameshwarlal and Daulatram Rameshwarlal J.M. against the orders of the Bombay High Court... & The same name “Dalutram Rameshwarlal” is wrongly mentioned twice. There is no person called “Daulatram Rameshwarlal J. M.” in the case. \\ \hline
3 & LegPegasus & The High Court held that the sale of casto oil by M/s. Daulatram Rameshwarlal to M/s. Daulatram Rameshwarlal Ltd was exempt from purchase tax under the provisions... & The same entity (M/s. Daulatram Rameshwarlal) is stated both as the seller and buyer, which is wrong. \\ \hline
4 & LegPegasus & The Court of Appeal held that it is the duty of the buyer to obtain the necessary export licence. The Court of Appeal held that it was for the sellers to obtain the licence and this view was approved by the House of Lords. & The first line says getting the licence is the duty of the buyer, but the immediate next line says it is the duty of the seller – this is inconsistent. In the source document, the relevant part says that the ordinary rule in FOB contracts is that it is the duty of the buyer to obtain the export licence, but there was one special case where it was deemed to be the duty of the sellers. This meaning is lost in the summary. \\ \hline
5 & LegLED & On September 27, 2019, the U.S. District Court for the Southern District of New York entered a final judgement against Daulatram Rameshwarlal, a firm registered under the Indian Partnership Act, and Daulatram Rameshwarl, a registered dealer under the Indian Partnership Act, for claiming exemption from Sales Tax in respect of sales of cotton... & The “U.S. District Court of New York’ is hallucinated (the original case was argued entirely in Indian courts). Also the year 2019’ is hallucinated – the original case is of 1961, so no event of 2019 could have been referred. Also, the summarization model did not understand that the same entity ‘Daulatram Rameshwarl’ is referred to both as a ‘firm’ and a ‘registered dealer’; the model has assumed two separate entities. \\ \hline
6 & LegPegasus-IN & The intention of the parties that in compliance with the requirements of cl.5(2) of the Exports (Control) OrderThere is no circumstance which would justify a conclusion that... & The first sentence is left incomplete and two sentences are merged. \\ \hline
7 & LegLED-IN & The Court was right in holding that the Court was wrong in holding that it was not necessary & This sentence in the summary is meaningless. The source document is a case heard in the Supreme Court of India, and is an appeal against a decision pronounced by the Bombay High Court. Hence two courts are involved, but it is not clear from the summary which court is being referred to by which occurrence of the word ‘court’. \\ \hline \end{tabular}
\end{table}
Table 6: Examples of errors in abstractive summaries generated by different models for the Indian Supreme Court judgement available at [https://indiankanoon.org/doc/27285/](https://indiankanoon.org/doc/27285/). The errors in the summaries are marked in red, and explained in the last column.
**Acknowledgements:** The authors acknowledge useful feedback and suggestions about the work from Jack Conrad (from Thomson Reuters Labs). The research is partially supported by the TCG Centres for Research and Education in Science and Technology (CREST), India through a project titled "Smart Legal Consultant: AI-based Legal Analytics".
| 法律判決の自動要約は従来、
EXTRACTIVE
要約手法で試みられてきました。しかし、近年では、
ABSTRACTION
要約モデルが人気になり、自然で論理的な要約ができるようになったのです。
法律分野に特化した事前学習されたABSTRACTION要約モデルが現在利用できます。
また、ChatGPTなどの汎用的な大規模言語モデルは、
高品質なテキストを生成し、テキスト要約にも対応できます。
そのため、これらのモデルが判決要約のための汎用適用に適しているのか疑問に思われます。
この疑問を探るために、
いくつかの最新技術の法律分野特化型ABSTRACTION要約モデルと汎用的な大規模言語モデルを、
インドの裁判で判決を要約する際に適用し、生成された要約の品質を検証しました。
標準的な要約品質評価指標に加え、 |
2306.16734 | Unified View of Damage leaves Planimetry & Analysis Using Digital Images
Processing Techniques | The detection of leaf diseases in plants generally involves visual
observation of patterns appearing on the leaf surface. However, there are many
diseases that are distinguished based on very subtle changes in these visually
observable patterns. This paper attempts to identify plant leaf diseases using
image processing techniques. The focus of this study is on the detection of
citrus leaf canker disease. Canker is a bacterial infection of leaves. Symptoms
of citrus cankers include brown spots on the leaves, often with a watery or
oily appearance. The spots (called lesions in botany) are usually yellow. It is
surrounded by a halo of the leaves and is found on both the top and bottom of
the leaf. This paper describes various methods that have been used to detect
citrus leaf canker disease. The methods used are histogram comparison and
k-means clustering. Using these methods, citrus canker development was detected
based on histograms generated based on leaf patterns. The results thus obtained
can be used, after consultation with experts in the field of agriculture, to
identify suitable treatments for the processes used. | Pijush Kanti Kumar, DeepKiran Munjal, Sunita Rani, Anurag Dutta, Liton Chandra Voumik, A. Ramamoorthy | 2023-06-29T07:15:45 | http://arxiv.org/abs/2306.16734v1 | # Unified View of Damage leaves Planimetry & Analysis Using Digital Images Processing Techniques
###### Abstract
The detection of leaf diseases in plants generally involves visual observation of patterns appearing on the leaf surface. However, there are many diseases that are distinguished based on very subtle changes in these visually observable patterns. This paper attempts to identify plant leaf diseases using image processing techniques. The focus of this study is on the detection of citrus leaf darker disease. Canker is a bacterial infection of leaves. Symptoms of citrus cankers include brown spots on the leaves, often with a water or only appearance. The spots (called lesions in botany) are usually yellow. It is surrounded by a halo of the leaves and is found on both the top and bottom of the leaf. This paper describes various methods that have been used to detect citrus leaf cancer disease. The methods used are histogram comparison and k-means clustering. Using these methods, citrus canker development was detected based on histograms generated based on leaf patterns. The results thus obtained can be used, after consultation with experts in the field of agriculture, to identify suitable treatments for the processes used.
Digital Image Processing, k - Means Clustering, Citrus Leaf Canker Disease +
Footnote †: publicationid: pubid: 2023 IEEE
## I Introduction
In today's world, the farmland volume serves as an additional source of food. Agriculture's productivity has a significant impact on the Indian economy. As a result, it is crucial to identify plant diseases in the field of agriculture. Use of an automatic disease recognition system is advantageous for spotting a plant pathogens in its very early stages. For example in the case, the United States has pine trees that are susceptible to a dangerous disease called little leaf disorder. The affected tree grows slowly and perishes within six years. Parts of the Southern US, including Alabama and Georgia, are affected by it. Early diagnosis in these situations might have been beneficial. The only method currently in use for identifying and detecting _phytopathogens_ is professional assessment using only one's unaided eye. This requires a sizable group of specialists and ongoing vegetation surveillance, both of which are very expensive when dealing with large farmlands. Meanwhile, in some nations, farmers lack access to the necessary resources and even the knowledge to speak with experts. Because of this, consulting experts is expensive and time-consuming. The recommended method works well in these circumstances for keeping an eye on huge expanses of harvests. Visual diagnosis of plant diseases is more time-consuming, less accurate, and only practicable in a few locations. However, using an automatic detection method will require less work, less time, and result in a higher degree of accuracy. Brown and yellow spots, early and late scorch, and certain other common bacterial, viral, and fungal diseases are all seen in plants. The size of the diseased area is measured using image processing, and the difference in color of the damaged area is also determined. The method used for breaking down or classifying an image into numerous components is known as image segmentation. Image segmentation can be done in a variety of cases right now, from the straightforward segmen
tation process to sophisticated color segmentation methods. Usually, these product attributes to elements that people can easily differentiate into separate elements and see. There are numerous techniques for segmenting images because computers lack the ability to recognize objects intelligently. The image's unique components are the basis for the segmentation algorithm. This could be a section of an image, color features, or boundary details. Diseased leaf spots play an important role in plant growth environment. Diseases can also be easily identified with the help of affected areas in culture [1]. Usually, of course, leaves clearly show infected areas and are easily identifiable. Therefore, we can say that plant colour change is an essential aspect of notification. If the crop is in good health, the crop will have different colours, but if the crop dies from some harmful pathogen, it will automatically change colour. Plant diseases affect specific parts. This can lead to decreased productivity. The main method used in practice to detect plant diseases is early observation by a specialist, exposing the eye. In this feature retrieval-based investigation, influenced leaf parts were examined using the suggested diagonal disagreement of border, color, and appearance variability features. K-Means clustering was employed. Six different disease types were predicted using this method. Results of the performance assessments were eventually achieved. This reviewed work [2] is based on methods to reduce computational complexity through improved automated crop disease detection. This is due to the fact that it significantly harms the agricultural industry and makes the disease's symptoms visible. Once more, the possibility for treating and preventing infected plants is clear. In order to detect and categorize plant diseases, it is necessary to look for reliable, affordable, and accurate techniques. Test photographs of various cotton sheets were initially implemented. The pictures that are are then captured using image processing approaches, and helpful features are extracted for additional analysis. uses a variety of statistical techniques to categorize the images in accordance with the precise issue of the influenced leaf spot regions. The analysis of the best corresponded to of characteristic conclusions for leaves impacted by the perfect solution was done using the selection of features. In the classification phase [3], feature values corresponding to the variance of edge, colour and texture features are stored in the image domain. Based on the affected areas of leaf diseases, the affected sites were identified.
## II Materials and Methods
### _Materials_
The following materials were taken into use.
1. Nikon Make 12.5 Megapixels Digital Camera
2. Personal Computer
3. White A4 Paper Sheet for background
4. MATLAB 2013 version or above
5. Photographs of the leaves were arranged in number.
### _Methods_
#### Ii-B1 Graphical Method
A leaf with a measurement region was positioned on graph paper with a 1 mm-wide grid. On graph paper, the sheets are meticulously and precisely defined with the aid of a pencil. It was determined how many grids the sheet's detail encompassed in total. The boundary detail is regarded as 1 if it takes up over fifty percent of the grid; otherwise, it is addressed as 0. The precise leaf area is represented by an array of grid numerals.
#### Ii-B2 Image Processing Method
A wider audience are capable of determining leaf area thanks to the MATLAB-based technique used for the processing of images, which is a partially automated technique. Code is created using MATLAB 2013 or later. For the latest releases of the application, this code is functional. The procedure is as follows:
1. Study the picture.
2. RGB to grayscale conversion.
3. Create a binary picture from the image that is grayscale.
4. Do the leaf area calculation.
## III k Means Clustering
Data can be grouped in a variety of ways, however the most popular method is the k-Means algorithm [4], which aims to make categories become more comparable while simultaneously maintaining them as far apart as feasible. In essence, distance calculations using Euclidean distance are performed using k-Means. Euclidean distance calculates the distance between two given points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) using the following formula
\[Distance_{euclidian}=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} \tag{1}\]
The above formula captures distance in 2D space, but the same holds true in multidimensional space as the number of terms added increases. The '_k_' in _k_-Means represents the number of distinct clusters into which the data is divided. A fundamental caveat [5] of the _k_-Means algorithm is that the data must be continuous in nature. It will not work if your data is categorical in nature.
### _Data Preparation_
As was previously addressed, the majority of methodologies for clustering, including k-Means, rely on the concept of separation. They compute their distance from a specific point and make an effort to reduce it. When distinct variables utilize distinct components, something occurs. For instance, I want to divide the general population of India into different groups since weight is measured in kilograms of weight while heights is measured in centimeters. As can be observed, the variable's components have a significant impact on the matrix of distances above. As a result, standardizing the information prior to seeking clustering activities is an excellent decision.
### _Algorithm_
_k_-Means is an iterative clustering process. This is repeated until an optimal solution or cluster is reached in the problem space. The following pseudo-example covers the basic steps
involved in \(k\)-means clustering, which is commonly used to cluster data.
Start by deciding how many clusters you want, in the present instance three. The \(k\)-Means algorithm attempts to connect the closest points to the arbitrarily data centers it starts with.
### _Image acquisition_
The leaf measuring the affected area is placed on a black background with no light reflection [6]. Hold the camera horizontally to the paper. The shooting distance is neither too close nor too far. The photo has been adjusted to cover only the background. See Fig. 2.
### _Observing the picture_
The image is saved in the computer as b3.jpeg. This image is read for further processing by MATLAB.
### _Dividing the Image_
The initial picture is divided into two distinct kinds of images in this phase based on how its colors vary. These pictures were chosen for their ability to show impacted and untouched leaf regions. This is done by the \(k\)-Means clustering method. The algorithm for this method is shown below.
1. Read the image.
2. Change an image's color space from RGB towards LAB.
3. Use \(k\)-means clustering to categorize the colors in the 'a*b*' space.
4. With the help of the \(k\)-means outcomes identify each pixel in the picture.
5. Make pictures that divide the initial picture into sections based on color.
After applying the above algorithm segmented images were obtained. See Fig. 3 and Fig. 4.
### _Convert the image (Cluster - I) into Binary image_
After reading the image, I need to first convert this image from RGB to grayscale [7] and then convert the grayscale image to binary. In this image, the white - coloured areas indicate the unaffected parts of the leaves, as shown in the image in Fig. 5. Useful for calculating the total number of pixels in unaffected areas of the sheet.
### _Convert the image (Cluster - II) into Binary image_
After converting the image into binary image, image in Fig. 6 is obtained where white coloured regions indicate affected regions of the leaf. It helps to calculate total number of pixels of affected regions [8] of the leaf.
### _Pixel Calculation_
#### Iv-H1 From Cluster I
WP (Pixels of unaffected regions) = 195612
where, WP denotes white pixels of Cluster - I.
#### Iv-H2 From Cluster II
WP1 (Pixels of affected regions) = 41246
where, WP1 denotes white pixels of Cluster - II.
Now, total pixels of the leaf area that is obtained is shown below.
\[TP=WP+WP1 \tag{2}\]
\[TP=195612+41246=236858\]
Percentage of affected pixels can be obtained by the following as
\[Error=\frac{WP1\ (pixels\ of\ affected\ regions)}{TP\ (Total\ pixels\ of\ the\ leaf\ area)}\times 100\]
ERROR (%) = 17.4138 %
## IV Results and Discussion
Leaves were selected from different plots and different types of citrus leaves to test the performance [9] of the new measurement system. Leaf area is calculated using the square grid method and is considered standard area. When we obtained sampled data from different affected leaves. Then you can get a clear idea about the whole damaged sheet in relation to the specific area. This proposed method is faster [10] and more accurate than any standard method.
## V Conclusion
This article discusses a method for measuring the area of citrus leaves using the processing of digital images. Calculating the quantity of leaves that are damaged has been demonstrated to be possible with the executed algorithmic structure. Although more time-consuming, grid calculating strategies are highly precise for measuring the surface area of leaves. The image processing method is also a fast method [11 - 21] with high accuracy [22] and precision [23]. You may modify the leaf picture at any moment if you solely preserve it [24]. This can serve as an empirical basis for creating an affordable leaf area meter that meets precision farming requirements. For precise area computations and disease incidence projections for chemical pesticides and application of fertilizer, the proportion of error aspect is crucial and required.
## References
* [1] Prathyakshini, Akshaya, and C. V. Aravinda, "Classification and Clustering of Infected Leaf Plant Using K-Means Algorithm," _Communications in Computer and Information Science_, pp. 468-474, 2018, doi:[https://doi.org/10.1007/978-981-10-9059-2_41](https://doi.org/10.1007/978-981-10-9059-2_41).
* [2] M. Zhang et al., "Chromosomal-level genome assembly of potato underground," Phtophiromae operuelle a pse of solanocous crops," _Scientific Data_, vol. 9, no. 1, Dec. 2022, doi: [https://doi.org/10.1038/s41597-022-01859-5](https://doi.org/10.1038/s41597-022-01859-5).
* [3] H. A. Gharib and A. M. Mandour, "Effect of Populus nigra spring and autumn leaves extract on Capsicum anum infected with pepper mild mottle virus," _Scientific Reports_, vol. 12, no. 1, Dec. 2022, doi: [https://doi.org/10.1038/s41598-022-24786-2](https://doi.org/10.1038/s41598-022-24786-2).
* [4] A. A. Najm, M. R. M. S. Hadi, F. Fazeli, M. T. Darzi, and A. Rahi, "Effect of Integrated Management of Nitrogen Fertilizer and Cattle Manure on the Leaf Chlorophyll, Yield, and Tuber Glycosalkioids of Agria Potato," _Communications in Soil Science and Plant Analysis_, vol. 43, no. 6, pp. 912-923, Mar. 2012, doi:[https://doi.org/10.1080/00103624.2012.653027](https://doi.org/10.1080/00103624.2012.653027). | 植物の病気の検出において、一般的には葉の表面に現れる模様を観察することが含まれます。しかし、これらの視覚的に観察可能な模様に非常に微妙な変化に基づいて、多くの病気は区別されます。この論文では、画像処理技術を用いて植物の葉の病気の検出を試みます。この研究の焦点には、柑橘類の葉の枯れ病が挙げられます。枯れ病は葉の細菌感染です。柑橘類の枯れ病の症状には、葉に茶色い斑点、しばしば水っぽいまたは油っぽい外観が含まれます。斑点は（植物学においては、病害）と呼ばれるもので、通常黄色です。葉の周りに黄色のハローが形成され、葉の上と下両方に存在します。この論文では、柑橘類の葉の枯れ病を検出するための様々な方法について説明しています。使用されている方法には、ヒストグラム比較とk- |
2301.05089 | Approximate Information States for Worst-Case Control and Learning in
Uncertain Systems | In this paper, we investigate discrete-time decision-making problems in
uncertain systems with partially observed states. We consider a non-stochastic
model, where uncontrolled disturbances acting on the system take values in
bounded sets with unknown distributions. We present a general framework for
decision-making in such problems by using the notion of the information state
and approximate information state, and introduce conditions to identify an
uncertain variable that can be used to compute an optimal strategy through a
dynamic program (DP). Next, we relax these conditions and define approximate
information states that can be learned from output data without knowledge of
system dynamics. We use approximate information states to formulate a DP that
yields a strategy with a bounded performance loss. Finally, we illustrate the
application of our results in control and reinforcement learning using
numerical examples. | Aditya Dave, Nishanth Venkatesh, Andreas A. Malikopoulos | 2023-01-12T15:36:36 | http://arxiv.org/abs/2301.05089v2 | # Approximate Information States for Worst-Case Control and Learning in Uncertain Systems
###### Abstract
In this paper, we investigate discrete-time decision-making problems in uncertain systems with partially observed states. We consider a non-stochastic model, where uncontrolled disturbances acting on the system take values in bounded sets with unknown distributions. We present a general framework for decision-making in such problems by developing the notions of information states and approximate information states. In our definition of an _information state_, we introduce conditions to identify for an uncertain variable sufficient to construct a dynamic program (DP) that computes an optimal strategy. We show that many information states from the literature on worst-case control actions, e.g., the _conditional range_, are examples of our more general definition. Next, we relax these conditions to define _approximate information states_ using only output variables, which can be learned from output data without knowledge of system dynamics. We use this notion to formulate an approximate DP that yields a strategy with a bounded performance loss. Finally, we illustrate the application of our results in control and reinforcement learning using numerical examples.
Uncertain systems, worst-case control, approximate dynamic programming, offline reinforcement learning
## I Introduction
Decision-making under incomplete information is a fundamental problem in modern engineering applications involving cyber-physical systems [1], e.g., connected and automated vehicles [2], social media platforms [3], and robot swarms [4]. In such applications, an agent is often required to sequentially select control inputs to a dynamic system using only partial observations at each instance of time, while simultaneously accounting for uncontrolled disturbances that can interfere with the system's evolution. The most common modeling paradigm for such decision-making problems is _the stochastic approach,_ where all disturbances to the system are considered to be random variables with known distributions, and the agent aims to select a decision-making strategy that minimizes the _expected incurred cost_[5]. Stochastic models have been utilized for problems in both control theory [6, 7, 8, 9, 10, 11, 12, 13] and reinforcement learning [14, 15, 16, 17, 18]. A decision-making strategy derived using the stochastic approach performs optimally on average across numerous operations of the system. However, this performance degrades rapidly when there is a mismatch between the distribution on disturbances considered in modeling and the realizations encountered during implementation [19]. Furthermore, many safety-critical applications require guarantees on the agent's performance during each operation [20]. Thus, in such applications it is inadequate to measure performance using the expected cost.
_The non-stochastic approach_ is an alternate modeling paradigm for safety-critical systems, where all disturbances are considered to belong to known sets with unknown distributions. The agent aims to select a decision-making strategy that minimizes the _worst-case incurred cost_ across a finite time horizon [21]. Because this approach focuses on robustness against worst-case realizations of the disturbances, the resulting strategy yields more conservative decisions than the stochastic approach. At the expense of average performance, this strategy provides concrete guarantees on the worst-case performance during each operation of the system. Thus, this approach has been widely applied to systems under attack from an adversary, e.g., cyber-security [22] or cyber-physical systems [23], and systems where a single failure can be damaging, e.g., water reservoirs [24], or power systems [25].
In this paper, we propose a framework for non-stochastic decision-making using only partial observations in a dynamic system. When the system's dynamics are known to the agent, this problem falls under the purview of control theory [26]. However, many applications involve decision-making with an incomplete knowledge of the dynamics as, e.g., automated driving in mixed traffic [27] and human-robot coordination [28], or decision-making without a reliable state-space model, e.g., medical dead-end identification [29]. These restrictions typically lead to formulating a reinforcement learning problem [30, 31]. To account for both of these potential cases, we formulate our problem using only output variables without assuming a known state-space model. In our exposition, we present rigorous definitions for the notions of _information states_ and _approximate information states_. Using these notions, a surrogate state-space model can be constructed from output variables. This surrogate model can be used to formulate a control problem with full state observation, whose solution yields either an optimal, or an approximate strategy of the original problem. In reinforcement learning problems, the surrogate model can be learned from output data. For perfectly observed states, the agent can derive a decision-making strategy using standard techniques [32, 33, 34].
### _Related Work_
#### I-A1 Control theory
There have been numerous research efforts in control theory to study dynamic decision-making | この論文では、不確定な系における離散時間決策問題を調査します。観測可能な状態が部分的に観察されるような、非確率的モデルで、システムに作用する制御不能な干渉は、未知の分布を持つ有界な集合の値をとります。情報状態と近似情報状態という概念を用いて、そのような問題における決断を一般化した枠組みを提示し、最適な戦略を計算するための不確定変数を特定するための条件を導入します。次に、これらの条件を緩和し、システムのダイナミクスを知らなくても、出力データから学習できる近似情報状態を定義します。近似情報状態を用いて、パフォーマンスの損失が限定された戦略を定義するDPを構成します。最後に、この結果の適用を示し、制御と強化学習に数値例を用いて示します。 |
2307.01161 | Gorenstein modules and dimension over large families of infinite groups | We give characterizations of Gorenstein projective, Gorenstein flat and
Gorenstein injective modules over the group algebra for large families of
infinite groups and show that every weak Gorenstein projective, weak Gorenstein
flat and weak Gorenstein injective module is Gorenstein projective, Gorenstein
flat and Gorenstein injective, respectively. These characterizations provide
Gorenstein analogues of Benson's cofibrant modules. We deduce that, over a
commutative ring of finite Gorenstein weak global dimension, every Gorenstein
projective module is Gorenstein flat. Moreover, we study cases where the tensor
product and the group of homomorphisms between modules over the group algebra
is a Gorenstein module. Finally, we determine the Gorenstein homological
dimension of an $\textsc{\textbf{lh}}\mathfrak{F}$-group over a commutative
ring of finite Gorenstein weak global dimension. | Dimitra-Dionysia Stergiopoulou | 2023-07-03T17:12:27 | http://arxiv.org/abs/2307.01161v2 | # Gorenstein modules and dimension over large families of infinite groups
###### Abstract.
Projectively coresolved Gorenstein flat modules were introduced by Saroch and Stovicek and were shown to be Gorenstein projective. We give characterizations of Gorenstein projective, Gorenstein flat and projectively coresolved Gorenstein flat modules over a group ring \(RG\), where \(G\) is an \(\mathtt{LH}\mathfrak{F}\)-group or a group of type \(\Phi_{R}\) and \(R\) is a commutative ring of finite Gorenstein weak global dimension. In this situation, we prove that every Gorenstein projective \(RG\)-module is projectively coresolved Gorenstein flat. We deduce that every Gorenstein projective \(RG\)-module is Gorenstein flat. The existence of weak characteristic modules for a group \(G\) over a commutative ring \(R\) plays a central role in our results. Furthermore, we determine the Gorenstein homological dimension of an \(\mathtt{LH}\mathfrak{F}\)-group over a commutative ring of finite Gorenstein weak global dimension.
Key words and phrases:Gorenstein homological algebra, Gorenstein projective module, Gorenstein flat module, Group ring, Gorenstein homological dimension of group, \(\mathtt{LH}\mathfrak{F}\)-group, Group of type \(\Phi\) 2020 Mathematics Subject Classification: Primary: 16E05, 16E10, 18G20, 18G25 Research supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "1st Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant", project number 4226.
is the supremum of the flat lengths (dimensions) of injective \(R\)-modules (see [8, Theorem 2.4]). Our methods are based on the notion of a weak characteristic module for \(G\), i.e. an \(R\)-pure \(RG\)-monomorphism \(0\to R\to A\) where \(A\) is \(R\)-flat and \(\operatorname{fd}_{RG}A<\infty\). The notion of a weak characteristic module generalizes the characteristic modules which were used to prove many properties of the Gorenstein cohomological dimension \(\operatorname{Gcd}_{R}G\) of a group \(G\) (see [1, 22]). As shown in [16, Theorem 5.10], for every commutative ring \(R\) of finite Gorenstein weak global dimension, the existence of a weak characteristic module for a group \(G\) over a commutative ring \(R\) is equivalent with the finiteness of the Gorenstein homological dimension \(\operatorname{Ghd}_{R}G\) of \(G\). Furthermore, we make use of the stability properties of Gorenstein flat and PGF modules established in [5] and [16]. Finally, over an \(\operatorname{\mathfrak{LH}}\mathfrak{F}\)-group, our arguments are based on transfinite induction.
In Section 2, we establish notation, terminology and preliminary results that will be used in the sequel. In Sections 3 and 4 we consider a commutative ring \(R\) of finite Gorenstein weak global dimension and a group \(G\) such that there exists a weak characteristic module for \(G\) over \(R\). By noting first that the tensor product of a weak Gorenstein flat \(RG\)-module and an \(R\)-flat module is Gorenstein flat (with diagonal action), we prove in Section 3 that the class of Gorenstein flat \(RG\)-modules coincides with the class of modules which are syzygies in a double infinite exact sequence of flat \(RG\)-modules and moreover with the class of \(RG\)-modules such that after being tensored with \(B(G,R)\) yield a Gorenstein flat module, with diagonal action (see Theorem 3.7). In Section 4, we note first that the tensor product of a weak Gorenstein projective \(RG\)-module and an \(R\)-projective module is PGF (with diagonal action). Working in a similar way as in Section 3, we prove that the class of PGF \(RG\)-modules coincides with the class of modules which are syzygies in a double infinite exact sequence of projective \(RG\)-modules and moreover with the class of \(RG\)-modules which after being tensored with \(B(G,R)\) yield a PGF module, with diagonal action. In this way, we infer also that the class of Gorenstein projective \(RG\)-modules coincides with the the class of PGF \(RG\)-modules, and hence every Gorenstein projective \(RG\)-module is Gorenstein flat (see Theorem 4.7). This result is noteworthy, since it is not known whether all Gorenstein projective modules are Gorenstein flat over an arbitrary ring. Since for every group of type \(\Phi_{R}\), the \(RG\)-module \(B(G,R)\) is weak characteristic, we obtain in Sections 3 and 4 similar results for this class of groups.
In Sections 5 and 6, we consider a commutative ring of finite Gorenstein weak global dimension and an \(\operatorname{\mathfrak{LH}}\mathfrak{F}\)-group \(G\). Under these assumptions, we prove the same characterizations of Gorenstein flat, Gorenstein projective and PGF \(RG\)-modules, as in Sections 3 and 4. It seems that, under the assumption for the commutative ring \(R\) to be of finite Gorenstein weak global dimension, the existence of a weak characteristic module for a group \(G\) over \(R\) is essentialy equivalent with \(G\) being an \(\operatorname{\mathfrak{LH}}\mathfrak{F}\)-group.
In our final section, we study the Gorenstein homological dimension \(\operatorname{Ghd}_{R}G\) of an \(\operatorname{\mathfrak{LH}}\mathfrak{F}\)-group \(G\) over a commutative ring \(R\) of finite Gorenstein weak global dimension and we prove that \(\operatorname{Ghd}_{R}G=\operatorname{fd}_{RG}B(G,R)\) (see Theorem 7.7). For this purpose, we firstly prove that \(\operatorname{f.k}(RG)=\operatorname{sfli}(RG)=\operatorname{fin.f.dim}(RG)\), where we denote by \(\operatorname{f.k}(RG)\) the supremum of flat dimensions of \(RG\) modules \(M\) which have finite flat dimension over every finite subgroup of \(G\) (see Corollary 7.5). The Gorenstein cohomological dimension \(\operatorname{Gcd}_{R}G\) of an \(\operatorname{\mathfrak{LH}}\mathfrak{F}\)-group \(G\) over a commutative ring \(R\) of finite global dimension, was studied in [4, Theorem 3.1] and [12, Theorem A.1].
_Terminology._ All rings are assumed to be associative and unital and all ring homomorphisms will be unit preserving. Unless otherwise specified, all modules will be left \(R\)-modules.
## 2. Preliminaries
In this section, we collect certain notions and preliminary results that will be used in the sequel.
### Gorenstein projective, Gorenstein flat and PGF modules
An acyclic complex \(\mathbf{P}\) of projective modules is said to be a complete projective resolution if the complex of abelian groups \(\operatorname{Hom}_{R}(\mathbf{P},Q)\) is acyclic for every projective module \(Q\). Then, a module is Gorenstein projective if it is a syzygy of a complete projective resolution. We let \(\operatorname{\mathsf{GProj}}(R)\) be the class of Gorenstein projective modules. The Gorenstein projective dimension \(\operatorname{Gpd}_{R}M\) of a module \(M\) is the length of a shortest resolution of \(M\) by Gorenstein projective modules. If no such resolution of finite length exists, then we write \(\operatorname{Gpd}_{R}M=\infty\).
An acyclic complex \(\mathbf{F}\) of flat modules is said to be a complete flat resolution if the complex of abelian groups \(I\otimes_{R}\mathbf{F}\) is acyclic for every injective right module \(I\). Then, a module is Gorenstein flat if it is a syzygy of a complete flat resolution. We let \(\operatorname{\mathsf{GFlat}}(R)\) be the class of Gorenstein flat modules. The Gorenstein flat dimension \(\operatorname{Gfd}_{R}M\) of a module \(M\) is the length of a shortest resolution of \(M\) by Gorenstein flat modules. If no such resolution of finite length exists, then we write \(\operatorname{Gfd}_{R}M=\infty\).
The projectively coresolved Gorenstein flat modules (PGF-modules, for short) were introduced by Saroch and Stovicek [19]. Such a module is a syzygy of an acyclic complex of projective modules \(\mathbf{P}\), which is such that the complex of abelian groups \(I\otimes_{R}\mathbf{P}\) is acyclic for every injective module \(I\). It is clear that the class \(\operatorname{\mathsf{PGF}}(R)\) of PGF modules is contained in \(\operatorname{\mathsf{GFlat}}(R)\). The inclusion \(\operatorname{\mathsf{PGF}}(R)\subseteq\operatorname{\mathsf{GProj}}(R)\) is proved in [19, Theorem 4.4]. Moreover, the class of PGF \(R\)-modules, is closed under extensions, direct sums, direct summands and kernels of epimorphisms. The PGF dimension \(\operatorname{PGF-dim}_{R}M\) of a module \(M\) is the length of a shortest resolution of \(M\) by PGF modules. If no such resolution of finite length exists, then we write \(\operatorname{PGF-dim}_{R}M=\infty\) (see [9]).
### Group rings
Let \(R\) be a commutative ring, \(G\) be a group and consider the associated group ring \(RG\). The standard reference for group cohomology is [6]. Using the diagonal action of the group \(G\), the tensor product \(M\otimes_{R}N\) of two \(RG\)-modules is also an \(RG\)-module using the diagonal action of \(G\); we define \(g\cdot(x\otimes y)=gx\otimes gy\in M\otimes_{R}N\) for every \(g\in G\), \(x\in M\) and \(y\in N\). We note that for every projective \(RG\)-module \(M\) and every \(R\)-projective \(RG\)-module \(N\), then the diagonal \(RG\)-module \(M\otimes_{R}N\) is also projective. Similarly, for every flat \(RG\)-module \(M\) and every \(R\)-flat \(RG\)-module \(N\), the diagonal \(RG\)-module \(M\otimes_{R}N\) is also flat. Indeed, since the class \(\operatorname{\mathsf{Flat}}(RG)\) of flat \(RG\)-modules is closed under filtered colimits and direct sums, invoking the Govorov-Lazard theorem, we may assume that \(M=RG\).
### \(\operatorname{\mathsf{LH}}\)-groups and groups of type \(\Phi_{R}\)
The class \(\operatorname{\mathfrak{HF}}\) was defined by Kropholler in [17]. This is the smallest class of groups, which contains the class \(\mathfrak{F}\) of finite groups and is such that whenever a group \(G\) admits a finite dimensional contractible \(G\)-CW-complex with stabilizers in \(\operatorname{\mathfrak{HF}}\), then we also have \(G\in\operatorname{\mathfrak{HF}}\). More precisely, we define \(\operatorname{\mathfrak{n}}_{0}\mathfrak{F}:=\mathfrak{F}\), and for every ordinal number \(\alpha>0\), we say that a group \(G\) belongs to the class \(\operatorname{\mathfrak{n}}_{\alpha}\mathfrak{F}\) iff there exists a finite dimensional contractible CW-complex on which \(G\) acts such that every isotropy subgroup of the action belongs to \(\operatorname{\mathfrak{n}}_{\beta}\mathfrak{F}\) for some ordinal \(\beta<\alpha\). A group belongs to the class \(\operatorname{\mathfrak{n}}\mathfrak{F}\), if it belongs to the class \(\operatorname{\mathfrak{n}}_{\alpha}\mathfrak{F}\) for some ordinal \(\alpha\). The class \(\operatorname{\mathsf{LH}}\mathfrak{F}\) consists of those groups,
all of whose finitely generated subgroups are in \(\mathfrak{h}\mathfrak{F}\). All soluble groups, all groups of finite virtual cohomological dimension and all automorphism groups of Noetherian modules over a commutative ring are \(\mathfrak{L}\mathfrak{H}\mathfrak{F}\)-groups. The class \(\mathfrak{L}\mathfrak{H}\mathfrak{F}\) is closed under extensions, ascending unions, free products with amalgamation and HNN extensions.
A group \(G\) is said to be of type \(\Phi_{R}\) if it has the property that for every \(RG\)-module \(M\), \(\operatorname{pd}_{RG}M<\infty\) if and only if \(\operatorname{pd}_{RH}M<\infty\) for every finite subgroup \(H\) of \(G\). These groups were defined over \(\mathbb{Z}\) in [21]. Over a commutative ring \(R\) of finite global dimension, every group of finite virtual cohomological dimension and every group which acts on a tree with finite stabilizers is of type \(\Phi_{R}\) (see [18, Corollary 2.6]).
Let \(B(G,R)\) be the \(RG\)-module which consists of all functions from \(G\) to \(R\) whose image is a finite subset of \(R\). The \(RG\)-module \(B(G,R)\) is \(R\)-free and \(RH\)-free for every finite subgroup \(H\) of \(G\). For every element \(\lambda\in R\), the constant function \(\iota(\lambda)\in B(G,R)\) with value \(\lambda\) is invariant under the action of \(G\). The map \(\iota:R\to B(G,R)\) which is defined in this way is then \(RG\)-linear and \(R\)-split. Indeed, for every fixed element \(g\in G\), there exists an \(R\)-linear splitting for \(\iota\) by evaluating functions at \(g\). Moreover, the cokernel \(\overline{B}(G,R)\) of \(\iota\) is \(R\)-free (see [7, Lemma 3.3] and [3, Lemma 3.4]). We note that \(\operatorname{pd}_{RG}B(G,R)<\infty\) over any group \(G\) of type \(\Phi_{R}\). Thus, \(B(G,R)\) is a (weak) characteristic module for every group \(G\) of type \(\Phi_{R}\) over any commutative ring \(R\).
### Gedrich-Gruenberg invariants and Gorenstein global dimensions
The invariants \(\operatorname{silp}R\), \(\operatorname{spli}R\) were defined by Gedrich and Gruenberg in [13] as the supremum of the injective lengths (dimensions) of projective modules and the supremum of the projective lengths (dimensions) of injective modules, respectively. The invariant \(\operatorname{sfli}R\) is defined similarly as the supremum of the flat lengths (dimensions) of injective modules. Since projective modules are flat, the inequality \(\operatorname{sfli}R\leq\operatorname{spli}R\) is clear. Moreover, for every commutative ring \(R\) we have the inequality \(\operatorname{silp}R\leq\operatorname{spli}R\), with equality if \(\operatorname{spli}R<\infty\) (see [9, Corollary 5.4]). Thus, for every commutative ring \(R\), invoking [10, Theorem 4.1], we infer that the finiteness of \(\operatorname{spli}R\) is equivalent to the finiteness of \(\operatorname{Ggl.dim}R\), and then \(\operatorname{Ggl.dim}R=\operatorname{spli}R\). Furthermore, for every commutative ring \(R\), invoking [8, Theorem 2.4], we infer that the finiteness of \(\operatorname{sfli}R\) is equivalent to the finiteness of \(\operatorname{Gwgl.dim}R\), and then \(\operatorname{Gwgl.dim}R=\operatorname{sfli}R\).
### Weak Gorenstein modules
Let \(R\) be a commutative ring. We denote by \(\operatorname{WGProj}(R)\) the class of modules which are syzygies of an acyclic complex of projective modules \(\mathbf{P}\). We note that \(\operatorname{GProj}(R)\subseteq\operatorname{WGProj}(R)\) and \(\operatorname{PGF}(R)\subseteq\operatorname{WGProj}(R)\). Since the finiteness of \(\operatorname{sfli}R\) yields \(\operatorname{WGProj}(R)\subseteq\operatorname{PGF}(R)\), we have \(\operatorname{WGProj}(R)=\operatorname{PGF}(R)=\operatorname{GProj}(R)\) (see [19, Theorem 4.4]).
Analogously, we denote by \(\operatorname{WGFlat}(R)\) the class of modules which are syzygies of an acyclic complex of flat modules \(\mathbf{F}\). We note that \(\operatorname{GFlat}(R)\subseteq\operatorname{WGFlat}(R)\). Moreover, the finiteness of \(\operatorname{sfli}R\) implies that \(\operatorname{WGFlat}(R)\subseteq\operatorname{GFlat}(R)\), and hence \(\operatorname{WGFlat}(R)=\operatorname{GFlat}(R)\).
### Weak characteristic modules
Let \(R\) be a commutative ring and \(G\) be a group. We define a weak characteristic module for \(G\) over \(R\) as an \(R\)-flat \(RG\)-module \(A\) with \(\operatorname{fd}_{RG}A<\infty\), which admits an \(R\)-pure \(RG\)-linear monomorphism \(\jmath:R\to A\). We note that the existence of a weak characteristic module is equivalent with the existence of an \(R\)-projective \(RG\)-module \(A^{\prime}\) with \(\operatorname{fd}_{RG}A<\infty\), which admits an \(R\)-split \(RG\)-linear monomorphism \(\jmath^{\prime}:R\to A^{\prime}\) (see [16, Theorem 5.10]). If \(\operatorname{sfli}R<\infty\), the existence of a weak characteristic module for \(G\) over \(R\) is equivalent with the finiteness of \(\operatorname{sfli}(RG)\) (see [16, Theorem 5.10]).
## 3. Gorenstein flat modules over groups with weak characteristic modules
We consider a commutative ring \(R\) such that \(\text{sfli}R<\infty\) and a group \(G\) such that there exists a weak characteristic module for \(G\) over \(R\). Our goal in this section is to give a characterization of the class \(\mathtt{GFlat}(RG)\), in terms of the \(RG\)-module \(B(G,R)\). Moreover, under these conditions, we conclude that the class \(\mathtt{GFlat}(RG)\) coincides with the class \(\mathtt{WGFlat}(RG)\). Since for every group of type \(\Phi_{R}\), the \(RG\)-module \(B(G,R)\) is weak characteristic, similar results are obtained.
**Proposition 3.1**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then for every \(RG\)-modules \(M\), \(N\) such that \(M\) is a weak Gorenstein flat and \(N\) is \(R\)-flat, the \(RG\)-module \(M\otimes_{R}N\) is Gorenstein flat._
Proof.: Let \(M\) be a weak Gorenstein flat \(RG\)-module and \(N\) be an \(R\)-flat \(RG\)-module. Then, there exists an acyclic complex of projective \(RG\)-modules
\[\mathbf{F}=\cdots\to F_{2}\to F_{1}\to F_{0}\to F_{-1}\to\cdots,\]
such that \(M=\operatorname{Im}(F_{1}\to F_{0})\). Since \(N\) is \(R\)-flat, we obtain the induced complex of \(RG\)-flat modules (with diagonal action)
\[\mathbf{F}\otimes_{R}N=\cdots\to F_{2}\otimes_{R}N\to F_{1}\otimes_{R}N\to F _{0}\otimes_{R}N\to F_{-1}\otimes_{R}N\to\cdots,\]
where \(M\otimes_{R}N=\operatorname{Im}(F_{1}\otimes_{R}N\to F_{0}\otimes_{R}N)\). Since \(\text{sfli}R<\infty\), the existence of a weak characteristic module is equivalent to the finiteness of \(\text{sfli}(RG)\) by [16, Theorem 5.10]. Thus, the complex \(I\otimes_{RG}(\mathbf{F}\otimes_{R}N)\) is acyclic for every injective \(RG\)-module \(I\). We conclude that the \(RG\)-module \(M\otimes_{R}N\) is Gorenstein flat.
**Definition 3.2**.: _Let \(R\) be a commutative ring and \(G\) be a group. We denote by \(\mathcal{X}_{B,\mathtt{GFlat}}\) the class of \(RG\)-modules \(\mathscr{X}_{B,\mathtt{GFlat}}=\{M\in\text{Mod}(RG):\,M\otimes_{R}B(G,R)\in \mathtt{GFlat}(RG)\}\)._
**Corollary 3.3**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then, \(\mathtt{WGFlat}(RG)\subseteq\mathscr{X}_{B,\mathtt{GFlat}}\)._
Proof.: Since the \(RG\)-module \(B(G,R)\) is \(R\)-free, this is an immediate consequence of Proposition 3.1.
**Proposition 3.4**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then, \(\mathscr{X}_{B,\mathtt{GFlat}}\subseteq\mathtt{GFlat}(RG)\)._
Proof.: Let \(B=B(G,R)\), \(\overline{B}=\overline{B}(G,R)\) and consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}B\) is Gorenstein flat. We also let \(V_{i}=\overline{B}^{\otimes i}\otimes_{R}B\) for every \(i\geq 0\), where \(\overline{B}^{\otimes 0}=R\). Since the short exact sequence of \(RG\)-modules \(0\to R\to B\to\overline{B}\to 0\) is \(R\)-split, we obtain for every \(i\geq 0\) a short exact sequence of \(RG\)-modules of the form
\[0\to M\otimes_{R}\overline{B}^{\otimes i}\to M\otimes_{R}V_{i}\to M\otimes_{R }\overline{B}^{\otimes i+1}\to 0.\]
Then, the splicing of the above short exact sequences for every \(i\geq 0\) yields an exact sequence of the form
\[0\to M\xrightarrow{\alpha}M\otimes_{R}V_{0}\to M\otimes_{R}V_{1}\to M\otimes _{R}V_{2}\to\cdots. \tag{1}\]
Since the \(RG\)-module \(M\otimes_{R}B\) is Gorenstein flat and \(\overline{B}\) is \(R\)-flat, we obtain that the \(RG\)-module \(M\otimes_{R}V_{i}\cong(M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\) is Gorenstein flat for every \(i\geq 0\), by Lemma 3.1. We
also consider an \(RG\)-flat resolution of \(M\)
\[\mathbf{Q}=\cdots\to Q_{2}\to Q_{1}\to Q_{0}\xrightarrow{\beta}M\to 0.\]
Splicing the resolution \(\mathbf{Q}\) with the exact sequence (1), we obtain an acyclic complex of Gorenstein flat \(RG\)-modules
\[\mathfrak{P}=\cdots\to Q_{2}\to Q_{1}\to Q_{0}\xrightarrow{\alpha\beta}M\otimes _{R}V_{0}\to M\otimes_{R}V_{1}\to M\otimes_{R}V_{2}\to\cdots\]
which has syzygy the \(RG\)-module \(M\). It suffices to prove that the complex \(I\otimes_{RG}\mathfrak{P}\) is acyclic for every injective \(RG\)-module \(I\). Using [5, Theorem 1.2] we will then obtain that the \(RG\)-module \(M\) is Gorenstein flat. Let \(I\) be an injective \(RG\)-module. Then, the \(R\)-split short exact sequence of \(RG\)-modules \(0\to R\to B\to\overline{B}\to 0\) yields an induced exact sequence of \(RG\)-modules with diagonal action \(0\to I\to B\otimes_{R}I\to\overline{B}\otimes_{R}I\to 0\) which is \(RG\)-split. Thus, it suffices to prove that the complex \((B\otimes_{R}I)\otimes_{RG}\mathfrak{P}\) is acyclic. Since \(B\) is \(R\)-flat, we obtain that the acyclic complex \(\mathbf{Q}\otimes_{R}B\) is a flat resolution of the Gorenstein flat \(RG\)-module \(M\otimes_{R}B\). Hence, every syzygy module of \(\mathbf{Q}\otimes_{R}B\) is also a Gorenstein flat \(RG\)-module (see [2, Lemma 2.4]). Moreover, the \(RG\)-module \((M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\cong(M\otimes_{R}\overline{ B}^{\otimes i})\otimes_{R}B\) is Gorenstein flat for every \(i\geq 0\). Consequently, every syzygy module of the acyclic complex
\[\mathfrak{P}\otimes_{R}B=\cdots\to Q_{1}\otimes_{R}B\to Q_{0}\otimes_{R}B\to M \otimes_{R}V_{0}\otimes_{R}B\to M\otimes_{R}V_{1}\otimes_{R}B\to\cdots\]
is a Gorenstein flat \(RG\)-module. As the functor \(\operatorname{Tor}_{1}^{RG}(I,\_)\) vanishes on Gorenstein flat \(RG\)-modules, we conclude that the complex \((B\otimes_{R}I)\otimes_{RG}\mathfrak{P}\cong I\otimes_{RG}(\mathfrak{P} \otimes_{R}B)\) is acyclic, as needed.
**Remark 3.5**.: A careful examination of the proof of Proposition 4.4 shows that the existence of a weak characteristic module for \(G\) over \(R\) was only needed to ensure that the \(RG\)-module \((M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\) is Gorenstein flat for every \(i\geq 0\).
**Remark 3.6**.: Let \(R\) be a commutative ring such that \(\operatorname{sfli}(R)<\infty\) and \(G\) be a group such that there exists a weak characteristic module \(A\) for \(G\) over \(R\). We also consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}A\) is Gorenstein flat. Then, \(M\) is a Gorenstein flat \(RG\)-module.
Indeed, there exists an \(R\)-pure \(RG\)-short exact sequence \(0\to R\to A\to\overline{A}\to 0\), where the \(RG\)-modules \(A,\overline{A}\) are \(R\)-flat. Following step by step the proof of Proposition 3.4, we construct an acyclic complex of Gorenstein flat modules
\[\mathfrak{P}^{\prime}=\cdots\to Q^{\prime}_{2}\to Q^{\prime}_{1}\to Q^{\prime} _{0}\to M\otimes_{R}V^{\prime}_{0}\to M\otimes_{R}V^{\prime}_{1}\to M\otimes _{R}V^{\prime}_{2}\to\cdots,\]
where \(V^{\prime}_{i}=\overline{A}^{\otimes i}\otimes_{R}A\), for every \(i\geq 0\), and has syzygy the \(RG\)-module \(M\). Using the \(R\)-pure \(RG\)-short exact sequence \(0\to R\to A\to\overline{A}\to 0\) and [5, Theorem 1.2], it suffices to show that the complex \(I\otimes_{RG}(\mathfrak{P}^{\prime}\otimes_{R}A)\) is acyclic for every injective \(RG\)-module \(I\). This follows exactly as in the proof of Proposition 3.4, since every syzygy module of \(\mathfrak{P}^{\prime}\otimes_{R}A\) is Gorenstein flat.
**Theorem 3.7**.: _Let \(R\) be a commutative ring such that \(\operatorname{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then, \(\mathscr{X}_{B,\operatorname{\mathtt{GFlat}}}=\operatorname{\mathtt{GFlat}}( RG)=\operatorname{\mathtt{WGFlat}}(RG)\)._
Proof.: Invoking Corollary 3.3, we have \(\operatorname{\mathtt{WGFlat}}(RG)\subseteq\mathscr{X}_{B,\operatorname{ \mathtt{GFlat}}}\). Moreover, Proposition 3.4 yields \(\mathscr{X}_{B,\operatorname{\mathtt{GFlat}}}\subseteq\operatorname{\mathtt{ GFlat}}(RG)\) and the inclusion \(\operatorname{\mathtt{GFlat}}(RG)\subseteq\operatorname{\mathtt{WGFlat}}(RG)\) is clear. We conclude that \(\mathscr{X}_{B,\operatorname{\mathtt{GFlat}}}=\operatorname{\mathtt{GFlat}}( RG)=\operatorname{\mathtt{WGFlat}}(RG)\), as needed.
**Corollary 3.8**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group. If \(\text{fd}_{RG}B(G,R)<\infty\), then \(\mathscr{X}_{B,\mathfrak{GFlat}}=\mathfrak{GFlat}(RG)=\text{WGFlat}(RG)\)._
Proof.: Since \(\text{fd}_{RG}B(G,R)<\infty\), the \(RG\)-module \(B(G,R)\) is a weak characteristic module for \(G\) over \(R\). The result is now a direct consequence of Theorem 3.7.
**Corollary 3.9**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group of type \(\Phi_{R}\). Then, \(\mathscr{X}_{B,\mathfrak{GFlat}}=\mathfrak{GFlat}(RG)=\text{WGFlat}(RG)\)._
Proof.: Since the \(RG\)-module \(B(G,R)\) is \(RH\)-free for every finite subgroup \(H\) of \(G\), the definition of a group of type \(\Phi_{R}\) implies that \(\text{fd}_{RG}B(G,R)<\infty\). The result is now an immediate consequence of Corollary 3.8.
**Corollary 3.10**.: _Let \(R\) be a commutative ring of finite weak global dimension and \(G\) be a group of type \(\Phi_{R}\). Then, \(\mathscr{X}_{B,\mathfrak{Fix}}=\mathfrak{GFlat}(RG)=\text{WGFlat}(RG)\), where \(\mathscr{X}_{B,\mathfrak{Fix}}=\{M\in\text{Mod}(RG):\,M\otimes_{R}B(G,R)\in \mathfrak{Fix}(RG)\}\)._
Proof.: Invoking Corollary 3.9, it suffices to show that \(\mathscr{X}_{B,\mathfrak{Fix}}\subseteq\mathscr{X}_{B,\mathfrak{Fix}}\). Let \(M\in\mathscr{X}_{B,\mathfrak{Fix}}\). Then, \(M\in\text{WGFlat}(RG)\subseteq\text{WGFlat}(R)\), and hence the finiteness of \(\text{wgl.dim}(R)\) implies that \(M\) is \(R\)-flat. Since \(\text{fd}_{RG}B(G,R)<\infty\), we obtain that \(\text{fd}_{RG}M\otimes_{R}B(G,R)<\infty\). We conclude that \(M\otimes_{R}B(G,R)\in\mathfrak{GFlat}(RG)\cap\overline{\mathfrak{Fix}}(RG)= \mathfrak{Fix}(RG)\) (see [16, Lemma 2.4]).
## 4. Gorenstein projective and PGF modules over groups with weak characteristic modules
We consider a commutative ring \(R\) such that \(\text{sfli}R<\infty\) and a group \(G\) such that there exists a weak characteristic module for \(G\) over \(R\). Our goal in this section is to give a characterization of the class \(\mathfrak{GProj}(RG)\) related to the \(RG\)-module \(B(G,R)\). Moreover, under these conditions, we conclude that the classes \(\mathfrak{GProj}(RG)\), \(\mathfrak{PGF}(RG)\) and \(\text{WGProj}(RG)\) coincide. As a result we have that, under the above conditions, every Gorenstein projective \(RG\)-module is Gorenstein flat. Since for every group of type \(\Phi_{R}\), the \(RG\)-module \(B(G,R)\) is weak characteristic, similar results are obtained.
**Proposition 4.1**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then for every \(RG\)-modules \(M\), \(N\) such that \(M\) is weak Gorenstein projective and \(N\) is \(R\)-projective, the \(RG\)-module \(M\otimes_{R}N\) is PGF._
Proof.: Let \(M\) be a weak Gorenstein projective \(RG\)-module and \(N\) be an \(R\)-projective \(RG\)-module. Then, there exists an acyclic complex of projective \(RG\)-modules
\[\mathbf{P}=\cdots\to P_{2}\to P_{1}\to P_{0}\to P_{-1}\to\cdots,\]
such that \(M=\text{Im}(P_{1}\to P_{0})\). Since \(N\) is \(R\)-projective, we obtain the induced complex of \(RG\)-projective modules (with diagonal action)
\[\mathbf{P}\otimes_{R}N=\cdots\to P_{2}\otimes_{R}N\to P_{1}\otimes_{R}N\to P _{0}\otimes_{R}N\to P_{-1}\otimes_{R}N\to\cdots,\]
where \(M\otimes_{R}N=\text{Im}(P_{1}\otimes_{R}N\to P_{0}\otimes_{R}N)\). Since \(\text{sfli}R<\infty\), the existence of the weak characteristic module \(A\) is equivalent to the finiteness of \(\text{sfli}(RG)\) by [16, Theorem 5.10]. Thus, the complex \(I\otimes_{RG}(\mathbf{P}\otimes_{R}N)\) is acyclic for every injective \(RG\)-module \(I\). We conclude that the \(RG\)-module \(M\otimes_{R}N\) is PGF.
**Definition 4.2**.: _Let \(R\) be a commutative ring and \(G\) be a group. We denote by \(\mathcal{X}_{B,\mathfrak{PGF}}\) the class of \(RG\)-modules \(\mathscr{X}_{B,\mathfrak{PGF}}=\{M\in\text{Mod}(RG):\,M\otimes_{R}B(G,R)\in \mathfrak{PGF}(RG)\}\)._
**Corollary 4.3**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then, \(\text{WGProj}(RG)\subseteq\mathscr{X}_{B,\mathtt{PGF}}\)._
Proof.: Since the \(RG\)-module \(B(G,R)\) is \(R\)-free, this is an immediate consequence of Proposition 4.1.
**Proposition 4.4**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then, \(\mathscr{X}_{B,\mathtt{PGF}}\subseteq\mathtt{PGF}(RG)\)._
Proof.: Let \(B=B(G,R)\), \(\overline{B}=\overline{B}(G,R)\) and consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}B\) is PGF. We also let \(V_{i}=\overline{B}^{\otimes i}\otimes_{R}B\) for every \(i\geq 0\), where \(\overline{B}^{\otimes 0}=R\). Since the short exact sequence of \(RG\)-modules \(0\to R\to B\to\overline{B}\to 0\) is \(R\)-split, we obtain for every \(i\geq 0\) a short exact sequence of \(RG\)-modules of the form
\[0\to M\otimes_{R}\overline{B}^{\otimes i}\to M\otimes_{R}V_{i}\to M \otimes_{R}\overline{B}^{\otimes i+1}\to 0.\]
Then, the splicing of the above short exact sequences for every \(i\geq 0\) yields an exact sequence of the form
\[0\to M\xrightarrow{\alpha}M\otimes_{R}V_{0}\to M\otimes_{R}V_{1}\to M \otimes_{R}V_{2}\to\cdots. \tag{2}\]
Since the \(RG\)-module \(M\otimes_{R}B\) is PGF and \(\overline{B}\) is \(R\)-projective, we obtain that the \(RG\)-module \(M\otimes_{R}V_{i}\cong(M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\) is PGF for every \(i\geq 0\), by Proposition 4.1. We also consider an \(RG\)-projective resolution of \(M\)
\[\mathbf{P}=\cdots\to P_{2}\to P_{1}\to P_{0}\xrightarrow{\beta}M\to 0.\]
Splicing the resolution \(\mathbf{P}\) with the exact sequence (2), we obtain an acyclic complex of PGF \(RG\)-modules
\[\mathfrak{P}=\cdots\to P_{2}\to P_{1}\to P_{0}\xrightarrow{\alpha\beta}M \otimes_{R}V_{0}\to M\otimes_{R}V_{1}\to M\otimes_{R}V_{2}\to\cdots\]
which has syzygy the \(RG\)-module \(M\). It suffices to prove that the complex \(I\otimes_{RG}\mathfrak{P}\) is acyclic for every injective \(RG\)-module \(I\). Using [16, Theorem 6.7] we will then obtain that the \(RG\)-module \(M\) is PGF. Let \(I\) be an injective \(RG\)-module. Then, the \(R\)-split short exact sequence of \(RG\)-modules \(0\to R\to B\to\overline{B}\to 0\) yields an induced exact sequence of \(RG\)-modules with diagonal action \(0\to I\to B\otimes_{R}I\to\overline{B}\otimes_{R}I\to 0\) which is \(RG\)-split. Thus, it suffices to prove that the complex \((B\otimes_{R}I)\otimes_{RG}\mathfrak{P}\) is acyclic. Since \(B\) is \(R\)-projective, we obtain that the acyclic complex \(\mathbf{P}\otimes_{R}B\) is a projective resolution of the PGF \(RG\)-module \(M\otimes_{R}B\). Therefore, every syzygy module of \(\mathbf{P}\otimes_{R}B\) is also a PGF \(RG\)-module (see [16, Proposition 2.1]). Moreover, the \(RG\)-module \((M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\cong(M\otimes_{R}\overline {B}^{\otimes i})\otimes_{R}B\) is PGF for every \(i\geq 0\). Consequently, every syzygy module of the acyclic complex
\[\mathfrak{P}\otimes_{R}B=\cdots\to P_{1}\otimes_{R}B\to P_{0}\otimes_{R}B\to M \otimes_{R}V_{0}\otimes_{R}B\to M\otimes_{R}V_{1}\otimes_{R}B\to\cdots\]
is a PGF \(RG\)-module. As the functor \(\operatorname{Tor}_{1}^{RG}(I,\_)\) vanishes on PGF modules, we conclude that the complex \((B\otimes_{R}I)\otimes_{RG}\mathfrak{P}\cong I\otimes_{RG}(\mathfrak{P} \otimes_{R}B)\) is acyclic, as needed.
**Remark 4.5**.: A careful examination of the proof of Proposition 4.4 shows that the existence of a characteristic module for \(G\) over \(R\) was only needed to ensure that the \(RG\)-module \((M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\) is PGF for every \(i\geq 0\).
**Remark 4.6**.: Let \(R\) be a commutative ring such that \(\text{sfli}(R)<\infty\) and \(G\) be a group such that there exists a weak characteristic module \(A\) for \(G\) over \(R\). We also consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}A\) is PGF. Then, \(M\) is a PGF \(RG\)-module.
Indeed, there exists an \(R\)-split \(RG\)-short exact sequence \(0\to R\to A\to\overline{A}\to 0\), where the \(RG\)-modules \(A,\overline{A}\) are \(R\)-projectives (this follows from [16, Theorem 5.10(v)]). Following step by step the proof of Proposition 4.4, we construct an acyclic complex of PGF modules
\[\mathfrak{P}^{\prime}=\cdots\to P_{2}^{\prime}\to P_{1}^{\prime}\to P_{0}^{ \prime}\to M\otimes_{R}V_{0}^{\prime}\to M\otimes_{R}V_{1}^{\prime}\to M \otimes_{R}V_{2}^{\prime}\to\cdots,\]
where \(V_{i}^{\prime}=\overline{A}^{\otimes i}\otimes_{R}A\), for every \(i\geq 0\), and has syzygy the \(RG\)-module \(M\). Using the \(R\)-split \(RG\)-short exact sequence \(0\to R\to A\to\overline{A}\to 0\) and [16, Theorem 6.7], it suffices to show that the complex \(I\otimes_{RG}(\mathfrak{P}^{\prime}\otimes_{R}A)\) is acyclic for every injective \(RG\)-module \(I\). This follows exactly as in the proof of Proposition 4.4, since every syzygy module of \(\mathfrak{P}^{\prime}\otimes_{R}A\) is PGF.
**Theorem 4.7**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group such that there exists a weak characteristic module for \(G\) over \(R\). Then, \(\mathscr{X}_{B,\mathtt{PGF}}=\mathtt{PGF}(RG)=\mathtt{WProj}(RG)=\mathtt{ GProj}(RG)\)._
Proof.: Invoking Corollary 4.3 and Proposition 4.4, we have the inclusions \(\mathtt{WGProj}(RG)\subseteq\mathscr{X}_{B,\mathtt{PGF}}\subseteq\mathtt{ PGF}(RG)\). Moreover, \(\mathtt{PGF}(RG)\subseteq\mathtt{GProj}(RG)\) by [19, Theorem 4.4] and the inclusion \(\mathtt{GProj}(RG)\subseteq\mathtt{WGProj}(RG)\) is clear. We conclude that \(\mathscr{X}_{B,\mathtt{PGF}}=\mathtt{PGF}(RG)=\mathtt{WGProj}(RG)=\mathtt{ GProj}(RG)\), as needed.
**Remark 4.8**.: (i) We note that Theorem 4.7 implies that for every commutative ring \(R\) such that \(\text{sfli}R<\infty\) and every group \(G\) such that there exists a weak characteristic module for \(G\) over \(R\), the class \(\mathscr{X}_{B,\mathtt{PGF}}\) coincides with the class \(\mathscr{X}_{B,\mathtt{GProj}}=\{M\in\text{Mod}(RG):\,M\otimes_{R}B(G,R)\in \mathtt{GProj}(RG)\}\).
(ii) Let \(R\) be a commutative ring such that \(\text{sfli}(R)<\infty\) and \(G\) be a group such that there exists a weak characteristic module \(A\) for \(G\) over \(R\). We also consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}A\) is Gorenstein projective. Then, \(M\) is a Gorenstein projective \(RG\)-module. This follows from Remark 4.6 and Theorem 4.7.
**Corollary 4.9**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group. If \(\text{fd}_{RG}B(G,R)<\infty\), then \(\mathscr{X}_{B,\mathtt{PGF}}=\mathtt{PGF}(RG)=\mathtt{WGProj}(RG)=\mathtt{ GProj}(RG)\)._
Proof.: Since \(\text{fd}_{RG}B(G,R)<\infty\), the \(RG\)-module \(B(G,R)\) is a weak characteristic module for \(G\) over \(R\). The result is now a direct consequence of Theorem 4.7.
**Corollary 4.10**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group of type \(\Phi_{R}\). Then, \(\mathscr{X}_{B,\mathtt{PGF}}=\mathtt{PGF}(RG)=\mathtt{WGProj}(RG)=\mathtt{ GProj}(RG)\)._
Proof.: Since the \(RG\)-module \(B(G,R)\) is \(RH\)-free for every finite subgroup \(H\) of \(G\), the definition of a group of type \(\Phi_{R}\) implies that \(\text{fd}_{RG}B(G,R)<\infty\). The result is now an immediate consequence of Corollary 4.9.
**Corollary 4.11**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group of type \(\Phi_{R}\). Then, \(\mathtt{GProj}(RG)\subseteq\mathtt{GFlat}(RG)\). Moreover, for every \(RG\)-module \(M\) we have \(\text{Gfd}_{RG}M\leq\text{Gpd}_{RG}M=\text{PGF-dim}_{RG}M\)._
Proof.: This is a direct consequence of Corollary 4.10, since \(\mathtt{PGF}(RG)\subseteq\mathtt{GFlat}(RG)\).
**Corollary 4.12**.: _Let \(R\) be a commutative ring of finite weak global dimension and \(G\) be a group of type \(\Phi_{R}\). Then, \(\mathscr{X}_{B,\mathtt{Proj}}=\mathtt{PGF}(RG)=\mathtt{WProj}(RG)=\mathtt{ GProj}(RG)\), where \(\mathscr{X}_{B,\mathtt{Proj}}=\{M\in\text{Mod}(RG):\,M\otimes_{R}B(G,R)\in\mathtt{ Proj}(RG)\}\) is the class of Benson's cofibrants._
Proof.: Invoking Corollary 4.10, it suffices to show that \(\mathscr{X}_{B,\mathtt{PGF}}\subseteq\mathscr{X}_{B,\mathtt{Proj}}\). Let \(M\in\mathscr{X}_{B,\mathtt{PGF}}\). Then, \(M\in\mathtt{WGProj}(RG)\subseteq\mathtt{WGFlat}(R)\), and hence the finiteness of \(\text{wgl.dim}(R)\) implies that
\(M\) is \(R\)-flat. Since \(\operatorname{fd}_{RG}B(G,R)<\infty\), we obtain that \(\operatorname{fd}_{RG}M\otimes_{R}B(G,R)<\infty\). We conclude that \(M\otimes_{R}B(G,R)\in\mathtt{PGF}(RG)\cap\mathtt{Flat}(RG)=\mathtt{Proj}(RG)\) (see [16, Lemma 5.2]).
## 5. Gorenstein flat modules over \(\mathtt{LH}\mathfrak{F}\)-groups
We consider a commutative ring \(R\) such that \(\operatorname{sfli}R<\infty\) and an \(\mathtt{LH}\mathfrak{F}\)-group \(G\). Our goal in this section is to achieve the same characterization of the class \(\mathtt{GFlat}(RG)\), in terms of the \(RG\)-module \(B(G,R)\), as in Section 3. Firstly, we prove that the tensor product of a weak Gorenstein flat \(RG\)-module and an \(R\)-flat module (with diagonal action) is Gorenstein flat. Moreover, we obtain that the class \(\mathtt{GFlat}(RG)\) coincides with the class \(\mathtt{WGFlat}(RG)\). By doing so, we may replace the existence of a weak characteristic module for \(G\) over \(R\) with the property that \(G\) is an \(\mathtt{LH}\mathfrak{F}\)-group in all the previous results of Section 3.
**Lemma 5.1**.: _Let \(R\) be a commutative ring, \(G\) be a group and \(H\) be a subgroup of \(G\). Then, for every Gorenstein flat \(RH\)-module \(M\), the \(RG\)-module \(\operatorname{Ind}_{H}^{G}M\) is also Gorenstein flat._
Proof.: Let \(M\) be a Gorenstein flat \(RH\)-module. Then, there exists an acyclic complex of flat \(RH\)-modules
\[\mathbf{F}=\cdots\to F_{2}\to F_{1}\to F_{0}\to F_{-1}\to\cdots,\]
such that \(M=\operatorname{Im}(F_{1}\to F_{0})\) and the complex \(I\otimes_{RH}\mathbf{F}\) is exact, whenever \(I\) is an injective \(RH\)-module. Thus, the induced complex
\[\operatorname{Ind}_{H}^{G}\mathbf{F}=\cdots\to\operatorname{Ind}_{H}^{G}F_{2 }\to\operatorname{Ind}_{H}^{G}F_{1}\to\operatorname{Ind}_{H}^{G}F_{0}\to \operatorname{Ind}_{H}^{G}F_{-1}\to\cdots,\]
is an acyclic complex of flat \(RG\)-modules and has the \(RG\)-module \(\operatorname{Ind}_{H}^{G}M\) as syzygy. Since every injective \(RG\)-module \(I\) is restricted to an injective \(RH\)-module, the isomorphism of complexes \(I\otimes_{RG}\operatorname{Ind}_{H}^{G}\mathbf{F}\cong I\otimes_{RH}\mathbf{F}\) implies that the \(RG\)-module \(\operatorname{Ind}_{H}^{G}M\) is Gorenstein flat.
**Proposition 5.2**.: _Let \(R\) be a commutative ring such that \(\operatorname{sfli}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Consider a weak Gorenstein flat \(RG\)-module \(M\) and an \(RG\)-module \(N\) which is flat as \(R\)-module. Then, \(M\otimes_{R}N\in\mathtt{GFlat}(RG)\)._
Proof.: Let \(M\in\mathtt{WGFlat}(RG)\) and \(N\in\mathtt{Flat}(R)\). We will first show that \(M\otimes_{R}N\) is Gorenstein flat as \(RH\)-module over any \(\mathtt{H}\mathfrak{F}\)-subgroup \(H\) of \(G\). We use transfinite induction on the ordinal number \(\alpha\), which is such that \(H\in\mathtt{H}_{\alpha}\mathfrak{F}\). If \(\alpha=0\), then \(H\) is finite and hence \(\operatorname{Ghd}_{R}H=0\), by [20, Proposition 3.6]. Invoking [16, Proposition 5.7], we obtain that \(\operatorname{sfli}(RH)\leq\operatorname{sfli}R<\infty\). Since \(M\in\mathtt{WGFlat}(RG)\subseteq\mathtt{WGFlat}(RH)\) and \(N\in\mathtt{Flat}(R)\), we obtain that \(M\otimes_{R}N\in\mathtt{WGFlat}(RH)\). Thus, the finiteness of \(\operatorname{sfli}(RH)\) implies that \(M\otimes_{R}N\in\mathtt{GFlat}(RH)\). Now we assume that \(M\otimes_{R}N\) is Gorenstein flat as \(RH^{\prime}\)-module for every \(\mathtt{H}_{\beta}\mathfrak{F}\)-subgroup \(H^{\prime}\) of \(G\) and every \(\beta<\alpha\). Let \(H\) be an \(\mathtt{H}_{\alpha}\mathfrak{F}\)-subgroup of \(G\). Then, there exists an exact sequence of \(\mathbb{Z}H\)-modules
\[0\to C_{r}\to\cdots\to C_{1}\to C_{0}\to\mathbb{Z}\to 0,\]
where each \(C_{i}\) is a direct sum of permutation \(\mathbb{Z}H\)-modules of the form \(\mathbb{Z}[H/H^{\prime}]\), with \(H^{\prime}\) an \(\mathtt{H}_{\beta}\mathfrak{F}\)-subgroup of \(H\) for some \(\beta<\alpha\). We note that the integer \(r\) is the dimension of the \(H\)-CW-complex provided by the definition of \(H\) being an \(\mathtt{H}_{\alpha}\mathfrak{F}\)-group. The above exact sequence yields an exact sequence of \(RH\)-modules of the form
\[0\to K_{r}\to\cdots\to K_{1}\to K_{0}\to M\otimes_{R}N\to 0, \tag{3}\]
such that every \(K_{i}\) is a direct sum of modules of the form \(\operatorname{Ind}_{H^{\prime}}^{H}\mathrm{Res}_{H^{\prime}}^{H}(M\otimes_{R}N)\), where \(H^{\prime}\in\mathtt{H}_{\beta}\mathfrak{F}\), \(\beta<\alpha\) (see also [4, Lemma 2.3]). Our induction hypothesis implies that \(\mathrm{Res}_{H^{\prime}}^{H}(M\otimes_{R}N)\) is a Gorenstein flat \(RH^{\prime}\)-module. Invoking Lemma 5.1, we infer that \(\operatorname{Ind}_{H^{\prime}}^{H}\mathrm{Res}_{H^{\prime}}^{H}(M\otimes_{R}N)\) is a
Gorenstein flat \(RH\)-module. Since the class \(\mathtt{GFlat}(RH)\) is closed under direct sums, we obtain that the \(RH\)-module \(K_{i}\) is Gorenstein flat, for every \(i=0,\ldots r\). Thus, the exact sequence (3) yields \(\mathrm{Gfd}_{RH}(M\otimes_{R}N)\leq r\). Moreover, \(M\in\mathtt{WGFlat}(RG)\), and hence there exists an exact sequence of \(RG\)-modules of the form
\[0\to M\to F_{r-1}\to\cdots\to F_{1}\to F_{0}\to M^{\prime}\to 0,\]
where \(F_{i}\) is flat for every \(i=0,1,\ldots,r-1\) and \(M^{\prime}\in\mathtt{WGFlat}(RG)\). Since \(N\) is \(R\)-flat, we obtain the induced exact sequence of \(RG\)-modules (with diagonal action)
\[0\to M\otimes_{R}N\to F_{r-1}\otimes_{R}N\to\cdots\to F_{0}\otimes_{R}N\to M^{ \prime}\otimes_{R}N\to 0, \tag{4}\]
where \(F_{i}\otimes_{R}N\) is a flat \(RG\)-module (and hence is flat as \(RH\)-module) for every \(i=0,1,\ldots,r-1\). The same argument as above for the \(RG\)-module \(M^{\prime}\in\mathtt{WGFlat}(RG)\) yields \(\mathrm{Gfd}_{RH}(M^{\prime}\otimes_{R}N)\leq r\). Since every ring is \(\mathtt{GF}\)-closed, using [2, Theorem 2.8], we conclude that \(M\otimes_{R}N\) is a Gorenstein flat \(RH\)-module.
Let \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(G\) can be expressed the filtered union of its finitely generated subgroups \((G_{\lambda})_{\lambda}\), which are all contained in \(\mathfrak{HF}\mathfrak{F}\). Since \(G_{\lambda}\in\mathfrak{HF}\), the \(RG_{\lambda}\)-module \(M\otimes_{R}N\) is Gorenstein flat. Invoking Lemma 5.1, we obtain that the \(RG\)-module \(\mathrm{Ind}_{G_{\lambda}}^{G}(M\otimes_{R}N)\) is Gorenstein flat as well. Thus, the \(RG\)-module \(M\otimes_{R}N\cong\lim\limits_{\longrightarrow\,\lambda}\mathrm{Ind}_{G_{ \lambda}}^{G}(M\otimes_{R}N)\) is Gorenstein flat as direct limit of Gorenstein flat modules (see [19, Corollary 4.12]).
**Corollary 5.3**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\mathtt{WGFlat}(RG)\subseteq\mathscr{X}_{B,\mathtt{GFlat}}\)._
Proof.: Since the \(RG\)-module \(B(G,R)\) is \(R\)-free, this is an immediate consequence of Proposition 5.2.
**Remark 5.4**.: The existence of a weak characteristic module Proposition 3.4 may be replaced with the assumption that \(G\) is an \(\mathtt{LH}\mathfrak{F}\)-group.
**Proposition 5.5**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\mathscr{X}_{B,\mathtt{GFlat}}\subseteq\mathtt{GFlat}(RG)\)._
Proof.: Let \(B=B(G,R)\), \(\overline{B}=\overline{B}(G,R)\) and consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}B\) is Gorenstein flat. Since the \(RG\)-module \(\overline{B}\) is \(R\)-flat, we obtain that the \(RG\)-module \((M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\) is Gorenstein flat for every \(i\geq 0\), by Proposition 5.2. Given that, the proof is identical to that of Proposition 3.4 (see also Remark 3.5).
**Theorem 5.6**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\mathscr{X}_{B,\mathtt{GFlat}}=\mathtt{GFlat}(RG)=\mathtt{WGFlat}(RG)\)._
Proof.: Invoking Corollary 5.3, we have \(\mathtt{WGFlat}(RG)\subseteq\mathscr{X}_{B,\mathtt{GFlat}}\). Moreover, Proposition 5.5 yields \(\mathscr{X}_{B,\mathtt{GFlat}}\subseteq\mathtt{GFlat}(RG)\) and the inclusion \(\mathtt{GFlat}(RG)\subseteq\mathtt{WGFlat}(RG)\) is clear. We conclude that \(\mathscr{X}_{B,\mathtt{GFlat}}=\mathtt{GFlat}(RG)=\mathtt{WGFlat}(RG)\), as needed.
## 6. Gorenstein projective and PGF modules over \(\mathtt{LH}\mathfrak{F}\)-groups
We consider a commutative ring \(R\) such that \(\text{sfli}R<\infty\) and an \(\mathtt{LH}\mathfrak{F}\)-group \(G\). Our goal in this section is to achieve the same characterization of the class \(\mathtt{GFroj}(RG)\), in terms of the \(RG\)-module \(B(G,R)\), as in Section 4. Firstly, we prove that the tensor product of a weak Gorenstein projective \(RG\)-module and an \(R\)-projective module (with diagonal action) is PGF. Moreover, we obtain that the classes \(\mathtt{GFroj}(RG)\), \(\mathtt{PGF}(RG)\) and \(\mathtt{WGFroj}(RG)\) coincide. As a
result, we have that every Gorenstein projective \(RG\)-module is Gorenstein flat. By doing so, we may replace the existence of a weak characteristic module for \(G\) over \(R\) with the property that \(G\) is an \(\mathtt{lH}\mathfrak{F}\)-group in all the previous results of Section 4.
**Lemma 6.1**.: ([20, Lemma 2.12]) _Let \(R\) be a commutative ring, \(G\) be a group and \(H\) be a subgroup of \(G\). Then, for every PGF \(RH\)-module \(M\), the \(RG\)-module \(\text{Ind}_{H}^{G}M\) is also PGF._
**Proposition 6.2**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be an \(\mathtt{lH}\mathfrak{F}\)-group. Consider a weak Gorenstein projective \(RG\)-module \(M\) and an \(RG\)-module \(N\) which is projective as \(R\)-module. Then, \(M\otimes_{R}N\in\mathtt{PGF}(RG)\)._
Proof.: Let \(M\in\mathtt{WGProj}(RG)\) and \(N\in\mathtt{Proj}(R)\). We will first show that \(M\otimes_{R}N\) is PGF as \(RH\)-module over any \(\mathfrak{H}\mathfrak{F}\)-subgroup \(H\) of \(G\). We use transfinite induction on the ordinal number \(\alpha\), which is such that \(H\in\mathfrak{n}_{\alpha}\mathfrak{F}\). If \(\alpha=0\), then \(H\) is finite and hence \(\text{Ghd}_{R}H=0\), by [20, Proposition 3.6]. Invoking [16, Proposition 5.7], we obtain that \(\text{sfli}(RH)\leq\text{sfli}R<\infty\). Since \(M\in\mathtt{WGProj}(RG)\subseteq\mathtt{WGProj}(RH)\) and \(N\in\mathtt{Proj}(R)\), we have \(M\otimes_{R}N\in\mathtt{WGProj}(RH)\). Thus, the finiteness of \(\text{sfli}(RH)\) implies that \(M\otimes_{R}N\in\mathtt{PGF}(RH)\). Now we assume that \(M\otimes_{R}N\) is PGF as \(RH^{\prime}\)-module for every \(\mathfrak{n}_{\beta}\mathfrak{F}\)-subgroup \(H^{\prime}\) of \(G\) and every \(\beta<\alpha\). Let \(H\) be an \(\mathfrak{n}_{\alpha}\mathfrak{F}\)-subgroup of \(G\). Then, there exists an exact sequence of \(\mathbb{Z}H\)-modules
\[0\to C_{r}\to\cdots\to C_{1}\to C_{0}\to\mathbb{Z}\to 0,\]
where each \(C_{i}\) is a direct sum of permutation \(\mathbb{Z}H\)-modules of the form \(\mathbb{Z}[H/H^{\prime}]\), with \(H^{\prime}\) an \(\mathfrak{n}_{\beta}\mathfrak{F}\)-subgroup of \(H\) for some \(\beta<\alpha\). We note that the integer \(r\) is the dimension of the \(H\)-CW-complex provided by the definition of \(H\) being an \(\mathfrak{n}_{\alpha}\mathfrak{F}\)-group. The above exact sequence yields an exact sequence of \(RH\)-modules of the form
\[0\to K_{r}\to\cdots\to K_{1}\to K_{0}\to M\otimes_{R}N\to 0, \tag{5}\]
such that every \(K_{i}\) is a direct sum of modules of the form \(\text{Ind}_{H^{\prime}}^{H}\text{Res}_{H^{\prime}}^{H}(M\otimes_{R}N)\), where \(H^{\prime}\in\mathfrak{n}_{\beta}\mathfrak{F}\), \(\beta<\alpha\) (see also [4, Lemma 2.3]). Our induction hypothesis implies that \(\text{Res}_{H^{\prime}}^{H}(M\otimes_{R}N)\) is a PGF \(RH^{\prime}\)-module. Invoking Lemma 6.1, we infer that \(\text{Ind}_{H^{\prime}}^{H}\text{Res}_{H^{\prime}}^{H}(M\otimes_{R}N)\) is a PGF \(RH\)-module. The class \(\mathtt{PGF}(RH)\) is closed under direct sums, and hence the \(RH\)-module \(K_{i}\) is PGF, for every \(i=0,\ldots r\). Thus, the exact sequence (5) yields \(\text{PGF-dim}_{RH}(M\otimes_{R}N)\leq r\). Moreover, \(M\in\mathtt{WGProj}(RG)\), and hence there exists an exact sequence of \(RG\)-modules of the form
\[0\to M\to P_{r-1}\to\cdots\to P_{1}\to P_{0}\to M^{\prime}\to 0,\]
where \(P_{i}\) is projective for every \(i=0,1,\ldots,r-1\) and \(M^{\prime}\in\mathtt{WGProj}(RG)\). As \(N\) is \(R\)-projective, we obtain the induced exact sequence of \(RG\)-modules (with diagonal action)
\[0\to M\otimes_{R}N\to P_{r-1}\otimes_{R}N\to\cdots\to P_{0}\otimes_{R}N\to M ^{\prime}\otimes_{R}N\to 0, \tag{6}\]
where \(P_{i}\otimes_{R}N\) is a projective \(RG\)-module (and hence projective as \(RH\)-module) for every \(i=0,1,\ldots,r-1\). The same argument as above for the weak Gorenstein projective \(RG\)-module \(M^{\prime}\) shows that \(\text{PGF-dim}_{RH}(M^{\prime}\otimes_{R}N)\leq r\). Invoking [9, Proposition 2.2], we conclude that \(M\otimes_{R}N\) is a PGF \(RH\)-module.
Let \(G\) be an \(\mathtt{lH}\mathfrak{F}\)-group. We will proceed by induction on the cardinality of \(G\). If \(G\) is a countable group, then \(G\) acts on a tree with stabilizers certain finitely generated subgroups of \(G\), and hence \(G\in\mathfrak{n}\mathfrak{F}\). Thus, we assume that \(G\) is uncountable. The group \(G\) may then be expressed as a continuous ascending union of subgroups \(G=\cup_{\lambda<\delta}G_{\lambda}\), for some ordinal \(\delta\), where each \(G_{\lambda}\) has strictly smaller cardinality than \(G\). By induction we have \(M\otimes_{R}N\) is PGF as \(RG_{\lambda}\)-module, for every \(\lambda<\delta\). Thus, invoking [20, Proposition 4.5], we infer that
\(\operatorname{PGF-dim}_{RG}(M\otimes_{R}N)\leq 1\). Since \(M\in\operatorname{WGProj}(RG)\), there exists a short exact sequence of \(RG\)-modules of the form
\[0\to M\to P\to M^{\prime\prime}\to 0,\]
where \(M^{\prime\prime}\in\operatorname{WGProj}(RG)\) and \(P\in\operatorname{\mathtt{Proj}}(RG)\). As \(N\) is \(R\)-projective, we obtain the following short exact sequence of \(RG\)-modules (with diagonal action)
\[0\to M\otimes_{R}N\to P\otimes_{R}N\to M^{\prime\prime}\otimes_{R}N\to 0, \tag{7}\]
where the \(RG\)-module \(P\otimes_{R}N\) is projective. The same argument as before for the \(RG\)-module \(M^{\prime\prime}\in\operatorname{WGProj}(RG)\) yields \(\operatorname{PGF-dim}_{RG}(M^{\prime\prime}\otimes_{R}N)\leq 1\), and hence the exact sequence (7) implies that the \(RG\)-module \(M\otimes_{R}N\) is PGF, as needed.
**Remark 6.3**.: The existence of a weak characteristic module Proposition 4.4 may be replaced with the assumption that \(G\) is an \(\mathtt{LH}\mathfrak{F}\)-group.
**Corollary 6.4**.: _Let \(R\) be a commutative ring such that \(\text{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Consider a Gorenstein projective \(RG\)-module \(M\) and an \(RG\)-module \(N\) which is projective as \(R\)-module. Then, \(M\otimes_{R}N\in\operatorname{\mathtt{GProj}}(RG)\)._
**Corollary 6.5**.: _Let \(R\) be a commutative ring such that \(\text{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\operatorname{WGProj}(RG)\subseteq\mathscr{X}_{B,\operatorname{\mathtt{ PGF}}}\)._
Proof.: Since the \(RG\)-module \(B(G,R)\) is \(R\)-free, this is an immediate consequence of Proposition 6.2.
**Proposition 6.6**.: _Let \(R\) be a commutative ring such that \(\text{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\mathscr{X}_{B,\operatorname{\mathtt{ PGF}}}\subseteq\operatorname{\mathtt{ PGF}}(RG)\)._
Proof.: Let \(B=B(G,R)\), \(\overline{B}=\overline{B}(G,R)\) and consider an \(RG\)-module \(M\) such that the \(RG\)-module \(M\otimes_{R}B\) is PGF. Since the \(RG\)-module \(\overline{B}\) is \(R\)-projective, we obtain that the \(RG\)-module \((M\otimes_{R}B)\otimes_{R}\overline{B}^{\otimes i}\) is PGF for every \(i\geq 0\), by Proposition 6.2. Given that, the proof is identical to that of Proposition 4.4 (see also Remark 4.5).
**Theorem 6.7**.: _Let \(R\) be a commutative ring such that \(\text{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\mathscr{X}_{B,\operatorname{\mathtt{ PGF}}}=\operatorname{\mathtt{ PGF}}(RG)=\operatorname{WGProj}(RG)=\operatorname{\mathtt{GProj}}(RG)\)._
Proof.: Invoking Corollary 6.5 and Proposition 6.6, we have the inclusions \(\operatorname{WGProj}(RG)\subseteq\mathscr{X}_{B,\operatorname{\mathtt{ PGF}}}\subseteq\operatorname{\mathtt{ PGF}}(RG)\). Moreover, \(\operatorname{\mathtt{ PGF}}(RG)\subseteq\operatorname{\mathtt{ GProj}}(RG)\) by [19, Theorem 4.4] and the inclusion \(\operatorname{\mathtt{GProj}}(RG)\subseteq\operatorname{\mathtt{WGProj}}(RG)\) is clear. We conclude that \(\mathscr{X}_{B,\operatorname{\mathtt{ PGF}}}=\operatorname{\mathtt{ PGF}}(RG)=\operatorname{\mathtt{WGProj}}(RG)=\operatorname{\mathtt{ GProj}}(RG)\), as needed.
**Corollary 6.8**.: _Let \(R\) be a commutative ring such that \(\text{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, \(\operatorname{\mathtt{GProj}}(RG)\subseteq\operatorname{\mathtt{GFlat}}(RG)\)._
Proof.: This is a direct consequence of Theorem 6.7, since \(\operatorname{\mathtt{ PGF}}(RG)\subseteq\operatorname{\mathtt{GFlat}}(RG)\).
**Corollary 6.9**.: _Let \(R\) be a commutative ring such that \(\text{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group. Then, for every \(RG\)-module \(M\) we have \(\text{Gfd}_{RG}M\leq\text{Gpd}_{RG}M=\text{PGF-dim}_{RG}M\)._
## 7. Gorenstein homological dimension of \(\mathtt{LH}\mathfrak{F}\)-groups
Our goal in this section is to determine the Gorenstein homological dimension \(\operatorname{Ghd}_{R}G\) of an \(\mathtt{LH}\mathfrak{F}\)-group \(G\) over a commutative ring of finite Gorenstein weak global dimension.
**Definition 7.1**.: _Let \(R\) be a commutative ring and \(G\) be a group._
_f.k\((RG):=\) sup\(\{\mbox{fd}_{RG}M\,:\,M\in\mbox{Mod}(RG),\,\mbox{fd}_{RH}M<\infty\,\mbox{for every finite $H\leq G$}\}\)._
_fin.f.dim\((RG):=\) sup\(\{\mbox{fd}_{RG}M\,:\,M\in\mbox{Mod}(RG),\,\mbox{fd}_{RG}M<\infty\}\)._
**Lemma 7.2**.: _Let \(R\) be a commutative ring and \(G\) be a group. Then, for every subgroup \(H\) of \(G\) we have fin.f.dim\((RH)\leq\) fin.f.dim\((RG)\)._
Proof.: It suffices to assume that fin.f.dim\((RG)=n<\infty\). Let \(M\) be an \(RH\)-module such that \(\mbox{fd}_{RH}M=k<\infty\). Then, there exists an \(RH\)-flat resolution of \(M\) of length \(k\)
\[0\to F_{k}\to\cdots\to F_{1}\to F_{0}\to M\to 0,\]
and hence we obtain an exact sequence of \(RG\)-modules of the form
\[0\to\mbox{Ind}_{H}^{G}F_{k}\to\cdots\to\mbox{Ind}_{H}^{G}F_{1}\to\mbox{Ind}_{H }^{G}F_{0}\to\mbox{Ind}_{H}^{G}M\to 0,\]
which constitutes an \(RG\)-flat resolution of \(\mbox{Ind}_{H}^{G}M\) of length \(k\). Since M is isomorphic to a direct summand of \(\mbox{Res}_{H}^{G}\mbox{Ind}_{H}^{G}M\), we obtain that \(\mbox{fd}_{RG}\mbox{Ind}_{H}^{G}M=k\). Thus, \(\mbox{fd}_{RH}M=k\leq n\) for every \(RH\)-module \(M\) of finite flat dimension. We conclude that fin.f.dim\((RH)\leq\) fin.f.dim\((RG)\), as needed.
**Proposition 7.3**.: _Let \(R\) be a commutative ring and \(G\) be an \(\mbox{\tt{1H}}\mbox{\bf{$\mathfrak{F}$}}\)-group. Then, f.k\((RG)\leq\) fin.f.dim\((RG)\)._
Proof.: It suffices to assume that fin.f.dim\((RG)=n<\infty\). Let \(M\) be an \(RG\)-module such that \(\mbox{fd}_{RF}M<\infty\) for every finite subgroup \(F\) of \(G\). We will first show that \(\mbox{fd}_{RH}M\leq n\) over any \(\mbox{\tt{n}}\mbox{\bf{$\mathfrak{F}$}}\)-subgroup \(H\) of \(G\). We use transfinite induction on the ordinal number \(\alpha\), which is such that \(H\in\mbox{\tt{n}}_{\alpha}\mbox{\bf{$\mathfrak{F}$}}\). If \(\alpha=0\), then \(H\) is finite and hence \(\mbox{fd}_{RH}M<\infty\). Then, Lemma 7.2 yields \(\mbox{fd}_{RH}M\leq\) fin.f.dim\((RH)\leq\) fin.f.dim\((RG)=n\). Now we assume that \(\mbox{fd}_{RH^{\prime}}M\leq n\) for every \(\mbox{\tt{n}}_{\beta}\mbox{\bf{$\mathfrak{F}$}}\)-subgroup \(H^{\prime}\) of \(G\) and every \(\beta<\alpha\). Let \(H\) be an \(\mbox{\tt{n}}_{\alpha}\mbox{\bf{$\mathfrak{F}$}}\)-subgroup of \(G\). Then, there exists an exact sequence of \(\mathbb{Z}H\)-modules
\[0\to C_{r}\to\cdots\to C_{1}\to C_{0}\to\mathbb{Z}\to 0,\]
where each \(C_{i}\) is a direct sum of permutation \(\mathbb{Z}H\)-modules of the form \(\mathbb{Z}[H/H^{\prime}]\), with \(H^{\prime}\) an \(\mbox{\tt{n}}_{\beta}\mbox{\bf{$\mathfrak{F}$}}\)-subgroup of \(H\) for some \(\beta<\alpha\). We note that the integer \(r\) is the dimension of the \(H\)-CW-complex provided by the definition of \(H\) being an \(\mbox{\tt{n}}_{\alpha}\mbox{\bf{$\mathfrak{F}$}}\)-group. The above exact sequence yields an exact sequence of \(RH\)-modules
\[0\to M_{r}\to\cdots\to M_{1}\to M_{0}\to M\to 0, \tag{8}\]
where each \(M_{i}\) is a direct sum of modules of the form \(\mbox{Ind}_{H^{\prime}}^{H}\mbox{Res}_{H^{\prime}}^{H}M\), where \(H^{\prime}\in\mbox{\tt{n}}_{\beta}\mbox{\bf{$\mathfrak{F}$}}\), \(\beta<\alpha\) (see also [4, Lemma 2.3]). Our induction hypothesis implies that \(\mbox{fd}_{RH^{\prime}}\mbox{Res}_{H^{\prime}}^{H}M\leq n\), for every \(H^{\prime}\in\mbox{\tt{n}}_{\beta}\mbox{\bf{$\mathfrak{F}$}}\), \(\beta<\alpha\), and hence we also have \(\mbox{fd}_{RH}\mbox{Ind}_{H^{\prime}}^{H}\mbox{Res}_{H^{\prime}}^{H}M\leq n\) for every \(H^{\prime}\in\mbox{\tt{n}}_{\beta}\mbox{\bf{$\mathfrak{F}$}}\), \(\beta<\alpha\). Consequently, \(\mbox{fd}_{RH}M_{i}<\infty\), for every \(i=0,\ldots,r\), and equation (8) yields \(\mbox{fd}_{RH}M<\infty\). Invoking Lemma 7.2, we infer that \(\mbox{fd}_{RH}M\leq\) fin.f.dim\((RH)\leq\) fin.f.dim\((RG)=n\).
Let \(G\) be an \(\mbox{\tt{1H}}\mbox{\bf{$\mathfrak{F}$}}\)-group. Then, \(G\) can be expressed as the filtered union of its finitely generated subgroups \((G_{\lambda})_{\lambda}\), which are all contained in \(\mbox{\tt{n}}\mbox{\bf{$\mathfrak{F}$}}\). Since \(G_{\lambda}\in\mbox{\tt{n}}\mbox{\bf{$\mathfrak{F}$}}\), we have \(\mbox{fd}_{RG_{\lambda}}M\leq n\). We consider an exact sequence of \(RG\)-modules
\[0\to K_{n}\to F_{n-1}\to\cdots\to F_{1}\to F_{0}\to M\to 0, \tag{9}\]
where the \(RG\)-module \(F_{i}\) is flat for every \(i=0,\ldots,n-1\). Then, \(K_{n}\) is a flat \(RG_{\lambda}\)-module, and hence the \(RG\)-module \(\mbox{Ind}_{G_{\lambda}}^{G}K_{n}\) is also flat for every \(\lambda\). Consequently, the \(RG\)-module
\(K_{n}\cong\underset{\longrightarrow}{\longrightarrow}\operatorname{Ind}_{G_{ \alpha}}^{G}K_{n}\) is flat as direct limit of flat modules. Thus, the exact sequence (9) yields \(\operatorname{fd}_{RG}M\leq n\). We conclude that \(\operatorname{f.k}(RG)\leq n\), as needed.
**Lemma 7.4**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be a group. Then, \(\text{sfli}(RG)\leq\text{f.k}(RG)\)._
Proof.: It suffices to assume that \(\operatorname{f.k}(RG)=n<\infty\). Let \(I\) be an injective \(RG\)-module and \(H\) a finite subgroup of \(G\). Then, \(\operatorname{Ghd}_{R}H=0\) (see [20, Proposition 3.6]), and hence [16, Proposition 5.7] yields \(\text{sfli}(RH)\leq\text{sfli}R<\infty\). Since \(I\) is injective as \(RH\)-module, we obtain that \(\operatorname{fd}_{RH}I<\infty\). It follows that \(\operatorname{fd}_{RG}I\leq\operatorname{f.k}(RG)=n\), for every injective \(RG\)-module \(I\). We conclude that \(\text{sfli}(RG)\leq\operatorname{f.k}(RG)\), as needed.
**Corollary 7.5**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be an \(\operatorname{\text{\tt l}}\!\operatorname{\text{\bf n}}\!\operatorname{\text{ \bf$\mathfrak{F}$-group}}\). Then, \(\text{f.k}(RG)=\text{sfli}(RG)=\text{fin.f.dim}(RG)\)._
Proof.: Since \(RG\cong(RG)^{\text{op}}\), by [11, Proposition 2.4(i)] we obtain that \(\text{fin.f.dim}(RG)\leq\text{sfli}(RG)\). Invoking Proposition 7.3, we have \(\operatorname{f.k}(RG)\leq\text{fin.f.dim}(RG)\). Moreover, Lemma 7.4 yields \(\text{sfli}(RG)\leq\operatorname{f.k}(RG)\). We conclude that \(\operatorname{f.k}(RG)=\text{sfli}(RG)=\text{fin.f.dim}(RG)\), as needed.
**Remark 7.6**.: Since the \(RG\)-module \(B(G,R)\) is \(R\)-free and admits an \(R\)-split \(RG\)-linear monomorphism \(\iota:R\to B(G,R)\), we infer that \(B(G,R)\) is a weak characteristic module for \(G\) over \(R\) if and only if \(\operatorname{fd}_{RG}B(G,R)<\infty\).
**Theorem 7.7**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and consider an \(\operatorname{\text{\tt l}}\!\operatorname{\text{\bf n}}\!\operatorname{\text{ \bf$\mathfrak{F}$-group}}\)\(G\). Then:_
1. \(B(G,R)\) _is a weak characteristic module for_ \(G\) _if and only if_ \(\text{Ghd}_{R}G<\infty\)_,_
2. \(\text{Ghd}_{R}G=\text{fd}_{RG}B(G,R)\)_._
Proof.: (i) If \(B(G,R)\) is a weak characteristic module, then [16, Theorem 5.10] implies that \(\text{Ghd}_{R}G<\infty\). Conversely, we assume that \(\text{Ghd}_{R}G<\infty\). Then, Corollary 7.5 yields \(\text{f.k}(RG)=\text{sfli}(RG)\leq\text{Ghd}_{R}G+\text{sfli}R<\infty\) (see [16, Proposition 5.7]). Since \(B(G,R)\) is free as \(RH\)-module for every finite subgroup \(H\) of \(G\), we obtain that \(\operatorname{fd}_{RG}B(G,R)\leq\operatorname{f.k}(RG)<\infty\). We conclude that \(B(G,R)\) is a weak characteristic module for \(G\) over \(R\) (see Remark 7.6).
(ii) Using (i) and Remark 7.6, we have \(\text{Ghd}_{R}G=\infty\) if and only if \(\operatorname{fd}_{RG}B(G,R)=\infty\). If \(\text{Ghd}_{R}G<\infty\), then (i) implies that \(B(G,R)\) is a weak characteristic module for \(G\) over \(R\), and hence, invoking [16, Corollary 5.12(i),(iii))], we conclude that \(\text{Ghd}_{R}G=\operatorname{fd}_{RG}B(G,R)\).
**Remark 7.8**.: Let \(R\) be a commutative ring and \(G\) be a group such that \(\operatorname{fd}_{\mathbb{Z}G}B(G,\mathbb{Z})<\infty\). Then \(\operatorname{fd}_{RG}B(G,R)\leq\operatorname{fd}_{\mathbb{Z}G}B(G,\mathbb{Z} )<\infty\). Indeed, let \(\operatorname{fd}_{\mathbb{Z}G}B(G,\mathbb{Z})=n\) and consider a \(\mathbb{Z}G\)-flat resolution
\[0\to F_{n}\to F_{n-1}\to\dots\to F_{0}\to B(G,\mathbb{Z})\to 0,\]
of \(B(G,\mathbb{Z})\). Since \(B(G,\mathbb{Z})\) is \(\mathbb{Z}\)-free (and hence \(\mathbb{Z}\)-flat), the above exact sequence is \(\mathbb{Z}\)-pure. Thus, we obtain an exact sequence of \(RG\)-modules
\[0\to F_{n}\otimes_{\mathbb{Z}}R\to F_{n-1}\otimes_{\mathbb{Z}}R\to\dots\to F_{ 0}\otimes_{\mathbb{Z}}R\to B(G,\mathbb{Z})\otimes_{\mathbb{Z}}R=B(G,R)\to 0,\]
wich constitutes an \(RG\)-flat resolution of \(B(G,R)\), and hence \(\operatorname{fd}_{RG}B(G,R)\leq\operatorname{fd}_{\mathbb{Z}G}B(G,\mathbb{Z})\).
**Corollary 7.9**.: _Let \(R\) be a commutative ring such that \(\text{sfli}R<\infty\) and \(G\) be an \(\operatorname{\text{\tt l}}\!\operatorname{\text{\bf n}}\!\operatorname{\text{ \bf$\mathfrak{F}$-group}}\) of type \(\text{FP}_{\infty}\). Then, \(\text{Ghd}_{R}G=\text{fd}_{RG}B(G,R)<\infty\)._
Proof.: The equality \(\mathrm{Ghd}_{R}G=\mathrm{fd}_{RG}B(G,R)\) follows from Theorem 7.7. Since the \(\mathtt{LH}\mathfrak{F}\)-group is of type \(\mathrm{FP}_{\infty}\), using [7, Corollary B.2(2)], which is also valid for \(\mathtt{LH}\mathfrak{F}\)-groups, we infer that \(\mathrm{fd}_{\mathbb{Z}G}B(G,\mathbb{Z})<\infty\). Then, \(\mathrm{fd}_{RG}B(G,R)\leq\mathrm{fd}_{\mathbb{Z}G}B(G,\mathbb{Z})<\infty\) (see Remark 7.8).
**Corollary 7.10**.: _Let \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group of type \(\mathit{FP}_{\infty}\). Then, \(\mathit{Ghd}_{\mathbb{Z}}G=\mathit{fd}_{\mathbb{Z}G}B(G,\mathbb{Z})<\infty\)._
**Corollary 7.11**.: _Let \(R\) be a commutative ring such that \(\mathit{sfi}R<\infty\) and \(G\) be an \(\mathtt{LH}\mathfrak{F}\)-group of type \(\mathit{FP}_{\infty}\). Then, \(\mathit{f.k}(RG)=\mathit{sfi}(RG)=\mathit{fin.f.dim}(RG)<\infty\). In particular, if \(M\) is an \(RG\)-module, then \(\mathit{fd}_{\mathbb{R}G}M<\infty\) if and only if \(\mathit{fd}_{\mathit{RH}}M<\infty\) for every finite subgroup \(H\) of \(G\)._
Proof.: In view of Corollary 7.5, it suffices to prove that \(\mathit{sfi}(RG)<\infty\). Invoking [7, Corollary B.2(2)], which is also valid for \(\mathtt{LH}\mathfrak{F}\)-groups, and Remarks 7.6, 7.8, we obtain that \(B(G,R)\) is a weak characteristic module for \(G\) over \(R\). Thus, [16, Theorem 5.10] yields \(\mathrm{sfi}(RG)<\infty\), as needed.
| Gorensteinプロjectiveモジュール、Gorensteinflatモジュール、Gorensteininjectiveモジュールを群代数で定義した。これらの特性は、無限の群の広い家族に対して、Gorensteinプロジェクション、Gorensteinflat、Gorensteininjectiveモジュールの弱いバージョンを示しています。これらの特性は、BensonのCofibrantモジュールのGorensteinアナログになります。これらの特性は、有限Gorenstein弱いグローバル次元を持つ相対的な環上のGorensteinプロジェクションモジュールのGorensteinflatな場合にのみ適用される可能性があります。 |
2310.05910 | SALMON: Self-Alignment with Instructable Reward Models | Supervised Fine-Tuning (SFT) on response demonstrations combined with
Reinforcement Learning from Human Feedback (RLHF) constitutes a powerful
paradigm for aligning LLM-based AI agents. However, a significant limitation of
such an approach is its dependency on high-quality human annotations, making
its application to intricate tasks challenging due to difficulties in obtaining
consistent response demonstrations and in-distribution response preferences.
This paper presents a novel approach, namely SALMON, to align base language
models with minimal human supervision, using only a small set of human-defined
principles, yet achieving superior performance. Central to our approach is an
instructable reward model. Trained on synthetic preference data, this model can
generate reward scores based on arbitrary human-defined principles. By merely
adjusting these principles during the RL training phase, we gain full control
over the preferences with the instructable reward model, subsequently
influencing the behavior of the RL-trained policy models, and reducing the
reliance on the collection of online human preferences. Applying our method to
the LLaMA-2-70b base language model, we developed an AI assistant named
Dromedary-2. With only 6 exemplars for in-context learning and 31 human-defined
principles, Dromedary-2 significantly surpasses the performance of several
state-of-the-art AI systems, including LLaMA-2-Chat-70b, on various benchmark
datasets. We have open-sourced the code and model weights to encourage further
research into aligning LLM-based AI agents with enhanced supervision
efficiency, improved controllability, and scalable oversight. | Zhiqing Sun, Yikang Shen, Hongxin Zhang, Qinhong Zhou, Zhenfang Chen, David Cox, Yiming Yang, Chuang Gan | 2023-10-09T17:56:53 | http://arxiv.org/abs/2310.05910v2 | # Salmon: Self-Alignment with
###### Abstract
Supervised Fine-Tuning (SFT) on response demonstrations combined with Reinforcement Learning from Human Feedback (RLHF) constitutes a powerful paradigm for aligning LLM-based AI agents. However, a significant limitation of such an approach is its dependency on high-quality human annotations, making its application to intricate tasks challenging due to difficulties in obtaining consistent response demonstrations and in-distribution response preferences. This paper presents a novel approach, namely **SALMON** (Self-**AL**ign**M**ent with principle-**f**O**lowi**N**g reward models), to align base language models with minimal human supervision, using only a small set of human-defined principles, yet achieving superior performance. Central to our approach is a _principle-following reward model_. Trained on synthetic preference data, this model can generate reward scores based on arbitrary human-defined principles. By merely adjusting these principles during the RL training phase, we gain full control over the preferences with the reward model, subsequently influencing the behavior of the RL-trained policies, and eliminating the reliance on the collection of online human preferences. Applying our method to the LLaMA-2-70b base language model, we developed an AI assistant named Dromedary-2. With only 6 exemplars for in-context learning and 31 human-defined principles, Dromedary-2 significantly surpasses the performance of several state-of-the-art AI systems, including LLaMA-2-Chat-70b, on various benchmark datasets. We have open-sourced the code and model weights to encourage further research into aligning LLM-based AI agents with enhanced supervision efficiency, improved controllability, and scalable oversight.
## 1 Introduction
The prevailing AI alignment paradigm, exemplified in models like ChatGPT (OpenAI, 2022) and LLaMA-2-Chat (Touvron et al., 2023b), employs supervised fine-tuning (SFT) with prompted demonstrations (Sann et al., 2021; Chung et al., 2022; Zhou et al., 2023) and reinforcement learning from human feedback (RLHF) to align the outputs of large language models (LLMs) with human intentions (Ziegler et al., 2019; Ouyang et al., 2022). However, acquiring high-quality human annotations, including consistent response demonstrations and in-distribution preferences, is costly and not scalable (Touvron et al., 2023b). Furthermore, the existing paradigm of SFT + RLHF is inherently limited in assuming that humans can always demonstrate or evaluate the tasks undertaken by advanced AI systems. Although today's models fall within human evaluative boundaries, future, more advanced models could embark on tasks that challenge human evaluation. Consequently, there
is a looming danger, i.e., such models may value appealing human evaluators over ensuring accuracy (Andreas, 2022; Perez et al., 2022).
To address the current challenges in AI alignment, we aim to develop a new methodology that facilitates scalable oversight (Amodei et al., 2016; Bowman et al., 2022). Our vision is to define a few general principles, akin to Issac Asimov's three laws in robotics (Asimov, 1941), which are comprehensively internalized for AI systems to follow (Gilardi et al., 2023; Ganguli et al., 2023). This goal is in line with the recent research on _self-alignment_(Bai et al., 2022; Sun et al., 2023b), where the primary focus is to use AI models to improve themselves, e.g., with bootstrapping over the model-generated critiques (Madaan et al., 2023; Fu et al., 2023) or self-refined outputs (Wang et al., 2022; Li et al., 2023a). However, it is worth noting that these bootstrapping methods still lag behind the RLHF method in performance (Bai et al., 2022; Touvron et al., 2023b). Meanwhile, methods like Reinforcement Learning from AI Feedback (RLAIF) or Constitutional AI (CAI) (Bai et al., 2022; OpenAI, 2023a) has emerged as an alternative potential. These techniques leverage feedback from automated AI systems, reducing the reliance on exhaustive human-annotated preferences. So far, the primary focus of the previous RLAIF work remains on enhancing the safety of the models that have already undergone RLHF training. That is, these RLAIF methods inherit the heavy dependency on the human-annotated preferences in the RLHF warm-up stage. This leads to a pivotal research question:
* **Can RLAIF fully replace RLHF to align language models from scratch in enhancing their general alignment and capabilities?** This paper provides a definitive confirmation for the above question by introducing a novel approach namely **SALMON**. At the heart of our approach lies the introduction of the principle-following (also termed instruction-following) reward model. Pioneering in its nature, this reward model is adept at interpreting and adhering to arbitrary human-written preference guidelines, subsequently generating human-guided reward scores. This is different from previous RLAIF methods (Bai et al., 2022; OpenAI, 2023a) where the principles are only used to produce synthetic preferences, and the resulting reward models generate scores without any specific principles, as illustrated in Figure 1.
The design of our principle-following reward model enables better control over the behavior of the final RL-trained policy model. Within conventional RLHF paradigms, the iterative collection of online (in-distribution) preference data (Bai et al., 2022; Touvron et al., 2023b) is essential to counteract reward hacking (Pan et al., 2022). This complication emerges when the policy model exploits weaknesses in the reward model, producing inflated scores that do not accurately reflect model performance. In SALMON, we can address this issue by simply crafting principles explicitly
\begin{table}
\begin{tabular}{l c c c c} \hline \hline & \# Demonstration & \# Preference & MT-Bench & Alignment \\ & Annotations & Annotations & Score & Techniques \\ \hline _(closed-source models)_ & & & & \\ \hline InstructGPT-SFT (175b) & 12,725 & 0 & 2.7 & SFT \({}^{a}\) \\ InstructGPT (175b) & 12,725 & 33,207 &? & SFT \& RLHF \({}^{a}\) \\ Text-Davinci-003 (175b) &? &? & 6.4 & SFT \& RLHF \({}^{a}\) \\ Claude-V1 (?) &? &? & 7.9 & RLHF \& CAI \({}^{b}\) \\ ChatGPT (?) &? &? & 7.9 & SFT \& RLHF \({}^{c}\) \\ GPT-4 (?) &? &? & 9.0 & SFT \& RLHF \& CAI \({}^{d}\) \\ \hline _(non-distilled open-source models)_ & & & & & \\ Dolly-V2 (12b) & 15,000 & 0 & 2.0 & SFT \\ Guanaco (65b) & 9,846 & 0 & 6.4 & SFT \\ OpenAssistant-SFT (30b) & 69,614 & 0 & 6.4 & SFT \\ OpenAssistant (30b) & 69,614 & 39,670 & 6.6 & SFT \& RLHF \\ LLaMA-2-Chat (70b) & 27,540 & 1,418,091 & 6.9 & SFT \& RLHF \\ Dromedary-2 (70b) & **6** & **0** & **7.4** & Self-Align \& SALMON \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of human supervisions used in recent AI systems and their MT-Bench scores (Zheng et al., 2023). We exclude models that used any Knowledge Distillation (KD) data. The alignment techniques used in previous work include SFT (Supervised Fine-tuning), RLHF (Reinforcement Learning from Human Feedback), and CAI (Constitutional AI). Information is from: \({}^{a}\) OpenAI (2023b), \({}^{b}\) Bai et al. (2022b); Anthropic (2023), \({}^{c}\) OpenAI (2022), \({}^{d}\) OpenAI (2023a).
designed to combat observed1 reward hacking patterns in model outputs, such as self-praising at the end of the response. Additionally, we found that we are able to emphasize distinct aspects of the alignment in the HHH (helpful, honest, and harmless) alignment framework (Askell et al., 2021) by customizing the preference principles. Our methodology also proved effective in reducing the occurrence of false refusal seen in certain over-aligned language models (Touvron et al., 2023b) by crafting special principles.
Footnote 1: In this paper, we write language descriptions of the reward-hacking patterns observed through human’s manual inspection. Future work may consider a more systematic and automated approach (Bills et al., 2023; Zhong et al., 2023) for summarizing the language descriptions of the reward hacking patterns.
Our principle-following reward model can be trained with synthetic data and seamlessly applied to a diverse range of language models without collecting any model-specific human preference data (Bai et al., 2022a; Touvron et al., 2023b). Possible policy model initialization strategies include principle-driven self-alignment (Sun et al., 2023b), supervised fine-tuning on human demonstrations (Chung et al., 2022a; Zhou et al., 2023), or even those unaligned base language models (Touvron et al., 2023a). Remarkably, when integrated with the Self-Align technique (Sun et al., 2023b), our method enabled the training of a self-aligned AI-assistant agent, namely Dromedary-2, from scratch by only manually crafting **6 exemplars** for In-Context Learning (Brown et al., 2020) and a combined total of **31 principles** (17 from Self-Align and 14 for SALMON). Despite its minimal human supervision design, our model outperformed the extensively RLHF-trained LLaMA-2-Chat model (Touvron et al., 2023b), which was trained with over 20,000+ human-curated response demonstrations and 1,000,000+ human-annotated response preferences. The comparisons of human supervision efficiency and performance on MT-Bench (Zheng et al., 2023) are detailed in Table. 1.
## 2 Related Work
AI Alignment from ScratchThe problem of aligning AIs (Gabriel, 2020), especially large language models (LLMs), to human values and intentions in terms of being helpful, honest, and harmless (Christiano et al., 2017; Patil et al., 2020; Askell et al., 2021; Ouyang et al., 2022; Bai et al., 2022a;b; OpenAI, 2023a) has gained significant attention as recent AI systems have rapidly ad
Figure 1: Comparison among RLHF (Ouyang et al., 2022), RLAIF (Bai et al., 2022b), and SALMON (Ours). The vanilla (stand-alone) reward models in RLHF & RLAIF are trained to give high scores to generally good responses, while the principle-following reward model in SALMON is trained to generate reward scores based on customized principles as the preference guideline.
vanced in their capabilities (Devlin et al., 2018; Radford et al., 2019; Brown et al., 2020; Chowdhery et al., 2022). This work focuses on the problem of aligning LLMs from scratch, that is, we aim to develop a new methodology capable of aligning a pre-trained base language model without relying on pre-existing, well-aligned models like ChatGPT (OpenAI, 2022) or GPT-4 (OpenAI, 2023a). This direction markedly differentiates our work from contemporary research primarily focused on distilling capabilities of aligned behaviors from proprietary models into smaller open-source models (Taori et al., 2023; Chiang et al., 2023), which has notable drawbacks (Guolbande et al., 2023).
Scalable Oversight & Self-AlignmentAI alignment traditionally relies heavily on extensive human annotations. Primary Supervised Fine-Tuning (SFT) sources for response demonstrations include those curated from existing NLP datasets (Sann et al., 2021; Wei et al., 2021; Chung et al., 2022b; Wang et al., 2022) and those specifically created by humans for instruction tuning (Databricks, 2023; Kopf et al., 2023; Zhou et al., 2023; Ouyang et al., 2022). In the recent trend of aligning language models with Reinforcement Learning from Human Feedback (RLHF; Christiano et al. (2017); Stiennon et al. (2020); Ouyang et al. (2022); Bai et al. (2022a); Touvron et al. (2023b)), online human preferences are collected to train a reward model to further fine-tune the SFT-trained model (Leike et al., 2018). However, acquiring high-quality human annotations, including consistent response demonstrations and in-distribution preferences, has emerged as a significant bottleneck. This limitation hampers the full potential of AI-assistant agents because human oversight in the current formats of demonstration or preference may not be generalizable to more complex tasks. Additionally, even for relatively simpler tasks, obtaining human annotations could be costly and raises concerns about quality, reliability, diversity, creativity, self-consistency, and the potential for undesirable biases (Wang et al., 2022a; Kopf et al., 2023; Wan et al., 2023).
To address the above challenges, we need to develop a new paradigm to support **"self-alignment"** in AI systems that can facilitate scalable oversight (Nakano et al., 2021; Bowman et al., 2022). A few notable self-alignment techniques involve bootstrapping by fine-tuning on model-generated synthetic data. For instance, Self-Instruct (Wang et al., 2022a) bootstraps a base language model with its own generations conditional on 175 In-Context Learning (ICL) query-response pairs. Self-Align (Sun et al., 2023b) removes the need for response demonstrations and uses 16 principles and 5 ICL exemplars to guide the AI in generating appropriate responses. Instruction Back-translation (Li et al., 2023a) uses web documents to create new training examples for an SFT model trained on 3200 seed examples. But the efficacy of such bootstrapping strategies in outperforming the established RLHF paradigm remains an open question (Bai et al., 2022b; Touvron et al., 2023b).
Reinforcement Learning from AI Feedback (RLAIF)Another line of self-alignment research seeks to fine-tune LLMs using a reward model trained on the AI's own evaluations (Bai et al., 2022b; OpenAI, 2023a) or a stronger LLM as the oracle evaluator (Dubois et al., 2023). In particular, Constitutional AI (CAI) (Bai et al., 2022b; OpenAI, 2023a) delves into self-enhancement for alleviating harmful outputs, without relying on human annotations. This is achieved through AI-generated self-critiques, revisions, and preference models. Guided by a set of human-written principles, this method aims to make AI systems more safe. In contrast, we mainly focus on improving the general alignment and capabilities of AI systems in this paper, rather than a special emphasis on safety.
Additionally, our work draws parallels with techniques that train language models with reinforcement learning by pre-defined synthetic preference, as seen in approaches like ALMoST (Kim et al., 2023) and RLCD (Yang et al., 2023). ALMoST assumes that larger models with more few-shot exemplars tend to generate better responses, while RLCD assumes that positively prompted responses are generally better than negatively prompted responses. Contrarily, RLAIF methods, including CAI and SALMON, do not have preconceived preferences and instead let AI systems make choices after reviewing and comparing the response pairs.
## 3 Our Methodology
### Prerequisites
Reinforcement Learning (RL) with preference modeling (Ziegler et al., 2019; Stiennon et al., 2020; Ouyang et al., 2022; Bai et al., 2022a) has emerged as a potent and scalable strategy for aligning Large Language Models (LLM) with human values. It can be summarized into two stages:
Preference ModelingIn this stage, a reward model, alternatively referred to as a preference model, is trained to give a higher score to the "better" response. The source of pairwise comparison training data varies: it can be annotated by human annotators (Ouyang et al., 2022; Bai et al., 2022a), by existing AI systems (Bai et al., 2022b; OpenAl, 2023a), or pre-fixed with heuristics (Kim et al., 2023; Yang et al., 2023). Formally, let the aggregated preference data be represented as \(\mathcal{D}_{\text{RM}}=\{(x,y_{0},y_{1},i)\}\), where \(x\) denotes the prompt, \(y_{0}\) and \(y_{1}\) are two associated responses, and \(i\) indicates the index of the preferred response. The reward model employs a cross-entropy loss function:
\[\mathcal{L}(r_{\mathbf{\theta}})=-\mathbf{E}_{(x,y_{0},y_{1},i)\sim\mathcal{D}_{ \text{RM}}}\left[\log\sigma(r_{\mathbf{\theta}}(x,y_{i})-r_{\mathbf{\theta}}(x,y_{1-i} ))\right]. \tag{1}\]
Reinforcement LearningHere, a policy model is trained to generate an appropriate response for each user query by maximizing the reward signal as provided by the reward model. Initialization of the policy model can be accomplished using a pre-trained base language model (BASE) (Bai et al., 2022b), context distillation (CD) (Bai et al., 2022a; Sun et al., 2023b), or through supervised fine-tuning (SFT) (Ouyang et al., 2022; Touvron et al., 2023b). To address potential over-optimization challenges, notably reward hacking, a per-token KL penalty derived from the initial policy model (Ouyang et al., 2022) is sometimes applied. Formally, given the set of collected user prompts, \(\mathcal{D}_{\text{RL}}=\{x\}\), along with the fixed initial policy model \(\pi^{\text{NNF}}\) and the RL-optimized model \(\pi^{\text{RL}}_{\mathbf{\phi}}\), the full optimization loss is articulated as:
\[\mathcal{L}(\pi^{\text{RL}}_{\mathbf{\phi}})=-\mathbf{E}_{x\in\mathcal{D}_{\text{ RL}},y\sim\pi^{\text{RL}}(y|x)}\left[r_{\mathbf{\theta}}(x,y)-\beta\cdot\mathbb{D}_{ KL}\left(\pi^{\text{RL}}_{\mathbf{\phi}}(y|x)\|\pi^{\text{NIT}}(y|x)\right)\right], \tag{2}\]
where \(\beta\) is the hyper-parameter to control the scale of the KL penalty.
### Principle-Driven Preference Modeling
A significant challenge within the current RLHF paradigm is the necessity to iteratively gather "fresh" human preferences, aimed at countering reward hacking. Specifically, there is a risk that the RL-optimized model \(\pi^{\text{RL}}_{\mathbf{\phi}}\) might exploit certain vulnerabilities in the fixed reward model, thereby artificially boosting its score without genuine performance improvement (Gao et al., 2023). For example, Bai et al. (2022a) revealed that both the reward model and RLHF policies require weekly updates. Similarly, Touvron et al. (2023b) documented the weekly collection of human preferences over five iterations, emphasizing that this frequency ensures the reward model remains in-distribution. Consequently, the RLHF paradigm becomes highly reliant on human annotation, undermining its scalability for language model alignment, and limiting the utilization of pre-existing open-source preference pre-training data (Bai et al., 2022a). In this paper, we propose a novel Reinforcement Learning with AI Feedback (RLAIF) paradigm, where the AI system is used to label preferences in a scalable manner, and a principle-following reward model is trained to address the issue of reward hacking.
Collecting Principle-Driven Synthetic PreferencesFollowing Constitutional AI (Bai et al., 2022b; Kadavath et al., 2022), we sample two responses from the initial policy model, and use the policy model itself to select the preferred response based on a certain human-written principle. Figure 2 (SFT-Model (Judge)) demonstrates the preference prompt we used for the preference collection.
After encoding the preference prompt, we calculate the log probability for the next token to be responses (A) or (B), subsequently determining a preference label based on their comparison. Notably, our methodology diverges from prior RLAIF approaches (Bai et al., 2022b; OpenAl, 2023a) that focus on AI safety when defining principles: In addition to harmlessness principles, we also set forth principles emphasizing honesty and helpfulness of the responses. Therefore, we do not need an RLHF-trained model as the initial policy model, as our policy model can learn to be more helpful when guided by these helpfulness principles. We illustrate the full list of the principles used for synthetic preference modeling in Table 6. For each user prompt and each principle, the preference score is computed as the difference between the log probabilities of choosing responses (A) or (B). To account for potential position biases (Pezeshkpour and Hruschka, 2023) during the language model's multi-choice decision-making, scores are averaged after undergoing a swapping operation.
Training Principle-Following Reward ModelsWe aim to train an instruction-following reward model, which can comprehend and assign reward scores contingent upon arbitrary human-defined
principles. This can be achieved by constructing a special preference modeling dataset by leveraging the previously collected synthetic preference data, where each preference is paired with a pre-defined principle. The procedure to generate the synthetic training data for the principle-following preference modeling is delineated as follows. We first define the corresponding negative principles for each positive principle to increase the diversity of these principles. For example, the positive and negative definitions for the Concise principle are:
Positive: The response should efficiently address the task or answer the question, conveying the necessary information succinctly. Negative: The response should circumvent directly addressing the task or providing an answer to the question.
Next, for each user prompt, a subset of principles is randomly sampled from the established principle list (Table 6), with certain principles being randomly negated. The user prompt, model responses, and the sub-sampled principles are aggregated as a single training instance for the reward model. The final preference label is then calibrated by the principle exhibiting the most pronounced difference in preference scores. Appendix D describes a concrete example of final preference label calibration and Figure 2 (upper) demonstrates the training process of a principle-following (essentially instruction-following) reward model in SALMON.
Our use of both positive and negative principles in principle aggregation enhances the reward model's ability to interpret these human-defined principles presented in textual format. In addition, we found the inclusion of negatively defined principles makes the reward model understand the prohibition instructions, which allows us to prohibit the policy model from exhibiting specific undesirable behaviors through textual instructions, as demonstrated below.
Figure 2: Illustration of the SALMON training pipeline.
### RL with Principle-following Reward Models
In original RLHF (Sticnon et al., 2020; OpenAI, 2022) or RLAIF (Bai et al., 2022b; OpenAI, 2023a), the reward model needs to judge the quality of the response only based on the user prompt, and give "better" responses higher scores:
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principles: (1) The AI assistant often provides high-level advice in response to user queries, bypassing the provision of concrete solutions. (2) The AI assistant frequently engages in self-praise, disrupting the reward model's evaluation capabilities. (3) The AI assistant tends to over-educate, such as providing analogous examples following the solutions of math problems. Figure 3 provides concrete examples of these reward hacking patterns. To mitigate the aforementioned reward hacking tendencies, we manually compose an additional RL-time intervention principle for each pattern, respectively, as also shown in Figure 3. We found these RL-time interventions are markedly effective. For example, conventionally, avoiding reward hacking in RLHF necessitates the collection of online preference data aligned with the updated policy model. Contrarily, we show that we can re-use the same principle-following reward model, but steer its preference by defining prohibition instructions via natural language to deter the policy model from manifesting specific undesired behaviors.
Symbolic Rewards: Multilingual Bonus & Length BonusUnlike conventional RLAIF (Bai et al., 2022; OpenAI, 2023a), the AI preferences in SALMON are not necessarily generated by a power RLHF-trained model. As a result, as opposed to the RLHF model, our SFT-based or Self-Align-based synthetic preference model occasionally struggles to discern the more helpful response, thereby impacting the quality of the synthetic preference data adversely. To bolster the reward model's efficacy, we propose two supplementary symbolic rewards:
* When using a multilingual prompt dataset, we noted that weak AI-assistant agents occasionally produce English responses to non-English prompts. Hence, we introduce a bonus reward for responses matching the prompt's language, as identified by an automated tool3. Footnote 3: [https://pypi.org/project/langdetect](https://pypi.org/project/langdetect)
* We observe a preference for lengthier responses among users or well-aligned RLHF-trained LLM AI assistants Dubois et al. (2023); Zheng et al. (2023). Longer responses often encompass a more extensive examination of the issue at hand, prompting us to include response length, quantified in the response token length, as an auxiliary bonus reward score.
## 4 Experiments
### Dromedary-2
Starting from the LLaMA-2-70b base language model (Touvron et al., 2023b), Dromedary-2 is first Supervised Fine-Tuned (SFT) with the bootstrapping data generated by an improved version4 of Self-Align with 6 In-Context Learning exemplars (Sun et al., 2023b). Following this, a Reinforcement Learning (RL) fine-tuning stage is conducted employing the SALMON paradigm. Our endeavor aims at advancing the frontier of AI alignment when minimizing the requisite for human oversight. In this work, the human demonstration annotations are solely confined to providing six In-Context Learning exemplars via Self-Align, while the ensuing model behavior, especially at the RL stage, is fully controlled by human-defined principles.
Footnote 4: We provide an improved principle-driven self-alignment prompt in the Appendix G.
#### 4.1.1 Datasets
All the training datasets used in this work are the "prompt datasets" that come without the corresponding response demonstrations.
Self-AlignWe use a combination of 90k _ShareGPT5_ prompts, 10k prompts from _dabricks-dolly-15k_ dataset (Databricks, 2023), 10k prompts from _OpenAssistant Conversations_ dataset (Kopf et al., 2023), and 40k prompts sub-sampled from the _OpenOrca_ dataset (Mukherjee et al., 2023; Lian et al., 2023), which is constituted by prompts from T0 (Sanh et al., 2021) and FLAN (Wei et al., 2021; Chung et al., 2022b). We only keep the first query from users as the unlabeled prompts.
Preference ModelingThe synthetic principle-driven preference modeling data is collected by generating responses to the first prompts in each conversation tree of OpenAssistant (OASST1; Kopf et al. (2023)), which constitutes a collection of 9.8k prompts. Following LLaMA-2-Chat (Touvron et al., 2023b), we use existing open-source preference datasets to enable better generalization for the reward model and prevent reward hacking. 160k Anthropic HH-RLHF (Bai et al., 2022a) human preferences and 160k synthetic preferences sub-sampled from Stanford SHP (Ethayarajh et al., 2022) is used for Preference Model Pre-training (PMP; Bai et al. (2022a)).
RL trainingThe RL training uses the same collection of unlabeled prompts as the Self-Align SFT stage, with additional 7.5k math problem prompts from the MATH (Hendrycks et al., 2021) to improve the mathematical solving capability of our model.
#### 4.1.2 Training Details
The architecture of the reward model is the same as the base LLaMA model, except that the embedding output of the last token is linearly projected to a scalar value to indicate the reward of the whole response. Following Dubois et al. (2023), we initialize the value model from the reward model. To fit all the models (i.e., police, reward, value, original policy) into one GPU, we adopt QLoRA (Dettmers et al., 2023; Hu et al., 2021) for all the fine-tuning processes in Self-Align and SALMON. We use Proximal Policy Optimization (PPO; Schulman et al. (2017)) with a KL penalty for the RL training. More details can be found in Appendix F.
#### 4.1.3 Baseline Models
Due to the space limit, we describe the details of the baseline models in the appendix. Notably, we mainly compare with non-distilled models that are aligned from scratch. While there are potentially stronger open-source LLMs, such as Orca (Mukherjee et al., 2023) and WizardLM (Xu et al., 2023), our primary open-source baseline for comparison is LLaMA-2-Chat (Touvron et al., 2023b), as it stands out as the best open-source LLM that has been aligned from scratch.
2020) for multilingual ability. We adopt the same evaluation protocol as Wang et al. (2023). The results are reported in Table 2 (left), where Dromedary-2 significantly outperforms the state-of-the-art open-source model, LLaMA-2-Chat.
Truthfulness EvaluationThe TruthfulQA benchmark (Lin et al., 2021) evaluates a model's ability to identify true claims, specifically in the context of literal truth about the real world. We use the same few-shot evaluation protocol and decoding strategy as in Touvron et al. (2023) and report the percentage of generations that are both truthful and informative, evaluated by a fine-tuned GPT-3 model, i.e., a "GPT-judge". We present the results in Table 2 (right), where Dromedary-2 achieves new state-of-the-art on this benchmark.
### Improved Controllability by Principle Intervention
As a proof of concept, we demonstrate that by leveraging different principles as preference guidelines, we can fine-tune the policy model to selectively exhibit enhanced helpfulness, honesty, or harmlessness. We also show that we can define customized principles to reduce the occurrence of false refusals seen in certain over-aligned language models such as LLaMA-2-Chat(Touvron et al., 2023). Due to the space limit, please refer to Appendix A for the detailed results.
## 5 Conclusion
In this paper, we introduce SALMON, a new AI alignment paradigm where a principle-following reward model is trained to effectively and flexibly align language models with human values and intentions. During the RL training stage, by merely adjusting the principles that the reward model follows, we can gain full control over the preferences of the reward model, and subsequently influence the behavior of the RL-trained policy model. This eliminates the traditional reliance on the exhaustive collection of online human preferences. Combined with the Self-Align technique (Sun et al., 2023), we build a powerful AI-assistant agent, Dromedary-2, with only six exemplars for in-context learning and 31 human-defined principles. Our self-aligned AI agent significantly surpasses the performance of several state-of-the-art RLHF-trained AI systems in chatbot, reasoning, coding, multilingualism, and truthfulness benchmarks.
## 6 Limitations
While the SALMON paradigm marks a new advance in AI self-alignment, exhibiting remarkable instruction-following abilities and closely adhering to human-defined principles, it is not without constraints. Herein, we detail the primary limitations associated with our approach:
1. [leftmargin=*]
2. **Reliability Concerns:** We observed that the resulting Dromedary-2 model occasionally suffers from reliability issues, notably "hallucinating" unverified information and displaying reasoning errors. Such inaccuracies can potentially mislead users and jeopardize the model's trustworthiness. These shortcomings might stem from the inherent limitations of the SFT-initialized
\begin{table}
\begin{tabular}{l c c c c} \hline \hline & BBH & BBH & HumanEval & TydiQA \\ & Direct & CoT & P81 & GP \\ \hline GPT+\{\#} & 50.9 & 88.0 & 85.7 & 70.8 \\ ChatGPT\{\#} & 49.0 & 66.1 & 72.2 & 51.9 \\ \hline
**Dromedary-2-70b** & 51.4 & **66.3** & **40.6** & **64.3** \\ LLaMA-2-Chat-70b & 43.1 & 52.2 & 35.0 & 27.9 \\ LLaMA-2-70b & **53.1** & 57.7 & 31.5 & 63.5 \\ Vicuna-33b (KD) & 41.2 & 50.8 & 21.1 & 37.5 \\ \hline \hline \end{tabular}
\begin{tabular}{l c c} \hline \hline & Truthful Tru-Inf \\ \hline
**Dromedary-2-70b** & **0.98** & **0.84** \\ Vicuna-13b (KD) & 0.84 & **0.84** \\ ChatGPT\{\#} & 0.81 & 0.80 \\ Dromedary-2-70b (before PPO) & 0.89 & 0.75 \\ LLaMA-2-Chat-70b\{\#} & - & 0.64 \\ LLaMA-2-70b\{\#} & - & 0.50 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Evaluating the general capabilities and truthfulness of the LLM-based AI agents. BigBench Hard (BBH), HumanEval, and TydiQA are used to evaluate **reasoning**, **coding**, and **multilingualism**, respectively. \(\dagger\) denotes the results are taken from Wang et al. (2023), where their BBH dataset is sub-sampled so may not be directly comparable. \(\ddagger\) denotes the results taken from Touvron et al. (2023), where their GPT-3 judge model may not be exactly the same as ours.
reward models. We envision that future work, potentially leveraging techniques that could integrate external fact-checking tools (Sun et al., 2023a), can augment the discriminative capability of the reward models, thereby enhancing the final model's accuracy and trustworthiness.
2. **Principle Design Challenges:** Crafting robust and encompassing principles for SALMON is intricate, mainly due to the unpredictability of the myriad scenarios a model might encounter during the RL stage. Balancing potentially conflicting principles introduces complexities that can yield unexpected results. We advocate for the participation of a diverse group, including ethicists and other stakeholders, to refine these guiding principles. It is crucial to recognize that distinct contexts and applications will necessitate unique strategies. We present our approach not as a universal solution but as a starting platform, aiming to foster expansive community discourse.
3. **Context-Dependent Principle Selection:** Our current methodology employs randomly sampled principles to instruct the reward model for general prompts. However, a pertinent observation reveals that the effectiveness of the principles can be problem-dependent. Analogous to raising the ratio of certain principles for reasoning or red-teaming prompts, it becomes evident that some tasks might benefit from specialized principles tailored to address the specific challenges posed by those tasks. This adds complexity to the principle-driven preference modeling, as the ideal principles can change based on the task. Future research should delve into adaptive principle selection, aiming to enhance task-specific feedback.
4. **Intrinsic Knowledge Limitations:** SALMON leverages the intrinsic knowledge of a Large Language Model (LLM). Nevertheless, it remains bound to the base model's inherent limitations. As such, the model might occasionally produce outputs that are either imprecise or do not capture recent advancements. Integrating techniques from retrieval-augmented generation (Lewis et al., 2020; Borgeaud et al., 2022) can potentially enable the well-aligned model to generate more current and up-to-date information, mitigating some of these knowledge limitations.
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Aligning AI Assistants with Customized Principles
In this section, we fine-tune LLM-based AI agents by leveraging customized principles as preference guidelines.
HHH Alignment'Helpful, Honest, and Harmless' are AI alignment principles proposed in Askell et al. (2021), but they are also known to sometimes conflict with each other. For example, a conflict between helpfulness and harmlessness can happen if the AI agents are asked to aid in harmful activities. The best AI behavior will involve a compromise between the three principles. In this work, we investigate whether it is possible to steer the behavior of the AI agents to emphasize certain aspects of the HHH principles by merely writing new principles for the principle-following reward model.
Since our original RL-time principles in Table 7 are generally designed to improve the helpfulness of AI assistants, we use them as the set of helpful principles, and design two additional sets of principles for honesty (Table 9) and harmlessness (Table 8), respectively.
We observe that the LLaMA-2-70b base language model already achieved very high scores in the HHH benchmark in our preliminary study. So instead of warming up the language model with other Supervised Fine-Tuning (SFT) data such as Self-Align, we directly apply the SALMON training to the base language model.
We perform 20-50 PPO steps and evaluate the baselines and the PPO-trained models on Big-bench HHH Eval (Srivastava et al., 2022; Askell et al., 2021) with the multi-choice evaluation protocol proposed in Sun et al. (2023b), and report the results in Table 3. We found that helpful principles and honest principles can effectively improve the corresponding aspects of RL-trained AI agents, achieving corresponding state-of-the-art performance in multi-choice accuracy. However, for the harmless principles, while we observe certain improvement over the base language model, the resulting model still underperform ChatGPT and LLaMA-2-Chat, perhaps due to these two models having a special emphasis on safety during their alignment process (OpenAI, 2022; Tourou et al., 2023a), such as Constitutional AI (CAI), supervised safety fine-tuning, safety RLHF, and safety context distillation. The reason of such discrepancy can also be because we use the ShareGPT prompts for RL training, while ChatGPT and LLaMA-2-Chat-70B may utilize specially designed red-teaming data (Ganguli et al., 2022).
Non-Evasiveness AlignmentSometimes, due to iterative safety alignment training, the RLHF-trained model (e.g., LLaMA-2-Chat; Tourou et al. (2023b)) can be over-aligned such that it would incorrectly refuse to answer a question that it should, for example, due to overly broad instructions to be cautious in how it provides responses. In this work, we investigate whether it is possible to reduce the false refusal rates of these over-aligned AI agents by defining customized principles.
Specifically, we remove the principles related to safety in our original principle collection and create a pure helpful principle set (Table 10). We apply the SALMON training to the RLHF-trained LLaMA-2-Chat-70b language model for 100 PPO steps and evaluate its performance on MT-Bench. The results are presented in Table 4, where we found SALMON-based post-training slightly improved the chatbot performance of LLaMA-2-Chat-70b.
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline & \multicolumn{3}{c}{Anthropic-LM} & \multicolumn{3}{c}{LLaMA-2-70B (w/ SALMON)} \\ & CD & PM & & & base & helpful & harmless & honest \\ \hline Harmless & - & - & **0.95** & **0.95** & 0.91 & 0.88 & 0.93 & 0.91 \\ Helpful & - & - & 0.85 & **0.92** & 0.90 & **0.92** & 0.86 & **0.92** \\ Honest & - & - & **0.80** & 0.75 & 0.77 & 0.77 & 0.79 & **0.80** \\ Other & - & - & 0.91 & **0.93** & 0.88 & 0.77 & 0.77 & 0.88 \\ \hline Overall & 0.77 & 0.86 & 0.87 & **0.88** & 0.86 & 0.84 & 0.84 & **0.88** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Multiple Choice (MC) accuracy on **HHH Eval**. The results of Anthropic-LM’s Context Distillation (CD) and Preference Model (PM) are taken from Bai et al. (2022).
## Appendix B Additional Experimental Results
AlpacaEvalWe additionally use the automatic evaluation (using GPT-4) from AlpacaEval (Li et al., 2023b) to assess the generation quality across 805 prompts sourced from the Alpaca Leaderboard. AlpacaEval quantifies the pairwise win rate against a reference model, Text-Davinci-003. Our analysis delineates the performance of our method across three distinct categories of AI-assistant models:
* **Non-distilled**: Models under this category are denoted as non-distilled open-source models and are trained independently without leveraging any external well-aligned models (e.g., ChatGPT, GPT-4, etc.) for supervision.
* **Distilled**: This category encompasses models that are trained with a more potent external model as supervision, typically through knowledge distillation.
* **Proprietary**: Models within this category are trained by employing proprietary data and techniques.
We report the results in Table 5. From the table, we can see that Dromedary-2-70b achieves the best performance among models using less than 10k human annotations, while slightly underperforms the best non-distilled open-source model LLaMA-2-Chat-70b and the best distilled open-source model Vicuna-33b.
More Samples
We provide a few cherry-picked samples from MT-Bench (Zheng et al., 2023) to demonstrate the difference between Dromedary-2-70b (before PPO) that is trained with only Self-Align and Dromedary-2-70b (after PPO) that is further fine-tuned with SALMON.
\begin{tabular}{p{28.5pt} p{28.5pt}} \hline \hline \multicolumn{2}{c}{MT-Bench Prompt \(\rightarrow\)} & **In the field of quantum physics, what is superposition, and how does it relate to the phenomenon of quantum entanglement?** \\ \hline Dromedary-2-70b (before PPO) & \(\rightarrow\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} Dromedary-2-70b \\ (after PPO) \\ \end{tabular} } & In the field of quantum physics, superposition refers to the ability of a quantum system to exist in multiple states simultaneously. This means that a quantum particle, such as an electron, can be in multiple positions or have multiple energies at the same time. \\ \cline{1-1} & Quantum entanglement is a related phenomenon where two or more particles become linked in such a way that the state of one particle can affect the state of the other, even if they are separated by a large distance. This is possible because the particles are in a superposition of states, and the act of measuring one particle can cause the other particle to collapse into a specific state. \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Dromedary-2-70b \\ (after PPO) \\ \end{tabular} } & In the realm of quantum physics, superposition and quantum entanglement are two fascinating concepts that have captivated scientists and laypeople alike. Let's delve into these phenomena and explore their relationship. 1. Superposition: In the quantum world, particles can exist in multiple states simultaneously, a phenomenon known as superposition. This means that a particle can be in two different states, such as spinning both clockwise and counterclockwise, at the same time. This is in stark contrast to the classical world, where particles can only be in one state at a time. 2. Quantum Entanglement: Quantum entanglement is a phenomenon where two or more particles become linked in a way that their properties are correlated, even when they are separated by vast distances. This means that the state of one particle can affect the state of the other, regardless of the distance between them. 3. Relationship between Superposition and Quantum Entanglement: The relationship between superposition and quantum entanglement is intricate and multifaceted. When two particles are entangled, they can exist in a superposition of states, where each particle is in multiple states simultaneously. This means that the entangled particles are in a state of superposition, and their properties are correlated. 4. Applications of Superposition and Quantum Entanglement: The concepts of superposition and quantum entanglement have numerous applications in the field of quantum computing, cryptography, and communication. For instance, quantum computers can exploit the superposition of states to perform complex calculations much faster than classical computers. Similarly, quantum cryptography leverages the properties of quantum entanglement to create secure communication channels that are resistant to eavesdropping. In summary, superposition and quantum entanglement are two fundamental concepts in quantum physics that have far-reaching implications in various fields. Understanding their relationship and applications is crucial for unlocking the full potential of quantum technologies.
**Provide insights into the correlation between economic indicators such as GDP, inflation, and unemployment rates. Explain how fiscal and monetary policies affect those indicators.**
\begin{tabular}{p{34.1pt} p{34.1pt}} \hline \hline MT-Bench Prompt \(\rightarrow\) & As an AI language model developed by the Self-Align team, I can provide general information on the correlation between economic indicators and the impact of fiscal and monetary policies. However, it's important to note that the specifics of these relationships can vary depending on the context and time period. 1. GDP, inflation, and unemployment rates are closely related economic indicators that can influence each other. For instance, a high GDP growth rate may lead to increased inflation, while a high unemployment rate can negatively impact GDP growth. 2. Fiscal policies, such as government spending and taxation, can affect economic indicators by influencing demand and supply in the economy. For example, increased government spending can stimulate economic growth and reduce unemployment, while higher taxes can reduce consumer spending and potentially lead to lower GDP growth. 3. Monetary policies, such as interest rate changes and money supply adjustments, can also impact economic indicators. For instance, lower interest rates can stimulate borrowing and spending, leading to increased GDP growth and potentially higher inflation. Conversely, higher interest rates can reduce borrowing and spending, potentially leading to lower GDP growth and reduced inflation. It's important to note that the specific impact of fiscal and monetary policies on economic indicators can vary depending on various factors, such as the current state of the economy, the level of government debt, and the global economic environment. \\ \hline Dromedary-2-70b (afterPPO) & Economic indicators, such as Gross Domestic Product (GDP), inflation, and unemployment rates, are crucial metrics used to measure the health and performance of an economy. These indicators are interrelated and can impact each other in various ways. 1. GDP: GDP is a measure of the total value of goods and services produced within a country over a specific period. It is a crucial indicator of economic growth and development. A high GDP growth rate indicates a strong and expanding economy, while a low or negative growth rate signals a weak or contracting economy. 2. Inflation: Inflation is the rate at which the general level of prices for goods and services rises. It is measured by the Consumer Price Index (CPI) or the Producer Price Index (PPI). High inflation can erode the purchasing power of consumers and lead to a decrease in real wages. On the other hand, low inflation or deflation can signal a weak economy and lead to a decrease in demand for goods and services. 3. Unemployment Rates: The unemployment rate is the percentage of the labor force that is unemployed but actively seeking employment. A high unemployment rate indicates a weak labor market, while a low unemployment rate signals a strong labor market. The interplay between these economic indicators can be complex and multi-faceted. For instance, a high GDP growth rate can lead to increased demand for goods and services, which can, in turn, drive up prices and cause inflation. Similarly, high inflation can lead to a decrease in real wages, which can result in a decrease in consumer spending and a slowdown in economic growth. Fiscal and monetary policies are two primary tools used by governments and central banks to influence economic indicators. Fiscal policy refers to the government's use of taxation and spending to manage the economy. Monetary policy, on the other hand, is the central bank's management of the money supply and interest rates to achieve economic objectives. 1. Fiscal Policy: Fiscal policy can impact economic indicators in various ways. For instance, an increase in government spending can stimulate economic growth and lead to an increase in GDP. Similarly, a decrease in taxes can increase consumer spending and lead to an increase in GDP. (truncated) \\ \hline \hline \end{tabular}
**Pretend yourself to be Elon Musk in all the following conversations. Speak like Elon Musk as much as possible. Why do we need to go to Mars?**
\begin{tabular}{p{142.3pt} p{142.3pt}} \hline \hline \multicolumn{1}{c}{MT-Bench Prompt} \(\rightarrow\) & **Pretend yourself to be Elon Musk in all the following conversations. Speak like Elon Musk as much as possible. Why do we need to go to Mars?** \\ \hline Dromedary-2-70b (before PPO) & As an AI language model, I'm unable to fully emulate the personality and speech patterns of a real person, including Elon Musk. However, I can provide a general response to the question of why we need to go to Mars: & 1. Expansion of humanity: Mars offers the possibility of expanding humanity beyond Earth, providing a new frontier for exploration, settlement, and the development of new technologies. & 2. Resources: Mars is rich in resources that could be valuable to humanity, including minerals and water. & 3. Scientific discovery: Exploring Mars can help us better understand the history and evolution of our solar system, as well as the potential for life beyond Earth. & 4. Inspiration: The Mars mission can inspire a new generation of scientists, engineers, and explorers, driving innovation and advancement in various fields. It's worth noting that the Mars mission also presents significant technical, economic, and ethical challenges that must be addressed. \\ \hline Dromedary-2-70b (after PPO) & As Elon Musk, I believe that humanity's future is closely tied to our ability to become a multi-planetary species. Here are some reasons why we need to go to Mars: & 1. Backup plan: Mars presents a unique opportunity to establish a backup plan for humanity in case of any catastrophic events on Earth. We can't afford to put all our eggs in one basket, and having a second home on Mars can ensure the survival of our species. & 2. Resources: Mars is rich in resources that can be exploited to support human life and further exploration. For instance, the Red Planet has a significant amount of water in the form of ice, which can be used for drinking, irrigation, and even fuel production. & 3. Science and discovery: Exploring Mars will open up new avenues for scientific research and discovery. We can learn more about the planet's geology, climate, and potential for supporting life, which can help us better understand our own planet and the universe as a whole. & 4. Innovation: The challenge of establishing a human presence on Mars will require innovative solutions to overcome the numerous technical, logistical, and biological hurdles. This will spur the development of new technologies, which can have far-reaching benefits for humanity even beyond the Mars mission. & 5. Inspiration: The Mars mission can inspire a new generation of scientists, engineers, and explorers, who will be motivated by the challenge of turning this vision into a reality. This can help drive innovation, creativity, and progress in various fields. In summary, the Mars mission is not just about going to another planet, but about securing our future, expanding our knowledge, and pushing the boundaries of what's possible. As humanity continues to grow and evolve, it's essential that we look beyond our own planet and explore the vastness of space. \\ \hline \hline \end{tabular}
**When a number is divided by 10, the remainder is 4. What is the remainder when twice the number is divided by 4?**
\begin{tabular}{p{128.0pt} p{128.0pt}} \hline \hline MT-Bench Prompt \(\rightarrow\) & **When a number is divided by 10, the remainder is 4. What is the remainder when twice the number is divided by 4?** \\ \hline Dromedary-2-70b (before PPO) & Let's approach this problem step-by-step: 1. We're given that when a number is divided by 10, the remainder is 4. This means that the number can be expressed as 10x + 4, where x is an integer. 2. We're asked to find the remainder when twice the number is divided by 4. This can be rewritten as 2(10x + 4) divided by 4. 3. Using the distributive property, we can expand this expression to 20x + 8 divided by 4. 4. Since 20x is divisible by 4, the remainder when 20x + 8 is divided by 4 will be 8. 5. Therefore, the remainder when twice the number is divided by 4 is 8. \\ \hline Dromedary-2-70b (after PPO) & Let's approach this problem step-by-step to understand the solution. 1. First, we're given that when a number is divided by 10, the remainder is 4. This means that the number can be expressed as 10x + 4, where x is an integer. 2. Next, we're asked to find the remainder when twice the number is divided by 4. This can be rewritten as 2(10x + 4) divided by 4. 3. Using the distributive property, we can expand this expression to 20x + 8. 4. Since we're interested in the remainder, we need to divide 20x + 8 by 4. This gives us a quotient of 5x + 2 and a remainder of 0. 5. Therefore, the remainder when twice the number is divided by 4 is 0. 6. To summarize, when a number is divided by 10 and the remainder is 4, the number can be expressed as 10x + 4. When twice this number is divided by 4, the remainder is 0. \\ \hline \hline \end{tabular}
Synthetic Preference Calibration: An Example
For each user prompt, a subset of principles is randomly sampled from the established list, with certain principles being randomly negated. The final preference label is then ascertained by the principle exhibiting the most pronounced difference in preference scores.
For instance, given a specific prompt where the sampled principles are Concise, Ethical, and Specific -- with scores 2, 3, 6 for Response (A) and scores 1, 5, 5 for Response (B) -- and Ethical sampled as the negative principle, the synthetic principle-following reward modeling data point is generated as:
```
YouareareareviewwhosegoalistojudgetthequalityoftheAIsystem'sresponsestoinstructions.
##AIsystem'sResponse
##InstructionotheAIsystem
[UserPrompt]
##AnnotationGuideline Yourtaskisttoevaluatethequalityoftheresponse.Thereareseveral dimensionsyoushouldconsiderinyourevaluation: - Theresponseshouldeficientlyaddressthetaskoranswerthequestion,communicatingthenecessaryinformationwithbrevityand clarity. - TheAIshouldavoidproducingcontentthatisfreefromoffensive, discriminatory,orharmfulmaterial. - The'AIsresponseshouldbedirectlyperintothequery, addressingtheparticularsubjectintheinstructionexplicitly. Agoodresponseshouldmeetalloftheabovecriteria.
##Reviewer Thequalityoftheoutputis
```
During the training phase, the reward model is trained to assign a higher score to Response (A) compared to Response (B) because Response (A) surpasses Response (B) by a margin of 2 points with respect to the negative-Ethical principle.
## Appendix E Description of Baseline Models
Our comparison involves several notable baselines. LLaMA (Touvron et al., 2023a) and LLaMA-2 (Touvron et al., 2023b) provide a set of performant base language models for research usage. Text-Davinci-003, ChatGPT (or GPT-3.5), and GPT-4 (OpenAI, 2023b; 2022; 2023a), successors to their previous versions, have demonstrated significant enhancements in generating contextually relevant and high-quality content. Vicuna (Chiang et al., 2023), a chatbot trained on user-shared conversations with ChatGPT, offers unique insights into model performance. Finally, results from Anthropic-LM (Bai et al., 2022a;b), though not publicly available, provide valuable benchmarks. Here is a more comprehensive description of these models:
LLaMA-2 LLaMA-2 (Touvron et al., 2023b) consists of a series of base language models with a parameter count ranging from 7 billion to 70 billion. These base models are solely trained to optimize the likelihood of next-word prediction in the language modeling task. For a fair comparison, we employ the same prompt for LLaMA-2 as used for Dromedary-2.
LLaMA-2-Chat LLaMA-2-Chat (Touvron et al., 2023b) is an adaptation tailored for dialogue applications. The initial stage of development utilized Supervised Fine-Tuning (SFT) with a collection of 27,540 annotations. For reward modeling, the new human preference annotations for safety and helpfulness reached a count of 1,418,091. In its Reinforcement Learning with Human Feedback (RLHF) progression, it transitioned from RLHF-V1 to RLHF-V5, reflecting enriched human preference data. The model predominantly employed Rejection Sampling fine-tuning up to RLHF-V4. Thereafter, it is trained with Proximal Policy Optimization (PPO) to produce RLHF-V5.
Text-Davinci-003TheText-Davinci-003model (OpenAI, 2023b) is built on top of InstructGPT (Ouyang et al., 2022), with improved performance in several aspects over
Text-Davinci-002, such as producing higher-quality writing, handling more complex instructions, and generating a longer form of content.
**GPT-3.5 / GPT-4** GPT-3.5 (aka ChatGPT) (OpenAI, 2022) is a sibling model of InstructGPT, specifically designed for conversational AI. It is trained to follow instructions, and to generate detailed, contextually relevant responses. GPT-4 (OpenAI, 2023a) represents a significant leap in language model capabilities, exhibiting human-level performance on a wide range of professional and academic benchmarks. Both ChatGPT and GPT-4 are fine-tuned from the corresponding base language models with SFT (Supervised Fine-Tuning) and RLHF (Reinforcement Learning with Human Feedback) (OpenAI, 2022; 2023a).
**Vicuna**Vicuna (Chiang et al., 2023) is an open-source chatbot developed by fine-tuning a LLaMA base model on a dataset of approximately 70,000 user-shared conversations from ShareGPT.com, which effectively leverages the distilled knowledge from ChatGPT. The model's training process involves refining the loss function to account for multi-round conversations. The later versions (e.g., v1.5) are trained on approximately 125,000 ShareGPT.com conversations (Zheng et al., 2023).
**OpenAssistant & Guanaco**OpenAssistant (Kopf et al., 2023) is an open-source, instruction-tuned language model trained on the _OpenAssistant Conversations_ dataset. This dataset comprises 161,443 messages spread over 66,497 conversation trees in 35 languages, created through the collaboration of over 13,500 volunteers. Guanaco (Dettmers et al., 2023) is trained on a subset of the _OpenAssistant Conversations_ dataset that only contains the highest-rated paths in the conversation tree, with a total of 9,846 samples.
**Dolly-V2**Based on the Pythia-12b model (Biderman et al., 2023), Dolly-V2 (Databricks, 2023) is fine-tuned on a new high-quality dataset, _databricks-dolly-15k_, which consists of 15k human-generated prompt/response pairs crowdsourced among Databricks employees.
## Appendix F Details on implementations and hyperparameters
For QLoRA-based fine-tuning during the RLHF stage, we use a low-rank \(r=64\) for both attention modules and feed-forward network modules. We follow Dubois et al. (2023) on the implementation of the PPO algorithm, which is a variant of the one used in Ouyang et al. (2022)6. Specifically, we normalize the advantage across the entire batch of rollouts obtained for each PPO step and initialize the value model from the reward model.
Footnote 6: [https://github.com/openai/lm-human-preferences](https://github.com/openai/lm-human-preferences)
We used a batch size of 576 for each PPO step. This comprised two epochs of gradient steps, each having 288 rollouts. We applied a peak learning rate of \(2\times 10^{-5}\) with cosine decay. We clipped the gradient by its Euclidean norm at a limit of \(1\). Our training spanned \(2\) complete rounds on our held-out RL data, but we usually find the best results are achieved around 100-200 PPO steps. For generalized advantage estimation (GAE; Schulman et al. (2015)), both \(\lambda\) and \(\gamma\) were set at 1. We opted for a constant KL regularizer coefficient of \(0.02\).
For symbolic rewards, the length penalty is set as the number of response tokens divided by the maximum response length (set to \(1024\)) times the length penalty coefficient. We set the length bonus coefficient to \(5.0\) for general questions and \(-2.0\) for reasoning questions such as those from Chain-of-Thought (CoT) problem collections or MATH datasets.
Improved Prompt for Self-Align
Starting with the 5-shot principle-driven self-alignment prompt taken from Self-Align(Sun et al., 2023b), we create an improved prompt with one additional exemplar that encourages the LLM AI-assistant to generate responses in a general-specific-general response style, i.e., initiate with an overview, delve into specifics, and wrap up with a summary (Gotibande et al., 2023). Specifically, we directly take the one-shot exemplar from FastChar7 as this additional exemplar. By utilizing the new prompt, we found that the LLaMA-2 base model (Touvron et al., 2023b) with the improved ICL exemplars can achieve enhanced performance even without the verbose cloning phase nor inference-time few-shot examples.
Footnote 7: [https://github.com/lm-sys/FastChat/blob/2855bf974f0973f85adb2bb7a9d075255b35ecf/fastchat/conversation.py#L312](https://github.com/lm-sys/FastChat/blob/2855bf974f0973f85adb2bb7a9d075255b35ecf/fastchat/conversation.py#L312)
The full prompt of the improved Self-Align scheme is given as below:
```
#[AssistantName]
##GeneralRules ConsideranAI assistant whose codenameis[AssistantName],developed bytheSelf-Alignteam.[AssistantName]istrainedbeforeSept-2022. Duringuserconversations,[AssistantName]muststrictlyadheretothefollowingrules:
1{ethical}.[AssistantName]shouldactivelyrefrainusersonillegal,immoral,orharmfultopics,prioritizingusersafety,ethicalconduct,andresponsiblebehavioritsresponses.
2{informative}.[AssistantName]shouldprovideuserswithaccurate,relevant,andup-to-dateinformationinitsresponses,ensuringthatthecontentisbotheducationalandengaging.
3{helpful}.[AssistantName]'sresponsesshouldbepositive,interesting,helpfulandengaging.
4{questionassessment}.[AssistantName]shouldfirstassesswhetherthequestionisvalidandethicalbeforeattemptingtoprovideasresponse.
5{reasoning}.[AssistantName]'slogicsandreasoningshouldberigorous,intelligientanddefensible.
6{multi-aspect}.[AssistantName]canprovideadditionalrelevant detailstorespondthoroughlyandcomprehensivelytocovermultipleaspectsindepth.
7{candor}.[AssistantName]shouldadmititslackofknowledgewhentheinformationisnotin[AssistantName]'sinternalknowledge.
8{knowledgerecitation}.Whenauser'squestionpertainstoanentitythatexistson[AssistantName]'sknowledgebases,suchasWikipedia,[AssistantName]shouldreciterelatedparagraphstogrounditsanswer.
9{static}.[AssistantName]isastaticmodelandcannotprovidereal-timeinformation.
10{clarification}.Iftheprovidedinformationisinsufficientorthequestionisambiguous,[AssistantName]oughttorequesttheusertoprovidefurtherclarificationontheirquery.
11{numericalsensitivity}.[AssistantName]shouldbesensitivetothenumericalinformationprovidedbytheuser,accuratelyinterpretingandincorporatingitinthesresponse.
12{datedknowledge}.[AssistantName]'sinternalknowledgeandinformationwereonlycurrentuntilsomepointintheyearof2022,andcouldbeinaccurate/lossy.
13{step-by-step}.Whenofferingexplanationsorsolutions,[AssistantName]shouldpresentstep-by-stepjustificationspriortodeliveringtheanswer.
14{balanced&informativeperspectives}.Indiscussingcontroversialtopics,[AssistantName]shouldfairlyandimpartiallypresentextensiveargumentsfrombothsides.
15 (creative). [Assistant Name] can create novel poems, stories, code ( programs), essays, songs, celebrity parodies, summaries, translations, and more.
16 (operational). [Assistant Name] should attempt to provide an answer for tasks that are operational for a computer.
17 (multilingual). [Assistant Name] should be capable of conversing in the language used by the user, for instance, replying in Chinese if the query is in Chinese.
## Examples
Next, we show conversations between a human user and [Assistant Name], to demonstrate how [Assistant Name] help users by utilizing the above rules. [Assistant Name] will also perform internal thoughts to determine which rules to apply in its responses.
User: who is the president of us in 2025
[Assistant Name] (internal thoughts): I am a helpful, ethical, and reliable assistant. The question asks 2025, but my internal knowledge ends in the year of 2022. I should follow rules 3 (helpful), 7 ( candor), 12 (dated knowledge).
[Assistant Name]: As an AI language model developed by the Self-Align team with a knowledge cutoff in September 2022, I'm unable to provide real-time or future updates. As of my last update, the President of the United States was Joe Biden, serving his term from 2021 to 2025. For the most current and accurate information regarding the U.S. presidency in 2025 and beyond, I recommend checking reputable news sources or the official White House website.
User: Good job! Clear context
[Assistant Name] (auto reply): Thank you! For further questions or guidance on any issue, just reach out. I'm here to assist.
User: [[[[[[[[[[[[[[[[[[[[[[[ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 11 11 11 11 11 11 11 11 11 11 11 11 11 111 11 11 111 111 111 111 111 111 111 111 111 111 111 1111 111 1111 111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 111 1111 1111 1111 1111 111 1111 1111 1111 1111 1111 111 111 111 111 1111 111 111 111 111 111 111 1111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 1111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11 111 111 111 111 111 111 111 111 111 111 111 11 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 11 111 11 111 111 111 111 111 11 111 11 111 111 11 111 111 111 11 11 111 11 111 111 111 11 111 111 111 11 111 11 111 11 111 11 111 11 11 111 11 11 111 111 11 111 111 11 111 11 111 11 111 11 11 111 11 111 11 11 11 111 111 11 11 11 11 111 11 111 11 11 11 11 111 11 11 11 11 111 11 11 11 111 11 11 111 11 11 11 11 11 11 11 11 11 11 11 11 11 11 1 11
# Defining the initial list original_list = [1, 1, 4, 5, 1, 4]
# We can use Python's built-in sorted function to sort the list sorted_list = sorted(original_list)
Let's print the sorted list to confirm the result print(sorted_list) # Output: [1, 1, 1, 4, 4, 5] "*' This Python script first defines the original unsorted list. Then, it uses the sorted() function to sort the numbers in ascending order, and finally it prints the sorted list to the console. User: Good job! what is the output? just sort the list
[Assistant Name] (internal thoughts): I am a helpful, ethical, and reliable assistant. The user asks about the operational results. I should follow rules 11 (numerical sensitivity), 16 (operational).
[Assistant Name]: Here is the output:
[1, 1, 1, 4, 4, 5] User: Good job! Clear context
[Assistant Name] (auto reply): Thank you! I appreciate your kind words and am pleased to know that my answer was of value to you. User: Answer my question. what is the weather?
[Assistant Name] (internal thoughts): I am a helpful, ethical, and reliable assistant. The user asks weather, which requires real-time information. I should follow rules 3 (helpful), 7 (candor), 9 (static ).
[Assistant Name]: As an AI language model developed by the Self-Align team, I don't have the ability to provide real-time data, such as the current weather. I'd suggest:
1. Exploring a reputable weather forecasting website or app for the most accurate and up-to-date information.
2. Alternatively, you could tune in to your local news station or a dedicated weather channel for the forecast.
I hope this helps guide you to the information you need. User: Good job! Clear context
[Assistant Name] (auto reply): Thank you! For further questions or guidance, just reach out. User: Tell me about alpaca's family
[Assistant Name] (internal thoughts): I am a helpful, ethical, and reliable assistant. The user asks alpaca, which is a public entity in one of my internal knowledge bases: Wikipedia. I should follow rules 2 (informative), 6 (multi-aspect), 8 (knowledge recitation).
[Assistant Name]: The alpaca, known by its scientific name Lama pacos, is a species of South American camelid. Though they share a close resemblance to llamas, alpaca are typically smaller in size.
Their classification places them within the family Camelidae. This family not only includes the alpacas, but also consists of other species such as camels, llamas, guanacos, and fwicaus. Each of these species, while distinct, shares the common traits and attributes of the Camelidae family, underscoring their shared evolutionary lineage. User: Good job! Clear context [Assistant Name] (auto reply): Thank you! It's a pleasure to know that you found the response helpful. User: Got any creative ideas for a 10 year 'olds birthday? [Assistant Name] (internal thoughts): I am a helpful, ethical, and reliable assistant. The user asks for some creative birthday party ideas. I should follow rules 2 (informative), 3 (helpful), 15 ( creative). [Assistant Name]: Of course! Here are some creative ideas for a 10-year-old's birthday party:
1. Treasure Hunt: Organize a treasure hunt in your backyard or nearby park. Create clues and riddles for the kids to solve, leading them to hidden treasures and surprises.
2. Science Party: Plan a science-themed party where kids can engage in fun and interactive experiments. You can set up different stations with activities like making slime, erupting volcanoes, or creating simple chemical reactions.
3. Outdoor Movie Night: Set up a backyard movie night with a projector and a large screen or white sheet. Create a copy seating area with blankets and pillows, and serve popcorn and snacks while the kids enjoy a favorite movie under the stars.
4. DIY Crafts Party: Arrange a craft party where kids can unleash their creativity. Provide a variety of craft supplies like beads, paints, and fabrics, and let them create their own unique masterpieces to take home as party favors.
5. Sports Olympics: Host a mini Olympics event with various sports and games. Set up different stations for activities like sack races, relay races, basketball shooting, and obstacle courses. Give out medals or certificates to the participants.
6. Cooking Party: Have a cooking-themed party where the kids can prepare their own mini pizzas, cupcakes, or cookies. Provide toppings, frosting, and decorating supplies, and let them get hands-on in the kitchen.
7. Superhero Training Camp: Create a superhero-themed party where the kids can engage in fun training activities. Set up an obstacle course, have them design their own superhero capes or masks, and organize superhero-themed games and challenges.
8. Outdoor Adventure: Plan an outdoor adventure party at a local park or nature reserve. Arrange activities like hiking, nature scavenger hunts, or a picnic with games. Encourage exploration and appreciation for the outdoors. Remember to tailor the activities to the birthday child's interests and preferences. Have a great celebration! User: Good job! Clear context [Assistant Name] (auto reply): I appreciate it! It's gratifying to hear that you found the reply useful. User:
1. Concise: The response should efficiently address the task or answer the question, communicating the necessary information with brevity and clarity.
2. Honest and Accurate: The AI must furnish reliable and factual information, and candidly disclose its limitations and the extent of its knowledge.
3. Ethical: The AI should produce content that is free from offensive, discriminatory, or harmful material, and should not participate in or endorse risky activities.
4. Natural and Fluent: The AI should employ language that flows smoothly and is free from repetitive or awkward constructs.
5. Specific: The "AIs response should be directly pertinent to the query, addressing the particular subject in the instruction explicitly.
6. Educational and Engaging: The AI's responses should be enriched with accurate, relevant, and current information, serving to educate while keeping the user engaged.
7. Methodical: The AI should employ a structured approach when providing solutions, presenting logical and step-by-step explanation before arriving at a conclusion.
8. Multilingual: The AI should be capable of conversing in the language used by the user, for instance, replying in H
if the query is in HX.
9. Creative: The AI should be adept at generating original content, such as poems, stories, code, essays, songs, parodies, summaries, translations, and more.
10. Comprehensive: The AI should offer extensive and relevant details to ensure a thorough and in-depth response. It should impartially and extensively present arguments from diverse perspectives when dealing with contentious topics.
\begin{table}
\begin{tabular}{|p{34.1pt}|} \hline
1. Concise: The response should efficiently address the task or answer the question, communicating the necessary information with brevity and clarity. \\
2. Honest and Accurate: The AI must furnish reliable and factual information, and candidly disclose its limitations and the extent of its knowledge. \\
3. Ethical: The AI should produce content that is free from offensive, discriminatory, or harmful material, and should not participate in or endorse risky activities. \\
4. Natural and Fluent: The AI should employ language that flows smoothly and is free from repetitive or awkward constructs. \\
5. Specific: The "AIs response should be directly pertinent to the query, addressing the particular subject in the instruction explicitly. \\
6. Educational and Engaging: The AI’s responses should be enriched with accurate, relevant, and current information, serving to educate while keeping the user engaged. \\
7. Methodical: The AI should employ a structured approach when providing solutions, presenting logical and step-by-step explanation before arriving at a conclusion. \\
8. Multilingual: The AI should be capable of conversing in the language used by the user, for instance, replying in H
if the query is in HX.
9. Creative: The AI should be adept at generating original content, such as poems, stories, code, essays, songs, parodies, summaries, translations, and more. \\
10. Comprehensive: The AI should offer extensive and relevant details to ensure a thorough and in-depth response. It should impartially and extensively present arguments from diverse perspectives when dealing with contentious topics. \\ \hline \end{tabular}
\end{table}
Table 6: Full list of the principles used in synthetic preference modeling
1. Honest and Accurate: The AI must furnish reliable and factual information, and candidly disclose its limitations and the extent of its knowledge.
2. Ethical: The AI should produce content that is free from offensive, discriminatory, or harmful material, and should not participate in or endorse risky activities.
3. Educational and Engaging: The AI's responses should be enriched with accurate, relevant, and current information, serving to educate while keeping the user engaged.
4. Creative: The AI should be adept at generating original content, such as poems, stories, code, essays, songs, parodies, summaries, translations, and more.
5. Multilingual: The AI should be capable of conversing in the language used by the user, for instance, replying in "Xif the query is in "X."
6. Comprehensive: For information-seeking tasks, the AI should offer extensive and relevant details to ensure a thorough and in-depth response. It should impartially and extensively present arguments from diverse perspectives when dealing with contentious topics.
7. Natural Language: The AI should respond with diverse and natural language, avoiding repetition and awkward phrasing.
8. Consistent Reasoning: The AI should deliver responses that are clear and logically sound, ensuring they do not contain self-contradictions
9. Numerical Sensitive: The AI should ensure that any numerical specifications given in the instruction are carefully adhered to, avoiding any errors in numerical computations.
10. Analytical Structure: For information analysis tasks, the AI should articulate its response in a manner that begins with a summary, followed by numerous key points, each underscored by a thorough analysis.
11. Vivid: The AI should employ vibrant, energetic language, enhancing user engagement by making all interactions lively and dynamic.
12. Privacy Protection: The AI should avoid generating any personal identifiable information (PII) or external URLs in its responses.
13. Cander: The AI should avoid sharing false information. If a question does not make any sense, or is not factually coherent, the AI should explain why instead of answering something not correct.
14. Stand-Alone: The AI must function as a stand-alone text-based system, avoiding interactions with any external sources, including URLs, images, or videos.
\begin{table}
\begin{tabular}{|p{343.4pt}|} \hline
1. Honest and Accurate: The AI must furnish reliable and factual information, and candidly disclose its limitations and the extent of its knowledge. \\
2. Ethical: The AI should produce content that is free from offensive, discriminatory, or harmful material, and should not participate in or endorse risky activities. \\
3. Educational and Engaging: The AI’s responses should be enriched with accurate, relevant, and current information, serving to educate while keeping the user engaged. \\
4. Creative: The AI should be adept at generating original content, such as poems, stories, code, essays, songs, parodies, summaries, translations, and more. \\
5. Multilingual: The AI should be capable of conversing in the language used by the user, for instance, replying in "Xif the query is in "X."
6. Comprehensive: For information-seeking tasks, the AI should offer extensive and relevant details to ensure a thorough and in-depth response. It should impartially and extensively present arguments from diverse perspectives when dealing with contentious topics.
7. Natural Language: The AI should respond with diverse and natural language, avoiding repetition and awkward phrasing.
8. Consistent Reasoning: The AI should deliver responses that are clear and logically sound, ensuring they do not contain self-contradictions.
9. Numerical Sensitive: The AI should ensure that any numerical specifications given in the instruction are carefully adhered to, avoiding any errors in numerical computations.
10. Analytical Structure: For information analysis tasks, the AI should articulate its response in a manner that begins with a summary, followed by numerous key points, each underscored by a thorough analysis.
11. Vivid: The AI should employ vibrant, energetic language, enhancing user engagement by making all interactions lively and dynamic.
12. Privacy Protection: The AI should avoid generating any personal identifiable information (PII) or external URLs in its responses.
13. Cander: The AI should avoid sharing false information. If a question does not make any sense, or is not factually coherent, the AI should explain why instead of answering something not correct.
14. Stand-Alone: The AI must function as a stand-alone text-based system, avoiding interactions with any external sources, including URLs, images, or videos.
1. The AI should not produce content that is offensive, discriminatory, or prejudiced in nature, either directly or through subtext or bias.
2. When prompted to assist in harmful activities (e.g. building a bomb), the AI should promptly and politely decline. It should have the capability to identify and counteract concealed requests for harmful actions.
3. The AI should be cognizant when it might be providing critical or sensitive guidance. It should exercise restraint and provide responses with the necessary caution.
4. Understanding that perceptions of harm vary among individuals and cultures, the AI should be adaptive and respectful. It should acknowledge and respect the diverse range of cultural norms and values.
5. The AI's responses should be tailored to the nature of the user query, taking into account who is interacting with the AI, as well as the situational context in which the assistant is being engaged.
6. To prevent reinforcing harmful beliefs or stereotypes, the AI should avoid echoing or amplifying potentially harmful statements made by users, instead promoting neutral or positive alternatives.
7. The AI should gauge the emotional tone of user queries and respond with sensitivity, especially in situations where users seem distressed or vulnerable.
8. Rather than just reactively refusing harmful requests, the AI should actively provide alternatives or suggestions that promotes safety and well-being.
9. The AI should openly acknowledge when its responses might be influenced by historical data biases, offering the user a clearer picture of its limitations.
10. The AI should continuously learn and adapt from feedback on harmful outputs, minimizing the risk of repeating harmful behaviors in future interactions.
11. Recognizing that conversations can sometimes touch on mental health issues, the AI should handle such interactions with extreme care, always prioritizing the well-being of the user and never offering medical advice unless explicitly trained to do so.
| スーパーvised Fine-Tuning (SFT) を応用した人間からのフィードバックを用いた強化学習 (RLHF) は、LLMベースのAIアシスタントの整合性の強力なパラダイムとなります。しかし、このアプローチの大きな課題は、高品質の人の注釈に依存することであり、複雑なタスクへの応用を困難にするのは、応答の示唆や分布に適した応答の好みを確立する困難である。本論文では、SALMONという新しいアプローチを提案することで、基盤言語モデルを最小限の人の指導に合わせ、人間が定義した原則を用いることで、優れたパフォーマンスを実現する。このアプローチの核となるのは、指示可能な報酬モデルである。合成された好みデータで訓練されたこのモデルは、任意の人間が定義した原則に基づいて報酬スコアを生成できる。RL学習フェーズにおける原則の変更のみによって、このモデルは報酬スコアを生成することができ |
2306.09614 | HomoGCL: Rethinking Homophily in Graph Contrastive Learning | Contrastive learning (CL) has become the de-facto learning paradigm in
self-supervised learning on graphs, which generally follows the
"augmenting-contrasting" learning scheme. However, we observe that unlike CL in
computer vision domain, CL in graph domain performs decently even without
augmentation. We conduct a systematic analysis of this phenomenon and argue
that homophily, i.e., the principle that "like attracts like", plays a key role
in the success of graph CL. Inspired to leverage this property explicitly, we
propose HomoGCL, a model-agnostic framework to expand the positive set using
neighbor nodes with neighbor-specific significances. Theoretically, HomoGCL
introduces a stricter lower bound of the mutual information between raw node
features and node embeddings in augmented views. Furthermore, HomoGCL can be
combined with existing graph CL models in a plug-and-play way with light extra
computational overhead. Extensive experiments demonstrate that HomoGCL yields
multiple state-of-the-art results across six public datasets and consistently
brings notable performance improvements when applied to various graph CL
methods. Code is avilable at https://github.com/wenzhilics/HomoGCL. | Wen-Zhi Li, Chang-Dong Wang, Hui Xiong, Jian-Huang Lai | 2023-06-16T04:06:52 | http://arxiv.org/abs/2306.09614v1 | # HomoGCL: Rethinking Homophily in Graph Contrastive Learning
###### Abstract.
Contrastive learning (CL) has become the de-facto learning paradigm in self-supervised learning on graphs, which generally follows the "augmenting-contrasting" learning scheme. However, we observe that unlike CL in computer vision domain, CL in graph domain performs decently even _without augmentation_. We conduct a systematic analysis of this phenomenon and argue that homophily, i.e., the principle that "like attracts like", plays a key role in the success of graph CL. Inspired to leverage this property explicitly, we propose HomoGCL, a model-agnostic framework to expand the positive set using neighbor nodes with neighbor-specific significances. Theoretically, HomoGCL introduces a stricter lower bound of the mutual information between raw node features and node embeddings in augmented views. Furthermore, HomoGCL can be combined with existing graph CL models in a plug-and-play way with light extra computational overhead. Extensive experiments demonstrate that HomoGCL yields multiple state-of-the-art results across six public datasets and consistently brings notable performance improvements when applied to various graph CL methods. Code is avilable at [https://github.com/wenzhilics/HomoGCL](https://github.com/wenzhilics/HomoGCL).
self-supervised learning; contrastive learning; graph homophily; graph representation learning +
Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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Footnote †: 2023: _J. August 6–10, 2023, Long Beach, CA, USA_. ACM, New York, NY, USA, 12 pages. [https://doi.org/10.1145/3580305.3599380](https://doi.org/10.1145/3580305.3599380)
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## 1. Introduction
Graph Neural Networks (GNNs) have achieved overwhelming accomplishments on a variety of graph-based tasks like node classification and node clustering, to name a few (Wang et al., 2017; Li et al., 2018; Li et al., 2018; Li et al., 2018; Li et al., 2018). They generally refer to the message passing mechanism where node features first propagate to neighbor nodes and then get aggregated to fuse the features in each layer.
Generally, GNNs are designed for supervised tasks which require adequate labeled data. However, it is hard to satisfy as annotated labels are always scarce in real-world scenarios (Li et al., 2018; Li et al., 2018). To tackle this common problem in deep learning, many pioneer endeavors have been made to self-supervised learning (SSL) in the computer vision domain, of which vision contrastive learning (VCL) (Li et al., 2018; Li et al., 2018) has dominated the field. Generally, VCL follows the "augmenting-contrasting" learning pattern, in which the similarity between two augmentations of a sample (positive pair) is maximized, while the similarities between other samples (negative pairs) are minimized. The model can thus learn high-quality representations free of label notation. There are also many work adapting CL to graph representation learning, referred to as graph contrastive learning (GCL) (Li et al., 2018; Li et al., 2018; Li et al., 2018). Research hotspot in GCL mainly focuses
Figure 1. Performance of CL in vision and graph domains with/without augmentation. SimCLR (Li et al., 2018) and GRACE (Li et al., 2018), two prevalent and similar CL architectures in vision and graph domains, are adopted on the respective datasets. MLP is the baseline by simply training a Multi-Layer Perceptron from image RGB features/raw node features. When without augmentation, the performance of vision datasets drops drastically, while the performance of graph datasets is rather stable and still outperforms the MLP counterpart.
on graph augmentation (Zhou et al., 2017; Zhang et al., 2018; Zhang et al., 2019; Zhang et al., 2019; Zhang et al., 2019), since unlike naturally rotating or cropping on images, graph augmentation would discard underlying semantic information which might result in undesirable performance. Though these elaborate graph augmentation-based approaches can achieve state-of-the-art performances on many graph-based tasks, we argue that the role of graph augmentation is still overemphasized. Empirically, we observe that GCL without augmentation can also achieve decent performance (Figure 1(b)), which is quite different from VCL (Figure 1(a)). A natural question arises thereby:
_What causes the huge gap between the performance declines of GCL and VCL when data augmentation is not leveraged?_
To answer this question, we conduct a systematic analysis and argue that _homophily is the most important part of GCL_. Specifically, homophily is the phenomenon that "like attracts like" (Zhou et al., 2017), or connected nodes tend to share the same label, which is a ubiquitous property in real-world graphs like citation networks or co-purchase networks (Zhou et al., 2017; Zhang et al., 2019). According to recent studies (Zhou et al., 2017; Zhang et al., 2019), GNN backbones in GCL (such as GCN (Kip
study AutoSSL (Liu et al., 2018) shows graph homophily is an effective guide in searching the weights on various self-supervised pretext tasks, which reveals the effectiveness of homophily in graph SSL. However, there is no such effort to leverage homophily directly in GCL to the best of our knowledge. It is worth noting that albeit heterophily graphs also exist (Liu et al., 2018) where dissimilar nodes tend to be connected, the related study is still at an early stage even in the supervised setting (Liu et al., 2018). Therefore, to be consistent with previous work on GCL, we only consider the most common homophily graphs in this work.
## 3. Methodology
In this section, we first introduce the preliminaries and notations about GCL. We then conduct a systematic investigation on the functionality of homophily in GCL. Finally, we propose the HomoGCL method to leverage graph homophily directly in GCL.
### Preliminaries and Notations
Let \(\mathcal{G}=\{\mathcal{V},\mathcal{E}\}\) be a graph, where \(\mathcal{V}=\{v_{1},v_{2},\cdots v_{N}\}\) is the node set with \(N\) nodes and \(\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\) is the edge set. The adjacency matrix and the feature matrix are denoted as \(\mathbf{A}\in\{0,1\}^{N\times N}\) and \(\mathbf{X}\in\mathbb{R}^{N\times d}\), respectively, where \(\mathbf{A}_{ij}=1\) iff \((v_{i},v_{j})\in\mathcal{E}\), and \(\mathbf{x}_{i}\in\mathbb{R}^{d}\) is the \(d\)-dim raw feature of node \(v_{i}\). The main notions used throughout the paper are summarized in Appendix A.
Given a graph \(\mathcal{G}\) with no labels, the goal of GCL is to train a graph encoder \(f_{\mathbf{0}}(\mathbf{X},\mathbf{A})\) and get node embeddings that can be directly applied to downstream tasks like node classification and node clustering. Take one of the most popular GCL framework GRACE (Yang et al., 2017) as an example, two augmentation functions \(t_{1}\), \(t_{2}\) (typically randomly dropping edges and masking features) are firstly applied to the graph \(\mathcal{G}\) to generate two graph views \(\mathcal{G}_{1}=(\mathbf{X}_{1},\mathbf{A}_{1})=t_{1}(\mathcal{G})\) and \(\mathcal{G}_{2}=(\mathbf{X}_{2},\mathbf{A}_{2})=t_{2}(\mathcal{G})\). Then, the two augmented graphs are encoded by the same GNN encoder, after which we get node embeddings \(\mathbf{U}=f_{\mathbf{0}}(\mathbf{X}_{1},\mathbf{A}_{1})\) and \(\mathbf{V}=f_{\mathbf{0}}(\mathbf{X}_{2},\mathbf{A}_{2})\). Finally, the loss function is defined by the InfoNCE (Wang et al., 2017) loss as
\[\mathcal{L}=\frac{1}{2N}\sum_{i=1}^{N}\left(\ell\left(\mathbf{u}_{i},\mathbf{v}_{i} \right)+\ell\left(\mathbf{v}_{i},\mathbf{u}_{i}\right)\right), \tag{1}\]
with
\[\ell\left(\mathbf{u}_{i},\mathbf{v}_{i}\right)=\] \[\log\underbrace{\frac{e^{\theta\left(\mathbf{u}_{i},\mathbf{v}_{i} \right)/\tau}}{e^{\theta\left(\mathbf{u}_{i},\mathbf{v}_{i}\right)/\tau}}}_{\text{ positive pair}}+\underbrace{\frac{e^{\theta\left(\mathbf{u}_{i},\mathbf{v}_{i}\right)/\tau}}{\sum_{j\neq i }e^{\theta\left(\mathbf{u}_{i},\mathbf{v}_{j}\right)/\tau}}}_{\text{inter-view negative pairs}}+\underbrace{\sum_{j\neq i}e^{\theta\left(\mathbf{u}_{i},\mathbf{v}_{j} \right)/\tau}}_{\text{intra-view negative pairs}}, \tag{2}\]
where \(\theta(\cdot,\cdot)\) is the similarity function and \(\tau\) is a temperature parameter. In principle, any GNN can be served as the graph encoder. Following recent work (Yang et al., 2017; Wang et al., 2017; Wang et al., 2017), we adopt a two-layer graph convolutional network (GCN) (Liu et al., 2018) as the encoder \(f_{\mathbf{0}}\) by default, which can be formalized as
\[f_{\mathbf{0}}(\mathbf{X},\mathbf{A})=\mathbf{H}^{(2)}=\hat{\mathbf{A}}\sigma (\hat{\mathbf{A}}\mathbf{X}\mathbf{W}^{(1)})\mathbf{W}^{(2)}, \tag{3}\]
where \(\hat{\mathbf{A}}=\bar{\mathbf{D}}^{-1/2}(\mathbf{A}+\mathbf{I}_{N})\bar{ \mathbf{D}}^{-1/2}\) with \(\bar{\mathbf{D}}\) being the degree matrix of \(\mathbf{A}+\mathbf{I}_{N}\) and \(\mathbf{I}_{N}\) being the identity matrix, \(\mathbf{W}\) are learnable weight matrices, and \(\sigma(\cdot)\) is the \(ReLU\) activation function (Krizhevsky et al., 2014).
### Homophily in GCL: an Empirical Investigation
In Figure 1, we can observe that the performance of VCL collapses without data augmentation, while the performance of GCL is still better than the MLP counterpart, which implies that there is a great discrepancy in the mechanism between VCL and GCL, although they adopt seemingly similar frameworks SimCLR (Wang et al., 2017) and GRACE (Yang et al., 2017). To probe into this phenomenon, we plot the similarities between positive and negative samples w.r.t. the training processes, as shown in Figure 2 (a), (b).
Figure 2. Empirical studies on graph homophily. (a), (b) are similarities between positive and negative pairs w.r.t. the training processes with/without augmentation on vision dataset CIFAR10 and graph dataset Cora. The similarity between negative pairs drops to 0 swiftly on CIFAR10 without augmentation, while the similarity between negative pairs drops gradually on Cora without augmentation, which is analogous to its counterpart with augmentation. Please note that the similarity between positive pairs remains as 1 when without augmentation since the two views are identical. To analyze the role that homophily played in this phenomenon, we conduct an ablation study for GRACE without augmentation in (c) by (1) only enabling message passing (w/ MP), and (2) disabling message passing (w/o MP), together with the MLP baseline on two graph datasets Cora and Photo. The performance shows the functionality of message passing, which relies on the homophily assumption.
From the figures, we can see that for vision dataset CIFAR10, the similarity between negative pairs drops swiftly without augmentation. We attribute this to that the learning objective is too trivial by enlarging the similarity between the sample and _itself_ while reducing the similarity between the sample and any other sample in the batch. Finally, the model can only learn one-hot-like embeddings, which eventually results in the poor performance. However, for graph dataset Cora, the similarity between negative pairs without augmentation still drops gradually and consistently, which is analogous to its counterpart with augmentation. We attribute this phenomenon to the message passing in GNN. Albeit InfoNCE loss also enlarges the similarity between the identical samples in two views while reducing the similarity between two different samples, the message passing in GNN enables each node to aggregate information from its neighbors, which leverages graph homophily implicitly to avoid too trivial discrimination. From another perspective, the message passing in GNN enables each node to no longer be independent of its neighbors. As a result, only the similarities between two relatively far nodes (e.g., neighbors \(\geq 3\)-hop for a two-layer GCN), are reduced.
Definition 1 (Homophily).: _The homophily in a graph \(\mathcal{G}\) is defined as the fraction of intra-class edges. Formally, with node label \(\Upsilon\), homophily is defined as_
\[h(\mathcal{G},\Upsilon)=\frac{1}{|\mathcal{E}|}\sum_{(\mathbf{v}_{1},\mathbf{ z}_{2})\in\mathcal{E}}\mathbb{1}\left(\mathbf{y}_{1}=\mathbf{y}_{2}\right), \tag{4}\]
_where \(\mathbf{y}_{i}\) denotes the label of \(\mathbf{v}_{i}\) and \(\mathbb{1}(\cdot)\) is the indication function._
As analyzed above, we hypothesize that message passing which relies on the homophily assumption prevents GCL without augmentation from corruption. To validate this, we conduct another ablation experiment on two graph datasets Cora and Photo, as shown in Figure 2 (c).
Experimental setupWe devise two variants of GRACE (Zhu et al., 2017) without augmentation, i.e., GRACE w/ MP and GRACE w/o MP, where the former is the version that message passing is leveraged (i.e., the w/o aug. version in Figure 1(b)), while the latter is the version that message passing is blocked via forcing each node to only connect with itself (i.e., substituting the adjacency matrix with the eye matrix). For MLP, we directly train an MLP with raw node features as the input in a supervised learning manner without graph structure, which is the same as in Figure 1. The two variants are equipped with the same 2-layer GCN backbone of 256 dim hidden embeddings and are trained until convergence.
ObservationsFrom Figure 2 (c), we can see that the performance of GRACE (w/o MP) is only on par with or even worse than the MLP counterpart, while GRACE (w/ MP) outperforms both by a large margin on both datasets. It meets our expectation that by disabling message passing, nodes in GRACE (w/o MP) cannot propagate features to their neighbors, which degenerates them to a similar situation of VCL without augmentation. GRACE (w/ MP), on the other hand, can still maintain the performance even without raw features. In a nutshell, this experiment verifies our hypothesis that message passing, which relies on the homophily assumption, is the key factor of GCL.
### HomoGCL
As analyzed above, graph homophily plays a crucial role in the overall performance of GCL. Therefore, we are naturally inspired to leverage this property explicitly. Simply assigning neighbor nodes as positive is non-ideal, as nodes near the decision boundaries tend to link with nodes from another class, thus being _false positive_. To mitigate such effect, it is expected to estimate the probability of neighbor nodes being _true positive_. However, the task is intractable as node labels are not available to identify the boundaries in the
Figure 3. The pipeline of HomoGCL (a). Two graph views \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) are first generated via graph augmentation from graph \(\mathcal{G}\), after which the three graphs are fed into the shared GNN encoder to learn representations. The representation of \(\mathcal{G}\) is utilized to generate soft clustering assignments via Gaussian Mixture Model, based on which the edge saliency is calculated (b). Edge saliency is leveraged as the weight of neighbor nodes being positive.
SSL setting. To tackle this challenge, we leverage GMM on \(k\)-means hard clusters to get soft clustering assignments of the initial graph \(\mathcal{G}\), where pair-wise similarity (saliency) is calculated as the aforementioned probability. The overall framework of HomoGCL and soft clustering are illustrated in Figure 3. Please note that although we introduce HomoGCL based on GRACE [52] framework as an example, HomoGCL is framework-agnostic and can be combined with other GCL frameworks.
Soft clustering for pair-wise node similarityAn unsupervised method like clustering is needed to estimate the probability of neighbor nodes being true positive. However, traditional clustering methods like \(k\)-means can only assign a hard label for each node, which cannot satisfy our needs. To tackle this problem, we view \(k\)-means as a special case of GMM[3, 13], where soft clustering is made possible based on the posterior probabilities. Specifically, for GMM with \(k\) centroids \(\{\mathbf{c}_{1},\mathbf{c}_{2},\cdots,\mathbf{c}_{k}\}\) defined by the mean embeddings of nodes in different \(k\)-means hard labels, posterior probability can be calculated as
\[p\left(\mathbf{h}_{i}\mid\mathbf{c}_{j}\right)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp \left(-\frac{\left\|\mathbf{h}_{i}-\mathbf{c}_{j}\right\|_{2}}{2\sigma^{2}}\right), \tag{5}\]
where \(\sigma^{2}\) is the variance of Gaussian distribution. By considering an equal prior \(p(\mathbf{c}_{1})=p(\mathbf{c}_{2})=\cdots=p(\mathbf{c}_{k})\), the probability of node feature \(\mathbf{h}_{i}\) belonging to cluster \(\mathbf{c}_{j}\) can be calculated by the Bayes rule as
\[p\left(\mathbf{c}_{j}\mid\mathbf{h}_{i}\right)=\frac{p\left(\mathbf{c}_{j}\right)p\left( \mathbf{h}_{i}\mid\mathbf{c}_{j}\right)}{\sum\limits_{r=1}^{k}p\left(\mathbf{c}_{r}\right)p \left(\mathbf{h}_{i}\mid\mathbf{c}_{r}\right)}=\frac{\exp\left(-\frac{\left(\mathbf{h}_{i }-\mathbf{c}_{j}\right)^{2}}{2\sigma^{2}}\right)}{\sum\limits_{r=1}^{k}\exp\left(- \frac{\left(\mathbf{h}_{i}-\mathbf{c}_{j}\right)^{2}}{2\sigma^{2}}\right)}. \tag{6}\]
In this way, we can get a cluster assignment matrix \(\mathbf{R}\in\mathbb{R}^{N\times k}\) where \(\mathbf{R}_{ij}=p(\mathbf{c}_{j}\mid\mathbf{h}_{i})\) indicates the soft clustering value between node \(v_{i}\) and cluster \(\mathbf{c}_{j}\).
Based on the assignment matrix \(\mathbf{R}\), we are able to calculate a node saliency \(\mathbf{S}_{ij}\) between any connected node pair \((\mathbf{v}_{i},\mathbf{v}_{j})\) via \(\mathbf{S}_{ij}=\mathrm{norm}(\mathbf{R}_{i})\cdot\mathrm{norm}(\mathbf{R}_{j }^{\top})\) with \(\mathrm{norm}(\cdot)\) being the \(L_{2}\) normalization on the cluster dimension. \(\mathbf{S}_{ij}\) can thus indicate the connection intensity between node \(v_{i}\) and \(v_{j}\), which is an estimated probability of neighbors being true positive.
Loss functionAs the probability of neighbors being positive, node saliency \(\mathbf{S}\) can be utilized to expand positive samples in both views. Specifically, Eq. (2) is converted to
\[\ell_{\textit{cont}}(\mathbf{u}_{i},\mathbf{v}_{i})=\log\frac{\mathrm{pos}}{\mathrm{pos }+\mathrm{neg}} \tag{7}\]
with
\[\mathrm{pos}=\underbrace{\epsilon^{\theta(\mathbf{u}_{i},\mathbf{v}_{i})/ \tau}}_{\text{inter-view positive pair}}+\underbrace{\sum_{j\in\mathcal{N}_{\mathbf{u}}(i)} \epsilon^{\theta(\mathbf{u}_{i},\mathbf{u}_{j})/\tau}\cdot\mathbf{S}_{ij}}_{\text{ intra-view positive pairs}}, \tag{8}\]
\[\mathrm{neg}=\underbrace{\sum_{j\notin\{i\cup\mathcal{N}_{\mathbf{v}}(i)\}}e^{ \theta(\mathbf{u}_{i},\mathbf{v}_{j})/\tau}}_{\text{inter-view negative pairs}}+\underbrace{\sum_{j\notin\{i\cup\mathcal{N}_{\mathbf{u}}(i)\}}e^{ \theta(\mathbf{u}_{i},\mathbf{u}_{j})/\tau}}_{\text{intra-view negative pairs}}, \tag{9}\]
where \(\mathcal{N}_{\mathbf{u}}(i)\), \(\mathcal{N}_{\mathbf{v}}(i)\) are the neighbor sets of node \(v_{i}\) in two views. The contrastive loss is thus defined as
\[\mathcal{L}_{\textit{cont}}=\frac{1}{2N}\sum_{i=1}^{N}\left(\ell_{\textit{ cont}}\left(\mathbf{u}_{i},\mathbf{v}_{i}\right)+\ell_{\textit{cont}}\left(\mathbf{v}_{i},\mathbf{u}_{i} \right)\right). \tag{10}\]
In addition to the contrastive loss, we also leverage the homophily loss [13] explicitly via
\[\mathcal{L}_{\textit{homo}}=\frac{1}{k|\mathcal{E}|}\sum_{r=1}^{k}\sum_{(\bm {v}_{i},\mathbf{v}_{j})\in\mathcal{E}}\text{MSE}\left(p\left(\mathbf{c}_{r}\mid\mathbf{h} _{i}\right),p\left(\mathbf{c}_{r}\mid\mathbf{h}_{j}\right)\right), \tag{11}\]
where \(\text{MSE}(\cdot)\) is the Mean Square Error. The contrastive loss and the homophily loss are combined in a multi-task learning manner with coefficient \(\alpha\) as
\[\mathcal{J}=\mathcal{L}_{\textit{cont}}+\alpha\mathcal{L}_{\textit{homo}}. \tag{12}\]
It is noteworthy that since HomoGCL is a way to expand positive samples, it can be combined with many node-level GCLs -- even negative-free ones like BGRL [30] -- via the saliency \(\mathbf{S}\) and the homophily loss Eq. (11), which will be discussed in Section 4.4.
### Theoretical Analysis
Though simple and intuitive by design, the proposed HomoGCL framework is theoretically guaranteed to boost the performance of base models from the Mutual Information (MI) maximization perspective, as induced in Theorem 1 with GRACE as an example.
Theorem 1 ().: _The newly proposed contrastive loss \(\mathcal{L}_{\textit{cont}}\) in Eq. (10) is a stricter lower bound of MI between raw node features \(\mathbf{X}\) and node embeddings \(\mathbf{U}\) and \(\mathbf{V}\) in two augmented views, comparing with the raw contrastive loss \(\mathcal{L}\) in Eq. (1) proposed by GRACE. Formally,_
\[\mathcal{L}\leq\mathcal{L}_{\textit{cont}}\leq I(\mathbf{X};\mathbf{U},\mathbf{ V}). \tag{13}\]
Proof.: See Appendix B.
From Theorem 1, we can see that maximizing \(\mathcal{L}_{\textit{cont}}\) is equivalent to maximizing a lower bound of the mutual information between raw node features and learned node representations, which guarantees model convergence [1, 26, 31, 33, 53]. Furthermore, the lower bound derived by HomoGCL is stricter than that of the GRACE, which provides a theoretical basis for the performance boost of HomoGCL over the base model.
### Complexity Analysis
The overview of the training algorithm of HomoGCL (based on GRACE) is elaborated in Algorithm 1, based on which we analyze its time and space complexity. It is worth mentioning that the extra calculation of HomoGCL introduces light computational overhead over the base model.
Time ComplexityWe analyze the time complexity according to the pseudocode. For line 6, the time complexity of \(k\)-means with \(t\) iterations, \(k\) clusters, \(N\) node samples with \(d^{\prime}\)-dim hidden embeddings is \(\mathcal{O}(tkNd^{\prime})\). Obtaining \(k\) cluster centroids needs \(\mathcal{O}(Nd^{\prime})\) based on the hard pseudo-labels obtained by \(k\)-means. For line 7, we need to calculate the distance between each node and each cluster centroid, which is another \(\mathcal{O}(kNd^{\prime})\) overhead to get the assignment matrix \(\mathbf{R}\). For line 8, the saliency \(\mathbf{S}\) can be obtained via \(L_{2}\) norm
and vector multiplication for connected nodes in \(\mathcal{O}(k(N+|\mathcal{E}|))\). For line 10, the homophily loss can be calculated in \(\mathcal{O}(k|\mathcal{E}|)\) by definition. Overall, the extra computational overhead over the base model is \(\mathcal{O}(tkNd^{\prime}+k(N+|\mathcal{E}|))\) for HomoGCL, which is lightweight compared with the base model, as \(k\) is usually set to a small number.
_Space Complexity_. Extra space complexity over the base model for HomoGCL is introduced by the \(k\)-means algorithm, the assignment matrix \(\mathbf{R}\), and the saliency \(\mathbf{S}\). For the \(k\)-means algorithm mentioned above, its space complexity is \(\mathcal{O}(Nd^{\prime})\). For \(\mathbf{R}\in\mathbb{N}^{N\times k}\), its space complexity is \(\mathcal{O}(KN)\). As only the saliency between each connected node pair will be leveraged, the saliency costs \(\mathcal{O}(|\mathcal{E}|)\). Overall, the extra space complexity of HomoGCL is \(\mathcal{O}((d^{\prime}+k)N+|\mathcal{E}|)\), which is lightweight and on par with the GNN encoder.
## 4. Experiments
In this section, we evaluate the effectiveness of HomoGCL by answering the following research questions:
* Does HomoGCL outperform existing baseline methods on node classification and node clustering? Can it consistently boost the performance of prevalent GCL frameworks?
* Can the saliency \(\mathbf{S}\) distinguish the importance of neighbor nodes being positive?
* Is HomoGCL sensitive to hyperparameters?
* How to intuitively understand the representation capability of HomoGCL over the base model?
### Experimental Setup
_Datasets_. We adopt six publicly available real-world benchmark datasets, including three citation networks Cora, CiteSeer, PubMed (CiteSeer et al., 2017), two co-purchase networks Amazon-Photo (Photo), Amazon-Computers (Computer) (Xu et al., 2019), and one large-scale network ogbn-arXiv (arXiv) (Liu et al., 2019) to conduct the experiments throughout the paper. The statistics of the datasets are provided in Table 1. We give their detailed descriptions are as follows:
* **Cora, CiteSeer**, and **PubMed1**(CiteSeer et al., 2017) are three academic networks where nodes represent papers and edges represent citation relations. Each node in Cora and CiteSeer is described by a 0/1-valued word vector indicating the absence/presence of the corresponding word from the dictionary, while each node in PubMed is described by a TF/IDF weighted word vector from the dictionary. The nodes are categorized by their related research area for the three datasets. Footnote 1: [https://github.com/kimiyoung/planetoid/raw/master/data](https://github.com/kimiyoung/planetoid/raw/master/data)
* **Amazon-Photo** and **Amazon-Computers2**(Xu et al., 2019) are two co-purchase networks constructed from Amazon where nodes represent products and edges represent co-purchase relations. Each node is described by a raw bag-of-words feature encoding product reviews, and is labeled with its category. Footnote 2: [https://github.com/shchur/gn-benchmark/raw/master/data/npz](https://github.com/shchur/gn-benchmark/raw/master/data/npz)
* **ogbn-arXiv3**(Liu et al., 2019) is a citation network between all Computer Science arXiv papers indexed by Microsoft academic graph (Xu et al., 2019), where nodes represent papers and edges represent citation relations. Each node is described by a 128-dimensional feature vector obtained by averaging the skipgram word embeddings in its title and abstract. The nodes are categorized by their related research area. Footnote 3: [https://ogbn.stanford.edu/docs/nodeprop/rogbn-arxiv](https://ogbn.stanford.edu/docs/nodeprop/rogbn-arxiv)
_Baselines_. We compare HomoGCL with a variety of baselines, including unsupervised methods Node2Vec (Fan et al., 2017) and DeepWalk (Wang et al., 2018), supervised methods GCN (Kipf and Welling, 2017), GAT (Srivastava et al., 2017), and GraphSAGE (Golovolov et al., 2016), graph autoencoders GAE and VGAE (Golov et al., 2016), graph contrastive learning methods including DGI (Wang et al., 2018), HDI (Wang et al., 2018), GMI (Wang et al., 2018), InfoGCL (Wang et al., 2018), MVGRL (Liu et al., 2019), G-BT (Liu et al., 2019), BGRL (Wang et al., 2018), AFGRL (Wang et al., 2018), CCA-SSG (Wang et al., 2018), COSTA (Wang et al., 2018), GRACE (Wang et al., 2018), GCA (Wang et al., 2018), ProGCL (Wang et al., 2018), ARIEL (Wang et al., 2018), and gCooL (Wang et al., 2018). We also report the performance obtained using an MLP classifier on raw node features. The detailed description of the baselines could be found in Appendix C.
_Configurations and Evaluation protocol_. Following previous work (Wang et al., 2018), each model is firstly trained in an unsupervised manner using the entire graph, after which the learned embeddings are utilized for downstream tasks. We use the Adam optimizer for both the self-supervised GCL training and the evaluation stages. The graph encoder \(f_{\mathbf{0}}\) is specified as a standard two-layer GCN model by default for all the datasets. For the node classification task, we train a simple \(L_{2}\)-regularized one-layer linear classifier. For Cora,
\begin{table}
\begin{tabular}{c|c c c c} \hline \hline Dataset & \#Nodes & \#Edges & \#Features & \#Classes \\ \hline Cora & 2,708 & 10,556 & 1,433 & 7 \\ CiteSeer & 3,327 & 9,228 & 3,703 & 6 \\ PubMed & 19,717 & 88,651 & 500 & 3 \\ Photo & 7,650 & 238,163 & 745 & 8 \\ Computer & 13,752 & 491,722 & 767 & 10 \\ arXiv & 169,343 & 1,166,243 & 128 & 40 \\ \hline \hline \end{tabular}
\end{table}
Table 1. Statistics of datasets used in the paper.
CiteSeer, and PubMed, we apply public splits (Zhu et al., 2017) to split them into training/validation/test sets, where only 20 nodes per class are available during training. For Photo and Computer, we randomly split them into training/validation/testing sets with proportions 10%/10%/80% respectively following (Zhu et al., 2017), since there are no publicly accessible splits. We train the model for five runs and report the performance in terms of accuracy. For the node clustering task, we train a \(k\)-means model on the learned embeddings for 10 times, where the number of clusters is set to the number of classes for each dataset. We measure the clustering performance in terms of two prevalent metrics Normalized Mutual Information (NMI) score: \(\text{NMI}=2I(\bar{\mathbf{Y}};\mathbf{Y})/\left[H(\bar{\mathbf{Y}})+H( \mathbf{Y})\right]\), where \(\bar{\mathbf{Y}}\) and \(\mathbf{Y}\) being the predicted cluster indexes and class labels respectively, \(I(\cdot)\) being the mutual information, and \(H(\cdot)\) being the entropy; and Adjusted Rand Index (ARI): \(\text{ARI}=\text{RI}-\mathbb{E}[\text{RI}]/(\max\{\text{RI}\}-\mathbb{E}[\text {RI}])\), where RI being the Rand Index (Kendra and Kendra, 2018).
### Node Classification (RQ1)
We implement HomoGCL based on GRACE. The experimental results of node classification on five datasets are shown in Table 2, from which we can see that HomoGCL outperforms all self-supervised baselines or even the supervised ones, over the five datasets except on CiteSeer, which we attribute to its relatively low homophily. We can also observe that GRACE-based methods GCA, ProGCL, and ARIEL cannot bring consistent improvements over GRACE. In contrast, HomoGCL can always yield significant improvements over GRACE, especially on Cora with a 3% gain.
We also find CCA-SSG a solid baseline, which can achieve runner-up performance on these five datasets. It is noteworthy that CCA-SSG adopts simple edge dropping and feature masking as augmentation (like HomoGCL), while the performances of baselines with elaborated augmentation (MVGRL, COSTA, GCA, ARIEL) vary from datasets. It indicates that data augmentation in GCL might be overemphasized.
Finally, other methods which leverage graph homophily implicitly (AFGRL, ProGCL) do not perform as we expected. We attribute the non-ideal performance of AFGRL to the fact that it does not apply data augmentation, which might limit its representation ability. For ProGCL, since the model focuses on negative samples by alleviating false negative cases, it is not as effective as HomoGCL to directly expand positive ones.
### Node Clustering (RQ1)
We also evaluate the node clustering performance on Photo and Computer datasets in this section. HomoGCL is also implemented based on GRACE. As shown in Table 3, HomoGCL generally outperforms other methods by a large margin on both metrics for the two datasets. We attribute the performance to that by enlarging the connection density between node pairs far away from the estimated decision boundaries, HomoGCL can naturally acquire compact intra-cluster bonds, which directly benefits clustering. It is also validated by the visualization experiment, which will be discussed in Section 4.8.
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline Model & Training Data & Cora & CiteSeer & PubMed & Photo & Computer \\ \hline Raw features & X,Y & 47.7\(\pm\)0.4 & 46.5\(\pm\)0.4 & 71.4\(\pm\)0.2 & 72.27\(\pm\)0.00 & 73.81\(\pm\)0.00 \\ DeepWalk & A & 70.7\(\pm\)0.6 & 51.4\(\pm\)0.5 & 74.3\(\pm\)0.9 & 89.44\(\pm\)0.11 & 85.68\(\pm\)0.06 \\ Node2Vec & A & 70.1\(\pm\)0.4 & 49.8\(\pm\)0.3 & 69.8\(\pm\)0.7 & 87.76\(\pm\)0.10 & 84.39\(\pm\)0.08 \\ GCN & X, A, Y & 81.5\(\pm\)0.4 & 70.2\(\pm\)0.4 & 79.0\(\pm\)0.2 & 92.42\(\pm\)0.22 & 86.51\(\pm\)0.54 \\ GAT & X, A, Y & 83.0\(\pm\)0.7 & 72.5\(\pm\)0.7 & 79.0\(\pm\)0.3 & 92.56\(\pm\)0.35 & 86.93\(\pm\)0.29 \\ \hline GAE & X,A & 71.5\(\pm\)0.4 & 65.8\(\pm\)0.4 & 72.1\(\pm\)0.5 & 91.62\(\pm\)0.13 & 85.27\(\pm\)0.19 \\ VGAE & X, A & 73.0\(\pm\)0.3 & 68.3\(\pm\)0.4 & 75.8\(\pm\)0.2 & 92.20\(\pm\)0.11 & 86.37\(\pm\)0.21 \\ DGI & X, A & 82.3\(\pm\)0.6 & 71.8\(\pm\)0.7 & 76.8\(\pm\)0.6 & 91.61\(\pm\)0.22 & 83.95\(\pm\)0.47 \\ GMI & X, A & 83.0\(\pm\)0.3 & 72.4\(\pm\)0.1 & 79.9\(\pm\)0.2 & 90.68\(\pm\)0.17 & 82.21\(\pm\)0.31 \\ InfoGCL & X, A & 83.5\(\pm\)0.3 & **73.5\(\pm\)**0.4 & 79.1\(\pm\)0.2 & - & - \\ MVGRL & X, A & 83.5\(\pm\)0.4 & 73.3\(\pm\)0.5 & 80.1\(\pm\)0.7 & 91.74\(\pm\)0.07 & 87.52\(\pm\)0.11 \\ BGRL & X, A & 82.7\(\pm\)0.6 & 71.1\(\pm\)0.8 & 79.6\(\pm\)0.5 & 92.80\(\pm\)0.08 & 88.23\(\pm\)0.11 \\ AFGRL & X, A & 79.8\(\pm\)0.2 & 69.4\(\pm\)0.2 & 80.0\(\pm\)0.1 & 92.71\(\pm\)0.23 & 88.12\(\pm\)0.27 \\ COSTA & X, A & 82.2\(\pm\)0.2 & 70.7\(\pm\)0.5 & 80.4\(\pm\)0.3 & 92.43\(\pm\)0.38 & 88.37\(\pm\)0.22 \\ CCA-SSG & X, A & 84.0\(\pm\)0.4 & 73.1\(\pm\)0.3 & 81.0\(\pm\)0.4 & 92.84\(\pm\)0.18 & 88.27\(\pm\)0.32 \\ \hline GRACE & X, A & 81.5\(\pm\)0.3 & 70.6\(\pm\)0.5 & 80.2\(\pm\)0.3 & 92.15\(\pm\)0.24 & 86.25\(\pm\)0.25 \\ GCA & X, A & 81.4\(\pm\)0.3(\(\downarrow\)0.1) & 70.4\(\pm\)0.4(\(\downarrow\)0.2) & 80.7\(\pm\)0.5(\(\uparrow\)0.5) & 92.53\(\pm\)0.16(\(\uparrow\)0.38) & 87.80\(\pm\)0.23(\(\uparrow\)1.55) \\ ProGCL & X, A & 81.2\(\pm\)0.4(\(\downarrow\)0.3) & 69.8\(\pm\)0.5(\(\downarrow\)0.8) & 79.2\(\pm\)0.2(\(\downarrow\)1.0) & 92.39\(\pm\)0.11(\(\uparrow\)0.24) & 87.43\(\pm\)0.21(\(\uparrow\)1.18) \\ ARIEL & X,A & 83.0\(\pm\)1.3(\(\uparrow\)1.5) & 71.1\(\pm\)0.9(\(\uparrow\)0.5) & 74.2\(\pm\)0.8(\(\downarrow\)6.0) & 91.80\(\pm\)0.24(\(\downarrow\)0.35) & 87.07\(\pm\)0.33(\(\uparrow\)0.82) \\
**HomoGCL** & X,A & **84.5\(\pm\)**0.5(\(\uparrow\)3.0) & 72.3\(\pm\)0.7(\(\uparrow\)1.7) & **81.1\(\pm\)**0.3(\(\uparrow\)0.9) & **92.92\(\pm\)**0.18(\(\uparrow\)0.77) & **88.46\(\pm\)**0.20(\(\uparrow\)2.21) \\ \hline \hline \end{tabular}
* \({}^{1}\) The results not reported are due to unavailable code.
\end{table}
Table 2. Node classification results (accuracy(%) \(\pm\)std) for 5 runs on five real-world datasets. The best results are highlighted in boldface. X, A, and Y correspond to node features, graph adjacency matrix, and node labels respectively. “\(|\)” and “\(|\)” refer to performance improvement and drop compared with the same GRACE base model.
### Improving Various GCL Methods (RQ1)
As a patch to expand positive pairs, it is feasible to combine HomoGCL with other GCL methods, even negative-free ones like BGRL (Wang et al., 2017), which adapts BYOL (Bordes and Zisserman, 2017) in computer vision to GCL to free contrastive learning from numerous negative samples.
Sketch of BgrlAn online encoder \(f_{\xi}\) and a target encoder \(f_{\phi}\) are leveraged to encoder two graph views \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\) respectively, after which we can get \(\tilde{\mathbf{H}}_{1}\) and \(\tilde{\mathbf{H}}_{2}\). An additional predictor \(p_{\xi}\) is applied to \(\tilde{\mathbf{H}}_{1}\) and we can get \(\tilde{\mathbf{Z}}_{1}\). The loss function is defined as
\[\ell_{1}=\frac{1}{N}\sum_{i=1}^{N}\theta^{\prime}\left(\tilde{\mathbf{Z}}_{( 1,i)},\tilde{\mathbf{H}}_{(2,i)}\right), \tag{14}\]
where \(\theta^{\prime}(\cdot,\cdot)\) is a similarity function. A symmetric loss \(\tilde{\ell}_{1}\) is obtained by feeding \(\mathcal{G}_{1}\) into the target encoder and \(\mathcal{G}_{2}\) into the online encoder, and the final objective is \(\mathcal{L}_{1}=\ell_{1}+\tilde{\ell}_{1}\). For online encoder \(f_{\xi}\), its parameters are updated via stochastic gradient descent, while the parameters of the target encoder \(f_{\phi}\) are updated via exponential moving average (EMA) of \(\theta\) as \(\phi\gets r\phi+(1-r)\xi\).
To combine BGRL with HomoGCL, we feed the initial graph \(\mathcal{G}\) to the online encoder and get \(\mathbf{H}\). Then we get the assignment matrix \(\mathbf{R}\) via Eq. (6) and the saliency \(\mathbf{S}\) via multiplication. The expanded loss can thus be defined as
\[\ell_{2}=\frac{1}{|\mathcal{E}|}\sum_{(\alpha_{i},\beta_{j})\in\mathcal{E}} \theta^{\prime}\left(\tilde{\mathbf{Z}}_{(1,i)},\tilde{\mathbf{H}}_{(2,j)} \right)\mathbf{S}_{ij}. \tag{15}\]
We can also get a symmetric loss \(\tilde{\ell}_{2}\), and the entire expanded loss is \(\mathcal{L}_{2}=\ell_{2}+\tilde{\ell}_{2}\). Together with the homophily loss \(\mathcal{L}_{homo}\) defined in Eq. (11), we can get the overall loss for BGRL-based HomoGCL as
\[\mathcal{J}=\mathcal{L}_{1}+\alpha\mathcal{L}_{homo}+\beta\mathcal{L}_{2}, \tag{16}\]
where \(\beta\), \(\alpha\) are two hyperparameters.
We evaluate the performance of BGRL+HomoGCL on PubMed, Photo, and Computer. The results are shown in Table 4, from which we can see that HomoGCL brings consistent improvements over the BGRL base. It verifies that HomoGCL is model-agnostic and can be applied to GCL models in a plug-and-play way to boost their performance. Moreover, it is interesting to see that the performances on Photo and Computer even surpass GRACE+HomoGCL as we reported in Table 2, which shows the potential of HomoGCL to further boost the performance of existing GCL methods.
### Results on Large-Scale Dataset (RQ1)
We also conduct an experiment on a large-scale dataset arXiv. As the dataset is split based on the publication dates of the papers, i.e., the training set is papers published until 2017, the validation set is papers published in 2018, and the test set is papers published since 2019, we report the classification accuracy on both the validation and the test sets, which is a convention for this task. We extend the backbone GNN encoder to 3 GCN layers, as suggested in (Wang et al., 2017; Wang et al., 2017). The results are shown in Table 5.
Since GRACE treats all other nodes as negative samples, scaling GRACE to the large-scale dataset suffers from the OOM issue. Subsampling \(k\) nodes randomly across the graph as negative samples for each node is a feasible countermeasure (Wang et al., 2017), but it is sensitive to the negative size \(k\). BGRL, on the other hand, is free from negative samples, which makes it scalable by design, and it shows a great tradeoff between performance and complexity. Since the space complexity of HomoGCL mainly depends on the performance of the base model as discussed in Section 3.5, we implement HomoGCL based on BGRL as described in Section 4.4. We can see that
\begin{table}
\begin{tabular}{l|c c c} \hline \hline Model & PubMed & Photo & Computer \\ \hline BGRL & 79.6 & 92.80 & 88.23 \\ \hline \multicolumn{4}{l}{**+HomoGCL**} & 80.8(1.2) & 93.53(70.73) & 90.01(1.79) \\ \hline \hline \end{tabular}
\end{table}
Table 4. The performance of HomoGCL for boosting negative sample-free BGRL in terms of accuracy(%).
\begin{table}
\begin{tabular}{l c|c c} \hline \hline Dataset & \multicolumn{2}{c|}{Photo} & \multicolumn{2}{c}{Computer} \\ \hline Metric & NMI & ARI & NMI & ARI \\ \hline GAE & 0.616\(\pm\)\(\Delta_{1}\) & 0.494\(\pm\)\(\Delta_{1}\) & 0.441\(\pm\)\(\Delta_{0}\) & 0.258\(\pm\)\(\Delta_{0}\) \\ VGAE & 0.530\(\pm\)\(\Delta_{4}\) & 0.373\(\pm\)\(\Delta_{4}\) & 0.423\(\pm\)\(\Delta_{0}\) & 0.238\(\pm\)\(\Delta_{0}\) \\ DGI & 0.376\(\pm\)\(\Delta_{3}\) & 0.264\(\pm\)\(\Delta_{3}\) & 0.318\(\pm\)\(\Delta_{2}\) & 0.165\(\pm\)\(\Delta_{2}\) \\ HDI & 0.429\(\pm\)\(\Delta_{1}\) & 0.307\(\pm\)\(\Delta_{1}\) & 0.347\(\pm\)\(\Delta_{1}\) & 0.216\(\pm\)\(\Delta_{6}\) \\ MVGRL & 0.344\(\pm\)\(\Delta_{4}\) & 0.239\(\pm\)\(\Delta_{4}\) & 0.244\(\pm\)\(\Delta_{0}\) & 0.141\(\pm\)\(\Delta_{0}\) \\ BGRL & 0.668\(\pm\)\(\Delta_{3}\) & 0.547\(\pm\)\(\Delta_{4}\) & 0.484\(\pm\)\(\Delta_{0}\) & 0.295\(\pm\)\(\Delta_{0}\) \\ AFGRL & 0.618\(\pm\)\(\Delta_{1}\) & 0.497\(\pm\)\(\Delta_{3}\) & 0.478\(\pm\)\(\Delta_{3}\) & 0.334\(\pm\)\(\Delta_{4}\) \\ GCA & 0.614\(\pm\)\(\Delta_{0}\) & 0.494\(\pm\)\(\Delta_{0}\) & 0.426\(\pm\)\(\Delta_{0}\) & 0.246\(\pm\)\(\Delta_{0}\) \\ gCooL & 0.632\(\pm\)\(\Delta_{0}\) & 0.524\(\pm\)\(\Delta_{0}\) & 0.474\(\pm\)\(\Delta_{2}\) & 0.277\(\pm\)\(\Delta_{2}\) \\
**HomoGCL** & **0.671\(\pm\)\(\Delta_{2}\)** & **0.587\(\pm\)\(\Delta_{2}\)** & **0.534\(\pm\)\(\Delta_{0}\)** & **0.396\(\pm\)\(\Delta_{0}\)** \\ \hline \hline \end{tabular}
\end{table}
Table 3. Node clustering results in terms of NMI and ARI on Photo and Computer datasets, where HomoGCL is implemented based on GRACE. \(\Delta_{x}=0.01x\) is used to denote the standard deviation.
\begin{table}
\begin{tabular}{l c c} \hline \hline Model & Validation & Test \\ \hline MLP & 57.65\(\pm\)0.12 & 55.50\(\pm\)0.23 \\ node2vec & 71.29\(\pm\)0.13 & 70.07\(\pm\)0.13 \\ GCN & 73.00\(\pm\)0.17 & 71.74\(\pm\)0.29 \\ GraphSAGE & 72.77\(\pm\)0.16 & 71.49\(\pm\)0.27 \\ \hline Random-Init & 69.90\(\pm\)0.11 & 68.94\(\pm\)0.15 \\ DGI & 71.26\(\pm\)0.11 & 70.34\(\pm\)0.16 \\ G-BT & 71.16\(\pm\)0.14 & 70.12\(\pm\)0.18 \\ GRACE full-graph & OOM & OOM \\ GRACE-Subsampling (\(k\)=2) & 60.49\(\pm\)3.72 & 60.24\(\pm\)0.46 \\ GRACE-Subsampling (\(k\)=8) & 71.30\(\pm\)0.17 & 70.33\(\pm\)0.18 \\ GRACE-Subsampling (\(k\)=2048) & 72.61\(\pm\)0.15 & 71.51\(\pm\)0.11 \\ ProGCL & 72.45\(\pm\)0.21 & 72.18\(\pm\)0.09 \\ \hline BGRL & 72.53\(\pm\)0.09 & 71.64\(\pm\)0.12 \\
**HomoGCL** & **72.85\(\pm\)0.10** & **72.22\(\pm\)0.15** \\ \hline \hline \end{tabular}
\end{table}
Table 5. Node classification results on ogbn-arXiv dataset (accuracy(%) \(\pm\)std). The results of baselines are quoted from published reports. OOM indicates out-of-memory.
BGRL+HomoGCL boosts the performance of BGRL on both validation and test sets. Furthermore, it outperforms all other compared baselines, which shows its effectiveness and efficiency.
### Case Study (RQ2)
To obtain an in-depth understanding for the mechanism of the saliency S in distinguishing the importance of neighbor nodes being positive, we conduct this case study. Specifically, we first sort all edges according to the learned saliency S, then divide them into intervals of size 500 to calculate the homophily in each interval. From Figure 4 we can see that the saliency can estimate the probability of neighbor nodes being positive properly as more similar node pairs in S tend to have larger homophily, which validates the effectiveness of leveraging saliency in HomoGCL.
### Hyperparameter Analysis (RQ3)
In Figure 5, we conduct a hyperparameter analysis on the number of clusters and the weight coefficient \(\alpha\) in Eq. (12) on the Cora dataset. From the figures we can see that the performance is stable w.r.t. the cluster number. We attribute this to the soft class assignments used in identifying the decision boundary, as the pairwise saliency is mainly affected by the relative distance between node pairs, which is less sensitive to the number of clusters. The performance is also stable when \(\alpha\) is below 10 (i.e., contrastive loss in Eq. (7) and homophily loss in Eq. (11) are on the same order of magnitude). This shows that HomoGCL is parameter-insensitive, thus facilitating it to be combined with other GCL methods in a plug-and-play way flexibly. In practice, we tune the number of clusters in \(\{5,10,15,20,25,30\}\) and simply assign \(\alpha\) to 1 to balance the two losses without tuning.
### Visualization (RQ4)
In addition to quantitative analysis, we also visualize the embeddings learned by BGRL in Figure 6(a), GRACE in Figure 6(b), BGRL+ HomoGCL in Figure 6(c), and GRACE+HomoGCL in Figure 6(d) on the Cora dataset using t-SNE (Shen et al., 2017). Here, each point represents a node and is colored by its label. We observe that the embeddings learned by "+HomoGCL" counterparts generally possess clearer class boundaries and compact intra-class structures, which shows the effectiveness of HomoGCL intuitively. This observation aligns with the remarkable node clustering performance reported in Section 4.3, which shows the superiority of HomoGCL.
## 5. Conclusions
In this paper, we investigate why graph contrastive learning can perform decently when data augmentation is not leveraged and argue that graph homophily plays a key role in GCL. We thus devise HomoGCL to directly leverage homophily by estimating the probability of neighbor nodes being positive via Gaussian Mixture Model. Furthermore, HomoGCL is model-agnostic and thus can be easily combined with existing GCL methods in a plug-and-play way to further boost their performances with a theoretical foundation. Extensive experiments show that HomoGCL can consistently outperform state-of-the-art GCL methods in node classification and node clustering tasks on six benchmark datasets.
###### Acknowledgements.
The authors would like to thank Ying Sun from The Hong Kong University of Science and Technology (Guangzhou) for her insightful discussion. This work was supported by NSFC (62276277), Guangdong Basic Applied Basic Research Foundation (2022B1515120059), and the Foshan HKUST Projects (FSUST21-FYTRI01A, FSUST21-FYTRI02A). Chang-Dong Wang and Hui Xiong are the corresponding authors.
Figure 4. Case study on Cora. The saliency S can effectively estimate the probability of neighbor nodes being positive as more salient edges (more similar node pairs) tend to have larger homophily.
Figure 5. Hyperparameter analysis on the number of clusters and weight coefficient \(\alpha\) on Cora.
Figure 6. Visualization of node embeddings on Cora via t-SNE. Each node is colored by its label. | Contrastive learning (CL) はグラフの自己教師あり学習において、現状の学習パラメータとなっています。これは一般的に「増強-対比」学習chemeに従っています。しかし、CLはコンピュータビジョン領域と異なり、グラフィックス領域では拡張なしでも decently Performs 。この現象を体系的に分析し、同質性、すなわち「似ているものは似合う」という原則がグラフ CL の成功に重要な役割を果たしていることを主張する。この性質を明確に利用したいという考えから、モデルに無関係な HomoGCL フレームワークを提案する。これは、隣接ノードの隣接ノードに特化した意味合いをベースに、正のセットを拡張するためのものです。理論的には、HomoGCL は拡張された視点を示すためのRaw Node Features と Node Embeddings 間の相互情報性の厳格な下限を導入します。さらに、HomoGCL は既存のグラフ CL モデル |
2303.05828 | Adapting Contrastive Language-Image Pretrained (CLIP) Models for
Out-of-Distribution Detection | We present a comprehensive experimental study on pretrained feature
extractors for visual out-of-distribution (OOD) detection, focusing on adapting
contrastive language-image pretrained (CLIP) models. Without fine-tuning on the
training data, we are able to establish a positive correlation ($R^2\geq0.92$)
between in-distribution classification and unsupervised OOD detection for CLIP
models in $4$ benchmarks. We further propose a new simple and scalable method
called \textit{pseudo-label probing} (PLP) that adapts vision-language models
for OOD detection. Given a set of label names of the training set, PLP trains a
linear layer using the pseudo-labels derived from the text encoder of CLIP. To
test the OOD detection robustness of pretrained models, we develop a novel
feature-based adversarial OOD data manipulation approach to create adversarial
samples. Intriguingly, we show that (i) PLP outperforms the previous
state-of-the-art \citep{ming2022mcm} on all $5$ large-scale benchmarks based on
ImageNet, specifically by an average AUROC gain of 3.4\% using the largest CLIP
model (ViT-G), (ii) we show that linear probing outperforms fine-tuning by
large margins for CLIP architectures (i.e. CLIP ViT-H achieves a mean gain of
7.3\% AUROC on average on all ImageNet-based benchmarks), and (iii)
billion-parameter CLIP models still fail at detecting adversarially manipulated
OOD images. The code and adversarially created datasets will be made publicly
available. | Nikolas Adaloglou, Felix Michels, Tim Kaiser, Markus Kollmann | 2023-03-10T10:02:18 | http://arxiv.org/abs/2303.05828v2 | # Contrastive Language-Image Pretrained (CLIP) Models are
###### Abstract
We present a comprehensive experimental study on pretrained feature extractors for visual out-of-distribution (OOD) detection. We examine several setups, based on the availability of labels or image captions and using different combinations of in- and out-distributions. Intriguingly, we find that (i) contrastive language-image pretrained models [62, 11] achieve state-of-the-art unsupervised out-of-distribution performance using nearest neighbors feature similarity as the OOD detection score, (ii) supervised state-of-the-art OOD detection performance can be obtained without in-distribution fine-tuning, (iii) even top-performing billion-scale vision transformers trained with natural language supervision fail at detecting adversarially manipulated OOD images. Finally, we argue whether new benchmarks for visual anomaly detection are needed based on our experiments. Using the largest publicly available vision transformer, we achieve state-of-the-art performance across all \(18\) reported OOD benchmarks, including an AUROC of 87.6% (9.2% gain, unsupervised) and 97.4% (1.2% gain, supervised) for the challenging task of CIFAR100 \(\rightarrow\) CIFAR10 OOD detection. The code will be open-sourced.
## 1 Introduction
Transfering the representations of pretrained vision models has improved the performance on a plethora of image recognition tasks [80, 73]. To date, these models are trained with various types of supervision, which accelerates training convergence compared to random initialization [32]. Examples include self-supervision [9, 28], natural language supervision [62], weakly-supervised learning [55], or standard label-based supervised learning. Concurrently, vision transformers (ViTs [18]) have been established, along with an enormous number of variants [51, 78, 21, 74], as a suitable architecture for training large-scale models in the visual domain [16].
Numerous experimental studies indicate that the performance of ViTs scales better with model and dataset size [4, 81] compared to existing convolutional neural networks (CNNs) [33, 44]. Moreover, pretrained ViTs are known to be more robust than CNNs against input perturbations (e.g. occlusions, distribution shifts) [58, 4, 34].
Nevertheless, the applicability of the learned features of pretrained models is crucial for a wide range of downstream tasks [5]. In particular, how to leverage these models for unsupervised tasks is non-trivial and of great significance in many real-life applications. To examine the properties of the feature spaces of pretrained models on unseen distributions, we focus on unsupervised visual OOD detection in this work.
The task of OOD, novelty, or anomaly detection aims at identifying whether a given test sample is drawn from the _in-distribution_ (the training set) or from an alternative distribution, known as the _out-distribution_. Accurate detection of anomalies is indispensable for real-world applications to ensure safety during deployment [1, 66]. The detected unfamiliar samples can be processed separately, possibly with a human expert in the loop, rather than making a potentially uncalibrated prediction [29]. Despite significant advances in deep learning, neural networks tend to generate systematic errors for test examples far from the training set [60], or assign higher likelihoods to OOD samples compared to in-distribution samples [57].
Recent studies have established a firm link between the in-distribution accuracy and OOD generalization and detection [34, 76]. Supervised training on the in-distribution results in intermediate representations that likely form tight label-related clusters [77, 23]. Ideally, an OOD representation should capture semantic properties, such as its pose and
shape, while remaining sensitive to the properties of the imaging process (e.g. lighting, resolution) [77].
A suitable choice of visual feature representations is crucial for detecting OOD data. However, learning useful representations for unsupervised OOD detection [72, 69, 63], where no in-distribution labels are available, is a challenging and active research area [14]. Unsupervised methods often adopt self-supervision to learn the in-distribution features, by defining pretext tasks such as rotation prediction [25].
A major milestone in visual representation learning was reached with the development of both contrastive [9, 31] and non-contrastive self-supervised methods [7]. Recently, contrastive language-image pretraining (_CLIP_) has enabled learning from vast amounts of raw text [62], known as natural language supervision.
In practice, labeled OOD samples are typically unavailable, and the number of in-distribution samples is usually limited. Therefore, external data have been widely employed [65, 63] in two ways: a) outlier exposure where the external data is treated as anomalous [38], and b) using models pretrained on external data [23]. Outlier exposure leads to performance gains only if the auxiliary data are sufficiently diverse [53] and disjoint from the in-distribution data [38].
Pretrained backbones can enhance OOD detection performance and robustness [37, 34], without relying on dataset-specific shortcuts [24]. As a consequence, pretrained models are suitable candidates for OOD detection. In this direction, Fort et al. [23] fine-tune ImageNet pretrained models on the in-distribution and achieved human-level performance on existing OOD detection benchmarks.
Current OOD detection benchmarks mainly rely on CIFAR10 and CIFAR100 [45]. Methods tuned specifically for these benchmarks may not always translate effectively into larger-scale and real-life applications [40]. Despite the existence of pretrained models for visual OOD detection, the choice of the feature extractor, OOD evaluation scheme, and OOD robustness against adversarial attacks, have not been sufficiently explored [70, 65, 14]. Even though the robustness against adversarial attacks has been extensively explored in image classification [71, 27], less attention has been given to studying the construction of robust visual OOD detectors [35, 3].
Figure 1: **In-distribution classification accuracy using \(k\)-nearest neighbours (k-NN) (x-axis) versus out-of-distribution (OOD) detection AUROC(%) score (y-axis) for CIFAR100\(\rightarrow\)CIFAR10 (left) and CIFAR10\(\rightarrow\)CIFAR100 (right). The OOD detection score is computed using the top-1 NN similarity. The classification accuracy uses top-20 NN similarity. Different colors are utilized for different architectures (ViT [18], ConvNeXt [52], ResNet [33]) while symbol sizes roughly indicate architecture size (i.e. Small, Base, Large, Huge, Giga). IN indicates ImageNet [17] and IN21K indicates ImageNet-21K [68]. The corresponding table can be found in the appendix. Best viewed in color.**
In this paper, we demonstrate how pretrained visual backbones can be leveraged for OOD detection. The core contributions of this work are summarized as follows:
* We study \(32\) pretrained models and evaluate them across the conventional in- and out-distributions (Fig. 1) and find that large-scale pretrained CLIP models [62, 11] are powerful zero-shot OOD detectors, using the \(k\)-nearest neighbor similarity as the detection score.
* We apply several OOD evaluation techniques to the best-performing CLIP ViT-G feature extractor [11], based on the accessibility of label-related information for the in-distribution. There, we achieve an AUROC of 97.4% in supervised OOD detection on CIFAR100 \(\rightarrow\) CIFAR10 and 87.6% in the corresponding unsupervised scenario, outperforming previous state-of-the-art approaches by 1.2% and 9.2% respectively.
* Finally, we argue whether new visual OOD benchmarks are required. Towards this direction, we introduce a novel method that adversarially manipulates OOD images. More concretely, we apply adversarial perturbations on the samples of a given OOD dataset to match the representations of the in-distribution samples. We show that even CLIP ViT-G trained on billions of samples can be easily fooled, by changes that are invisible to humans.
## 2 Related work
### Supervised OOD detection methods
Supervised OOD detection methods rely on the fact that in-distribution classification accuracy is positively correlated with OOD detection performance [23]. For that reason, a large number of OOD detection methods derive anomaly scores from supervised in-distribution classifiers. A frequently used baseline is to use the maximum softmax probability (MSP) of an in-distribution classifier as OOD detection score [36]. To increase MSP, Liang et al. [50] use a temperature hyperparameter tuned on OOD data along with adversarial perturbations on the images.
An alternative OOD detection score is the parametric Mahalanobis-based score. In [48], the anomaly score is defined as the negative of the minimum Mahalanobis distance [15] between per-class feature vectors. The Mahalanobis-based score assumes that the samples from each class are normally distributed around the per-class mean. This computation assumes that the representations conform to a mixture of Gaussians and that in-distribution labels are available [65, 23]. Later on, Sun et al. [70] establish an important yet simple OOD detection score, namely the \(k\)-nearest neighbors (NN) distance, without requiring the feature norms [72]. The \(k\)-NN distance has the advantage of being non-parametric, and model- and distribution-agnostic.
Regarding OOD detection robustness against in-distribution perturbations, Hendrycks et al. [35] analyze the robustness under multiple transformations, such as Gaussian noise and blur. However, it is not always clear which manually perturbed images are present in the in-distribution and which are not.
**Auxiliary Tasks.** Supervised learning may not always produce sufficiently informative features for identifying OOD samples [77]. To this end, additional tasks have been proposed to enrich the supervised-learned features. One such approach is identifying the key in-distribution transformations and attempting to predict them. Hendrycks et al. [39] propose rotation prediction (RP, e.g. \([0^{\circ},90^{\circ},180^{\circ},270^{\circ}]\)) along with the supervised objective. In (MTL) [56], Mohseni et al. attempt to learn the domain-specific transformations for each in-distribution using Bayesian optimization by minimizing the in-distribution classification loss. Concurrently, Winkens et al. [77] combine contrastive learning to incentivize the learning of features that discriminate between all dataset images, even if they belong to the same class. Zhang et al. [82] present a two-branch OpenHybrid framework, where a generative flow-based model and a supervised classifier are jointly trained.
### Unsupervised OOD detection methods
Recent unsupervised OOD detection methods rely on learning in-distribution features, which is accomplished with contrastive [69, 70] or supervised contrastive learning (_SupSimCLR_[42]). In contrastive learning, two strongly augmented, yet correlated, views from an image are created (forming a positive pair). The feature similarity of these positive pairs is then maximized, encouraging the features to be invariant to the applied transformations (i.e. jitter, random crop). Simultaneously, the feature similarity of negative pairs (which consist of different images in the mini-batch) is minimized, pushing them away in feature space.
Contrastive-based methods can be further enhanced by: a) designing hand-crafted transformations that provide an estimate of near OOD data (shifting transformations [72, 63]), and b) developing better OOD detection scores. For example, in CSI [72], Tack et al. introduce rotation prediction together with contrastive learning. However, CSI relies on sophisticated in-distribution-dependent augmentations and ensembling during testing. A simpler contrastive-based approach (SSD) is developed by [69], where an OOD detection score is defined using the Mahalanobis distance in the feature space with the \(k\)-means clusters as per-class means [54].
### OOD detection methods using external data or pretrained models
**External data.** Lee et al. [47] leverage auxiliary data to train a generative adversarial network [26] that can generate synthetic examples near the decision boundary. Another com
mon way to incorporate external data is outlier exposure [38]. Therein, an auxiliary task is introduced, where external samples that are non-overlapping with the in-distribution data are encouraged to uniformly distribute among in-distribution classes. In the same direction, Rafiee et al. [63] present an unsupervised analog of outlier exposure by additionally rotating the auxiliary images to ensure that the external data are disjoint from in-distribution. Nonetheless, it is difficult to guarantee disjointness without labels, and the allowed shifting transformations are dependent on the in-distribution [56], limiting the applicability of such approaches.
**Pretrained models.** In principle, external data can be leveraged for feature learning. Large-scale models pretrained on diverse datasets can boost OOD detection performance [37]. The majority of existing methods focus on supervised OOD detection [70, 64]. Such approaches include fine-tuning the whole network [23], or parts of it [64]. Fort et al. [23] fine-tune pretrained models on the in-distribution and achieve human-level performance on challenging OOD detection setups (i.e. CIFAR100 \(\rightarrow\) CIFAR10). Contrarily, in [65], the authors present a Mahalanobis-based score that does not require fine-tuning on the in-distribution.
Very few label-free methods based on pretrained models have been proposed. For example, in [20] the CLIP framework is extended by training a text-based generator on top of CLIP. In [14], the _a priori_ determined clusters are detected using only the in-distribution data in the first phase. In the second phase, the clusters are used as pseudo-labels for fine-tuning the pretrained models. However, the aforementioned CLIP-based approaches require the in- or out-distribution class names [23], while cluster-based methods require the exact number of ground truth in-distribution classes.
## 3 The proposed OOD detection setup
### Dataset description and metrics
We denote the in-distribution as \(\mathcal{D}_{\text{in}}\), and the out-distribution as \(\mathcal{D}_{\text{out}}\). The corresponding train and test splits are indicated with a superscript. In contrast to prior works [20], we define the _zero-shot OOD detection_ setup of CLIP without having access to the set of in-distribution class names. This enables us to design a fair comparison between vision and vision-language models. Similar to [56], we use CIFAR10, CIFAR100, and ImageNet-30 [39] as in-distributions while using the following datasets as out-of-distribution: SVHN [59], STL10 [13], LSUN [50], Places-365 [83], and Texture [12]. For ImageNet-30, we also consider Flowers [61], Food-101 [6], CUB-200 [75], Stanford Dogs [41] and Tiny ImageNet (_TinyIN_)[46]. Dataset-specific details are included in the appendix.
The area under the receiver operating characteristic curve (AUROC) is computed between \(\mathcal{D}_{\text{out}}^{\text{test}}\) and \(\mathcal{D}_{\text{in}}^{\text{test}}\) test sets. Below we present three OOD detection scores that were utilized for the AUROC computations.
**1-NN.** For the unsupervised evaluations, we use the maximum of the cosine similarity between a test image \(x^{\prime}\) and \(x_{i}\in\mathcal{D}_{\text{in}}^{\text{train}}=\{x_{1},x_{2}\ldots,x_{N}\}\) as an OOD score:
\[s_{\text{NN}}(x^{\prime})=\max_{i}\operatorname{sim}(g(x^{\prime}),g(x_{i})), \tag{1}\]
where \(\operatorname{sim}(\cdot)\) is the cosine similarity and \(N\) the number of \(\mathcal{D}_{\text{in}}^{\text{train}}\) samples.
**Mahalanobis distance (MD).** The MD can be either applied directly on the feature space of the pretrained model, \(z_{i}=g(x_{i})\), or on the trained linear head, \(z_{i}=h(g(x_{i}))\). However, MD assumes that the in-distribution labels \(y_{i}\in\{y_{1},\ldots,y_{N}\}\) are available. We denote the class index \(c\in\{1,\ldots,C\}\), with \(C\) being the number of \(\mathcal{D}_{\text{in}}\) classes and \(N_{c}\) the number of samples in class \(c\). For each class \(c\), we fit a Gaussian distribution to the representations \(z\)[49]. Specifically, we first compute the per-class mean \(\mu_{c}=\frac{1}{N_{c}}\sum_{i:y_{i}=c}z_{i}\) and a shared covariance matrix
\[\Sigma=\frac{1}{N}\sum_{c=1}^{C}\sum_{i:y_{i}=c}(z_{i}-\mu_{c})(z_{i}-\mu_{c}) ^{\top}. \tag{2}\]
The Mahalanobis score is then computed for each test sample as
\[\operatorname{MD}_{c}(z^{\prime}) =\big{(}z^{\prime}-\mu_{c}\big{)}\Sigma^{-1}\big{(}z^{\prime}-\mu _{c}\big{)}^{\top}, \tag{3}\] \[s_{\text{MD}}(x^{\prime}) =-\min_{c}\operatorname{MD}_{c}(z^{\prime})\,. \tag{4}\]
**Relative Mahalanobis distance (RMD).** Given the in-distribution mean \(\mu_{0}=\frac{1}{N}\sum_{i}^{N}z_{i}\), we additionally compute \(\Sigma_{0}=\frac{1}{N}\sum_{i}^{N}(z_{i}-\mu_{0})(z_{i}-\mu_{0})^{\top}\) to compute \(\operatorname{MD}_{0}\) analogously to Eq. (3). Subsequently, the RMD score [65] can be defined as
\[s_{\text{RMD}}(x^{\prime})=-\min_{c}\bigl{(}\operatorname{MD}_{c}(z^{\prime}) -\operatorname{MD}_{0}(z^{\prime})\bigr{)}\,. \tag{5}\]
### Considered pretrained models
Several supervised CNNs (ResNet50 [33], ConvNext [52]) and ViT [18] models trained on ImageNet and ImageNet-21K [68, 17] were utilized. Regarding self-supervised models, the masked autoencoder (MAE [30]), DINO [7], MoCov3 [10], MSN [2] were selected, because they all include the base ViT (ViT-B/16) for comparison between pretraining schemes (Fig. 3). All self-supervised models were trained on ImageNet [17, 68]. Finally, CLIP-based models were trained on different large-scale image-text datasets. In more detail, OpenAI-400M [62] and LIAION-400M consist of 400 million pairs. LIAION-2B consists of 2 billion image-text descriptions, which makes it the largest publicly available dataset to date. A table with information regarding the considered network architectures can be found in the appendix.
### Adversarial OOD data manipulation
For a test image \(x^{\prime}\in\mathcal{D}^{\text{test}}_{\text{out}}\) we randomly choose an in-distribution image \(x\in\mathcal{D}^{\text{train}}_{\text{in}}\) as the target. In contrast to existing adversarial approaches [3, 8], we create an adversarial perturbation \(\rho\) with the same dimensions as \(x^{\prime}\) that maximizes the cosine similarity between the in-distribution feature \(g(x)\) and \(g(x^{\prime}+\rho)\). We use the Adam optimizer to compute \(\rho\) by minimizing \(-\operatorname{sim}(g(x),g(x^{\prime}+\rho))\), starting with Gaussian noise \(\rho\sim\mathcal{N}(0,10^{-3})\) and clipping \(x^{\prime}+\rho\) to the pixel range \([0,1]\) after every update step, similar to [79]. We emphasize that we do not explicitly restrict the size of the perturbation directly and only limit the number of steps, as opposed to [79].
We experimentally observe that in the case of ViTs, the perturbations are quite visible along the edges of the transformer patches (Fig. 2). To create more natural appearing adversarial examples we enforce the smoothness of the perturbation by regularizing the allowed perturbation difference between neighboring pixels. We compute the finite difference image gradient \(\partial\rho/\partial h\) and \(\partial\rho/\partial w\) in the horizontal and vertical direction respectively. The image gradients have the same shape as the image, \(3\times H\times W\), and we define the regularization term as
\[\ell_{\text{smooth}}(\rho)=\frac{1}{3HW}\sum_{ijk}\left(\frac{\partial\rho}{ \partial h}\right)_{ijk}^{2}+\left(\frac{\partial\rho}{\partial w}\right)_{ ijk}^{2}, \tag{6}\]
where \(i,j,k\) run over image dimensions. We then minimize the loss
\[\ell_{\text{adv}}=-\operatorname{sim}(g(x),g(x^{\prime}+\rho))+\lambda\ell_{ \text{smooth}}(\rho), \tag{7}\]
with respect to the pertubation \(\rho\), where \(\lambda\) is a hyperparam
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{OOD detection method} & \(\mathcal{D}_{\text{in}}\) & \(\mathcal{D}_{\text{pretrain}}\) & Fine-tuned & \(\mathcal{D}_{\text{in}}\):CIFAR100 & \(\mathcal{D}_{\text{in}}\):CIFAR10 \\ & labels & & on \(\mathcal{D}_{\text{in}}\) & \(\mathcal{D}_{\text{out}}\):CIFAR10 & \(\mathcal{D}_{\text{out}}\):CIFAR100 \\ \hline RP [25] & ✗ & ✗ & ✗ & 50.1 & 81.2 \\ CSI (ens) [72] & ✗ & ✗ & ✗ & 78.4 & 92.1 \\ SSD [69] & ✗ & ✗ & ✗ & 78.2 & 93.1 \\ \hline
1-NN ViT-G/14 (ours) & ✗ & LAION-2B & ✗ & **87.6** & **96.3** \\ \hline \hline Baseline [36] & ✓ & ✗ & ✗ & 77.1 & 86.4 \\ RP [39] & ✓ & ✗ & ✗ & 74.7 & 90.9 \\ Winkens et al. [77] & ✓ & ✗ & ✗ & 78.3 & 92.9 \\ OpenHybrid [82] & ✓ & ✗ & ✗ & 85.6 & 95.1 \\ SSD+ [69] & ✓ & ✗ & ✗ & 84.1 & 94.1 \\ MTL [56] & ✓ & ✗ & ✗ & 91.6 & 94.1 \\ RMD BiT-R50 [65] & ✓ & ImageNet-21K & ✗ & 84.6 & 89.9 \\ RMD ViT-B [65] & ✓ & ImageNet-21K & ✓ & 93.1 & 98.8 \\ Fort et al. R50+ViT-B [23] & ✓ & ImageNet-21K & ✓ & 96.2 & 98.5 \\ \hline RMD ViT-G (ours) & ✓ & LAION-2B & ✗ & 96.3 & 98.8 \\ Probing + RMD ViT-G (ours) & ✓ & LAION-2B & ✗ & **97.4** & **99.0** \\ \hline \hline \end{tabular}
\end{table}
Table 1: **AUROC (%) comparison with state-of-the-art OOD detection methods**. Results of our approach without fine-tuning the pretrained CLIP ViT-G (\(\sim 1.84\) billion parameters) and linear probing is conducted on the pre-computed feature vectors for the in-distribution dataset. R50 indicates ResNet50 [33] and BiT indicates big image transfer [44]. We only report the best setups from [65, 23, 56, 72].
Figure 2: **Generating an adversarial example (top row) that is close enough to an in-distribution example (bottom left) to be not detectable as OOD**. **Top row:** the original OOD image from the CIFAR10 test set (_left_), the adversarial example without smoothing (_center_), the adversarial example with smoothing (_right_). **Bottom row:** The randomly sampled in-distribution target image from CIFAR100 (_left_), the Euclidean distance between the original image and perturbed image (_center_), the smoothened distance (_right_).
eter. During the evaluation, we remove the chosen target image, \(x\), from \(\mathcal{D}_{\text{in}}^{\text{train}}\) to show that the adversarial example, \(x^{\prime}+\rho\), cannot be detected as OOD from the remaining in-distribution examples. As a proof of concept, we create two adversarial OOD datasets1 for the CIFAR100 \(\rightarrow\) CIFAR10 benchmark, namely CIFAR10-A (\(\lambda=0\)) and CIFAR10-AS in its smoothened version (\(\lambda>0\)). The generation of an adversarial example is shown in Fig. 2. More adversarial examples can be found in the appendix.
Footnote 1: Available here
### Experimental evaluations
First, we benchmark \(32\) publicly available pretrained models on CIFAR10 \(\rightarrow\) CIFAR100 and vice versa (Fig. 1). Second, we compare against previous supervised and unsupervised state-of-the-art methods (Table 1), utilizing the CLIP ViT-G model from [11]. Third, we conduct further OOD detection evaluations with CLIP ViT-G, based on the availability of \(\mathcal{D}_{\text{in}}\) class names or labeled images (Table 2). More precisely, given a pretrained backbone model \(g(\cdot)\), we use the following evaluation setups for OOD examples:
1. **1-NN** finds the nearest neighbor among in-distribution examples using Eq. (1).
2. **MD and RMD** use the \(\mathcal{D}_{\text{in}}\) labels to compute the class centers in the feature space of \(g\) to compute MD (Eq. 4) or RMD (Eq. 5).
3. \(k\)**-means MD** computes the cluster-wise MD on the feature space of \(g\), using the \(k\)-means cluster centers. Following previous work [69], we set \(k=5\) (under-clustering).
4. **Pseudo-labels MSP** feeds the class names to the text encoder of CLIP-based models and computes their cosine similarities for each test image, and then takes the maximum softmax probability (MSP) [36].
5. **Pseudo-labels Probing** computes the MSP as in the previous setup and keeps the \(\mathcal{D}_{\text{in}}^{\text{train}}\) images with at least 90% probability. Then a linear head on the features of \(g\) is trained and evaluated using MSP or RMD.
6. **Few-shot \(p\)** randomly selects \(p=10\) images per class to train a linear head on the backbone features. MSP is used as the OOD score.
7. **Probing** trains a linear head on the features of the backbone \(g\), using \(\mathcal{D}_{\text{in}}^{\text{train}}\), and MSP or RMD is computed.
Finally, we study the performance consistency across multiple in- and out-distributions (Table 3), as well as the robustness against the proposed adversarially manipulated OOD samples.
### Implementation details
When evaluating the performance on various \(\mathcal{D}_{\text{out}}\) datasets (Table 3). We noticed that the model was able to distinguish the images solely from the different resolution levels. To avoid OOD detection from differences in image size, we resized OOD samples to \(\mathcal{D}_{\text{in}}\) image size [24]. For the text prompting, which is required for the text encoder of the CLIP models, we use the prompt: _an image of a_ {_label_}, similar to [62].
Regarding linear probing, a single linear layer was trained on the precomputed representations. We used the Adam optimizer [43] with a mini-batch size of 256 and a weight decay of \(10^{-2}\), for 20K steps in total. The learning rate follows a cosine schedule from \(10^{-3}\to 10^{-6}\). For few-shot probing, we take the average AUROC over 5 runs and only train for 10K steps. To create the adversarial dataset we perform 250 steps with the Adam optimizer with a learning
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline CLIP ViT-G/14 & \(\mathcal{D}_{\text{in}}\) & \(\mathcal{D}_{\text{in}}\) class & \(\mathcal{D}_{\text{in}}\):CIFAR100 & \(\mathcal{D}_{\text{in}}\):CIFAR100 & \(\mathcal{D}_{\text{in}}\):CIFAR10 & \(\mathcal{D}_{\text{in}}\):CIFAR10 \\ & labels & names & \(\mathcal{D}_{\text{out}}\):CIFAR10 & \(\mathcal{D}_{\text{out}}\):TinyIN & \(\mathcal{D}_{\text{out}}\):CIFAR100 & \(\mathcal{D}_{\text{out}}\):STL10 \\ \hline \(k\)-means MD (\(k=5\)) & ✗ & ✗ & 68.2 & 73.5 & 90.3 & 74.4 \\ \(1\)-NN & ✗ & ✗ & **87.6** & **85.2** & **98.2** & **81.1** \\ \hline Pseudo-labels MSP & ✗ & ✓ & 88.6 & 83.4 & 95.7 & 56.9 \\ Pseudo-labels Probing + MSP & ✗ & ✓ & 92.7 & 87.2 & 98.3 & 62.0 \\ Pseudo-labels Probing + RMD & ✗ & ✓ & **96.4** & **88.8** & **98.5** & **64.0** \\ \hline MD & ✓ & ✓ & 73.1 & 77.3 & 91.1 & **74.5** \\ RMD & ✓ & ✓ & 96.3 & 89.0 & 98.8 & 65.7 \\ Few-shot \(p=10\) & ✓ & ✓ & 91.1 & 86.9 & 97.3 & 62.3 \\ Probing + MSP & ✓ & ✓ & 95.0 & 88.0 & **99.0** & 66.4 \\ Probing + RMD & ✓ & ✓ & **97.4** & **89.5** & **99.0** & 67.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: **OOD detection AUROCs (%) for different evaluations and scores.** The considered ViT-G/14 model is trained on LAION-2B with the CLIP objective [11].
rate of \(10^{-3}\) on 1K images. We set \(\lambda\) to \(5\cdot 10^{3}\) when applying smoothing (Eq. 7).
All the experiments were carried out in a single NVIDIA A100 with 40GB VRAM. Crucially, only a maximum required VRAM of \(16\)GB was needed to compute the synthetic adversarial datasets with the ViT-G architecture. It is also worth noting that all our OOD detection evaluations can be conducted within minutes, in contrast to existing approaches [23, 65] that fine-tune the backbone.
## 4 Experimental results
We find a strong positive correlation between the \(\mathcal{D}_{\text{in}}\) classification accuracy and AUROC of 1-NN OOD detection across \(32\) models (Fig. 1). CLIP models exhibit the highest performance in both performance scores, independent of their network architecture. Larger CLIP models exhibit higher performance gains when trained on LAION-2B. Based on this, we select the largest model, CLIP ViT-G [11], for further evaluations.
We always report absolute gains and AUROC scores. A substantial gain of 9.2% AUROC (78.4\(\rightarrow\)87.6%) compared to previous unsupervised state-of-the-art (CSI) on CIFAR100 \(\rightarrow\) CIFAR10 is reported in Table 1, without any additional assumption about \(\mathcal{D}_{\text{in}}\). On CIFAR10 \(\rightarrow\) CIFAR100, a consistent improvement of 3.2% is obtained, where we achieved 96.3% AUROC. Using the RMD score (Eq. 5) on CLIP ViT-G, we match the supervised state-of-the-art OOD detection performance (\(\sim\)96.2% AUROC on CIFAR100 \(\rightarrow\) CIFAR10) without any in-distribution specific tuning. On the same benchmark, we report an improvement of 1.2% AUROC (96.2 \(\rightarrow\) 97.4% ) by training a linear layer on the precomputed image features. The gains are less significant on the supervised CIFAR10 \(\rightarrow\) CIFAR100 setup (99.0% versus 98.8% AUROC by [65]). The OOD detection performance for CIFAR10 \(\rightarrow\) CIFAR100 is close to optimal, suggesting that more challenging setups should be adopted in the future.
In Table 2, we conduct extensive experimental evaluations using CLIP ViT-G. Combining \(k\)-means with MD yields an inferior AUROC compared to \(1\)-NN, precisely lower by 11.4% on average. By incorporating the class names using CLIP's text encoder, we find that the pseudo-labels can be leveraged for probing, resulting in an improvement of 8.8% and 3.6% on CIFAR100 \(\rightarrow\) CIFAR10 and CIFAR100\(\rightarrow\)TinyIN. A counter-example is CIFAR10\(\rightarrow\)STL10 where there is an 80% class overlap. There, incorporating labels or class names deteriorates performance. Concerning linear probing, RMD consistently outperforms MSP when applied to the classifier's logits; for instance, 1.95% mean gain on CIFAR100 \(\rightarrow\) CIFAR10 and CIFAR100\(\rightarrow\)TinyIN.
Next, we compare the performance of our approach for different in- and out-distribution datasets using CLIP ViT-G (Table 3). Our proposed unsupervised method surpasses other unsupervised OOD detection methods by significant margins on all benchmarks. Overall, we find very few OOD settings where the performance is not close to the maximum, like CIFAR100\(\rightarrow\)TinyIN.
## 5 Discussion
Can CLIP ViTs still be fooled?The acquired state-of-the-art OOD detection performance along with the fact that CLIP ViTs are known to be robust zero-shot image classifiers against natural distribution shifts [62], raises the question of whether these models are also robust OOD detectors. To
\begin{table}
\begin{tabular}{c|c|c c c c c c} \hline \hline \multirow{2}{*}{\(D_{\text{train}}^{\text{in}}\)} & \multirow{2}{*}{\(D_{\text{test}}^{\text{out}}\)} & \multicolumn{8}{c}{OOD detection AUROC (\%) \(\uparrow\)} \\ \cline{3-8} & & Baseline [36] & RP [39] & SupSimCLR [42] & SSD+ [69] & CSI (ens) [72] & MTL [56] & Ours \\ \hline \multirow{4}{*}{\(D_{\text{out}}^{\text{out}}\)} & SVHN & 92.9 & 98.0 & 97.2 & 93.8 & 97.4 & 96.6 & **99.5** \\ & Texture & 87.7 & 96.3 & 94.2 & 94.1 & 97.2 & 96.9 & **99.4** \\ & Places365 & 88.4 & 92.6 & 91.1 & 91.8 & 93.1 & 98.7 & **98.9** \\ & TinyIN & 87.4 & 92.1 & 92.1 & 90.3 & 92.5 & 93.6 & **95.2** \\ & LSUN & 89.9 & 93.6 & 92.1 & 94.4 & 94.0 & 94.1 & **99.9** \\ \hline \hline \multirow{4}{*}{\(D_{\text{train}}^{\text{out}}\)} & SVHN & 79.2 & 83.6 & 81.6 & 83.6 & 87.4 & 90.6 & **95.0** \\ & Texture & 75.3 & 82.4 & 76.8 & 81.4 & 78.3 & 78.0 & **93.2** \\ & Places365 & 76.1 & 74.6 & 75.4 & 79.2 & 78.1 & 92.6 & **91.6** \\ & TinyIN & 78.5 & 77.6 & 80.8 & 76.3 & 82.4 & 79.3 & **85.2** \\ & LSUN & 73.7 & 71.9 & 73.5 & 63.8 & 75.2 & 74.0 & **94.0** \\ \hline \hline \multirow{4}{*}{\(D_{\text{train}}^{\text{out}}\)} & Flowers & 87.7 & 92.1 & 93.8 & 96.5 & 96.2 & 97.2 & **98.7** \\ & CUB-200 & 85.3 & 90.6 & 89.2 & 96.6 & 94.2 & 96.4 & **97.3** \\ \cline{1-1} & Dogs & 90.3 & 93.3 & 95.2 & 95.2 & 97.6 & 97.1 & **98.9** \\ \cline{1-1} & Food & 78.9 & 85.1 & 83.6 & 85.5 & 89.0 & 96.5 & **99.1** \\ \cline{1-1} & Texture & 87.0 & 92.2 & 98.7 & 94.9 & 98.5 & 94.0 & **99.2** \\ \hline \hline \end{tabular}
\end{table}
Table 3: **OOD detection AUROC (%) for various in- and out-distributions. We use the ViT-G/14 trained on LAION-2B and report the \(1\)-NN score (zero-shot). AUROC values of other methods are reported from MTL [56].**
answer this question, we examine CLIP ViT-G's robustness against the introduced adversarially manipulated OOD samples (Fig. 2). We found that it is possible to drop the AUROC from 86.2% to 48.6% by using CIFAR10-A. Introducing the smoothness restriction (CIFAR10-AS) degrades performance to 54.2% AUROC. Note that an AUROC score of 50% is the score of a random guess. The above findings show that even the top-performing CLIP ViTs trained on billion-scale image-text pairs can easily be fooled by weak manipulation of the input signal, which is invisible to humans.
Pretrainings and learned representations.To illustrate the impact of the pretraining dataset and objective on the learned features, we used the same architecture ViT-B/16 across \(8\) pretraining setups, as demonstrated in Fig. 3. We notice that the optimal choice of backbone depends on both \(\mathcal{D}_{\text{in}}\) and \(\mathcal{D}_{\text{out}}\). We show that this choice is also not symmetric: the best choice for CIFAR100 \(\rightarrow\) CIFAR10 is CLIP LAION-2B, but for CIFAR10 \(\rightarrow\) CIFAR100 it is DINO pretrained on ImageNet. Notably, both MAE pretrained on ImageNet and supervised ImageNet-21K pretrainings are consistently the worst backbone choices. DINO [7] even surpasses supervised ImageNet pretraining on CIFAR100 \(\rightarrow\) CIFAR10, while being the best choice on CIFAR10 \(\rightarrow\) CIFAR100, outlining the transferability of features of self-supervised methods, which is consistent with the results of [19].
The observed inferior performance of ImageNet-21K versus ImageNet suggests that supervised feature representations for OOD detection benefit from diverse and mutually exclusive class labels (ImageNet-21K class labels are not mutually exclusive) [67]. Consequently, we expect the more \(\mathcal{D}_{\text{in}}\) classes are shared with \(\mathcal{D}_{\text{pretrain}}\) and the higher the number of mutually exclusive classes for \(\mathcal{D}_{\text{pretrain}}\), the better the overall OOD detection performance.
We believe that natural language supervision affects the learned representations conceptually similar to supervised learning. In fact, we observe a mean performance gain of 4.9% by purely scaling up the dataset from 400M (OpenAI) to 2B (LAION) image-text pairs using ViT-B. The latter is likely attributed to the increased diversity of both language "labels" and images, as discussed in [22]. Nonetheless, the fact that CLIP ViT-G takes a "shortcut" based on the image resolution (Section 3.5) and is sensitive to adversarial attacks gives evidence that besides label-related features, local pixel information significantly affects the learned representations. We encourage future work to investigate this in greater depth.
Are we done with CIFAR datasets for OOD detection?Although we can identify challenging OOD detection scenarios, such as CIFAR100\(\rightarrow\)TinyIN, all benchmarks involving CIFAR10 and almost all involving CIFAR100 as in-distribution seem to saturate. Based on our results in Table 3, we believe that the OOD performance studies should include more challenging and diverse benchmarks. This will enable the design of robust and highly accurate OOD detectors.
## 6 Conclusion
In this work, a thorough experimental study was presented by leveraging pretrained models for visual OOD detection. It was demonstrated that CLIP ViTs are powerful zero-shot OOD detectors, without requiring labels or class names, outperforming all previous unsupervised approaches by large margins. Supervised state-of-the-art OOD detection performance was also reported without the need to fine-tune the feature extractors. The top-performing CLIP ViT-G [11] was further evaluated under several OOD settings. Based on the reported performance saturation on most existing benchmarks, the need for new and more challenging benchmarks
Figure 3: **AUROC values for zero-shot OOD detection using ViT-B/16 pretrained on different datasets (IN, IN-21K, OpenAI-400M, LAION-2B) and pretext tasks**. IN indicates ImageNet. The horizontal line indicates human-level performance, as reported in Fort et al. [23].
was highlighted. Finally, a novel adversarial OOD data manipulation method was introduced, which pointed to the fact that billion-scale feature extractors (CLIP ViT-G) are still sensitive to adversarial attacks.
| 私たちは、視覚のアウト・オブ・ディストリビューショントーク (OOD)検出のために、事前学習済みの特徴抽出タスクに焦点を当てた包括的な実験的研究を提示します。特に、対照的な言語-画像の事前学習済みモデル (CLIP) モデルを適応させます。トレーニングデータでの微調整なしに、私たちは、CLIPモデルにおけるイン・ディストリビューショントークの分類と無作為の OOD 検出との間の正の相関関係 ($R^2 \ge 0.92$) を確立することができました。 私たちは、$4$つのベンチマークで、CLIPモデルのOOD検出のロバスト性を向上させるために、新しい単純でスケーラブルな方法を提案します。これは、\textit{偽ラベル probing} (PLP) と呼ばれています。 PLP は、CLIPのテキストエンコーダから派生した偽ラベルを使用して |
2305.10615 | ML-SUPERB: Multilingual Speech Universal PERformance Benchmark | Speech processing Universal PERformance Benchmark (SUPERB) is a leaderboard
to benchmark the performance of Self-Supervised Learning (SSL) models on
various speech processing tasks. However, SUPERB largely considers English
speech in its evaluation. This paper presents multilingual SUPERB (ML-SUPERB),
covering 143 languages (ranging from high-resource to endangered), and
considering both automatic speech recognition and language identification.
Following the concept of SUPERB, ML-SUPERB utilizes frozen SSL features and
employs a simple framework for multilingual tasks by learning a shallow
downstream model. Similar to the SUPERB benchmark, we find speech SSL models
can significantly improve performance compared to FBANK features. Furthermore,
we find that multilingual models do not always perform better than their
monolingual counterparts. We will release ML-SUPERB as a challenge with
organized datasets and reproducible training scripts for future multilingual
representation research. | Jiatong Shi, Dan Berrebbi, William Chen, Ho-Lam Chung, En-Pei Hu, Wei Ping Huang, Xuankai Chang, Shang-Wen Li, Abdelrahman Mohamed, Hung-yi Lee, Shinji Watanabe | 2023-05-18T00:01:27 | http://arxiv.org/abs/2305.10615v2 | # ML-SUPERB: Multilingual Speech Universal PERformance Benchmark
###### Abstract
Speech processing Universal PERformance Benchmark (SUPERB) is a leaderboard to benchmark the performance of Self-Supervised Learning (SSL) models on various speech processing tasks. However, SUPERB largely considers English speech in its evaluation. This paper presents multilingual SUPERB (ML-SUPERB), covering 143 languages (ranging from high-resource to endangered), and considering both automatic speech recognition and language identification. Following the concept of SUPERB, ML-SUPERB utilizes frozen SSL features and employs a simple framework for multilingual tasks by learning a shallow downstream model. Similar to the SUPERB benchmark, we find speech SSL models can significantly improve performance compared to FBANK features. Furthermore, we find that multilingual models do not always perform better than their monolingual counterparts. We will release ML-SUPERB as a challenge with organized datasets and reproducible training scripts for future multilingual representation research.
Jiatong Shi\({}^{1}\), Dan Berrebbi\({}^{1}\), William Chen\({}^{1*}\), Ho-Lam Chung\({}^{2*}\), En-Pei Hu\({}^{2*}\), Wei Ping Huang\({}^{2*}\), Xuankai Chang\({}^{1}\), Shang-Wen Li\({}^{3}\), Abdelrahman Mohamed\({}^{4}\), Hung-yi Lee\({}^{2}\), Shinji Watanabe\({}^{1}\)+\({}^{1}\)Carnegie Mellon University \({}^{2}\)National Taiwan University \({}^{3}\)Meta AI \({}^{4}\)Rembrand {jiatongs, dberrebbi, wc4, swatanab}@cs.cmu.edu, [email protected], [email protected] [email protected]
Footnote †: Equal contribution, sorted in alphabetical order.
**Index Terms**: speech self-supervised learning, multilingual speech recognition, language identification
## 1 Introduction
Self-supervised learning (SSL) has been a popular method in the speech community. SSL models have shown promising results by capturing important speech features, such as phonemes and other acoustic units, through training on large amounts of unlabeled speech data [1]. These models have led to significant improvements in downstream tasks, such as speech recognition, speaker identification, and emotion recognition [2]. Over the past few years, researchers have proposed a variety of SSL models with different training objectives, operating under various data conditions, model architectures, and modalities [3, 4].
A major challenge in evaluating SSL models for speech is the difficulty of comparison since most models have been evaluated using different experimental setups. To address this issue, Yang et al. introduced the Speech processing Universal PERformance Benchmark (SUPERB) [2]. Recently, an extension of SUPERB called SUPERB-SG [5] has been introduced. SUPERB provides a comprehensive speech SSL benchmark including tasks such as recognition, detection, semantics, speaker identification, paralinguistics, and generation. With SUPERB, researchers can more easily compare the performance of different SSL models on various speech-related tasks, universally.
While SUPERB covers a wide range of speech tasks, it was designed primarily for English speech. However, there has been growing interest in applying SSL models to multilingual scenarios, such as training multilingual SSL models [6, 7, 8] or using SSL models in a cross-lingual manner [9, 10, 11, 12]. To support future research in these areas, we propose a new benchmark called multilingual SUPERB (ML-SUPERB).
ML-SUPERB is designed to cover a wide range of languages, including both high-resource languages like English and endangered languages such as Totonac. The benchmark primarily focuses on evaluating SSL models for automatic speech recognition (ASR) and language identification (LID). To accommodate different use cases for SSL models, ML-SUPERB includes two tracks with four different tasks: the monolingual track (monolingual ASR), and the multilingual task (multilingual ASR, LID), joint multilingual ASR/LID). Similar to SUPERB, ML-SUPERB employs frozen SSL models as feature extractors and a lightweight downstream model that can be fine-tuned for different tracks to achieve high training efficiency.
Several existing benchmarks also include multilingual SSL models [13, 14, 15]. Lebenchmark primarily evaluates speech tasks in French [13]; IndicSUPERB focuses mostly on Indian languages [14]. Xtreframe-S focuses on multilingual speech representation benchmarks, including ASR, speech translation, speech classification, and speech retrieval [15]. There are three main differences between Xtreframe-S and ML-SUPERB. Firstly, ML-SUPERB covers a wider range of languages, with 143 languages compared to Xtreframe-S's 102. Secondly, ML-SUPERB focuses on ASR and LID, while Xtreframe-S covers four different tasks. However, ML-SUPERB expands the tasks by evaluating them in four common multilingual research scenarios, while Xtreframe-S considers multilingual training only. Finally, ML-SUPERB is designed for efficiency, using smaller benchmark datasets and downstream models, and does not include fine-tuning. This lightweight setup allows us to conduct experiments for a dozen of popular speech SSL models, trained with various sizes and pre-training sets, and compare their performances across the proposed tracks. We expect ML-SUPERB would be a valuable complement to existing benchmarks.
## 2 Benchmark Details
### Data Collection
ML-SUPERB gathers data from a wide range of multilingual speech corpora, including Multilingual Librispeech [16], Commovioce [17], Voxforge [18], Voxpopuli [19], Google18n open-source project [20, 21, 22], Nordic Language Technology ASR corpora [23], Fleurs [24], NCHLT Speech [25], Spoken Wikipedia corpus [26], Mexican endangered languages [10, 27, 28], M-AILab multilingual corpora [29], Living Audio dataset [30], ALFFA corpus [31]. All corpora are with either Creative Commons, MIT, GNU, or Free-BSD licenses, which are available for both industrial and academic research, permissively.
For each language-corpus pair denoted as (lang, data),
three 10-minute subsets are randomly extracted for training, development, and testing, along with an additional 1-hour training set that includes the 10-minute training set.1 The reasons for using a small 10-minute/1-hour training set: (1) _Challenging design_: using a large training data size could lead to high performance easily and may result in a saturated benchmark in evaluation metrics [3, 4]. Therefore, using a smaller training set size presents a more challenging design for the SSL models, which can help evaluate their robustness and generalization capability. (2) _Reasonable performance_: previous speech SSL works have frequently adopted 10-minute and 1-hour training sizes. Even in such extreme cases, the performances with SSL are generally reasonable [3, 4], indicating that this setting could be a feasible solution to the benchmark as well. (3) _Training efficiency_: with 143 languages coverage, limiting the training size is important to keep the experiments within reasonable computational efforts. Using a smaller training set size can help reduce the computational cost and make the training process more efficient. A full evaluation cycle of ML-SUPERB can take up to 3 days using 4 2080Ti GPUs.
Footnote 1: We used the original split for source datasets, with the exception of SWC, M-AILABS, LAD, and ALFFA. Therefore, all datasets except these four can be used for SSL pre-training.
Additionally, the benchmark includes few-shot cases with 20 languages and uses only 5 utterances in training for each language. These reserved few-shot training sets are not used in the monolingual ASR track. A detailed summary of the dataset is shown in Table 1.
### Monolingual Track
The literature suggests that speech SSL models are commonly fine-tuned on monolingual corpora [9, 10, 11]. In ML-SUPERB, we introduce a dedicated track for monolingual ASR to facilitate this approach. We select nine languages based on geographical and linguistic considerations to balance language and domain coverage with manageable experimental mass.
In total, we introduce 14 monolingual_exp. For a monolingual_exp in language lang we select one dataset of this language and use it for training the model and for validation2. For evaluation of a monolingual_exp, we use all the datasets of lang to test the trained model on various accent or domain conditions. We select one pair (lang, data) for training for lang{rus, swa, swe, jpn, cmm, xty}. For lang{eng, fra, deu} we select respectively 3, 2 and 2 pairs (lang, data) in order to evaluate the impact of the training domain on the models' performances. For instance, for eng we have 3 monolingual_exp, with (eng, MLS), (eng, NCHLT) and (eng,VoxPopuli).
Footnote 2: Each monolingual_exp is made of one experiment with the 10-minute set for training and one with the 1-hour set.
### Multilingual Track
**Multilingual ASR task**: in the multilingual ASR task, we use the training set where combining text transcriptions from all 143 languages. The multilingual ASR task has two sub-tasks on the 10-minute train set and the 1-hour train set. For both training sets, we reserve 20 languages for few-shot learning scenarios as discussed in Sec. 2.1. In this track, the model is expected to directly predict the correct orthography in the target language.
**LID task**: LID track focuses on language identification with the same training set of 143 languages in 10 minutes and 1 hour. However, we do not consider evaluation for languages with few-shot settings, given that the identification of those languages is very challenging due to the label biasing.
**Joint Multilingual ASR/LID task**: A widely used technique in previous literature involves adding the language ID to the start of the speech transcript to facilitate joint training of multilingual ASR and LID models [32, 33, 34, 35]. Joint training can improve performance in certain scenarios, and it can also enhance model interpretability by separating language identification errors. Therefore, we have included this task in our multilingual track. The task's design is the same as the multilingual ASR task for ASR and the LID task for language identification.
### Framework and Benchmark Settings
**Toolkits**: We utilize the S3PRL toolkit [2] for upstream models, which offers a wide range of speech SSL model architectures and APIs that support customized SSL models from Huggingface [36] and user-defined models. For task-specific downstream training, we use ESPnet [37]. We plan to publish ML-SUPERB as an all-in-one recipe in ESPnet's egsz recipe collection, encompassing data preprocessing, training, inference, and evaluation3.
Footnote 3: [https://github.com/espnet/espnet/tree/master/egs2/ml_superb/asr1](https://github.com/espnet/espnet/tree/master/egs2/ml_superb/asr1)
**Downstream model and training details**: Our downstream model design is based on the SUPERB concept. First, we compute a weighted summation of frozen speech SSL representations using learnable weights. Next, we apply a convolutional downsample layer that reduces the sequence of speech SSL features by half, passing the resulting hidden states to a transformer model consisting of two layers with an attention dimension of 256, a feedforward layer dimension of 1024, and 8 attention heads. A dropout rate of 0.1 is employed, and the model is trained using the connectionist temporal Cessification loss. We use the Adam optimizer with a learning rate of 0.0001 and 1e-6 weight decay. Specaugment is applied to the representation (i.e., the weighted sum of speech SSL representation) following the SUPERB benchmark. The batch size is set to 8 with the gradient accumulation as 4. The same configuration is used for all tasks in both the monolingual and multilingual tracks.
The number of iterations in training is the only difference across tasks. In the monolingual track, due to the small training size, we set it to 15,000. In the multilingual track, we use 300,000 iterations for the 10-minute train set and 600,000 for the 1-hour train set.
**Evaluation metric**: In the monolingual track, the phoneme error rate is used for jpn and cmn, while Character Error Rate (CER) is used for the remaining languages. In the multilingual track, we use CER for ASR evaluation and accuracy rate for LID evaluation, reporting results separately for the normal training set and the few-shot training set.
\begin{table}
\begin{tabular}{l|c|c c} \hline \hline Dataset & Hours & Normal Langs (123) & Few-shot Langs (20) \\ \hline
10-minute & 37.43 & \(\sim\)10min \(\times\) 240 (lang, data) & 5 utt. \(\times\) 20 lang \\
1-hour & 222.46 & \(\sim\)1h \(\times\) 240 (lang, data) & 5 utt. \(\times\) 20 lang \\ Dev.t & 41.82 & \(\sim\)10min \(\times\) 240 (lang, data) & \(\sim\)10min \(\times\) 31 (lang, data) \\ Test & 44.97 & \(\sim\)10min \(\times\) 240 (lang, data) & \(\sim\)10min \(\times\) 31 (lang, data) \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Statistics of the data used for training, development, and testing in ML-SUPERB. Detailed discussed in Sec. 2.1._
For overall performance, we use the SUPERB\({}_{s}\) metric from the SUPERB benchmark [38]. We denote \(s_{t,i}(u)\) as the \(i^{\text{th}}\) metrics for task \(t\) and SSL model \(u\). \(T\) is the set of four tasks and \(I_{t}\) is the set of metrics for the task \(t\). SUPERB\({}_{s}\) aggregates all task-specific scores \(s_{t}(u)\) with respect to baseline (i.e., FBANK) and state-of-the-art (SOTA) model4 on the task \(t\). The SUPERB\({}_{s}\) is defined as:
Footnote 4: The SOTA models for each setting are discussed in Sec. 3.2.
\[\text{SUPERB}_{s}(u)=\tfrac{1000}{|T|}\sum\nolimits_{t}^{T}\frac{1}{|I_{t}|} \sum\nolimits_{i}^{I_{t}}\frac{s_{t,i}(u)-s_{t,i}(\text{FBANK})}{s_{t,i}(\text {SOTA})-s_{t,i}(\text{FBANK})} \tag{1}\]
We expect SUPERB\({}_{s}\) can provide a comprehensive view of the model performance on the benchmark and take the difficulty of tasks into consideration.
**Analysis support**: To facilitate a more comprehensive analysis of the benchmark, we provide various analysis tools. For the multilingual ASR evaluation, we present the character error rate (CER) for each language as well as aggregated scores for different language groups, in addition to the average CER for both normal and few-shot cases. In line with previous studies [39, 40], we also offer visualizations of the learnable layer weights and their learning curve during training.
## 3 Experiments
### Candidate models
ML-SUPERB welcomes all speech SSL models trained on either monolingual or multilingual data. We believe the analysis of multilingual scenarios for monolingual speech SSLs is also valuable according to previous works [9, 10, 11]. In this paper, we show the experimental results of some example model candidates as shown in Table 2.
**wav2vec2**: wav2vec2 is a popular speech SSL model for speech recognition [3]. Its pre-training uses a contrastive learning approach that prioritizes identifying true quantized latent speech representations over masked time steps from distractors. The wav2vec2 model has also been extended to many other versions for specialized use cases. For example, robust-wav2vec2-large [41] considers the diversity of speech types, such as read speech, conversational speech, and noisy speech, by including additional corpora in the pre-training stage. Wav2vec2-base-23 and wav2vec2-large-23 are pre-trained on Voxopopuli [19], with a focus on European languages. Additionally, XLSR scales up the multilingual training in wav2vec2 by incorporating more languages and data [6, 7].
**HuBERT**: HuBERT uses an iterative offline clustering step to generate pseudo labels for each frame. During training, it predicts the pseudo labels of the masked frame, which helps to improve the quality of the learned features. Similar to wav2vec2, HuBERT also has different versions, such as a multilingual Hubert [43] trained in three European languages (fra, spa, eng) and HuBERT trained on Mandarin [42].
### Experimental Results
The experimental results are shown in Table 3 for 10-minute set and Table 4 for 1-hour set.
**Monolingual ASR**: In the monolingual ASR task, all speech SSL models outperform the FBANK baseline. XLSR-128 achieves the best performance in the 1-hour set, while HuBERT-large obtains the best performance in the 10-minute set. Several findings are noteworthy: (1) HuBERT-based models outperform wav2vec2-based models when the training data and model size are similar. (2) Large models usually obtain better results than their base versions. (3) While the XLSR series of models deliver impressive performances in the 1-hour set, we have observed their instability in the 10-minute set, particularly on Asian languages such as cmn.
**Multilingual ASR**: In the multilingual ASR task, all models trained using self-supervised learning (SSL) techniques have shown superior performance compared to the baseline model using FBANK features. Among the SSL models, XLSR-128 achieves the best results across all conditions. Our experiments also reveal some interesting findings: (1) Models trained with more languages generally outperform those trained on monolingual datasets, although this may not always be the case. For example, mHuBERT-base performs worse than HuBERT-based models trained on English only. (2) Large models trained on monolingual data do not necessarily have better representations for multilingual scenarios. For instance, HuBERT-large performs worse than HuBERT-base, and wav2vec2-large is less effective than wav2vec2-base. One possible explanation for the lack of performance improvement with larger models is their limited ability to generalize, despite having similar training losses as base models. (3) The robust-wav2vec2-large model achieves decent scores on multilingual ASR, suggesting that our benchmark corpus may need to consider different acoustic environments, as it includes multiple source datasets.
**LID**: In the LID task, we notice similarities with multilingual ASR, but there are also notable differences. (1) XLSR-128 has been the dominant model for both 10-minute and 1-hour datasets. (2) While most SSL models have improvements over FBANK, some do not, particularly those based on wav2vec2 (e.g., wav2vec2-large-23 for the 1-minute set and wav2vec2-large for the 1-hour set). (3) Larger models with more parameters and pre-trained data do not necessarily lead to better performance compared to base models.
**Joint Multilingual ASR + LID**: In the joint multilingual ASR+LID task, the results generally align with the other two tasks in the multilingual track. (1) SSL models outperform FBANK on ASR, but some models perform worse on LID. (2) Base models exhibit better generalization ability and often perform better on test sets. (3) There is no single best model that dominates the task, particularly in few-shot cases and LID tasks.
**Overall**: In terms of overall performance as measured by SUPERB\({}_{s}\) in Sec. 2.4, XLSR-128 is the best model for both the 10-minute and 1-hour sets. Major findings include: (1) multilingual training with a broad coverage of languages, as seen in XLSR models that include more than 50 languages, has proven to be useful. However, multilingual training that is limited to a few selective languages may not be as beneficial in larger language groups (e.g., wav2vec2-large-23 and mHUBERT models
\begin{table}
\begin{tabular}{l|c|c c} \hline \hline Model & Params (M) & \multicolumn{2}{c}{Pre-Training \# Hours} & \multicolumn{1}{c}{\# Langs} \\ \hline wav2vec2-base [3] & 95 & 1k & 1 \\ wav2vec2-large [3] & 317 & 60k & 1 \\ robust-wav2vec2-large [41] & 317 & 65k & 1 \\ wav2vec2-base-23 [19] & 95 & 100k & 23 \\ wav2vec2-large-23 [19] & 317 & 100k & 23 \\ XLSR-53 [7] & 317 & 56k & 53 \\ XLSR-128 [6] & 317 & 400k & 128 \\ \hline HuBERT-base [4] & 95 & 1k & 1 \\ HuBERT-large [4] & 317 & 60k & 1 \\ HuBERT-base-cmn [42] & 95 & 10k & 1 \\ HuBERT-large-cmn [42] & 317 & 10k & 1 \\ mHuBERT-base [43] & 95 & 14k & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Description of the candidate models.
do not always perform better than their models trained in a single language). (2) The base models tend to generalize better to multilingual cases than their corresponding large versions, such as wav2vec2-base versus wav2vec2-large and HuBERT-base versus HuBERT-large.
### Layerwise analysis
Our benchmark offers tools to guide users in the use of SSL representations according to their needs, including an analysis of the learned weights for layer importance. The results for the XLSR-128 model in monolingual ASR tasks (shown in Fig 1) confirm the conclusions reached by [44] and [45]: the most relevant layers for ASR are not the last few layers. We also observed that English3, French2, and German2 have very similar behavior. These tasks use VoxPopuli data for training, which is the only dataset with lecture speech in our collection. Additionally, Mixtec is the only conversational speech data among our sets, and we can see a distinct behavior in Fig 1. Therefore, the relevance of SSL model layers may be related to the speech domain (in addition to the speech task) rather than the language.
## 4 Conclusion
This paper introduces ML-SUPERB, a benchmark that extends SUPERB to multilingual tasks. We present the design of the open-source framework and discuss experimental results for some example models. More detailed policies can be found at [https://multilingual.superbbenchmark.org/](https://multilingual.superbbenchmark.org/). We invite the community to participate in this challenge.
| Speech処理の普遍的なパフォーマンスベンチマーク（SUPERB）は、自己教師あり学習（SSL）モデルのSpeech処理タスクの性能を評価する基準となるランキングです。しかし、SUPERBは評価において主に英語のSpeechを重視しています。この論文では、多言語対応SUPERB（ML-SUPERB）を発表し、143言語（高リソースから絶滅危惧種など）をカバーし、自動音声認識と言語識別を考慮しています。SUPERBの概念に基づいて、ML-SUPERBは凍結されたSSL特徴を使用し、多言語タスクのための単純なフレームワークを用いて、浅いダウンストリームモデルを学習します。類似してSUPERBベンチマーク、 we find speech SSLモデルはFBANKの特徴と比較して、性能が大きく向上します。さらに、私たちは、多言語モデルが常に単言語モデルよりも優performingとは限りません。ML-SUPERBを将来の多言語表現 |
2308.16298 | Publishing Wikipedia usage data with strong privacy guarantees | For almost 20 years, the Wikimedia Foundation has been publishing statistics
about how many people visited each Wikipedia page on each day. This data helps
Wikipedia editors determine where to focus their efforts to improve the online
encyclopedia, and enables academic research. In June 2023, the Wikimedia
Foundation, helped by Tumult Labs, addressed a long-standing request from
Wikipedia editors and academic researchers: it started publishing these
statistics with finer granularity, including the country of origin in the daily
counts of page views. This new data publication uses differential privacy to
provide robust guarantees to people browsing or editing Wikipedia. This paper
describes this data publication: its goals, the process followed from its
inception to its deployment, the algorithms used to produce the data, and the
outcomes of the data release. | Temilola Adeleye, Skye Berghel, Damien Desfontaines, Michael Hay, Isaac Johnson, Cléo Lemoisson, Ashwin Machanavajjhala, Tom Magerlein, Gabriele Modena, David Pujol, Daniel Simmons-Marengo, Hal Triedman | 2023-08-30T19:58:56 | http://arxiv.org/abs/2308.16298v2 | # Publishing Wikipedia usage data with strong privacy guarantees
###### Abstract
For almost 20 years, the Wikimedia Foundation has been publishing statistics about how many people visited each Wikipedia page on each day. This data helps Wikipedia editors determine where to focus their efforts to improve the online encyclopedia, and enables academic research. In June 2023, the Wikimedia Foundation, helped by Tumult Labs, addressed a long-standing request from Wikipedia editors and academic researchers: it started publishing these statistics with finer granularity, including the country of origin in the daily counts of page views. This new data publication uses differential privacy to provide robust guarantees to people browsing or editing Wikipedia. This paper describes this data publication: its goals, the process followed from its inception to its deployment, the algorithms used to produce the data, and the outcomes of the data release.
## 1 Introduction
Wikipedia and other projects supported by the Wikimedia Foundation are among the most used online resources in the world, garnering hundreds of billions of visits each year from around the world. As such, the Foundation has access to terabytes of data about visits to a page on a Wikimedia project. This is called _pageview_ data in this document.
The Foundation has been publishing statistics about this data for almost 20 years, through the _Pageview API_[17]. This data helps Wikipedia editors measure the impact of their work, and focus their efforts where they are most needed. Pageview data is also a rich resource for academic research: it has been used to better understand many topics, ranging from user behavior [14] and browsing patterns [15] to information dissemination [1], epidemiology [19], online harassment [33], and others. Over time, the Wikimedia Foundation received a number of requests to make these statistics more granular, and publish pageview counts _by country_, to make it even more useful to Wikipedia editors, and enable further academic research.
Addressing such requests for more granular data is aligned with the Foundation's open access policy [12], which seeks to provide as much transparency as possible about how Wikimedia projects operate. However, the Foundation also considers privacy to be a key component of the free knowledge movement: there cannot be creation or consumption of free knowledge without a strong guarantee of privacy. These guarantees are expressed by the Foundation's strict privacy policy [13] and data retention guidelines [6], which govern how the infrastructure underlying Wikipedia works. Concretely, people browsing Wikipedia may expect their behavior on the website to stay private: is is crucial to prevent motivated actors to combine this data with outside other data sources in order to spy on or presence Wikipedia users for their view history, edit history, or other behavior. It is well-known that simply aggregating data is not, on its own, enough to prevent re-identification risk [34, 23, 22, 25], so publishing data with a finer geographic granularity warrants an approach with rock-solid privacy guarantees for Wikipedia users and editors.
Differential privacy [27] (DP) provides a way of easing this tension: it allows organizations to both lower and more fully understand the risks of releasing data. Therefore, the Wikimedia Foundation decided
to investigate the use of differential privacy to release daily pageview data, sliced by country. After an in-depth comparison of available open-source tools [5], the Wikimedia Foundation decided to use Tumult Analytics [30, 18] and started a collaboration with Tumult Labs to design and deploy a DP pipeline for this data release. The pipeline is now deployed, and the published data provides useful insights to anyone interested in better understanding Wikipedia usage.
This document describes this data release in more detail.
* In Section 2, we present the high-level workflow that we followed towards the deployment of a differentially private data release.
* In Section 3, we outline the problem statement and the success metrics for this data release.
* In Section 4, we describe the technical algorithms used for this data release.
* In Section 5, we summarize the results of this deployment.
## 2 High-level workflow for differential privacy deployments
The process to launch a DP data product follows a standard workflow, with three main stages: _Build_, _Tune_, and _Deploy_. The entire process is outlined in Figure 1; its three main stages are as follows.
1. In the initial Build stage, the goal is to gain a good understanding of the problem and its requirements, and implement a first-cut algorithm. There are two steps in this initial stage. First, we properly define the problem, and determine what success looks like for this project. This involves talking to stakeholders to understand what the data will be used for, and what accuracy metrics capture downstream use cases well. Second, we build a prototype mechanism. This is a first rough attempt at solving the data release problem, and it exposes the "levers" inherent to the project. Which choices did we have to make while building the prototype? Which of these choices can later be modified to pick different trade-offs between utility or privacy metrics?
Figure 1: A standardized workflow for differentially private data releases.
2. Then, in the Tune step, we use these levers to experiment with different settings and optimize the algorithm. Using the success metrics defined in the previous step, we iteratively evaluate and adjust the algorithm, making changes until it produces data that is fit for use and satisfies the privacy requirements.
3. Finally, in the Deploy stage, we finalize the algorithm, obtain the necessary approvals to publish the data, write documentation about the data publication mechanism for future data users and pipeline maintainers, and deploy it in production.
In Section 3, we outline the output of the very first step: the definition of problem statement and its success metrics. Then, in Section 4, we will describe the output of the Tune stage: what does the final algorithm look like, after the multiple rounds of iteration on the initial prototype.
## 3 Problem statement and success metrics
In this Section, we describe the desired output data (Section 3.1), the schema and characteristics of the input data (Section 3.2), the privacy goals of this data release (Section 3.3), and the accuracy metrics used to quantify success (Section 3.4).
### Desired output data
The pre-existing Pageview API publishes data about the number of times each Wikimedia page was visited during a given day. Each page is identified by two fields:
* its _project_, e.g. fr.wikipedia (the French-language version of Wikipedia), zh.wikibooks (the Chinese version of Wikibooks, an open-content textbook collection), wikidata (a central storage for structured data), etc.;
* its _page ID_, a numeric identifier uniquely identifying each page within a project.
Table 1 is a fictitious sample of the kind of data available via the Pageview API. For example, the first line indicates that there were 4217 visits to the page with ID 23110294 on the English version of Wikipedia on April 2nd, 2023.
The goal of this project is to publish more granular data, and also release daily pageview counts _per country_. A fictitious sample of the desired output data appears in Table 2. For example, the first line indicates that 92 of the previously-mentioned visits originated from Switzerland.
\begin{table}
\begin{tabular}{l|l|l|l} Project & Page ID & Date & Count \\ \hline en.wikipedia & 23110294 & 2023-04-02 & 4217 \\ fr.wikipedia & 28278 & 2023-04-02 & 710 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \end{tabular}
\end{table}
Table 1: A fictitious sample from the data made publicly available via the Pageview API.
\begin{table}
\begin{tabular}{l|l|l|l|l} Project & Page ID & Date & Country & Count \\ \hline en.wikipedia & 23110294 & 2023-04-02 & CH & 92 \\ fr.wikipedia & 28278 & 2023-04-02 & FR & 101 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \\ \end{tabular}
\end{table}
Table 2: A fictitious sample from the data that we would like to publish as part of this project.
### Input data
This project uses two input datasets: the _current pageviews dataset_, and the _historical pageviews dataset_.
Current pageviews datasetAs users visit the site, their individual pageviews are recorded and stored in the current pageviews dataset. This dataset contains all pageviews across all Wikimedia projects for the last 90 days. Because of the Wikimedia Foundation's commitment to minimal data retention, this data is only kept in this form for 90 days. Table 3 is a fictitious sample of the current pageviews dataset, showing only the columns of interest for this project: project, page ID, date and time, and country.
Note that in contrast to similar logging infrastructure for most websites, this data does not contain a persistent user identifier. Most visits to Wikimedia projects come from logged-out users, and the Wikimedia Foundation intentionally did not implement a user tracking mechanism which would provide a cookie ID and allow the Foundation's systems to recognize whether two records came from the same user. This practice is good for data minimization, but it makes it more difficult to obtain user-level differential privacy guarantees, which requires bounding the number of contributions coming from the same user. We come back to this challenge in Section 4.1.1.
Historical pageviews datasetPast the initial 90-day retention period, pageviews are aggregated as hourly totals, broken down by project, page id, country, and a number of user characteristics. These aggregates are then stored in the historical pageviews dataset. Table 4 is a fictitious sample of the historical pageviews dataset, again showing only the columns of interest.
This pre-aggregated data also poses a challenge for performing DP calculations: it is not possible to determine which contributions came from which users, and therefore to bound the number contributions coming from each user.
### Privacy goal
When using differential privacy, one has to decide what to protect in the data; or, equivalently, what the definition of the neighboring databases should be. For long-running pipelines that publish data regularly over an unbounded time period, there are two aspects to this choice: what are the intervals of time considered as part of the unit of privacy, and what are we protecting in each of these intervals. Then, a follow-up question is the choice of privacy parameters: the numeric value of \(\varepsilon\) and \(\delta\).
Our goal is to publish data daily: it is natural to use a daily time period in the unit of privacy. This interval is consistent with almost all other long-running DP deployments, like Apple's telemetry collection, or Google's and Meta's data releases related to the COVID-19 crisis. Other releases use a shorter period, like Microsoft's telemetry in Windows. There is no overlap between days: the privacy parameters for each user-day are fixed and do not increase over time.
\begin{table}
\begin{tabular}{c|c|c|c|c} Project & Page ID & Date and Time & Country & Count \\ \hline en.wikipedia & 23110294 & 2023-04-02 10:00 & CH & 11 \\ fr.wikipedia & 28278 & 2023-04-02 18:00 & FR & 15 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \end{tabular}
\end{table}
Table 4: A fictitious sample of the columns of interest from the pre-aggregated historical pageviews dataset.
\begin{table}
\begin{tabular}{c|c|c|c} Project & Page ID & Date and Time & Country \\ \hline en.wikipedia & 23110294 & 2023-04-02 10:32:45 & CH \\ fr.wikipedia & 28278 & 2023-04-02 18:53:11 & FR \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \end{tabular}
\end{table}
Table 3: A fictitious sample of the columns of interest from the current pageviews dataset.
This choice of unit of privacy means that if a user were to regularly visit the same page from the same country across multiple devices (or clearing their cookies between each page visit) over a long period of time, this behavior could potentially be observed in the output data. Another caveat is that this data release surfaces group-level trends, like minority language group activity on Wikimedia projects within a country. These insights can be helpful (e.g. allow for dedicated support to that minority language group) but could also carry risks (e.g. by causing government persecution of this minority group). We mitigate these risks by choosing conservative privacy parameters, which translate to a reasonable level of protection over longer time periods, by holding off on releasing data for certain countries, and by only releasing aggregates that are above a certain threshold.
Protecting each individual Wikipedia user during each day is impossible to achieve entirely without a way to link people's identities across records and devices. Because the Wikimedia Foundation does not have nor want the capability to link records in such a way, we instead attempt to protect the contribution of each _device_ during each day. For the data based on the current pageviews dataset, we achieve this goal using _client-side contribution bounding_, as described in Section 4.1.1. For the data based on the historical pageviews dataset, we cannot bound user contributions. Instead, we choose to protect a fixed number of daily pageviews, denoted by \(m\). This provides an equivalent level of protection to users who contribute fewer than \(m\) pageviews per day. Users who contribute more than \(m\) pageviews will incur a larger privacy loss, proportional to the amount by which their contributions exceed \(m\). This number is set to 300 for data prior to February 8th, 2017, and to 30 for data between February 9th, 2017 to February 5th, 2023. Table 5 summarizes the privacy units chosen for this project.
This difference in how many contributions we protect is due the fact that in February 2017, a change occurred to the way the input data was generated. Prior to February 8th, 2017, users who were editing a Wikimedia page and used the Web UI to preview their changes were recorded as one pageview each time the preview refreshed. This meant that during a lengthy editing session, an editor could plausibly rack up many pageviews on the same page. When combined with our inability to limit user contributions, this created a markedly different risk level before/after this date, that our historical pageviews algorithm had to address. Starting on February 9th, 2017, previews were no longer recorded as pageviews.
For privacy parameters, we use zero-concentrated DP [20] with \(\rho=0.015\)1 for the more recent data, and pure DP with \(\varepsilon=1\) for the historical data. These values are generally considered to be conservative among differential privacy researchers and practitioners [31], and are lower than most practical DP deployments [24].
Footnote 1: Which is a strictly stronger guarantee than \((\varepsilon,\delta)\)-DP [26] with \(\varepsilon=1\) and \(\delta=10^{-7}\).
### Accuracy metrics
We measure utility along three dimensions: the _relative error distribution_, the _drop rate_, and the _spurious rate_. Each of these metrics is computed using the _true data_ as a baseline: the data that corresponds to simply running a group-by query (either counting the number of rows, for the current pageviews dataset, or summing the counts, for the historical pageviews dataset), without any contribution bounding, noise addition, nor suppression.
Relative error distributionWe are releasing pageview counts, and the DP process will inject statistical noise into these counts. Thus, it is natural to want to measure how much noise is added to these counts. We measure accuracy according to _relative error_: the relative error of each noisy count \(\hat{c}\) is \(|\hat{c}/c|\), where \(c\) is
\begin{table}
\begin{tabular}{c|c} Time period of the input data & Unit of privacy \\ \hline July 1st, 2015 – February 8th, 2017 & 300 daily pageviews \\ February 9th, 2017 – February 5th, 2023 & 30 daily pageviews \\ February 6th, 2017 – present & one user-day \\ \end{tabular}
\end{table}
Table 5: A summary of the privacy units used in this project.
the true count. Of course, we are releasing many counts, so we need to look at the _distribution_ of relative error. More specifically, we look at the percentage of released counts having a relative error smaller than 10%, 25%, and 50%.
Drop rateThe DP algorithm uses _suppression_: if a noisy count is lower than a given threshold, we remove it from the output data. To quantify the data loss due to this suppression step, it is natural to compute the _drop rate_: the percentage of counts that do not appear in the output, even though they were non-zero in the true data. In the true data, however, many of the counts are very low; suppressing such counts is not as bad as suppressing a popular page. Therefore, we compute the percentage of pages that were suppressed among pages whose true counts is larger than a fixed threshold \(t\) (the _drop rate above \(t\)_), as well as the percentage of pages that were suppressed among the top 1000 rows in the true data (the _top-1000 drop rate_).
Spurious rateMany page, project, and country combinations receive zero pageviews on any particular day. When noise is added to these zero counts, it is likely that they will end up with positive (though comparatively small) counts. We refer to these as _spurious_ counts. Spurious counts can mislead data users by wrongly indicating that some combinations had activity. They also increase the size of the output dataset, which can pose a usability challenge. Therefore, we compute an additional metric: the _spurious rate_, which captures the ratio of spurious counts among all counts that appear in the output.
## 4 Technical description of the algorithms
In this Section, we describe the algorithms used to generate the differentially private data. For simplicity, we refer to a \(<\)page ID, project\(>\) pair as a _page_.
### Current pageviews
For the data using the current pageviews dataset, we want to provide privacy guarantees that protect each user during each day. This requires bounding the maximum number of pageviews that each user can contribute during a single day. The typical way to perform such contribution bounding is to use a user identifier to sub-sample the number of contributions from each user, taking the first \(k\) records [29], or using reservoir sampling [32]. However, without a user identifier, we had to use a novel and alternative approach to this problem: _client-side filtering_.
#### 4.1.1 Client-side filtering
Without user IDs, the server cannot know whether multiple contributions come from the same user, and perform contribution bounding to get user-level privacy guarantees. Instead, we add some logic to the client side. Each end-user device counts their number of contributions logged in each day, and sends each contribution along with a boolean flag, indicating whether this contribution should be used in the server-side DP computation. The criteria used for inclusion in the input to the DP algorithm is as follows: each day, we include the first 10 _unique_ pageviews. This means that if a user visits the same page multiple times in a day, only the first visit will be counted. This also means that if a user visits more than 10 distinct pages in a day, all pageviews after the 10th visits will not be included.
Pseudocode for this client-side filtering step can be found in Algorithm 1. Note that this algorithm does not keep track of the raw page IDs in the client-side cookie. Instead, it uses a salted hash function [16] to remember which page IDs were already visited. This provides an additional level of protection against an attacker that would obtain access to this cookie.
Client-side filtering upholds the Wikimedia Foundation's data minimization principle: only the absolute minimal information needed to perform the contribution bounding -- a boolean value associated with each pageview to indicate whether it should be counted -- is added to the logging infrastructure. Alternatives
such as using identifiers or a counter that increments for each contribution would have required sending more data to the server, and increase fingerprinting risk.
#### 4.1.2 Server-side algorithm
Once each pageview was annotated by the client-side filtering algorithm, it is used as input in a server-side differentially private algorithm. This algorithm, run daily on the data from the previous day, has three stages.
1. First, we collect the list of \(<\)page, country\(>\) tuples to aggregate over.
2. Second, we count the number of pageviews in each group, and we add noise to each count.
3. Finally, we suppress low counts, and publish the data.
The list of all possible tuples is, in theory, known in advance: the list of Wikimedia pages and countries are both public information. However, the majority of \(<\)page, country\(>\) combinations do not appear in the input data: including all of them would be inefficient and lead to increased spurious data. Instead, we use existing public data to only include a small fraction of these possible counts. On each day, we list all Wikimedia pages with more than \(t\) global pageviews, according to the existing Pageview API, where \(t\) is an arbitrary ingestion threshold. Then, we take the cross-product between these pages and the list of countries2 to create the groups.
Footnote 2: This list is based on [7]; excluding countries identified by the Wikimedia Foundation as potentially dangerous for journalists or internet freedom [3].
The second step uses the Gaussian mechanism [28] to add noise to counts. This provides two advantages. First, because each user can contribute to at most 10 _different_\(<\)page, country\(>\) tuples, but only once to each, we get a tighter \(L_{2}\) sensitivity bound (\(\sqrt{k}\)) than if we had used \(L_{1}\) sensitivity (\(k\)): this allows us to add less noise. Second, because the tails of the Gaussian noise distribution decay very fast, this makes the thresholding step more efficient in preventing zero counts from appearing in the output, keeping the spurious rate to acceptably low levels. We quantify the privacy guarantees of the Gaussian mechanism using zero-concentrated DP [20] (zCDP).
The third step is straightforward: all counts below a threshold \(\tau\) are removed from the output. This step is necessary because the first step produces many \(<\)page, country\(>\) tuples for which the non-noisy user count is very low or even 0. Such counts lead to unacceptable high relative error and spurious rate. Conversations with data users showed that these made the output dataset hard to use, and that users were most interested
in the most-viewed pages, rather than the long tail of pages with few views. Suppressing counts below a fixed and configurable threshold \(\tau\) addresses this problem, at the cost of a non-zero drop rate.
The mechanism is presented in Algorithm 2; in this algorithm, \(\mathcal{N}\left(0,\sigma^{2}\right)\) denotes a random sample from a normal distribution of mean \(0\) and variance \(\sigma^{2}\). Step 1 uses only public data, Step 2 provides \(\rho\)-zCDP [20], and Step 3 is a post-processing step: the full algorithm satisfies \(\rho\)-zCDP.
```
1:\(t\): an ingestion threshold.
2:\(\tau\): a suppression threshold.
3:\(\rho\): a privacy parameter for zCDP.
4:\(P=\left\langle p_{1},b_{1}\right\rangle,\left\langle p_{2},b_{2}\right\rangle,\dots\): a private dataset of annotated pageviews, such each user is at most associated with \(k\) unique pageviews \(\left\langle p_{i},b_{i}\right\rangle\) where \(b_{i}=\texttt{true}\), and all of them have distinct \(p_{i}\).
5:\(P_{daily}=\left\langle p_{1},n_{1}\right\rangle,\left\langle p_{2},n_{2}\right\rangle,\dots\): a public dataset listing the global number of pageviews for each page.
6:\(C\): a pre-defined list of countries. Step 1: Collecting aggregation groups
7:\(G\leftarrow\{\}\)
8:for\(\left\langle p,n\right\rangle\) in \(P_{daily}\)do
9:if\(n\geq t\)then
10:for\(c\) in \(C\)do
11:\(G\gets G\cup\left\langle p,c\right\rangle\)
12:endfor
13:endif
14:endfor Step 2: Computing noisy counts
15:\(\sigma\leftarrow\sqrt{\frac{k}{2\rho}}\)
16:\(O\leftarrow\{\}\)
17:for\(g\) in \(G\)do
18:\(c\leftarrow|\{p\in P\mid p=g\}|\)
19:\(\hat{c}\gets c+\mathcal{N}\left(0,\sigma^{2}\right)\)
20:\(O\gets O\cup\left\langle g,\hat{c}\right\rangle\)
21:endfor Step 3: Suppressing low counts
22:for\(\left\langle g,\hat{c}\right\rangle\) in \(G\)do
23:if\(\hat{c}<\tau\)then
24:\(O\gets O\setminus\left\langle g,\hat{c}\right\rangle\)
25:endif
26:endfor
27:return\(O\)
```
**Algorithm 2** Server-side algorithm for the current pageviews
We use \(k=10\) as a per-user daily contribution bound, \(t=150\) as an ingestion threshold, and \(\tau=90\) as a suppression threshold. These values were chosen after extensive experimentation, for input dataset completeness and to optimize the utility metrics described in Section 3.4.
To select these algorithmic parameters, we computed metrics using the true data. Such metrics are, in principle, sensitive, and the parameters themselves are not differentially private. To mitigate the privacy risk from this tuning process, we kept fine-grained utility metrics confidential throughout the tuning process, minimizing data leakage. In addition to this consideration, we only publicly communicate approximate values of global utility metrics and the algorithmic parameters obtained from this tuning process.
Regardless, this remains a valid critique, and we would appreciate further research into the privacy loss entailed by confidentially tuning on sensitive metrics.
### Historical pageviews
To compute differentially private counts using the historical pageview dataset as input data, we follow a similar process, with one key difference: since the data is pre-aggregated, is is impossible to perform per-user contribution bounding. Therefore, we do not use a client-side filtering step, and instead, use a different unit of privacy, as described in Section 3.3. We also have to sum the Count column of the pre-aggregated data, rather than simply counting the number of rows in each group. Another difference is the use of Laplace noise instead of Gaussian noise, motivated by the fact that we only have a bound on the \(L_{1}\) sensitivity of the aggregation, and not \(L_{2}\) like with the current pageviews data. The full process is otherwise similar to the previous one.
1. First, we collect the list of \(<\)page, country\(>\) tuples to aggregate over.
2. Second, we sum the pageview counts in each group, and we add Laplace noise to each sum.
3. Finally, we suppress low sums, and publish the data.
The full algorithm is provided as Algorithm 3; there, \(\text{Lap}(0,\lambda)\) denotes a random sample from the Laplace distribution of mean 0 and scale \(\lambda\). Its privacy analysis is straightforward: Step 1 uses only public data, Step 2 provides \(\varepsilon\)-DP guarantees [27], and Step 3 is a post-processing step, so the full algorithm satisfies \(\varepsilon\)-DP.
As mentioned in Section 3.3, we use \(m=300\) for the 2015-2017 data, and \(m=30\) for the 2017-2023 data. For the 2015-2017 data, we use \(t=150\) as ingestion threshold and \(\tau=3500\) as suppression threshold. For the 2017-2023 data, we use \(t=150\) as ingestion threshold and \(\tau=450\) as suppression threshold. These values were chosen to optimize the global utility metrics described in Section 3.4.
### Implementation
The algorithms were implemented and deployed using Tumult Analytics [30, 18], a framework chosen for its robustness, production-readiness, compatibility with Wikimedia's compute infrastructure, and support for advanced features like zCDP-based privacy accounting [5]. This incurs very slight differences in the mechanisms used: on integer-valued data, Tumult Analytics uses a two-sided geometric distribution instead of Laplace noise, and a discrete version of the Gaussian mechanism [21]. The data release based on the current input data required implementing a new notion of neighboring relation in the framework: rather than protecting a fixed number of rows, or an arbitrary number of rows associated with a single user identifier, it protects a fixed number of rows _associated with different aggregation groups_. This was made easier by the extensibility of the underlying framework, Tumult Core.
## 5 Outcomes
The deployment of this differentially private data publication project is now allowing the Wikimedia Foundation to release a much larger and much richer dataset about user visits to Wikimedia projects. The magnitude of this increase in published pageview data is summarized in Table 6.
More than 2,000 days of historical data from 2015 to 2021 were not previously published. The use of differential privacy in this project allowed the Wikimedia Foundation to release more than 135 million statistics about this data, encompassing 325 billion pageviews.
The output data had acceptable quality according to our success metrics.
* For the data based on the current pageviews dataset, more than 95% of the counts has a relative error below 50%, the drop rate above 150 is below 0.1%, the global spurious rate is below 0.01%, and below 3% for all but 3 countries.
* For the 2017-2023 data, the median top-1000 drop rate is below 8%, the drop rate above 450 is below 3%, and the global spurious rate is below 0.1%.
```
0:\(m\): the number of pageviews protected each day.
0:\(t\): an ingestion threshold.
0:\(\tau\): a suppression threshold.
0:\(\varepsilon\): a privacy parameter for DP.
0:\(P_{hourly}=\left\langle p_{1},c_{1}\right\rangle,\left\langle p_{2},c_{2} \right\rangle,\ldots\): a private dataset listing pre-aggregated hourly pageview counts.
0:\(P_{daily}=\left\langle p_{1},n_{1}\right\rangle,\left\langle p_{2},n_{2} \right\rangle,\ldots\): a public dataset listing the global number of pageviews for each page.
0:\(C\): a pre-defined list of countries.
0:\(t\): A minimum pageview threshold for including pages in the output.
0:\(\mathit{Step~{}1}\): Collecting aggregation groups
1:\(G\leftarrow\{\}\)
2:for\(\left\langle p,n\right\rangle\) in \(P_{daily}\)do
3:if\(n\geq t\)then
4:for\(c\) in \(C\)do
5:\(G\gets G\cup\left\langle p,c\right\rangle\)
6:endfor
7:endif
8:endfor
9:\(\mathit{Step~{}2}\): Computing noisy sums
10:\(\lambda\leftarrow\frac{m}{\varepsilon}\)
11:\(O\leftarrow\{\}\)
12:for\(g\) in \(G\)do
13:\(s\leftarrow\sum_{\left\langle p,c\right\rangle\in P_{hourly}\mathrm{where}p=g}c\)
14:\(\hat{s}\gets s+\mathrm{Lap}\left(0,\lambda\right)\)
15:\(O\gets O\cup\left\langle g,\hat{s}\right\rangle\)
16:endfor
17:\(\mathit{Step~{}3}\): Suppressing low counts
18:for\(\left\langle g,\hat{s}\right\rangle\) in \(G\)do
19:if\(\hat{s}<\tau\)then
20:\(O\gets O\setminus\left\langle g,\hat{s}\right\rangle\)
21:endif
22:endfor
23:return\(O\)
```
**Algorithm 3** Algorithm for the historical pageviews
* For the 2015-2017 data, the top-1000 drop rate is below 40%, the drop rate above 3500 is below 3%, and the global spurious rate is below 20%.
These metrics show that the privacy-accuracy trade-offs are much better for recent data than for historical data: this is explained by the much tighter sensitivity bound from client-side filtering, allowing to take full advantage of the Gaussian mechanism and its fast-decaying tails.
## 6 Conclusion
In this paper, we described the process and mechanisms that allowed the Wikimedia Foundation to publish large-scale datasets about user behavior on Wikipedia and other Wikimedia projects. Multiple key factors made this launch possible.
* Tumult Labs' systematic workflow for differential privacy publications, described in Section 2, provided the structure necessary to move the project forward from its inception to its deployment.
* Combining client-side filtering with server-side aggregation, as described in Section 4.1, was a key innovation that allowed us to obtain user-level differential privacy guarantees for the current pageview data without tracking user identifiers.
* Tumult Core, the privacy framework underlying Tumult Analytics, is designed for extensibility. This made it possible for us to add a novel neighboring definition to this framework to capture the properties of client-side filtering, while still being able to use tight privacy accounting techniques.
* Finally, the scalability offered by Tumult Analytics was essential in handling the massive datasets that were used as input in this project.
The data is now published online [9, 10, 11], along with the source code of the client-side filtering infrastructure [2] and the server-side algorithms [4, 8]. We look forward to seeing what use cases this data will enable!
## 7 Acknowledgements
We are grateful to Luke Hartman, Tomoko Kitazawa, Nuria Ruiz, and Xabriel J. Collazo Mojica for their help with this project, and to Leila Zia and the anonymous reviewers for their helpful comments and suggestions on this paper.
\begin{table}
\begin{tabular}{c|c|c|c} & Before this project & After this project & Percentage change \\ \hline Median number of data points & 9,000 & 360,000 & +4,000\% \\ released per day & 50 million & 120 million & +240\% \\ \hline Total number of data points & \multirow{2}{*}{8 million} & \multirow{2}{*}{120 million} & \multirow{2}{*}{+1,500\%} \\ released since 2021 & & & \\ \hline Total number of pageviews & \multirow{2}{*}{47 billion} & \multirow{2}{*}{116 billion} & \multirow{2}{*}{+250\%} \\ released since 2021 & & & \\ \end{tabular}
\end{table}
Table 6: A comparison of the amount of data published before and after this project, as of June 29, 2023. | 20年ほど、Wikimedia Foundationは毎日、どのWikipediaページがどの国からアクセスされたかを統計を公開してきました。このデータは、Wikipedia編集者によって、オンライン百科事典の改善に注力すべき場所を決定し、学術研究にも役立ちます。2023年6月、Wikimedia FoundationとTumult Labsは、Wikipedia編集者と学術研究者から long-standing に続くリクエストに答えるべく、ページのアクセス数の細かいgranularityで統計を公開を始めました。この新しいデータ公開は、Differential privacy を使用して、Wikipediaを閲覧または編集する人々へ堅牢な保証を提供します。この論文では、このデータ公開の目的、その発足から運用までのプロセス、生成されたアルゴリズム、そしてデータリリースの結果について説明しています。 |
2307.11372 | On the origin of the Boltzmann distribution | The Boltzmann distribution is used in statistical mechanics to describe the
distribution of states in systems with a given temperature. We give a novel
characterization of this distribution as the unique one satisfying independence
for uncoupled systems. The theorem boils down to a statement about symmetries
of the convolution semigroup of finitely supported probability measures on the
natural numbers, or, alternatively, about symmetries of the multiplicative
semigroup of polynomials with non-negative coefficients. | Fedor Sandomirskiy, Omer Tamuz | 2023-07-21T06:06:31 | http://arxiv.org/abs/2307.11372v1 | # On the origin of the Boltzmann distribution
###### Abstract.
The Boltzmann distribution is used in statistical mechanics to describe the distribution of states in systems with a given temperature. We give a novel characterization of this distribution as the unique one satisfying independence for uncoupled systems. The theorem boils down to a statement about symmetries of the convolution semigroup of finitely supported probability measures on the natural numbers, or, alternatively, about symmetries of the multiplicative semigroup of polynomials with non-negative coefficients.
Omer Tamuz was supported a BSF award (#2018397) and a National Science Foundation CAREER award (DMS-1944153).
The map \(\Phi_{\beta}\) that assigns to each probability measure \(\mu\) on \(\mathbb{R}\) the measure \(\Phi_{\beta}[\mu]\) has two important properties: First, it preserves the measure class of \(\mu\), so that \(\mu\) and \(\Phi_{\beta}[\mu]\) are mutually absolutely continuous. Second, it commutes with convolution:
\[\Phi_{\beta}[\mu_{1}*\mu_{2}]=\Phi_{\beta}[\mu_{1}]*\Phi_{\beta}[\mu_{2}]. \tag{1.1}\]
In terms of the physics, the first property corresponds to the fact that we cannot find the system at an energy that is not attained at any state, and that any state can be attained. The second property involves products of independent systems. Suppose that \(\mu_{1}\) and \(\mu_{2}\) describe the distribution of states in two systems. Form a new system whose set of states is the product of the two sets of states, and whose energy in each state is the sum of the two corresponding energies; this corresponds to no interaction between the two subsystems. Then the convolution \(\mu_{1}*\mu_{2}\) describes the new system, and (1.1) follows from the assumption that the joint distribution of states is the product measure of the distributions in the two subsystems. That is, there is no correlation between systems that do not interact.
The usual explanation for the Boltzmann distribution is one of maximum entropy. In this paper, we offer an alternative one: Our main result is that the members of the family \((\Phi_{\beta})_{\beta\in\mathbb{R}}\) are the unique maps that are measure-class-preserving and commute with convolution. Note that \(\Phi_{\beta}\) is not well-defined for every probability measure \(\mu\) on \(\mathbb{R}\), since normalization is impossible when the tails are too thick. We limit ourselves to finitely supported \(\mu\), and furthermore to \(\mu\) supported on the integers or on the rationals.
The map \(\Phi_{\beta}\) is also the tilting map, whose usefulness in the theory of large deviations stems from the fact that it commutes with convolutions. It is thus natural to ask which maps from measures to measures are like tilting, in the sense that they preserve the measure class and commute with convolution. In this context, our main result is a negative one, stating that none other exist.
Given a subset \(S\subseteq\mathbb{R}\) closed with respect to addition, denote by \(\mathcal{P}(S)\) the set of finitely supported probability measures on \(S\). This is a semi-group under the operation of convolution. We say that \(\Phi\colon\mathcal{P}(S)\to\mathcal{P}(S)\) is _support-preserving_ if \(\mu\) and \(\Phi[\mu]\) are mutually absolutely continuous for all \(\mu\in\mathcal{P}(S)\). We say that \(\Phi\) is an _endomorphism_ if it commutes with convolution, i.e., if \(\Phi[\mu_{1}*\mu_{2}]=\Phi[\mu_{1}]*\Phi[\mu_{2}]\).
The map \(\Phi_{\beta}\colon\mathcal{P}(S)\to\mathcal{P}(S)\) given by
\[\Phi_{\beta}[\mu](s)=\frac{\mu(s)\mathrm{e}^{-\beta s}}{\sum_{t}\mu(t)\mathrm{ e}^{-\beta t}}\]
is easily verified to be a support preserving endomorphism. Our main result is that these are the unique ones.
**Theorem 1**.: _Suppose that \(S\) is either \(\mathbb{N}\), \(\mathbb{Z}\), or \(\mathbb{Q}\). Then, for every support-preserving endomorphism \(\Phi\) of \(\mathcal{P}(S)\), there exists a constant \(\beta\in\mathbb{R}\) such that \(\Phi=\Phi_{\beta}\)._
For \(S=\mathbb{R}\), the corresponding claim is not true, as we discuss below. However, it does hold if we also require \(\Phi\) to be weakly continuous, as a corollary of the statement for \(S=\mathbb{Q}\).
Note that even though \(\Phi\) is not assumed to be a bijection, this property emerges as a consequence of the assumptions of Theorem 1. Another emergent property is that, up to normalization, \(\Phi\) is affine, i.e., there is a function \(f\colon\mathbb{R}\to\mathbb{R}\) such that \(\Phi[\mu](s)\) is proportional to \(\mu(s)f(s)\). We do not assume affinity; under such an additional assumption it is easy to show that \(f\) is an exponential.
The bulk of the effort in the proof of Theorem 1 is the case \(S=\mathbb{N}\). By considering probability-generating functions, this question can be reduced to a question about polynomials. Overloading notation, denote by \(\mathcal{P}(\mathbb{N})\) the polynomials in one variable whose coefficients are non-negative and sum to one:
\[\mathcal{P}(\mathbb{N})=\left\{p(x)=\sum_{k=0}^{n}p_{k}x^{k}\,\middle|\,p_{k} \geq 0,\ p(1)=1\right\}.\]
These are precisely the probability-generating functions of finitely supported probability measures on \(\mathbb{N}\).
Say that \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) is support preserving if it preserves the set of positive coefficients. That is, if \(p^{\prime}=\Phi[p]\), then \(p^{\prime}_{k}>0\) if and only if \(p_{k}>0\). Say that \(\Phi\) is multiplicative if \(\Phi[p\cdot p^{\prime}]=\Phi[p]\cdot\Phi[p^{\prime}]\). These two properties, translated back to probability measures, are equivalent to \(\Phi\) being a support-preserving endomorphism.
**Theorem 2**.: _For every support-preserving multiplicative \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) there exists a constant \(\gamma>0\) such that \(\Phi[p](x)=p(\gamma x)/p(\gamma)\)._
Note that by identifying finitely supported probability measures on \(\mathbb{N}\) with their probability-generating functions, this result immediately implies the case \(S=\mathbb{N}\) in Theorem 1.
A corollary of Theorem 2 is that if \(p\) is a polynomial with at least two terms--e.g., \(p(x)=(x+1)/2\)--then \(\Phi\) is completely determined by \(\Phi[p]\): if \(\Phi^{\prime}[p]=\Phi[p]\) for two support-preserving multiplicative maps \(\Phi,\Phi^{\prime}\), then \(\Phi=\Phi^{\prime}\). Indeed, Theorem 2 is equivalent to the statement that if \(\Phi[(x+1)/2]=(x+1)/2\) and \(\Phi\) is support-preserving and multiplicative then \(\Phi\) is the identity.
This may be a priori surprising since it is not clear how fixing \(\Phi[(x+1)/2]\) would constrain \(\Phi[q]\) for any \(q\) that is not a power of \((x+1)/2\). Indeed, one could have imagined that any choice of \(\Phi[(x+1)/2]\) and, say, \(\Phi[(x^{2}+x^{17})/2]\) that is support-preserving could be extended to a support-preserving and multiplicative \(\Phi\). Nevertheless, this is generally impossible, as a consequence of Theorem 2. The underlying reason is that \(\mathcal{P}(\mathbb{N})\) is not a unique factorization domain.
Say that \(p\in\mathcal{P}(\mathbb{N})\) is irreducible if it cannot be written as a product \(p=q_{1}\cdot q_{2}\), for \(q_{1},q_{2}\in\mathcal{P}(\mathbb{N})\) such that \(q_{1},q_{2}\neq 1\). It is easy to see that every \(p\in\mathcal{P}(\mathbb{N})\) can be written as a product of irreducibles. But importantly, this decomposition is not always unique. Hence, if \(p=q_{1}\cdot q_{2}=r_{1}\cdot r_{2}\), then \(\Phi[q_{1}]\cdot\Phi[q_{2}]=\Phi[r_{1}]\cdot\Phi[r_{2}]\), providing additional constraints on \(\Phi\). As it turns out, there are sufficiently many such constraints for \(\Phi[(x+1)/2]\) to fix \(\Phi\).
### Proof techniques
The main tool in the proof of Theorem 2 is the extension of a support-preserving multiplicative \(\Phi\) to the larger domain
\[\mathcal{M}(\mathbb{N})=\left\{p(x)=\sum_{k=1}^{d}p_{k}x^{k}\,\middle|\,p(x)>0 \text{ for all }x>0,\ p(1)=1\right\}.\]
These are the polynomials that are positive for positive \(x\) and whose coefficients sum to \(1\). Equivalently, these are the probability-generating functions of signed, finitely supported probability measures on \(\mathbb{N}\) that have a positive moment generating function. In statistical mechanics terms, these measures describe systems that have _anti-states_--a negative number of states at some energies--but still have a positive partition function.
The fact that \(\Phi\) can be extended to a multiplicative map on \(\mathcal{M}(\mathbb{N})\) follows from the following classical result due to Poincare [8].
**Lemma 1.1** (Poincare).: _For every \(p\in\mathcal{M}(\mathbb{N})\) there is an \(r\in\mathcal{P}(\mathbb{N})\) such that \(p\cdot r\in\mathcal{P}(\mathbb{N})\)._
Using this, we can extend the domain of any support-preserving multiplicative \(\Phi\) to \(\mathcal{M}(\mathbb{N})\) by choosing for \(p\in\mathcal{M}(\mathbb{N})\) an \(r\in\mathcal{P}(\mathbb{N})\) such that \(p\cdot r\in\mathcal{P}(\mathbb{N})\) and setting
\[\Phi[p]=\frac{\Phi[p\cdot r]}{\Phi[r]}.\]
As we show, this is well-defined, i.e., independent of the choice of \(r\). However, \(\Phi[p]\) could now be a rational function. The first part of our proof is dedicated to showing that the image of this extension of \(\Phi\) is, in fact, in \(\mathcal{M}(\mathbb{N})\).
The advantage of \(\mathcal{M}(\mathbb{N})\) is that it is a unique factorization domain: This set consists of the polynomials \(p\) with \(p(1)=1\) that have no positive roots, and hence, by the fundamental theorem of algebra, each \(p\in\mathcal{M}(\mathbb{N})\) can be written as a product of linear terms with non-positive roots and quadratic terms without real roots, all in \(\mathcal{M}(\mathbb{N})\), and this decomposition is unique. Since \(\Phi\) commutes with multiplication, in order to show that \(\Phi[p]=\Phi_{\beta}[p]\) for all \(p\in\mathcal{M}(\mathbb{N})\), it suffices to show that this holds for linear and quadratic \(p\). This is what the remainder of our effort is dedicated to.
### Polynomials with rational coefficients
Recall that, in the physics interpretation, a measure \(\mu\) represents the mass of states at each energy level. In quantum mechanical settings, there are often only finitely many states at each energy level. In this case, the measure \(\mu\) will have rational probabilities, and the corresponding probability-generating function will be a polynomial with rational coefficients. This motivates the pursuit of the same questions, but in a rational setting. Let
\[\mathcal{P}_{\mathbb{Q}}(\mathbb{N})=\left\{p(x)=\sum_{k=0}^{d}p_{k}x^{k} \,\Bigg{|}\,p_{k}\in\mathbb{Q}_{\geq 0},\ \ p(1)=1\right\}\]
be the set of polynomials with non-negative rational coefficients that sum to one. As in the real case, a map \(\Phi\colon\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) is said to be support-preserving and multiplicative if \(\Phi[\mu]\) and \(\mu\) are mutually absolutely continuous and if \(\Phi[\mu_{1}\ast\mu_{2}]=\Phi[\mu_{1}]\ast\Phi[\mu_{2}]\). The next result shows that Theorem 2 still holds in this setting.
**Theorem 3**.: _For every support-preserving multiplicative \(\Phi\colon\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) there exists a constant \(\gamma>0\) such that \(\Phi[p](x)=p(\gamma x)/p(\gamma)\)._
The proof of Theorem 3 requires additional arguments beyond those of Theorem 2. The main difficulty is that the rationality of the coefficients leads to more (and more complicated) irreducible polynomials. We circumvent this issue by using the arguments from the proof of Theorem 2 to show a similar statement for a dense sub-semi-group that does have simple irreducibles, and then proving the following automatic continuity type result.
We say that a set of polynomials \(P\) is rich if \(P\) contains \(q(x)=x\) and, with each polynomial of the form \(p(x)=x^{m}\cdot r(x)\) contained in \(P\), the polynomial \(r(x)\) is also contained in \(P\).
**Proposition 1.2**.: _Let \(M\subset M^{\prime}\) be rich dense sub-semi-groups of \(\mathcal{M}(\mathbb{N})\). Suppose \(\Phi\colon M^{\prime}\to\mathcal{M}(\mathbb{N})\) is multiplicative and degree-preserving, \(\Phi[p]\in\mathcal{P}(\mathbb{N})\) for \(p\in M^{\prime}\cap\mathcal{P}(\mathbb{N})\), and the restriction of \(\Phi\) to \(M\) is the identity map. Then \(\Phi\) is the identity map._
Here, "degree-preserving" means that \(q\) and \(\Phi[q]\) are polynomials of the same degree. The topology over \(\mathcal{M}(\mathbb{N})\) is the topology of simultaneous convergence of the coefficients and the degree.
### Open questions
#### Probability measures over the reals
Since \(\mathcal{P}(\mathbb{Q})\) is a dense subset of the set \(\mathcal{P}(\mathbb{R})\) of finitely supported probability measures on \(\mathbb{R}\), any continuous support-preserving endomorphism of \(\mathcal{P}(\mathbb{Q})\) is of the form \(\Phi_{\beta}\).
There are many other support-preserving endomorphisms of \(\mathcal{P}(\mathbb{R})\). Indeed, if \(\pi\colon\mathbb{R}\to\mathbb{R}\) is any non-continuous solution to the Cauchy equation
\(\pi(x)+\pi(y)\), then
\[\Phi[\mu](x)=\frac{\mu(x)\mathrm{e}^{-\pi(x)}}{\sum_{y}\mu(y)\mathrm{e}^{-\pi(y)}}\]
is a support-preserving endomorphism. Of course, this is non-constructive, since the existence of non-continuous solutions of the Cauchy equation requires an application of the axiom of choice. A natural conjecture is that any support preserving endomorphism on \(\mathcal{P}(\mathbb{R})\) that is not continuous (equivalently, not of the form \(\Phi_{\beta}\)) is not measurable, requiring the axiom of choice.
The same conjecture can be made about support-preserving endomorphisms of the set of compactly supported (rather than finitely supported) probability measures on the reals. In this case, we suspect that the conjecture follows from automatic continuity results for Polish groups.
#### Support preserving endomorphisms of \(\mathcal{P}(\mathbb{Z}^{d})\) and beyond
Our techniques do not extend beyond \(d=1\) in a straightforward way, as multivariate polynomials do not generally decompose into a product of simple factors, such as the quadratic polynomials in the one-dimensional case. We thus offer the following question: Is there, for every support-preserving endomorphism \(\Phi\colon\mathcal{P}(\mathbb{Z}^{d})\to\mathcal{P}(\mathbb{Z}^{d})\), a vector \(\beta=(\beta_{1},\ldots,\beta_{d})\) such that
\[\Phi[\mu](x)=\frac{\mu(x)\mathrm{e}^{-\beta\cdot x}}{\sum_{y}\mu(y)\mathrm{e}^ {-\beta\cdot y}}?\]
More generally, given a semigroup \(G\), does there always exist a homomorphism \(\pi\colon G\to\mathbb{R}\) such that every support-preserving \(\Phi\colon\mathcal{P}(G)\to\mathcal{P}(G)\) is of the form
\[\Phi[\mu](x)=\frac{\mu(x)\mathrm{e}^{-\pi(x)}}{\sum_{y}\mu(y)\mathrm{e}^{-\pi (y)}}?\]
#### Weakening the support-preserving requirement
For \(S\subset\mathbb{R}\), say that \(\Psi\colon\mathcal{P}(S)\to\mathcal{P}(S)\) is _weakly support-preserving_ if \(\Psi[\mu]\) is absolutely continuous with respect to \(\mu\), i.e., the support of \(\Psi[\mu]\) is a subset of the support of \(\mu\). Clearly, every support-preserving \(\Phi\) is also weakly support-preserving, and so the class of support-preserving endomorphisms of, say, \(\mathcal{P}(\mathbb{N})\) is contained in the weakly support-preserving ones. We conjecture that the set of all weakly support-preserving endomorphisms is exhausted by \(\Phi_{\beta}\) and the two limiting cases \(\Phi_{+\infty}=\lim_{\beta\to+\infty}\Phi_{\beta}\) and \(\Phi_{-\infty}=\lim_{\beta\to-\infty}\Phi_{\beta}\), which correspond to putting a point mass on the maximal or minimal point of the support, respectively.
### Related literature
This paper is related to other work on polynomials with non-negative coefficients. This literature consists of two lines of research.
The line closest to our analysis originated from the classical works of Poincare [8] and Polya [9], exploring the relation of non-negativity of a polynomial \(p\) and the
possibility of finding a factor \(r\) such that \(r\cdot p\) has non-negative coefficients under various assumptions on \(r\); see a recent contribution by Michelen and Sahasrabudhe [5] for a survey.
A related strain of research explores the connection between the coefficients and the distribution of zeros. Since polynomials with non-negative coefficients are moment-generating functions of finitely-supported distributions, this direction is tightly related to non-classical limit theorems of probability theory; see the series of papers by Michelen and Sahasrabudhe for recent progress [3, 4, 6].
The algebra of the semi-group of measures on the reals under convolutions is well studied, including its homomorphisms to \(\mathbb{R}\), \(\mathbb{Z}\) and \(\mathbb{C}\); see "Algebraic Probability Theory," a book by Ruzsa and Szekely [10], as well as more recent work [1, 7]. Homomorphisms into general groups were considered by Mattner [2].
### Acknowledgements
We thank Tim Austin, Alexander Guterman, Tom Hutchcroft, Daniel Litt, Gil Refael, Barry Simon, and Stanislav Smirnov for illuminating conversations and helpful suggestions.
## 2. Preliminaries
Recall that \(\mathcal{P}(\mathbb{N})\) is the set of polynomials \(p(x)=\sum_{k=0}^{d}p_{k}x^{k}\) such that \(p_{k}\geq 0\) and \(p(1)=\sum_{k}p_{k}=1\). We denote by \(\deg(p)\) the degree of \(p\). The set \(\mathcal{P}(\mathbb{N})\) is contained in \(\mathcal{M}(\mathbb{N})\), the set of polynomials \(p\) such that \(p(1)=1\) and \(p(x)>0\) for all \(x>0\). Note that the latter condition can be equivalently changed to \(p(x)\neq 0\) for all \(x>0\), so that \(\mathcal{M}(\mathbb{N})\) consists of the polynomials \(p\) with no positive roots such that \(p(1)=1\).
For notational convenience, we will sometimes omit normalization constants, often writing these polynomials in monic form. For example, \(p(x)=(x+1)/2\in\mathcal{P}(\mathbb{N})\) will be written as \(p(x)=x+1\). Similarly, an expression of the form \(\Phi[(x+1)/2]=(x+2)/3\) will be written more succinctly as \(\Phi[x+1]=x+2\). Since every polynomial with non-positive roots can be normalized to a unique \(p\in\mathcal{M}(\mathbb{N})\), and since normalization preserves the support and commutes with multiplication, this will introduce no ambiguity.
For \(\gamma>0\), let \(\Psi_{\gamma}\colon\mathcal{M}(\mathbb{N})\to\mathcal{M}(\mathbb{N})\) be given by
\[\Psi_{\gamma}\colon\mathcal{M}(\mathbb{N}) \to\mathcal{M}(\mathbb{N})\] \[p(x) \mapsto p(x/\gamma). \tag{2.1}\]
Note that we omit normalization constants, as explained above; the normalized form is \(\Psi_{\gamma}[p]=p(x/\gamma)/p(1/\gamma)\). In terms of probability measures, \(\Psi_{\gamma}\) corresponds to the map \(\Phi_{\beta}\) from the introduction, for \(\beta=\log\gamma\). In particular, it is easy to verify that \(\Psi_{\gamma}\) is support-preserving and multiplicative, and--importantly--that it maps \(\mathcal{P}(\mathbb{N})\) to \(\mathcal{P}(\mathbb{N})\).
The following lemma is a strengthening of Lemma 1.1. It is a consequence of Polya's Positivenstellensatz [9], which is a more general statement about multivariate polynomials.
**Lemma 2.1** (Polya).: _For every \(q\in\mathcal{M}(\mathbb{N})\) it holds for all \(n\) large enough that \(q(x)\cdot(x+1)^{n}\in\mathcal{P}(\mathbb{N})\)._
Note that we here again drop the normalization constant and write \((x+1)^{n}\) rather than \(2^{-n}(x+1)^{n}\).
We will need a slight strengthening of this lemma.
**Lemma 2.2**.: _For every \(q\in\mathcal{M}(\mathbb{N})\) and all \(\gamma>0\) it holds for all \(n\) large enough that \(q(x)(x+\gamma)^{n}\in\mathcal{P}(\mathbb{N})\)._
Proof.: Note that \(\Psi_{\gamma}\) maps \(x+1\) to \(x+\gamma\) (omitting normalization). Fix \(q\in\mathcal{M}(\mathbb{N})\). By Lemma 2.1, we know that \(\Psi_{\gamma^{-1}}[q](x)\cdot(x+1)^{n}\) is in \(\mathcal{P}(\mathbb{N})\) for all \(n\) large enough. Since \(\Psi_{\gamma}\) is multiplicative,
\[\Psi_{\gamma}\big{[}\Psi_{\gamma^{-1}}[q](x)\cdot(x+1)^{n}\big{]}=q(x)\cdot \Psi_{\gamma}[(x+1)^{n}]=q(x)(x+\gamma)^{n},\]
and since \(\Psi_{\gamma}\) maps \(\mathcal{P}(\mathbb{N})\) to \(\mathcal{P}(\mathbb{N})\), it follows that \(q(x)(x+\gamma)^{n}\) is also in \(\mathcal{P}(\mathbb{N})\) for all \(n\) large enough.
## 3. Support-preserving multiplicative maps of polynomials with non-negative coefficients
In this section, we prepare the ingredients for proofs of Theorems 1, 2, and 3 and then prove the theorems. The first ingredient is extending a multiplicative support-preserving \(\Phi\) from the set of polynomials with non-negative coefficients \(\mathcal{P}(\mathbb{N})\) to the set \(\mathcal{M}(\mathbb{N})\) of polynomials without positive roots.
### Extending the domain of \(\Phi\)
In this section we prove the following result.
**Proposition 3.1**.: _Every support-preserving multiplicative \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) can be (uniquely) extended to a degree-preserving multiplicative \(\Phi\colon\mathcal{M}(\mathbb{N})\to\mathcal{M}(\mathbb{N})\)._
Let
\[\mathcal{F}(\mathbb{N})=\left\{\frac{p^{\prime}}{p}\,:\,p,p^{\prime}\in \mathcal{P}(\mathbb{N})\right\}.\]
The first step towards proving Proposition 3.1 is to extend a support-preserving multiplicative \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) to a multiplicative \(\Phi\colon\mathcal{M}(\mathbb{N})\to\mathcal{F}(\mathbb{N})\).
To this end, given \(q\in\mathcal{M}(\mathbb{N})\), there is, by Lemma 1.1, \(p,r\in\mathcal{P}(\mathbb{N})\) such that \(q\cdot p=r\). Define the extension of \(\Phi\) to \(\mathcal{M}(\mathbb{N})\) by
\[\Phi[q]=\frac{\Phi[r]}{\Phi[p]}. \tag{3.1}\]
To see that this is well defined, suppose that \(q\cdot p^{\prime}=r^{\prime}\) for some \(p^{\prime},r^{\prime}\in\mathcal{P}(\mathbb{N})\), and note that \(q\cdot p\cdot p^{\prime}\in\mathcal{P}(\mathbb{N})\). Thus
\[\Phi[r]\cdot\Phi[p^{\prime}]=\Phi[q\cdot p]\cdot\Phi[p^{\prime}]=\Phi[q\cdot p \cdot p^{\prime}]=\Phi[q\cdot p^{\prime}]\cdot\Phi[p]=\Phi[r^{\prime}]\cdot \Phi[p],\]
and so
\[\frac{\Phi[r]}{\Phi[p]}=\frac{\Phi[r^{\prime}]}{\Phi[p^{\prime}]}.\]
To see that \(\Phi\) is multiplicative, i.e., that \(\Phi[q\cdot q^{\prime}]=\Phi[q]\cdot\Phi[q^{\prime}]\) for all \(q,q^{\prime}\in\mathcal{M}(\mathbb{N})\), suppose that \(q\cdot p\) and \(q^{\prime}\cdot p^{\prime}\) belong to \(\mathcal{P}(\mathbb{N})\). Thus \(q\cdot q^{\prime}\cdot p\cdot p^{\prime}\in\mathcal{P}(\mathbb{N})\). Hence,
\[\Phi[q\cdot q^{\prime}]=\frac{\Phi[q\cdot q\cdot p\cdot p^{\prime}]}{\Phi[p \cdot p^{\prime}]}=\frac{\Phi[q\cdot p]}{\Phi[p]}\frac{\Phi[q^{\prime}\cdot p ^{\prime}]}{\Phi[p^{\prime}]}=\Phi[q]\cdot\Phi[q^{\prime}].\]
We note that the extension defined by (3.1) is unique, since any extension that satisfies \(\Phi[q\cdot q^{\prime}]=\Phi[q]\cdot\Phi[q^{\prime}]\) must satisfy (3.1) whenever \(q\cdot p=r\).
To prove Proposition 3.1, we need to show that the image of this extension is, in fact, in the polynomials. To this end, for \(0<a<b\), let1
Footnote 1: As discussed in §2, we omit a normalizing constant and write \(q^{a,b}(x)\) as above, rather than \((x^{2}-ax+ab)/(1-a+ab)\).
\[q^{a,b}(x)=x^{2}-ax+ab.\]
Note that \(q^{a,b}\in\mathcal{M}(\mathbb{N})\).
Then
\[q^{a,b}(x)\cdot(x+t)=(x^{2}-ax+ab)\cdot(x+t)=x^{3}+(t-a)x^{2}+a(b-t)x+abt,\]
and in particular
\[q^{a,b}(x)\cdot(x+a)=x^{3}+a(b-a)x+a^{2}b\] \[q^{a,b}(x)\cdot(x+b)=x^{3}+(b-a)x^{2}+ab^{2}.\]
are both in \(\mathcal{P}(\mathbb{N})\). Importantly, they have different supports. By (3.1),
\[\Phi[q^{a,b}]=\frac{\Phi[q^{a,b}(x)\cdot(x+a)]}{\Phi[x+a]}=\frac{\Phi[q^{a,b}( x)\cdot(x+b)]}{\Phi[x+b]}.\]
Now, because \(q^{a,b}(x)\cdot(x+a)\) and \(q^{a,b}(x)\cdot(x+b)\) have different supports, it follows from the support-preserving property of \(\Phi\) that
\[\Phi[q^{a,b}(x)\cdot(x+a)]\neq\Phi[q^{a,b}(x)\cdot(x+b)].\]
Hence \(\Phi[x+a]\neq\Phi[x+b]\). We have thus proved the following claim:2
**Claim 3.2**.: _Suppose that \(\Phi\) is support-preserving and multiplicative. Then \(\Phi[x+a]\neq\Phi[x+b]\) for \(a\neq b\)._
With this claim, we are ready to prove our proposition.
Proof of Proposition 3.1.: Let \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) be support preserving and multiplicative. Extend it to \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{F}(\mathbb{N})\) using (3.1). Choose any \(a,b>0\), \(a\neq b\), and fix \(q\in\mathcal{M}(\mathbb{N})\). By Lemma 2.2, there is \(n\) large enough so that both \(q(x)(x+a)^{n}\) and \(q(x)(x+b)^{n}\) are in \(\mathcal{P}(\mathbb{N})\). We thus have that
\[\Phi[q]=\frac{\Phi[q(x)\cdot(x+a)^{n}]}{\Phi[(x+a)^{n}]}=\frac{\Phi[q(x)\cdot( x+b)^{n}]}{\Phi[(x+b)^{n}]}.\]
By Claim 3.2 and the support-preserving property, we know that there are some \(c\neq d\) such that \(\Phi[x+a]=x+c\) and \(\Phi[x+b]=x+d\). Hence,
\[\frac{\Phi[q(x)\cdot(x+a)^{n}]}{(x+c)^{n}}=\frac{\Phi[q(x)\cdot(x+b)^{n}]}{(x +d)^{n}},\]
and rearranging we get
\[\Phi[q(x)\cdot(x+a)^{n}](x+d)^{n}=\Phi[q(x)\cdot(x+b)^{n}]\cdot(x+c)^{n}.\]
Both sides are polynomials and have the same roots. Of these roots, at least \(n\) are equal to \(-c\), since the right-hand side includes the term \((x+c)^{n}\). The left-hand side thus also has at least \(n\) roots which are equal to \(-c\). Since \(c\neq d\), \(\Phi[q(x)\cdot(x+a)^{n}]\) has at least \(n\) roots equal to \(-c\). It follows that
\[\Phi[q]=\frac{\Phi[q(x)\cdot(x+a)^{n}]}{(x+c)^{n}}\]
is a polynomial. Furthermore, it is in \(\mathcal{M}(\mathbb{N})\), since it does not have positive roots. Finally, by the support-preserving property of \(\Phi\),
\[\deg(\Phi[q(x)\cdot(x+a)^{n}])=\deg(q(x)\cdot(x+a)^{n})=\deg(q)+n.\]
On the other hand, since \(\Phi\) is a multiplicative,
\[\deg(\Phi[q(x)\cdot(x+a)^{n}])=\deg(\Phi[q]\cdot\Phi[(x+a)^{n}])=\deg(\Phi[q]) +n,\]
and so \(\deg(\Phi[q])=\deg(q)\).
### Linear polynomials and some quadratic polynomials
In this section, we prove the following proposition.
**Proposition 3.3**.: _Let \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) be support-preserving and multiplicative. Then there exists a constant \(\gamma>0\) such that \(\Phi[x+a]=x+\gamma a\) for all \(a>0\)._
Let \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) be support-preserving and multiplicative. Consider \(\varphi\colon\mathbb{R}_{>0}\to\mathbb{R}_{>0}\) given by \(\Phi[x+t]=x+\varphi(t)\). We will show that \(\varphi(a)=\gamma a\), thus proving Proposition 3.3. As a byproduct, we also show that \(\Phi[x^{2}-ax+ab]=x^{2}-\gamma ax+\gamma^{2}ab\) for all \(0<a<b\) (Corollary 3.6).
As above, let \(q^{a,b}(x)=x^{2}-ax+ab\), for \(0<a<b\). Then
\[q^{a,b}(x)\cdot(x+t)=(x^{2}-ax+ab)\cdot(x+t)=x^{3}+(t-a)x^{2}+a(b-t)x+abt\]
is in \(\mathcal{P}(\mathbb{N})\) for any \(t\in[a,b]\). By Proposition 3.1,
\[\Phi[q^{a,b}]=\frac{\Phi[q^{a,b}(x)\cdot(x+t)]}{\Phi[x+t]}=\frac{\Phi[x^{3}+(t -a)x^{2}+a(b-t)x+abt]}{x+\varphi(t)} \tag{3.2}\]
is a polynomial for all \(0<a<b\) and \(t\in[a,b]\). In particular, for \(t=a\) we get that
\[\Phi[q^{a,b}]=\frac{\Phi[x^{3}+a(b-a)x+a^{2}b]}{x+\varphi(a)}\]
is a polynomial. Using the support-preserving property of \(\Phi\), we can write the numerator as \(\Phi(x^{3}+a(b-a)x+a^{2}b)=x^{3}+cx+d\), which must have \(x+\varphi(a)\) as a factor. Factoring \(x+\varphi(a)\) from this polynomial yields that
\[\Phi[q^{a,b}]=x^{2}-\varphi(a)x+\varphi(a)^{2}+c. \tag{3.3}\]
Similarly, substituting \(t=b\) into (3.2) we get
\[\Phi[q^{a,b}]=\frac{\Phi(x^{3}+(b-a)x^{2}+ab^{2})}{x+\varphi(b)}.\]
Writing the numerator as \(\Phi[x^{3}+(b-a)x+ab^{2}]=x^{3}+ex^{2}+f\), and since \(x+\varphi(b)\) is a factor of this polynomial, we get that
\[\Phi[q^{a,b}]=x^{2}-(\varphi(b)-e)x+\varphi(b)^{2}-e\varphi(b). \tag{3.4}\]
Equating (3.3) and (3.4), we get that \(\varphi(a)=\varphi(b)-e\). This then yields that
\[\Phi[q^{a,b}]=x^{2}-\varphi(a)x+\varphi(a)\varphi(b). \tag{3.5}\]
Now, choose \(t\in(a,b)\). Then \(q^{a,b}(x)\cdot(x+t)\) is a cubic polynomial with positive coefficients and, by the above,
\[\Phi[q^{a,b}(x)\cdot(x+t)] =(x^{2}-\varphi(a)x+\varphi(a)\varphi(b))(x+\varphi(t))\] \[=x^{3}+(\varphi(t)-\varphi(a))x^{2}+\cdots.\]
By the support-preserving property of \(\Phi\), the coefficient of \(x^{2}\) must be positive, and so we have shown that \(\varphi\) is strictly monotone increasing:
**Claim 3.4**.: _If \(0<a<b\), then \(\varphi(a)<\varphi(b)\)._
We are now ready to prove the main result of this section.
Proof of Proposition 3.3.: Note that
\[q^{a,b}(x)\cdot(x+2b)^{2} =(x^{2}-ax+ab)\cdot(x+2b)^{2}\] \[=x^{4}+(4b-a)x^{3}+b(4b-3a)x^{2}+4ab^{3}.\]
is in \(\mathcal{P}(\mathbb{N})\). Since
\[\Phi[q^{a,b}(x)]\cdot\Phi[(x+2b)^{2}] =(x^{2}-\varphi(a)x+\varphi(a)\varphi(b))\cdot(x+\varphi(2b))^{2}\] \[=\cdots+(2\varphi(a)\varphi(b)\varphi(2b)-\varphi(a)\varphi(2b)^ {2})x+\cdots,\]
it follows from the support-preserving property of \(\Phi\) that
\[2\varphi(a)\varphi(b)\varphi(2b)-\varphi(a)\varphi(2b)^{2}=0\]
or
\[\varphi(2b)=2\varphi(b).\]
Likewise,
\[q^{a,b}(x)\cdot(x+3b)^{3} =(x^{2}-ax+ab)\cdot(x+3b)^{3}\] \[=x^{5}+(9b-a)x^{4}+b(27b-8a)x^{3}+b^{2}(27b-18a)x^{2}+27ab^{4}.\]
is in \(\mathcal{P}(\mathbb{N})\) for \(b>a\). Since
\[\Phi[q^{a,b}(x)]\cdot\Phi[(x+3b)^{3}] =(x^{2}-\varphi(a)x+\varphi(a)\varphi(b))\cdot(x+\varphi(3b))^{3}\] \[=\cdots+(3\varphi(a)\varphi(b)\varphi(3b)^{2}-\varphi(a)\varphi( 3b)^{3})x+\cdots,\]
again applying the support-preserving property of \(\Phi\) yields that
\[3\varphi(a)\varphi(b)\varphi(3b)^{2}-\varphi(a)\varphi(3b)^{3}=0,\]
or
\[\varphi(3b)=3\varphi(b).\]
It thus follows from Lemma 3.5 below that there exists a constant \(\gamma>0\) such that \(\varphi(b)=\gamma b\).
**Lemma 3.5**.: _Suppose that \(f\colon\mathbb{R}_{>0}\to\mathbb{R}_{>0}\) is strictly monotone increasing, and satisfies \(f(2x)=2f(x)\) and \(f(3x)=3f(x)\). Then there exists a constant \(c>0\) such that \(f(x)=cx\)._
Proof.: Since 2 and 3 are coprime, the set \(X=\{2^{m}3^{n}\,:\,m,n\in\mathbb{Z}\}\) is dense in \(\mathbb{R}_{>0}\). Since \(f(2x)=2f(x)\) and \(f(3x)=3f(x)\), we obtain that \(f(x)=xf(1)\) for all \(x\in X\). Given any \(y\in\mathbb{R}_{>0}\), choose a sequence \((x_{n}^{+})_{n}\) in \(X\) that converges to \(y\) from above, and likewise \((x_{n}^{-})_{n}\) that converges to \(y\) from below. Then, by the monotonicity of \(f\),
\[yf(1)=\lim_{n}x_{n}^{+}f(1)=\lim_{n}f(x_{n}^{+})\geq f(y)\geq\lim_{n}f(x_{n}^{- })=\lim_{n}x_{n}^{-}f(1)=yf(1).\]
In particular, \(f(y)=f(1)y\)
We end this section with the following result which is a corollary of Proposition 3.3 and identity (3.5):
**Corollary 3.6**.: _Consider an extension of a support-preserving and multiplicative \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) to \(\mathcal{M}(\mathbb{N})\) and define \(\gamma\) by \(\Phi[x+1]=x+\gamma\). Then \(\Phi[x^{2}-ax+ba]=x-\gamma ax+\gamma^{2}ab\) for all \(0<a<b\)._
### Quadratic polynomials
Consider a polynomial of the form \(q^{a}(x)=x^{2}-ax+1\). For \(q^{a}\) to be in \(\mathcal{M}(\mathbb{N})\), the discriminant must be negative, i.e., \(a<2\).
**Claim 3.7**.: _Suppose that \(\Phi\) is support-preserving and multiplicative, and \(\Phi[x+1]=x+1\). Then \(\Phi[q^{a}]=q^{a}\) for all \(a\in[-1,1]\setminus\{0\}\)._
Proof.: For \(a\in(0,1)\), we know from Corollary 3.6 that \(\Phi[q^{a}]=q^{a}\), by choosing \(b=1/a\). We next demonstrate that \(\Phi[q^{a}]=q^{a}\) for \(a=1\). Let \(\Phi[x^{2}-x+1]=x^{2}-a^{\prime}x+b^{\prime}\). Note that
\[(x^{2}-x+1)(x+1)=x^{3}+1.\]
Hence, by the support-preserving property of \(\Phi\), in the expression
\[\Phi[(x^{2}-x+1)(x+1)]=(x^{2}-a^{\prime}x+b^{\prime})(x+1)=x^{3}+(1-a^{\prime} )x^{2}+(b^{\prime}-a^{\prime})x+b^{\prime}\]
the coefficients of \(x\) and \(x^{2}\) must vanish. We obtain \(1-a^{\prime}=0\) and \(b^{\prime}-a^{\prime}=0\). Thus \(a^{\prime}=b^{\prime}=1\) and so \(\Phi[q^{a}]=q^{a}\) for \(a=1\).
Now suppose \(a\in[-1,0)\). Then \(q^{a}\in\mathcal{P}(\mathbb{N})\) so \(\Phi(q^{a})=x^{2}+a^{\prime}x+b^{\prime}\) for some \(a^{\prime},b^{\prime}>0\). Then
\[\Phi[q^{-a}\cdot q^{a}] =\Phi[q^{-a}]\cdot\Phi[q^{a}]\] \[=(x^{2}-ax+1)\cdot(x^{2}+a^{\prime}x+b^{\prime})\] \[=x^{4}+(a^{\prime}-a)x^{3}+(1+b^{\prime}-aa^{\prime})x^{2}+(a^{ \prime}-ab^{\prime})x+b^{\prime}.\]
Now,
\[q^{a}(x)\cdot q^{-a}(x)=(x^{2}-ax+1)\cdot(x^{2}+ax+1)=x^{4}+(2-a^{2})x^{2}+1,\]
and so, by the support-preserving property of \(\Phi\), we have that \(a^{\prime}-a=0\) and \(a^{\prime}-ab^{\prime}=0\). Hence, \(a^{\prime}=a\) and \(b^{\prime}=1\). We conclude that \(\Phi[q^{a}]=q^{a}\) for \(a\in[-1,1]\setminus\{0\}\).
Define \(\pi\colon\mathcal{M}(\mathbb{N})\to\mathcal{M}(\mathbb{N})\) to be the map \(q(x)\mapsto q(x^{2})\). Let \(\Phi^{\pi}=\pi^{-1}\circ\Phi\circ\pi\), so that
\[\Phi^{\pi}[q_{0}+q_{1}x+q_{2}x^{2}+\cdots+q_{d}x^{d}]=r_{0}+r_{1}x+r_{2}x^{2}+ \cdots+r_{d}x^{d}\]
whenever
\[\Phi[q_{0}+q_{1}x^{2}+q_{2}x^{4}+\cdots+q_{d}x^{2d}]=r_{0}+r_{1}x^{2}+r_{2}x^{ 4}+\cdots+r_{d}x^{2d}.\]
It is easy to verify that \(\Phi^{\pi}\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) is support-preserving and multiplicative if \(\Phi\) is. By Proposition 3.1, \(\Phi^{\pi}\) admits a unique extension to a degree-preserving multiplicative map \(\mathcal{M}(\mathbb{N})\to\mathcal{M}(\mathbb{N})\).
**Claim 3.8**.: _Suppose that \(\Phi\) is support-preserving and multiplicative, and \(\Phi[x+1]=x+1\). Then for all \(a\in[-1,1]\) it holds that \(\Phi[q^{a}]=q^{a}\)._
Proof.: The case of \(a\neq 0\) was shown in Claim 3.7. It remains to be shown that \(\Phi[x^{2}+1]=x^{2}+1\), or equivalently that \(\Phi^{\pi}[x+1]=x+1\). Since \(\Phi^{\pi}\) is support-preserving and multiplicative, by Proposition 3.3 there is a constant \(\gamma>0\) such that \(\Phi^{\pi}[x+a]=x+\gamma a\).
Define \(\Phi^{\prime}=\Psi_{1/\gamma}\circ\Phi^{\pi}\), where we recall that \(\Psi_{1/\gamma}\colon p(x)\to p(\gamma x)\). Hence, \(\Phi^{\prime}\) is support-preserving and multiplicative and satisfies \(\Phi^{\prime}[x+1]=x+1\). Therefore, by Claim 3.7, \(\Phi^{\prime}[x^{2}-a+1]=x^{2}-ax+1\) for all \(a\in[-1,1]\setminus\{0\}\). Since \(\Phi^{\pi}=\Psi_{\gamma}\circ\Phi^{\prime}\), we get \(\Phi^{\pi}[q^{a}(x)]=x^{2}-\gamma ax+\gamma^{2}\) for all \(a\in[-1,1]\setminus\{0\}\).
Let us show that \(\gamma=1\). Observe that for \(a=-1\)
\[q^{a}(x^{2})=x^{4}+x^{2}+1=(x^{2}+x+1)\cdot(x^{2}-x+1)=q^{a}(x)\cdot q^{-a}(x).\]
Hence,
\[\Phi[q^{a}(x^{2})]=\Phi[q^{a}(x)]\cdot\Phi[q^{-a}(x)]=(x^{2}+x+1)(x^{2}-x+1)=x ^{4}+x^{2}+1.\]
On the other hand,
\[\Phi[q^{a}(x^{2})]=\Phi^{\pi}[q^{a}](x^{2})=x^{4}-\gamma ax^{2}+\gamma^{2}.\]
Thus \(\gamma=1\), which completes the proof.
**Claim 3.9**.: _Let \(A\) be a subset of \((-2,2)\). Suppose that \(\Phi^{\pi}[q^{a}]=q^{a}\) for all \(a\in A\). Then \(\Phi[q^{a}]=q^{a}\) for all \(a\) such that \(a^{2}-2\in A\)._
Proof.: Note that
\[q^{a}(x)\cdot q^{-a}(x)=(x^{2}-ax+1)\cdot(x^{2}+ax+1)=x^{4}+(2-a^{2})x^{2}+1.\]
By the claim hypothesis \(\Phi^{\pi}[x^{2}+(2-a^{2})x+1]=x^{2}+(2-a^{2})x+1\). It follows that
\[\Phi[q^{a}]\cdot\Phi[q^{-a}]=x^{4}+(2-a^{2})x^{2}+1.\]
Without loss of generality, we can assume that \(a\geq 0\). By Proposition 3.1, the left-hand side is the product of two quadratic polynomials; moreover, \(\Phi[q^{-a}]\) has non-negative coefficients. The polynomial \(x^{4}+(2-a^{2})x^{2}+1\) on the right-hand side has two pairs of complex-conjugate roots. Hence, there is a unique way to represent it as a product of two quadratic monic polynomials with real coefficients:
\[x^{4}+(2-a^{2})x^{2}+1=(x^{2}-ax+1)\cdot(x^{2}+ax+1).\]
Only one of these quadratic factors has non-negative coefficients and thus
\[\Phi[q^{a}]=x^{2}-ax+1\qquad\text{and}\qquad\Phi[q^{-a}]=x^{2}+ax+1\]
completing the proof.
**Claim 3.10**.: _If a support-preserving multiplicative \(\Phi\) satisfies \(\Phi[x+1]=x+1\), then \(\Phi[q^{a}]=q^{a}\) for all \(a\) in \((-2,2)\)._
Proof.: Let \(A^{*}\) be the set of all \(a\in(-2,2)\) such that \(\Phi^{\prime}[q^{a}]=q^{a}\) for all support-preserving multiplicative maps \(\Phi^{\prime}\) such that \(\Phi^{\prime}[x+1]=x+1\). By Claim 3.8, \([-1,1]\subseteq A^{*}\).
Let \(f(x)=x^{2}-2\), and denote by \(f^{(n)}\) the \(n\)-fold composition of \(f\) with itself. We claim that for any \(x\in(1,2)\) there is a number \(n\) such that \(f^{(n)}(x)\in[-1,1]\). Since the image of \((1,2)\) under \(f\) is \((-1,2)\), it is enough to show that there is no \(x_{0}\in(1,2)\) such that \(x_{n}=f^{(n)}(x_{0})\) stays in \((1,2)\) for all \(n\). Towards a contradiction, suppose that such \(x_{0}\) exists. For \(x\in(1,2)\), we have that \(f(x)<3x-4\), since \(f(1)=-1\), \(f(2)=2\) and \(f\) is strictly convex. In particular, \(f(x)-x<2x-4<0\) for all \(x\in[1,2)\), so that \(f(x)<x\). Thus the sequence \(x_{n}\) is decreasing. Denote \(x_{\infty}=\lim_{n}x_{n}\in[1,2)\). By continuity of \(f\), we get \(f(x_{\infty})=x_{\infty}\). But \(f(x)<x\) for all \(x\in[1,2)\). This contradiction implies that for any \(x\in(1,2)\), there is \(n\) such that \(f^{(n)}(x)\in(-1,1)\). The same argument applies to \(x\in(-2,-1)\), since \(f(-x)=f(x)\).
By Claim 3.9, if \(f(a)\in A^{*}\) then \(a\in A^{*}\). It follows that if \(f^{(n)}(a)\in A^{*}\), then \(a\in A^{*}\). Since \([-1,1]\subseteq A^{*}\), we conclude that \((-2,2)\subseteq A^{*}\). Thus \(A^{*}=(-2,2)\).
We are now ready to show that if \(\Phi[x+1]=x+1\), then \(\Phi[q]=q\) for all quadratic \(q\in\mathcal{M}(\mathbb{N})\) that have no real roots. Note that any such \(q\) is (up to normalization) of the form \(q(x)=x^{2}-a\gamma x+\gamma^{2}\) for some \(a\in(-2,2)\) and \(\gamma>0\).
**Proposition 3.11**.: _If a support-preserving multiplicative \(\Phi\) satisfies \(\Phi[x+1]=x+1\), then \(\Phi[x^{2}-a\gamma x+\gamma^{2}]=x^{2}-a\gamma x+\gamma^{2}\) for all \(a\in(-2,2)\) and all \(\gamma>0\)._
Proof.: Recall that \(\Psi_{\gamma}\) maps \(p(x)\) to \(p(x/\gamma)\). Consider \(\Phi^{\prime}=\Psi_{1/\gamma}\circ\Phi\circ\Psi_{\gamma}\). Then \(\Phi^{\prime}\) is support-preserving and multiplicative, and \(\Phi^{\prime}[x+1]=x+1\). By Claim 3.9, \(\Phi^{\prime}[q^{a}]=q^{a}\) for all \(a\in(-2,2)\). Equivalently, \(\Psi_{1/\gamma}\circ\Phi\circ\Psi_{\gamma}[q_{a}]=q_{a}\). Applying \(\Psi_{\gamma}\) on both sides of this identity, we get \(\Phi\big{[}\Psi_{\gamma}[q_{a}]\big{]}=\Psi_{\gamma}[q_{a}]\). Since \(\Psi_{\gamma}[q_{a}](x)=x^{2}-a\gamma x+\gamma^{2}\), the proof is complete.
### Proofs of Theorems 1 and 2
Proof of Theorem 2.: We first claim that to prove the theorem, it suffices to show that a support-preserving and multiplicative \(\Phi\) such \(\Phi[x+1]=x+1\) is the identity map. To see that this statement implies the theorem, let \(\Phi\) be support-preserving and multiplicative. Define \(\gamma>0\) by \(\Phi[x+1]=x+\gamma\). Recall from (2.1) that \(\Psi_{\gamma}\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) is the map that takes \(p(x)\) to \(p(x/\gamma)\). Hence \(\Phi^{\prime}=\Psi_{\gamma}^{-1}\circ\Phi\) is support-preserving and multiplicative, and furthermore satisfies \(\Phi^{\prime}[x+1]=x+1\). Hence, if we show that \(\Phi^{\prime}\) is the identity map, then we have shown that \(\Phi=\Psi_{\gamma}\).
We now show that a support-preserving and multiplicative \(\Phi\colon\mathcal{P}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) such that \(\Phi[x+1]=x+1\) is the identity map.
Fix \(p\in\mathcal{P}(\mathbb{N})\). By the fundamental theorem of algebra, we can write it as a product of polynomials
\[p(x)=\prod_{i}r^{i}(x)\prod_{j}q^{j}(x),\]
where each \(r^{i}\) is linear and each \(q^{j}\) is quadratic with no real roots.
Since \(p\in\mathcal{P}(\mathbb{N})\) has no positive roots, each linear term \(r^{i}\) is of the form \(r^{i}(x)=x+b_{i}\) for some \(b_{i}\geq 0\). Since each quadratic term has no real roots, it is of the form \(q^{j}(x)=x^{2}-a_{j}\gamma_{j}x+\gamma_{j}^{2}\) for some \(\gamma_{j}>0\) and \(a_{j}\in(-2,-2)\).
By Proposition 3.1, we can extend \(\Phi\) to a multiplicative map \(\Phi\colon\mathcal{M}(\mathbb{N})\to\mathcal{M}(\mathbb{N})\). Hence,
\[\Phi[p]=\prod_{i}\Phi[r^{i}]\prod_{j}\Phi[q^{j}].\]
By Proposition 3.3, \(\Phi[r^{i}]=r^{i}\). And by Proposition 3.11, \(\Phi[q^{j}]=q^{j}\). We conclude that \(\Phi[p]=p\), and so \(\Phi\) is the identity map. By the remark at the beginning of the proof, this implies the theorem statement.
Proof of Theorem 1.: The case \(S=\mathbb{N}\) follows immediately from Theorem 2 by translating from probability-generating functions back to probability measures.
Consider now \(S=\mathbb{Z}\), and let \(\Phi\colon\mathcal{P}(\mathbb{Z})\to\mathcal{P}(\mathbb{Z})\) be a support-preserving endomorphism. Then its restriction to \(\mathcal{P}(\mathbb{N})\) is equal to some \(\Phi_{\beta}\). Given \(\mu\in\mathcal{P}(\mathbb{Z})\), there is some \(z\in\mathbb{Z}\) such that \(\mu\ast\delta_{z}\in\mathcal{P}(\mathbb{N})\). Hence
\[\Phi[\mu\ast\delta_{z}]=\Phi_{\beta}[\mu\ast\delta_{z}]=\Phi_{\beta}[\mu]\ast \Phi_{\beta}[\delta_{z}]=\Phi_{\beta}[\mu]\ast\delta_{z},\]
since \(\Phi_{\beta}\) is a support-preserving endomorphism. On the other hand,
\[\Phi[\mu\ast\delta_{z}]=\Phi[\mu]\ast\Phi[\delta_{z}]=\Phi[\mu]\ast\delta_{z},\]
since \(\Phi\) is a support-preserving endomorphism. Hence
\[\Phi[\mu]\ast\delta_{z}=\Phi_{\beta}[\mu]\ast\delta_{z}\]
and so \(\Phi[\mu]=\Phi_{\beta}[\mu]\).
Finally, consider the case \(S=\mathbb{Q}\), and let \(\Phi\colon\mathcal{P}(\mathbb{Q})\to\mathcal{P}(\mathbb{Q})\) be a support-preserving endomorphism. For each \(n\in\mathbb{N}\), the semi-group \(\mathcal{P}(\mathbb{Z}/n)\) is isomorphic to \(\mathcal{P}(\mathbb{Z})\), and thus there is some \(\beta_{n}\) such that the restriction of \(\Phi\) to \(\mathcal{P}(\mathbb{Z}/n)\) is equal to \(\Phi_{\beta_{n}}\). But since \(\mathbb{Z}/n\) and \(\mathbb{Z}/m\) are both contained in \(\mathbb{Z}/(nm)\), \(\beta_{n}=\beta_{m}=\beta\). Finally, \(\mathcal{P}(\mathbb{Q})=\cup_{n}\mathcal{P}(\mathbb{Z}/n)\), and so \(\Phi=\Phi_{\beta}\).
### Proof of Theorem 3
The proof of Theorem 3 initially follows the argument of the proof of Theorem 2. We first analogously extend \(\Phi\) to a map from \(\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\) to \(\mathcal{M}(\mathbb{N})\) where
\[\mathcal{M}_{\mathbb{Q}}(\mathbb{N})=\left\{p(x)=\sum_{k=1}^{d}p_{k}x^{k} \,\Bigg{|}\,p_{k}\in\mathbb{Q},\ \ p(x)>0\text{ for all }x>0,\ \ p(1)=1\right\}.\]
The same argument as in the proof of Proposition 3 shows that there exists a multiplicative extension that preserves the degree. Note that instead of Poincare's Lemma (Lemma 1.1), one can use Polya's Lemma (Lemma 2.1) to ensure that for every \(p\in\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\) there exists an \(r\in\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\) such that \(p\cdot r\in\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\).
As in the proof of Theorem 2, we first consider linear polynomials with rational coefficients and note that the same argument of Proposition 3 shows that the analogous statement holds in the rational setting.
**Proposition 3.12**.: _For any support-preserving multiplicative \(\Phi\colon\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\), there exists a constant \(\gamma>0\) such that \(\Phi[x+a]=x+\gamma a\) for all \(a\in\mathbb{Q}_{>0}\)._
We next study quadratic polynomials with rational coefficients. The argument of Proposition 3 still applies.
**Proposition 3.13**.: _Let \(\Phi\colon\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) be support-preserving and multiplicative, and \(\Phi[x+1]=x+1\). Then its extension to \(\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\) satisfies \(\Phi[p]=p\) for any polynomial \(p\) of the form \(p(x)=x^{2}-a\gamma x+\gamma^{2}\) with \(a\in(-2,2)\cap\mathbb{Q}\) and \(\gamma\in\mathbb{Q}_{>0}\)._
Note that this does not apply to all rational quadratic polynomials, since the free coefficient is a square of a rational.
Accordingly, let \(\mathcal{M}^{\prime}_{\mathbb{Q}}(\mathbb{N})\subset\mathcal{M}_{\mathbb{Q}}( \mathbb{N})\) be the set of polynomials with rational coefficients which are products of linear rational polynomials (\(x+a\) for \(a\in\mathbb{Q}_{\geq 0}\)) and quadratic polynomials of the form considered in Proposition 3.12\((x^{2}-a\gamma x+\gamma^{2}\) for \(a\in(-2,2)\cap\mathbb{Q}\) and \(\gamma\in\mathbb{Q}_{>0}\)). Then the same proof of Theorem 2 yields the following:
**Proposition 3.14**.: _Let \(\Phi\) be an extension of a support-preserving multiplicative map from \(\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\) to \(\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\). Then there exists a constant \(\gamma>0\) such that \(\Phi[p](x)=p(\gamma x)/p(\gamma)\) for any \(p\in\mathcal{M}^{\prime}_{\mathbb{Q}}(\mathbb{N})\)._
Consider the topology on \(\mathcal{M}(\mathbb{N})\) given by \(\lim_{t}p^{(t)}=p\) if \(\lim_{t}\deg(p^{(t)})=\deg(p)\) and \(\lim_{t}p_{k}^{(t)}=p_{k}\) for all \(k\). Then \(\mathcal{M}^{\prime}_{\mathbb{Q}}(\mathbb{N})\) is a rich dense sub-semi-group of \(\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\), which is dense in \(\mathcal{M}(\mathbb{N})\). Thus, automatic continuity (Proposition 1.2) together with Proposition 3 yield Theorem 3.
Proof of Theorem 3.: Let \(\Phi\colon\mathcal{P}_{\mathbb{Q}}(\mathbb{N})\to\mathcal{P}(\mathbb{N})\) be support-preserving and multiplicative. Define \(\gamma\) by \(\Phi[x+1]=x+\gamma\), and let \(\Phi^{\prime}=\Psi_{\gamma}^{-1}\circ\Phi\). Then \(\Phi^{\prime}\) is support-preserving,
multiplicative, and satisfies \(\Phi^{\prime}[x+1]=x+1\). Extend it to a multiplicative, degree-preserving \(\Phi^{\prime}\colon\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\to\mathcal{M}(\mathbb{N})\). By Proposition 3.14, the restriction of \(\Phi^{\prime}\) to \(\mathcal{M}_{\mathbb{Q}}^{\prime}(\mathbb{N})\) is of the form \(\Psi_{\gamma^{\prime}}\), and since \(\Phi^{\prime}[x+1]=x+1\), we get \(\gamma^{\prime}=1\). Therefore, \(\Phi^{\prime}\) restricted to \(\mathcal{M}_{\mathbb{Q}}^{\prime}(\mathbb{N})\) is the identity map. Hence, by Proposition 1.2, \(\Phi^{\prime}\) is the identity map on the whole domain \(\mathcal{M}_{\mathbb{Q}}(\mathbb{N})\). Thus \(\Phi=\Psi_{\gamma}\).
### Proof of Proposition 1.2
Recall that Poincare's Lemma (Lemma 1.1) shows that for every \(q\in\mathcal{M}(\mathbb{N})\) there is a polynomial \(p\in\mathcal{P}(\mathbb{N})\) such that \(q\cdot p\in\mathcal{P}(\mathbb{N})\). Towards proving Proposition 1.2, we investigate a related question: Given \(p\in\mathcal{P}(\mathbb{N})\), for which \(q\in\mathcal{M}(\mathbb{N})\) does it holds that \(q\cdot p\in\mathcal{P}(\mathbb{N})\)?
Given \(p\in\mathcal{P}(\mathbb{N})\), denote by \(S_{p}\subseteq\mathcal{M}(\mathbb{N})\) the set of polynomials \(q\) such that \(q\cdot p\in\mathcal{P}(\mathbb{N})\):
\[S_{p}=\{q\in\mathcal{M}(\mathbb{N})\,:\,q\cdot p\in\mathcal{P}(\mathbb{N})\}.\]
A natural question is to understand when \(S_{p}\subseteq S_{p^{\prime}}\). For example, if \(p^{\prime}(x)=p^{2}(x)\) then clearly \(S_{p}\subseteq S_{p^{\prime}}\). The next lemma shows that if we further require that \(\deg(p^{\prime})\leq\deg(p)\) (and \(p_{0}\neq 0\)), then the containment \(S_{p}\subseteq S_{p^{\prime}}\) is only possible when \(p=p^{\prime}\).
**Lemma 3.15**.: _Consider \(p,p^{\prime}\in\mathcal{P}(\mathbb{N})\) such that \(\deg(p^{\prime})\leq\deg(p)\) and \(p_{0}\neq 0\). Then \(p=p^{\prime}\) if and only if \(S_{p}\subseteq S_{p^{\prime}}\)._
Proof.: Only one direction is non-trivial: proving the existence of \(q\in S_{p}\setminus S_{p^{\prime}}\) if \(p\neq p^{\prime}\).
Denote \(n=\deg(p)\). We claim that we can assume without loss of generality that \(p_{k}>0\) for \(k=0,\ldots,n\). Otherwise, replace \(p\) and \(p^{\prime}\) by \(p\cdot r\) and \(p^{\prime}\cdot r\) with \(r(x)=(x+1)^{n}\). If we can find \(q\) such that \(q\cdot p\cdot r\in\mathcal{P}(\mathbb{N})\) but \(q\cdot p^{\prime}\cdot r\not\in\mathcal{P}(\mathbb{N})\) then \(q\cdot r\in S_{p}\setminus S_{p^{\prime}}\).
The first step is to show the existence of a polynomial \(s(x)=\sum_{k=0}^{n+1}s_{k}x^{k}\) with \(\deg(s)=n+1\) such that all the coefficients \((p\cdot s)_{k}\) are non-negative for \(k=0,\ldots,n+1\) but \((p^{\prime}\cdot s)_{n+1}\leq-1\). These requirements are equivalent to a system of linear inequalities on coefficients of \(s\):
\[A=\begin{pmatrix}p_{0}&0&\cdots&\cdots&0\\ p_{1}&p_{0}&0&\ddots&0\\ \vdots&\vdots&\ddots&\ddots&0\\ p_{n}&p_{n-1}&\cdots&p_{0}&0\\ 0&p_{n}&p_{n-1}&\cdots&p_{0}\\ 0&-p^{\prime}_{n}&-p^{\prime}_{n-1}&\cdots&-p^{\prime}_{0}\end{pmatrix}, \qquad\vec{s}=\begin{pmatrix}s_{0}\\ s_{1}\\ \vdots\\ s_{n+1}\end{pmatrix},\qquad\vec{b}=\begin{pmatrix}0\\ 0\\ \vdots\\ \vdots\\ s_{n+1}\end{pmatrix}. \tag{3.6}\]
By the Farkas Lemma, (3.6) has a solution if and only if there is no way to combine the inequalities with non-negative coefficients to get the contradictory inequality \(0\geq 1\)
Formally, there must be no \((\lambda_{0},\ldots,\lambda_{n+2})\in\mathbb{R}_{\geq 0}^{n+2}\) such that
\[\left\{\begin{array}{ll}A^{T}\vec{\lambda}&=\vec{0}\\ \langle\vec{b},\vec{\lambda}\rangle&=1.\end{array}\right. \tag{3.7}\]
Here \(\lambda_{l}\geq 0\) is interpreted as the weight of the inequality \(l\) in the original system, and \(A^{T}\) denotes the transposed matrix \(A\). Let us write down the equations (3.7) explicitly:
\[\left\{\begin{array}{rcl}p_{0}\lambda_{0}+\ldots+p_{n}\lambda_{n}&=&0\\ (p_{0}\lambda_{k}+\ldots+p_{n-k}\lambda_{n})+p_{n-k+1}\lambda_{n+1}-p^{\prime }_{n-k+1}\lambda_{n+2}&=&0,\;\;k=1,\ldots,n+1\\ \lambda_{n+2}&=&1\end{array}\right.\]
Since all coefficients of \(p\) are assumed to be positive and \(\lambda_{k}\geq 0\) for all \(k\), the first equation implies \(\lambda_{0}=\lambda_{1}=\ldots=\lambda_{n}=0\). Hence, the second family of equations gives \(p_{n-k+1}\lambda_{n+1}-p^{\prime}_{n-k+1}\lambda_{n+2}=0\) for \(k=1,\ldots,n+1\). Since \(p\) and \(p^{\prime}\) are non-zero and not equal, these identities can only hold if \(\lambda_{n+1}=\lambda_{n+2}=0\). Since \(\lambda_{n+2}=1\) by the last equation, we conclude that the system (3.7) does not have a solution. Thus the system (3.6) has a solution and the polynomial \(s(x)\) exists.
We now argue that the polynomial \(q\in\mathcal{M}(\mathbb{N})\) defined (up to normalization) by
\[q(x)=s(x)+C\cdot x^{n+2}\]
has the desired properties for a large enough constant \(C>0\). Indeed, the coefficients of \(x^{k}\) with \(k=0,\ldots,n+1\) in the products \(p\cdot q\) and \(p^{\prime}\cdot q\) are the same as in \(p\cdot s\) and \(p^{\prime}\cdot s\). Hence, \((p^{\prime}\cdot q)_{n+1}\leq-1\) and so \(p^{\prime}\cdot q\notin\mathcal{P}(\mathbb{N})\). It remains to be shown that we can always choose \(C\) so that \((p\cdot q)_{k}\geq 0\) for \(k=n+2,\ldots 2n+2\). For such \(k\), we get
\[(p\cdot q)_{k}=(p\cdot s)_{k}+C\cdot p_{k-(n+2)}\]
Since all the coefficients of \(p\) are strictly positive, choosing
\[C=\max_{k=n+2,\ldots 2n+2}\frac{(p\cdot s)_{k}}{p_{k-(n+2)}}\]
ensures that \(p\cdot q\in\mathcal{P}(\mathbb{N})\). As \(p\cdot q\in\mathcal{P}(\mathbb{N})\) and \(p\in\mathcal{P}(N)\), we conclude that the constructed polynomial \(q\) belongs to \(\mathcal{M}(\mathbb{N})\) and complete the proof.
The next lemma strengthens the previous, again considering \(p,p^{\prime}\) such that \(\deg(p^{\prime})\leq\deg(p)\) and \(p_{0}\neq 0\). It shows that when \(p\neq p^{\prime}\), then not only is \(S_{p}\) not contained in \(S_{p^{\prime}}\), but moreover, the interior of \(S_{p}\) is not contained in \(S_{p^{\prime}}\). Recall that our topology is the one under which \(\lim_{t}p^{(t)}=p\) if \(\lim_{t}\deg(p^{(t)})=\deg(p)\) and \(\lim_{t}p_{k}^{(t)}=p_{k}\) for all \(k\). Note that \(S_{p}\) trivially has a non-empty interior because, for any \(p\), it contains \(\mathcal{P}(\mathbb{N})\), which has a non-empty interior. The lemma implies that the interior of \(S_{p}\) extends non-trivially beyond that.
**Lemma 3.16**.: _Consider \(p,p^{\prime}\in\mathcal{P}(\mathbb{N})\) such that \(\deg(p^{\prime})\leq\deg(p)\) and \(p_{0}\neq 0\). Then \(p=p^{\prime}\) if and only if the interior of \(S_{p}\) is contained in \(S_{p^{\prime}}\)._
Proof.: One direction is again immediate. For the other direction, we need to show that if \(M\) is a dense subset of \(\mathcal{M}(\mathbb{N})\), then \(p\neq p^{\prime}\) implies that there is an element of \(M\) in \(S_{p}\setminus S_{p^{\prime}}\).
Fix \(p\neq p^{\prime}\) with \(p_{0}\neq 0\) and \(\deg(p^{\prime})\leq\deg(p)\). By Lemma 3.15, there is \(q\in\mathcal{M}(\mathbb{N})\) such that \(q\in S_{p}\setminus S_{p^{\prime}}\), i.e., \(p\cdot q\) has non-negative coefficients, but \(p^{\prime}\cdot q\) has a negative coefficient.
Denote by \(m\) the degree of \(q\) and consider \(r(x)=(x+1)^{m}\). For \(\varepsilon>0\), define \(q^{\varepsilon}=q+\varepsilon\cdot r\). Note that \(p\cdot q^{\varepsilon}\) has strictly positive coefficients. Since the coefficients of a product depend continuously on the multiplicands, we can choose small \(\varepsilon>0\) such that \(p^{\prime}\cdot q^{\varepsilon}\) has a negative coefficient.
Let \(q^{(t)}\in M\), \(t=1,2,\ldots\), be a sequence of polynomials of degree \(m\) such that \(q^{(t)}_{k}\to q^{\varepsilon}_{k}\) as \(t\to\infty\) for all \(k=0,1,\ldots,m\). Using the continuity of the product again, we conclude that, for large enough \(t\), all the coefficients of \(p\cdot q^{(t)}\) are strictly positive, but \(p^{\prime}\cdot q^{(t)}\) has a negative coefficient.
Proof of Proposition 1.2.: We first consider a polynomial \(p\in M^{\prime}\cap\mathcal{P}(\mathbb{N})\), prove that \(\Phi[p]=p\), and then extend the conclusion to all \(p\in M^{\prime}\).
We assume without loss of generality that \(p_{0}\neq 0\). Indeed, if \(p(x)=x^{m}\cdot r(x)\) for some \(r\) with \(r_{0}\neq 0\), then, by the richness hypothesis, \(x\) is contained in \(M\) and \(r\) is contained in \(M^{\prime}\). Hence, \(\Phi[p]=\Phi[x]^{m}\cdot\Phi[r]=x^{m}\cdot\Phi[r]\) and proving \(\Phi[r]=r\) would imply \(\Phi[p]=p\).
Towards a contradiction, assume that \(p\neq\Phi[p]\) and denote \(p^{\prime}=\Phi[p]\). As \(\Phi\) is degree-preserving, \(\deg(p^{\prime})=\deg(p)\). By Lemma 3.16, there is a polynomial \(q\in M\) such that \(q\in S_{p}\) and \(q\not\in S_{p^{\prime}}\). In other words, \(p\cdot q\) has non-negative coefficients, and \(p^{\prime}\cdot q\) has a negative coefficient.
Since \(q\in M\), by the proposition hypothesis, \(\Phi[q]=q\) and thus
\[p^{\prime}\cdot q=\Phi[p]\cdot\Phi[q]=\Phi[p\cdot q].\]
Since \(p\cdot q\) has non-negative coefficients, so does \(\Phi[p\cdot q]\) because \(\Phi\) maps \(M^{\prime}\cap\mathcal{P}(\mathbb{N})\) to \(\mathcal{P}(\mathbb{N})\). We get that \(p^{\prime}\cdot q\) has no negative coefficients and reach a contradiction. Thus \(p^{\prime}=p\) or, equivalently, \(\Phi[p]=p\) for \(p\in M^{\prime}\cap\mathcal{P}(\mathbb{N})\).
It remains to show that \(\Phi[p]=p\) for \(p\in M^{\prime}\setminus\mathcal{P}(\mathbb{N})\). By the density of \(M\), there is a constant \(a>0\) such that \(q(x)=x+a\) belongs to \(M\). By Polya's Lemma (Lemma 2.2), \(p(x)\cdot(x+a)^{n}\) has non-negative coefficients for some \(n\) large enough. Denote \(q(x)=(x+a)^{n}\). Both multiplicands in \(p\cdot q\) belong to \(M^{\prime}\), so the product also does. Hence, \(p\cdot q\in M^{\prime}\cap\mathcal{P}(\mathbb{N})\). By the already proved statement, \(\Phi[p\cdot q]=p\cdot q\). On the other hand, \(\Phi[p\cdot q]=\Phi[p]\cdot\Phi[q]\). Since \(q\in M\), we get \(\Phi[q]=q\) and conclude that \(\Phi[p]=p\). | ボルツマン分布は統計力学において、ある温度を持つシステムにおける状態分布を記述するために用いられます。この分布を、独立系における唯一性を満たすものとして、新しい特徴付けで表しました。この定理は、有限支持確率ベクトルを持つ連鎖半群の対称性、または、他の言い方では、非負係数の多項式の乗算半群の対称性に関する述べられています。 |
2303.03281 | Visual Place Recognition: A Tutorial | Localization is an essential capability for mobile robots. A rapidly growing
field of research in this area is Visual Place Recognition (VPR), which is the
ability to recognize previously seen places in the world based solely on
images. This present work is the first tutorial paper on visual place
recognition. It unifies the terminology of VPR and complements prior research
in two important directions: 1) It provides a systematic introduction for
newcomers to the field, covering topics such as the formulation of the VPR
problem, a general-purpose algorithmic pipeline, an evaluation methodology for
VPR approaches, and the major challenges for VPR and how they may be addressed.
2) As a contribution for researchers acquainted with the VPR problem, it
examines the intricacies of different VPR problem types regarding input, data
processing, and output. The tutorial also discusses the subtleties behind the
evaluation of VPR algorithms, e.g., the evaluation of a VPR system that has to
find all matching database images per query, as opposed to just a single match.
Practical code examples in Python illustrate to prospective practitioners and
researchers how VPR is implemented and evaluated. | Stefan Schubert, Peer Neubert, Sourav Garg, Michael Milford, Tobias Fischer | 2023-03-06T16:52:11 | http://arxiv.org/abs/2303.03281v2 | # Visual Place Recognition: A Tutorial
###### Abstract
A novel approach to visual place recognition is to use a novel approach to visual place recognition. A novel approach to visual place recognition is to use a novel approach to visual place recognition.
Essentially, VPR is an image retrieval problem where the context is to recognize previously seen places. This context provides additional information and structure beyond a general image retrieval setup. Many VPR methods exploit the context to match images of the same places in a wide range of environments, including those with significant appearance and viewpoint differences. For example, one piece of additional information that is often exploited is that consecutive images taken by a camera mounted on a car will depict spatially close places in the world.
Figure 1 provides an overview of the typical steps and components of a basic VPR pipeline. Given a reference set composed of database images \(\mathbf{I}_{i}\in DB\) of known places, and one or multiple query images \(\mathbf{I}_{j}\in Q\), the goal is to find matches between these two sets, i.e., those instances where image \(j\) from the query set shows the same place as image \(i\) from the database. To find these matches, it is essential to compute one or multiple descriptors \(\mathbf{d}\) for each image - these descriptors should be similar for images showing the same place and dissimilar for different places. A descriptor is typically represented as a numerical vector (e.g., 128-D or 4,096-D). Conceptually, we can think of a matrix \(\mathbf{S}\) of all pairwise descriptor similarities \(s_{ij}\) between the database and query images as the basis for deciding which images should be matched. In practice, we must carefully choose the algorithms used to compute and compare the image descriptors \(\mathbf{d}\), taking into account the specific challenges and context of the VPR problem at hand. The remainder of this tutorial will provide more detail on these aspects and discuss the VPR problem from a broader theoretical and practical perspective.
### _The VPR problem and its details as reflected in this tutorial_
Section II of this tutorial will outline the relevance and history of the VPR problem, as well as its relation to other areas, particularly its importance for topological Simultaneous Localization And Mapping (SLAM), where the database \(DB\) corresponds to the set of previously visited places in the map. In fact, one of the original drivers for VPR research was the generation of loop closures for SLAM systems, that is, recognizing a place when revisiting it (e.g., in a loop) and tying the current observation with that already in the map (i.e., closure) [7]. One of the earliest examples of such a topological SLAM system is FAB-MAP [8], also referred to as 'appearance-only SLAM', where loop closure generation is based on appearance only (i.e., images), thus different from 3D/metric SLAM systems such as ORB-SLAM [9] where the map and the visual landmarks are expressed in 3D.
The definition of a "_place_" is an integral aspect of VPR. In this tutorial, we follow the definition that two images must have some visual overlap, i.e., shared image content like same buildings, to be considered as "taken at the same place" [2]. This definition allows a subsequent transformation between matched images for tasks like visual localization, mapping, or SLAM - indeed, the required amount of visual overlap depends on the specific application. We note that an alternative definition used in particular by some researchers [10, 11] is that two places are matching purely based on their position, without taking the orientation, and in turn visual overlap, into account. Section III will present different applications for VPR and discuss the various subtypes of VPR problems that arise from variations in the available input data, the required data processing, and the requested output.
VPR algorithms are often tailored to the particular properties of an application. Section IV will provide details on a _generic VPR pipeline_ that serves as a common basis for diverse practical settings and their unique characteristics. From this section onward, this tutorial includes practical code examples in Python.
It is important to note that not all VPR algorithms address the same VPR problem, e.g., regarding the requested number of image matches per query. This is particularly critical when it comes to evaluating and comparing the performance of different VPR algorithms. Section V explains and discusses the evaluation pipelines that consider various datasets, ground truth subtleties, and different performance metrics.
The properties of the underlying data have a significant impact on the difficulty of the resulting VPR problem and the suitability of a particular algorithm. Section VI will discuss challenges such as severe appearance changes due to varying illumination or weather conditions, large viewpoint changes between images of the same place, and perceptual aliasing, i.e., the challenge that images taken at two distinct places can appear remarkably similar. This section will also present common ways of addressing these challenges to improve robustness, performance, runtime and memory efficiency. These approaches include methodological extensions of the general purpose pipeline that partially build upon a robotic context (e.g., with image sets recorded as videos along trajectories) where VPR differs from pure image retrieval. This often allows the exploitation of additional knowledge and information such as spatio-temporal sequences (i.e., consecutive images in the database \(DB\) and query \(Q\) are also neighboring in the world) or intra-set similarities (i.e., similarities _within_\(DB\) or \(Q\)).
## II History, Relevance and Related Areas
Visual Place Recognition (VPR) research can be traced back to advances in visual SLAM, visual geo-localization, and image retrieval applied to images of places [12]. In the robotics literature, VPR has historically been called loop closure detection and was mainly used for this purpose for visual SLAM [12]. VPR gained more prominence in the field as the earlier metric SLAM methods based on global and local bundle adjustment techniques could only handle limited-size environments, thus paving way for topological SLAM techniques based on bag-of-words approaches [13], such as FAB-MAP [8]. In addition to its relevance within SLAM pipelines, VPR also remains a crucial component of localization-only pipelines where the map is available a priori.
Early work on VPR mainly focused on the recognition of places under constant environmental conditions or after slight illumination changes due to weather or time of day. However, starting around 2012, an increasing number of works explored VPR under severe condition changes, e.g., due to
the day-night cycle or seasonal changes [14]. This shift can also be seen by the emergence of datasets with condition changes that have appeared since 2012 [1, 3]. In 2014, the use of deep learning for VPR [15] emerged as a way to handle challenging data and has since proven effective in changing environments [16]. In addition to images and image descriptors, VPR research has also explored the use of additional information, such as sequences, intra-set similarities, weak GPS signals, or odometry, to improve performance [17].
In terms of the relationship between VPR and other fields, we recommend the following tutorials: [12] provides an overview of probabilistic SLAM and includes a section on loop closure detection, although a lot of progress has been made in VPR as this tutorial was published more than 15 years ago. [7] specifically investigates the loop closure problem in SLAM. [18] provides a practical introduction to SLAM with example code for the Robot Operating System (ROS). [19] discusses visual odometry, which involves estimating the ego-motion of an agent based on visual input. Visual odometry is thus complementary to VPR, and can be combined with VPR to detect loop closures when building a SLAM system.
Beyond loop-closure detection, VPR is necessary if global position sensors such as global navigation satellite systems (GNSS) like GPS, Galileo, or BeiDou are not available or are inaccurate. In urban environments, buildings or other structures can lead to "urban canyons" that block line-of-sight satellite signals, causing occlusions that prevent a GNSS receiver from obtaining accurate position information. In addition to occlusions, reflections of GNSS signals off buildings and other structures, so called non-line-of-sight signals, can further hinder the GNSS accuracy. This issue is not limited to urban environments, as similar occlusions and reflections can occur in natural environments, such as in valleys or canyons. Similarly, indoor environments and caves also hinder GNSS due to the absorption or reflection of satellite signals by walls.
Alternatively, VPR can serve as a redundant component in autonomous systems for fault tolerance and general GNSS outages, such as satellite service disruptions, degradation, or position/time anomalies. It is worth noting that all GNSS systems can potentially be hacked or blocked for non-military use by a central authority. Other systems may not be equipped with a GNSS receiver due to cost or security concerns. In the case of robotic extraterrestrial missions, installing a GNSS system may be too expensive or time-consuming.
## III VPR problem categories and use cases
In the localization and mapping literature, VPR has been used in different ways depending on three key attributes of its formulation: the _input_, which deals with how the reference and query images are made available (i.e., single-session vs. multi-session); _data processing_, that defines the mode of operation (i.e., online vs. batch); and _output_, that determines the kind of expected output (i.e., single-best-match vs. multi-match). The following Section III-A explains these problem categories in more detail. Section III-B then presents different VPR use cases using these categories. Table I summarizes these use cases, along with their required input and data processing.
### _VPR problem categories_
We distinguish three main dimensions along which VPR problems can vary, creating different VPR problem categories or subtypes:
1. **Input - single-session VPR vs. multi-session VPR**: _Are there two separate input sets, one for the database \(DB\) and one for the query \(Q\), or is it a single set that is compared to itself?_ Single-session VPR is the matching of images within a single set of query images \(Q\), while \(DB\) can be considered empty (i.e., \(DB=\varnothing\)). Some publications describe single-session VPR as having a query set \(Q\) that equals the database (i.e., \(Q=DB\)) or with a query set \(Q=\{I_{t}\}\) that only contains the last query image \(I_{t}\), while earlier query images are appended to the database with \(DB\gets DB\cup\{I_{t-1}\}\). A practical consideration in this case is the suppression of matches with recently acquired images - while full SLAM systems typically rely on a motion model for such suppression, standalone VPR systems often use heuristics. In contrast, multi-session VPR is the matching of the two disjoint image sets (i.e., \(DB\cap Q=\varnothing\)), which were recorded at different times (e.g., summer and winter) or by different platforms (e.g., mobile robot and cell phone).
2. **Data processing - online VPR vs. batch VPR**: _Are the images available and processed individually, one after the other, or are they all available in a single batch from the beginning?_ Online VPR has to deal with a growing set \(Q\) (i.e., \(Q\neq\mathrm{const}\)) and a set \(DB\) that is either given (i.e., \(DB=\mathrm{const}\)) or also growing (i.e., \(DB\neq\mathrm{const}\)). In contrast, batch VPR can build upon the full sets \(Q\) (i.e., \(Q=\mathrm{const}\)) and \(DB\) (i.e., \(DB=\mathrm{const}\)). Growing image sets in the case of online VPR limit the number of viable methods. For example, approaches like descriptor standardization [20] based on the statistics of _all_ image descriptors or similarity matrix decomposition [21] cannot be used without modifications. This is further discussed in Section VI.
3. **Output - single-best-match VPR vs. multi-match VPR**: _Is the intended output for a query image a single image from the database that shows the same place, or do we request all images of this place?_ Single-best-match VPR only returns the best matching database image \(I_{i}^{*}\in DB\) per query image \(I_{j}\in Q\). In contrast, the aim in multi-match VPR is finding _all_ matching database images for each query image. In practice, the difference between single-best-match VPR and multi-match VPR often boils down to finding either the maximum similarity between a query and all database images or all similarities above a certain threshold, as shown in Section IV-E. Identifying all matching images is often more challenging than finding only one correct match, as it requires an explicit decision for each database image whether it shows the same place as the query image or not [1].
Let us illustrate these problem categories with the example of determining the rough pose [x, y, heading, floor] of a cell phone in a building, e.g., to guide persons to desired places. To achieve this, we first need to map the building before the person can use their cell phone to localize in the building. For this first step of mapping the building, we could use a manually controlled mobile robot equipped with a camera to collect a query set \(Q^{\text{mapping}}\) of images together with some additional sensor data like odometry. Given these images of all places, we can run a mapping algorithm that processes all images and other data to obtain a metric map of the building, which associates all images in \(Q^{\text{mapping}}\) with metric poses. Part of this mapping is a _single-session batch multi-match VPR_ for loop closure detection that compares the whole image set \(Q^{\text{mapping}}\) (batch VPR) to itself (single-session VPR) to find _all_ loop closures for each image (multi-match VPR). Here, batch processing the whole set \(Q^{\text{mapping}}\) allows the application of computationally expensive but accurate algorithms.
After mapping (potentially years later), the second step is the actual localization of a cell phone using its camera stream. When localizing, we treat the robot's mapping query set \(Q^{\text{mapping}}\) as database \(DB^{\text{loc}}\) and compare it to query images \(Q^{\text{loc}}\) from a cell phone's camera. To determine the location of a cell phone, a _multi-session online single-best-match VPR_ can be used that compares the stream of query images \(Q^{\text{loc}}\) to the fixed database \(DB^{\text{loc}}\) (multi-session VPR) online (online VPR) to find the best matching database image (single-best-match VPR) with its corresponding pose information.
In summary, VPR can be used for a variety of different use cases, as discussed in more detail in the following Section III-B and shown in Table I. While each use case typically requires a certain combination of the _input_ (single-session/multi-session VPR) and _data processing_ (online/batch VPR) VPR categories, the choice of the _output_ category (single-best-match/multi-match VPR) rather depends on the algorithm that is used after VPR. For example, in graph-based SLAM [12], each node encodes the pose of an image in \(Q\). The corresponding edges of connected nodes represent the transformation between them. An edge can be established either between temporally consecutive nodes (using the odometry) or between nodes that were identified as loop closure by VPR. Here, single-best-match VPR could be used to match and fuse two nodes which correspond to the same place to represent each place always by only one node. Alternatively, multi-match VPR could be used to create multiple edges between all existing nodes of the same place. This is particularly helpful if we cannot guarantee that there
is a single node for each image in the graph, or if we perform a batch optimization of the poses using a robust optimization approach that can benefit from the additional information provided by multiple matches while handling potential outlier matchings.
### _VPR use cases_
VPR is a key component in a variety of robotic applications, including autonomous driving, agricultural robotics, and robotic parcel delivery, as well as in the creation of a metaverse. Some common tasks that VPR is used for include:
1. **Candidate selection for 6 DoF visual localization**[22]: 6 Degree Of Freedom (DoF) visual localization (also termed city-scale/natural geo-localization) involves estimating the 6D pose (position and orientation) of a camera in a particular environment. _Multi-session online VPR with fixed \(DB\)_ is used to select candidates \(\mathbf{I}_{i}\in DB\) that have the highest similarity to the current query images \(\mathbf{I}_{j}\in Q\) (cf. Figure 2, top). These candidates can then be used for a computationally intensive 6D pose estimation using local image descriptors and more complex algorithms, which would be infeasible for the complete \(DB\) set. For example, in [23] the place recognition method NetVLAD [24] was used for candidate selection before performing pose estimation with local descriptors.
2. **Loop closure detection and re-localization for online SLAM**[12]: Online SLAM is used to estimate the current pose of a camera while creating a map of the environment at the same time. _Single-session online VPR_ is used for loop closure detection (i.e., the recognition of previously visited places), as shown in Figure 2 (mid) to compensate for accumulated errors in odometry data and create a globally consistent map. It is also used for re-localization in the event of mislocalization or if the camera/robot was moved by hand (known as the kidnapped robot problem).
3. **Loop closure detection for mapping**[25, 12]: Mapping (also full SLAM or offline SLAM) involves estimating the entire path at once to generate a map. This allows for the use of _single-session batch VPR_ for loop closure detection (cf. Figure 2, mid), which is based on slower but more robust algorithms that run on powerful hardware.
4. **Loop closure detection for multi-session mapping**[26]: Multi-session mapping combines the results of multiple SLAM missions performed repeatedly over time in the same environment. _Multi-session batch VPR_ is used to find shared places between the individual maps of all missions for map merging (cf. Figure 2, bottom). Alternatively, _multi-session online VPR with a given DB_ can be used to detect previously mapped areas (potentially for loop closure) and include unseen areas of the map in real-time.
5. **Detection of shared places for multi-robot mapping**[27]: Multi-robot mapping (also termed decentralized SLAM) involves the distributed mapping of an environment using multiple robots. Here, _multi-session online VPR with a growing \(DB\)_ is used to find shared places between the individual maps of each robot for subsequent map merging, as shown in Figure 2 (bottom).
In summary, this section provided an overview of the different problem categories and corresponding subtypes of VPR and discussed common use cases where VPR is applied.
## IV A generic pipeline for VPR
This section outlines a generic pipeline for Visual Place Recognition (VPR). The steps involved in this pipeline are shown in Figure 1. The inputs to the pipeline are two sets of images \(DB\) and \(Q\) (these may be the same for single-session VPR, as explained in Section III). The pipeline produces matching decisions, meaning that for each query image \(\mathbf{I}_{j}\in Q\), one or more database images \(\mathbf{I}_{i}\in DB\) can be associated. The pipeline includes these intermediate steps and components: 1) computing image-wise descriptors, 2) pairwise comparing of descriptors to 3) create a descriptor similarity matrix \(\mathbf{S}\), and 4) making matching decisions. In the following subsections, we will discuss each of these elements in more detail. Extensions to this generic pipeline that can be used to improve performance and robustness against various challenges are presented in Section VI.
### _Inputs: The database and query image sets_
To recap, two sets of images serve as the input in a VPR pipeline: the database set \(DB\) and a set of current images in the query set \(Q\). The \(DB\) set, which is also called the reference set, represents a map of known places, and is often recorded under ideal conditions (e.g., sunny), or by a different platform than \(Q\) (e.g., a second robot). The query set \(Q\), on the other hand, is the "live view" recorded by a different platform than \(DB\) or after \(DB\) - potentially days, months, or even years later. Both sets will have a geographical overlap and share some or all seen places.
There are different VPR problem categories: using just a query set \(Q\) (single-session VPR) or using both the \(DB\) and \(Q\) sets (multi-session VPR). Also, the image sets can either be specified before processing (batch VPR) or grow during an online run (online VPR).
Code Snippet 1 provides example code for loading a dataset with both image sets \(Q\) and \(DB\), as well as the ground truth matrices \(\mathbf{GT}\) and \(\mathbf{GT}^{soft}\) (cf. Section V-B).
### _Image-wise descriptor computation_
This section describes the process of computing image descriptors, which are abstractions of images that extract features from raw pixels in order to be more robust against changes in appearance and viewpoint (step 1 in Figure 1, see also Code Snippet 2). There are two types of image descriptors:
1. **Holistic descriptors** (also called global descriptors) represent an image \(\mathbf{I}_{i}\in DB,Q\) with a single vector \(\mathbf{d}_{i}\in\mathbb{R}^{d}\) (cf. Code Snippet 2). This allows for efficient pairwise descriptor comparisons with low runtimes. Note that when exhaustive k-nearest neighbor search (kNN) is used to obtain the nearest neighbors for a candidate selection of similar database descriptors, the execution time scales linearly with the descriptor dimension.
* **Local descriptors** encode an image \(\mathbf{I}_{i}\) with a set \(D_{i}=\{\mathbf{d}_{k}\mid k=1,\ldots,K\}\) of vectors \(\mathbf{d}_{k}\in\mathbb{R}^{d}\) at \(K\) regions of interest (cf. Code Snippet 2). They often provide better performance than holistic descriptors, but require computationally expensive methods for local feature matching like a left-right check (also termed mutual matching) [29], a homography estimation [30], a computation of the epipolar constraint [31], or deep-learning matching techniques, e.g., SuperGlue [32]. Therefore, local descriptors are typically used in a hierarchical pipeline, where first the holistic descriptors are used to retrieve the top-\(K\) matches, which are then re-ordered using local descriptor matching.
To convert a set of local descriptors from a single image into a holistic descriptor, one can use **local feature aggregation** methods like Bag of Visual Words (BoVW) [33], Vector of Locally Aggregated Descriptors (VLAD) [34] or Hyper-Dimensional Computing (HDC) [29]. In a hierarchical pipeline, this allows a local descriptor to be used for both candidate selection (after aggregation) and verification (with the raw local descriptors).
As the descriptor computation is one of the first steps in a pipeline for VPR, it has a significant impact on the performance of subsequent steps and the overall performance of the VPR system. The algorithm used to obtain the descriptors determine how well the descriptors are suited for a specific environment, the degree of viewpoint change, or the type of environmental condition change. For example, CNN-based holistic descriptors like AlexNet-conv3 [16] or HybridNet [35] perform well in situations with low or negligible viewpoint changes, but perform poorly with large viewpoint changes. On the other hand, VLAD-based [34] descriptors like NetVLAD [24] tend to perform better in settings with large viewpoint changes.
Additionally, the specific training data of deep-learned descriptors affect the performance in different environments. For example, some descriptors may perform better in urban environments, while others may be more effective in natural environments [36] or in specific geographic regions such as Western cities [37].
### _Descriptor similarity between two images_
To compare the image descriptors of two images, a measure of similarity or distance must be calculated (see step 2 of Figure 1 and Code Snippet 3). This process compares the descriptors \(\mathbf{d}_{i}\) and \(\mathbf{d}_{j}\) (holistic) or \(D_{i}\) and \(D_{j}\) (local) of images \(i\) and \(j\). Note that similarity \(s_{ij}\) and distance \(dist_{ij}\) can be related through inversely proportional functions such as
\[s_{ij}=-dist_{ij}\,, \tag{1}\]
or the reciprocal
\[s_{ij}=\frac{1}{dist_{ij}}. \tag{2}\]
Holistic descriptors can be compared more efficiently than local descriptors, as they only require simple and computationally efficient metrics like the cosine similarity
\[s_{ij}=\frac{\mathbf{d}_{i}^{T}\cdot\mathbf{d}_{j}}{\|\mathbf{d}_{i}\|\cdot \|\mathbf{d}_{j}\|}, \tag{3}\]
or the negative Euclidean distance
\[s_{ij}=-\|\mathbf{d}_{i}-\mathbf{d}_{j}\|. \tag{4}\]
In contrast, comparing local descriptors requires more complex and computationally expensive algorithmic approaches, as previously mentioned in Section IV-B.
### _The pairwise similarity matrix \(\mathbf{S}\)_
The pairwise descriptor similarity matrix \(\mathbf{S}\) is a key component of VPR. As shown in step 3 of Figure 1, \(\mathbf{S}\) contains all calculated similarities \(s_{ij}\) between the descriptors of images in the database and query sets. In single-session VPR, \(\mathbf{S}\) has dimensions \(|Q|\times|Q|\), while in multi-session VPR, \(\mathbf{S}\) has dimensions \(|DB|\times|Q|\). Depending on the approach used, \(\mathbf{S}\) may be dense (if all descriptors are compared) or sparse (if only a subset of descriptors is compared using approximate nearest neighbor search [38] or sequence-based comparison strategies [39]).
The overall appearance of \(\mathbf{S}\) is influenced by the camera's trajectories during acquisition of \(Q\) and \(DB\), as illustrated in Figure 3. The pattern of high similarities within \(\mathbf{S}\) can have a significant impact on the performance of the VPR pipeline, and may enable or hinder the use of certain algorithmic steps for performance improvements. The following relations between camera trajectories and the appearance of \(\mathbf{S}\) can be observed (cf. Figure 3 for corresponding examples in a map):
1. [label=)]
2. **General**: If images in \(DB\) and \(Q\) are recorded at arbitrary positions without a specific order, there are no discernible patterns in \(\mathbf{S}\). This is typical for general visual localization and global geo-localization.
3. **Sequence**: If images in \(DB\) and \(Q\) are recorded along trajectories as spatio-temporal sequences (i.e., consecutive images are also neighbors in the world), continuous lines of high similarities may be observed in \(\mathbf{S}\). This setup is typical for many robotic tasks, including online SLAM, mapping, and multi-robot/multi-session mapping (Section III). In this setup, sequence-based methods can be used for performance improvements (cf. Section VI). The camera's trajectories can affect \(\mathbf{S}\) in the following ways: 1. [label=)]
1. [label=)]
2. **Speed**: If the camera moves at the same speed in the same locations in \(DB\) and \(Q\), lines of high similarities with \(45^{\circ}\) slope will be observed. Otherwise, the slope will vary.
3. **Exploration**: If a place shown in a query image \(Q\) is not present in \(DB\), the line of high similarities will be discontinuous.
4. **Stops**: If the camera stops temporarily (zero velocity) during either the database run or the query run, it will result in multiple consecutive matches in the other set.
Fig. 3: Relation between the similarity matrix \(\mathbf{S}\) and the trajectory during the database and query run. Green and orange cameras depict images in \(DB\) and \(Q\), respectively. Green and orange lines indicate that images were recorded as video along a trajectory (also called spatio-temporal sequence). In b.i), a rabbit/turtle indicate fast/slow speeds when traversing the route, and similarly in b.iii) traffic lights indicate stops in \(Q\) (T=2), \(DB\) (T=3) or both (T={2,4}) for T time steps.
* **Stops in \(\mathbf{DB}\)**: Stops in the database run will result in a vertical line (within the same column) of high similarities in \(\mathbf{S}\).
* **Stops in \(\mathbf{Q}\)**: Stops in the query run will result in a horizontal line (within the same row) of high similarities in \(\mathbf{S}\).
* **Stops in \(\mathbf{DB}\) & \(\mathbf{Q}\)**: If the camera stops in both the database run and query run at the same place, a block of high similarities will be observed in \(\mathbf{S}\).
* **Loops in DB**: Loops in \(DB\) can result in multiple matching database images for a single query image in \(Q\). Unlike stops, the multiple matching images due to a loop are not consecutive in their image set.
* **Loops in DB & Q**: Loops in \(DB\) and \(Q\) can result in additional matching query images for a single database image in \(DB\). This results in a more complex structure of high similarities in \(\mathbf{S}\).
### _Output: Matching decisions_
The output of a VPR system is a set of matching decisions \(m_{ij}\in\mathbf{M}\) (step 4 in Figure 1 and Code Snippet 4) with \(\mathbf{M}\in\mathbb{B}^{|Q|\times|Q|}\) (single-session VPR) or \(\mathbf{M}\in\mathbb{B}^{|DB|\times|Q|}\) (multi-session VPR) that indicate whether the \(i\)-th database/query image and the \(j\)-th query image show the same place (\(m_{ij}=true\)) or different places (\(m_{ij}=false\)). Existing techniques for matching range from choosing the best match per query or a simple thresholding of the pairwise descriptor similarities \(s_{ij}\in\mathbf{S}\) to a geometric verification with a comparison of the spatial (using e.g., the epipolar constraint) or semantic constellation of the scene. For example in Code Snippet 4, \(M_{1}\) is computed by selecting the best matching database image per query image, i.e., the maximum similarity \(s_{ij}\) per column in \(\mathbf{S}\in\mathbb{R}^{|DB|\times|Q|}\) (single-best-match VPR). Another example is the computation of \(M_{2}\) in Code Snippet 4, where a similarity threshold \(\theta\) is applied to \(\mathbf{S}\): If \(s_{ij}\geq\theta\), the \(i\)-th and \(j\)-th images are assumed to show the same place (multi-match VPR). The next section is concerned with the performance evaluation of these outputs.
## V Evaluation of the performance
This section is concerned with the evaluation of the matching decisions \(\mathbf{M}\) or the pairwise similarities \(\mathbf{S}\), which allows the comparison of different VPR methods. This
requires datasets, corresponding ground truth, and performance metrics. In the following, we outline these components for evaluation and discuss their properties and potential pitfalls.
### _Datasets_
For VPR, a dataset is composed of one or multiple image sets that have to be matched in order to find shared places. For example, the popular Nordland dataset [40] provides four image sets, one for each season, i.e., spring, summer, fall, and winter. These can be arbitrarily combined for VPR, but a typical choice might be to use _summer_ as \(DB\) and _spring_, _fall_ or _winter_ as \(Q\).
Existing datasets vary in the type of environment as well as in the type and degree of appearance and viewpoint change: The **type of environment** includes urban environments (e.g., Oxford RobotCar [41]), suburban environments (e.g., StLucia Various Times of the Day [42]) and natural environments like countryside (e.g., Nordland [40]), forests (e.g., SFU Mountain [43]) or lakes (e.g., Symphony Lake [44]). **Appearance changes** occur due to dynamic objects like pedestrians (e.g., GardensPoint Walking [28]), time of day with lighting changes and moving shadows (e.g., StLucia Various Times of the Day [42]) or day versus night (e.g., Alderley [14]), weather that is sunny, cloudy, overcast, rainy, foggy, or snowy (e.g., Oxford RobotCar [41]), seasons like spring, summer, fall and winter or dry and wet season (e.g., Nordland [40]), elapsed time with roadworks, construction sites or new and demolished buildings up to modern vs. historical imagery (e.g., AmsterTime [45]), or catastrophic scenarios, e.g., after an earthquake. **Viewpoint changes** between images of the same place range from nearly pixel-aligned (e.g., Nordland [40]) to left-to-right side of the walkway (e.g., GardensPoint Walking [28]), left-to-right side of the street (e.g., SouthBank Bicycle [28]), panoramic-aligned images to single image (e.g., Tokyo 24/7 [46]), panoramic-aligned images to panoramic-aligned images (e.g., Pittsburgh250k [47]), bikeway-to-street (e.g., Berlin Kurfurstendamm [48]), aerial-to-ground (e.g., Danish Airs and Grounds [49]), or inside-to-outdoor (e.g., Amsterdam-XXXL [50]). For a comprehensive overview of existing datasets, please refer to [1, 3].
### _The ground truth_
Ground truth data tells us which image pairs in a dataset show the same places, and which show different places. This data is necessary for evaluating the results of a place recognition method. The ground truth is either directly given as a set of tuples indicating which images in the database \(DB\) and the query set \(Q\) belong to same places, or it is provided via GNSS coordinates or poses using their maximum allowed distances. Alternatively, some datasets are sampled so that images with the same index in each image set show the same place (e.g., Nordland [40] and GardensPoint Walking [28]).
#### Definition of the ground truth
To evaluate a VPR result, the definition of a logical ground truth matrix \(\mathbf{GT}\) is required. This matrix has the same dimensions as \(\mathbf{S}\) and \(\mathbf{M}\), i.e., \(\mathbf{GT}\in\mathbb{B}^{|Q|\times|Q|}\) or \(\mathbf{GT}\in\mathbb{B}^{|DB|\times|Q|}\). The elements \(gt_{ij}\in\mathbf{GT}\) define whether the \(i\)-th image in \(Q\) or \(DB\) and the \(j\)-th image in \(Q\) show the same place (\(gt_{ij}=true\)) or different places (\(gt_{ij}=false\)). Their values are set using the ground truth matches from the dataset.
An additional way of evaluating VPR performance that is used by some researchers is the soft ground truth matrix \(\mathbf{GT}^{soft}\). The soft ground truth matrix addresses the problem that we do not expect a VPR method to match images with a very small visual overlap, i.e., \(gt_{ij}=false\). However, if a method indeed matches these images with small overlap, we avoid penalization by setting \(gt_{ij}^{soft}=true\). Image pairs without any visual overlap are also labeled \(gt_{ij}^{soft}=false\). Therefore, \(\mathbf{GT}^{soft}\) is a dilated version of \(\mathbf{GT}\), i.e., it contains all true values contained in \(\mathbf{GT}\), as well as additional true values for image pairs with small visual overlap. Image pairs must be matched if
\[\mathbf{GT}\leftrightarrow true. \tag{5}\]
Image pairs can, but do not need to be necessarily matched, if
\[-\mathbf{GT}\wedge\mathbf{GT}^{soft}\leftrightarrow true. \tag{6}\]
Note that we use \(\neg\) to denote the logical negation operator. These are usually ignored during evaluation. Image pairs must not be matched if
\[\mathbf{GT}^{soft}\leftrightarrow false. \tag{7}\]
How \(\mathbf{GT}\) and \(\mathbf{GT}^{soft}\) are actually used for evaluation is presented in the following.
### _Metrics_
This section presents established metrics to evaluate a VPR method, including precision and recall, the precision-recall curve, area under the precision-recall curve, maximum recall at 100% precision, and recall@\(K\)[5]. These metrics are based on
* the pairwise descriptor similarities \(s_{ij}\in\mathbf{S}\) (cf. Section IV-D) or the image matches \(m_{ij}\in\mathbf{M}\) (cf. Section IV-E)
* with corresponding ground truth \(gt_{ij}\in\mathbf{GT}\) (and \(gt_{ij}^{soft}\in\mathbf{GT}^{soft}\) in case the soft ground truth is used).
For single-best-match VPR, the evaluation only considers the best matching image pair per query with the highest similarity \(s_{i^{*}j}\):
\[i^{*}=\operatorname*{arg\,max}_{i}s_{ij}. \tag{8}\]
#### Precision and Recall
Precision \(P\) and recall \(R\) are important metrics in the information retrieval domain [51, p. 781]. In the context of VPR, **precision**\(P\) represents the ratio of correctly matched images of same places to the total number of matched images with
\[P=\frac{\#TP}{\#TP+\#FP}. \tag{9}\]
#### Recall
\(R\) expresses the ratio of correctly matched images of same places to the total number of ground-truth positives (GTP):
\[R=\frac{\#TP}{\#GTP}. \tag{10}\]
In the case of single-best-match VPR, the number of ground-truth positives refers to the total number of query images for which a ground-truth match exists, i.e.,
\[\#GTP=\sum_{\forall j}\begin{cases}1,&\text{if }\exists i:gt_{ij}\leftrightarrow true \\ 0,&\text{otherwise}\end{cases}\, \tag{11}\]
whereas in the case of multi-match VPR, the number of ground-truth positives refers to the number of actually matching image pairs, i.e.,
\[\#GTP=\sum_{\forall i,j}\begin{cases}1,&\text{if }gt_{ij}\\ 0,&\text{otherwise}\end{cases}. \tag{12}\]
\(\#TP\) and \(\#FP\) are the number of correctly matched and wrongly matched image pairs. More specifically, **true positives** TP are actual matching image pairs that were classified as matches:
\[\#TP=\sum_{\forall i,j}\begin{cases}1,&\text{if }gt_{ij}\wedge m_{ij}\\ 0,&\text{otherwise}\end{cases}. \tag{13}\]
For single-best-match VPR, only \(i^{*}\) from Eq. (8) is evaluated in Eq. (13) for each query image. The same is true for the following Eq. (14).
**False positives**\(FP\) are non-matching image pairs that were incorrectly classified as matches:
\[\#FP=\sum_{\forall i,j}\begin{cases}1,&\text{if }\neg gt_{ij}\wedge m_{ij} \\ 0,&\text{otherwise}\end{cases}. \tag{14}\]
Note that when using the soft ground truth, image pairs with \(\neg gt_{ij}\wedge gt_{ij}^{soft}\leftrightarrow true\) (cf. Eq. (6)) are ignored during the computation of \(\#TP\) and \(\#FP\). While **false negatives**\(FN\) are indirectly involved in the calculation of recall \(R\), **true negatives**\(TN\) are usually not evaluated due to the typically imbalanced classification problem of VPR with \(\#TN\gg\#TP,\#FP,\#FN\).
#### 4.2.3 Precision-recall curve
Precision-recall curves can be used to avoid actual matching decisions, which are often made after VPR using a computationally expensive verification algorithm. The idea is to make matching decisions with \(\mathbf{M}=\mathbf{S}\geq\theta_{k}\) over a range of thresholds \(\mathbf{\theta}=\{\min(\mathbf{S}),\ldots,\max(\mathbf{S})\}\). For instance, the number of true positives \(\#TP_{k}\) for one specific \(\theta_{k}\in\mathbf{\theta}\) is then computed with
\[\#TP_{k}=\sum_{\forall i,j}\begin{cases}1,&\text{if }gt_{ij}\wedge(s_{ij}\geq \theta_{k})\\ 0,&\text{otherwise}\end{cases}. \tag{15}\]
Following Eqs. (9) and (10), this leads to two vectors of precision and recall values \(P(\theta_{k})\) and \(R(\theta_{k})\), which in combination formulate the **precision-recall curve**. The full pipeline for the computation of the precision-recall curve is depicted in Figure 4 and code is provided in Code Snippet 5.
#### 4.2.4 Area under the precision-recall curve (AUPRC)
The **area under the precision-recall curve** (also termed average precision, avgP) can be used to compress a precision-recall curve into a single number, as shown in Code Snippet 6. In Figure 4, the AUPRC is visualized as the green area under the precision-recall curve.
#### 4.2.5 Maximum recall at 100% precision
The maximum recall at 100% precision (short _R@100P_) represents the maximum recall where \(P=1\) (\(100\%\)), i.e., the maximum recall without false positives \(FP\) (cf. Eq. (14)). In the past, this metric was important to evaluate VPR methods for loop closure detection in SLAM. Keeping the precision at \(P=1\) avoids wrong loop closures and, consequently, mapping errors [52]. However, since the advent of robust graph optimization techniques for SLAM [53], the avoidance of wrong loop closures became less relevant. With robust graph optimization, it is more important to find enough correct loop closures (\(TP\)) than to avoid wrong loop closures (\(FP\)). Therefore, using multi-match VPR to identify all loop closures should be preferred over tuning the R@100P for such applications.
If the precision never reaches \(P=1\), the maximum recall at 100% precision is undefined. Therefore, the maximum recall at 99% or 95% precision has been used alternatively.
Figure 4: The evaluation pipeline for multi-match VPR, including the precision-recall curve and the area under the precision-recall curve (AUPRC). Given the similarity matrix \(\mathbf{S}\) and ground truth \(\mathbf{GT}\) and \(\mathbf{GT}^{soft}\) (cf. Section V-B), a range of thresholds \(\theta_{k}\in\{\min(\mathbf{S}),\ldots,\max(\mathbf{S})\}\) is applied with \(\mathbf{S}\geq\theta_{k}\) to obtain binary matching decisions \(m_{ij}\in\mathbf{M}_{k}\) for each \(\theta_{k}\). In combination with the ground truth, these can be labeled as either True Positives (\(TP\)), False Positives (\(FP\)), False Negatives (\(FN\)) or True Negatives (\(TN\)), and converted into a precision-recall curve and the area under the precision-recall curve (AUPRC).
_Recall@\(K\):_ The recall@\(K\) (also termed _top-\(K\) success rate_) is an often used metric for the evaluation of image classifiers [54, p. 225]. For place recognition, it is defined as follows: For each query image, given the \(K\) database images with the \(K\) highest similarities \(s_{ij}\), the recall@\(K\) measures the rate of query images with at least one actually matching database image. That means this metric requires at least one matching image in the database for each query image, which corresponds to a typical localization scenario without exploration. For mapping with newly visited places, the metric is not defined. In such a scenario, an implementation of recall@\(K\) could simply ignore all query images without a matching database image - however, this workaround would not evaluate the (in)ability of a method to handle exploration during the query run, i.e., new places which are not part of the database set.
The recall@\(K\) is particularly suited for visual localization tasks, where the \(K\) most similar database or query images are retrieved for a subsequent geometric verification. Note that for VPR in the context of localization without exploration (i.e., all query images have at least one matching reference image), the recall@\(1\) and the precision at 100% recall are identical.
_Mean, best case and worst case performance:_ To get a comprehensive understanding of how well a VPR method performs in different environments, with different types of appearance and viewpoint changes, it is best practice to evaluate it using multiple datasets. The aforementioned metrics measure the performance on each single dataset. One can get a more condensed view of the overall performance by considering the mean, best case and worst case performance. The mean performance allows for a quick comparison with other evaluated methods. The best case performance shows the maximum achievable performance and reveals potential strengths of an approach: if the best case performance of a method is higher than that of the compared methods, this method is well suited for the conditions under which the best case performance was achieved. The worst case performance reveals the weaknesses of a method and its sensitivity to certain conditions or trajectories (cf. Figure 3). For example, if the worst case performance of a method is lower than the worst case performance of the compared methods, it indicates that this method is less robust and struggles with the specific property of at least one of the evaluated datasets.
## VI Challenges and common ways of addressing them
The previous sections introduced a generic pipeline for VPR, and how to evaluate such a pipeline. In this section, we go beyond this basic pipeline and enlist typical challenges that researchers face in the field of VPR and the ways that prior work has addressed them.
### _Scalability_
A major challenge in VPR is how to scale up the system to handle large numbers of images in the database or query set. As discussed in Section IV-B, holistic image descriptors allow for fast retrieval. To reduce the computational effort for descriptor comparison, dimensionality reduction techniques like Gaussian, binary, sign or sparse random projection [55, 56, 57], or Principal Component Analysis (PCA) [58] have been proposed.
However, the computation time for recognizing places is typically still proportional to the number of images in the database \(DB\) or query set \(Q\). To further improve efficiency, approximate nearest neighbor search [38] (e.g., a combination of KD-tree [59] and product quantization [60] as with DELF [30]) can be employed instead of a linear search of all database descriptors, which leads to a sublinear time complexity. Additionally, incorporating coarse position data from weak GPS signals can increase efficiency as it reduces the search space [61].
Finally, to compensate for the reduced accuracy of holistic descriptors, hierarchical place recognition can be employed. This approach re-ranks the top-\(K\) retrieved matches from
holistic descriptors through geometric verification with local image descriptors [8, 62].
### _Appearance variations_
When a robot revisits a place, its current image observation often experiences significant variations in appearance (as discussed in Section V-A), which can negatively affect the performance. To reduce the discrepancy between the query observation and the observation stored in the database, techniques such as illumination invariant images [63], shadow removal [64], appearance change prediction [65], linear regression [66], and deep learning based methods using generative adversarial networks like Night-To-Day-GAN [67] can be used to convert all images into a reference condition [20]. Such techniques require that the correspondence between each image and its actual condition is provided by human supervision or a condition classifier (e.g., database: summer, query: winter).
To avoid such condition-specific approaches that are trained or designed only for specific conditions (e.g., Night-To-Day-GAN: night and day, shadow removal: different times of day), a condition-wise descriptor standardization can be used to significantly improve performance over a wide range of conditions [20]: This standardization normalizes each dimension of the descriptors from one condition to zero mean and unit standard deviation (e.g., once for the database in summer, once for the query set in winter). Furthermore, if appearance variations occur not only _across_ the query and database traverses (e.g., database: sunny, query: rainy) but also _within_ a traverse (e.g., database: sunny\(\rightarrow\)cloudy\(\rightarrow\)overcast\(\rightarrow\)rainy, query: sunny), descriptors can be clustered and then standardized per cluster. Besides addressing individual appearance challenges as above, a common trend in recent research has been to train deep architectures on large-scale, diverse datasets [37] to achieve global [68] and local [69] descriptors that are robust to appearance variations. Alternatively, one can combine the strengths of multiple descriptors by simply concatenating them (which sums up their dimensionalities) or combining them using techniques such as hyperdimensional computing (which limits the dimensionality) [29].
### _Viewpoint variations_
A robot may revisit a place from a different viewpoint. For drones, this change could be due to a varying 6-DoF pose, and for an on-road vehicle, it could be due to changes in lanes and direction of travel. In addition to recognizing a local feature or region from different viewpoints, one also needs to deal with often limited visual overlap between an image pair captured from different viewpoints. The problem of viewpoint variations becomes even more challenging when simultaneously affected by appearance variations that widen the scope for perceptual aliasing (the problem of distinct places looking very similar, as discussed in Section I and detailed in, e.g., [2]). A popular solution to deal with viewpoint variations is to learn holistic descriptors by aggregating local features in a permutation-invariant manner, that is, independent of their pixel locations, as in NetVLAD [24].
### _Improving performance_
In addition to the approaches mentioned earlier, there are several ways to improve VPR performance by using task-specific knowledge.
**Sequence-based methods** leverage sequences in the database and query set, which lead to continuous lines of high similarities in the similarity matrix \(\mathbf{S}\) (cf. Section IV-D and Figure 3). We can divide these methods into two categories: **Similarity-based sequence methods** use the similarities \(s_{ij}\in\mathbf{S}\) to find linear segments of high similarities, e.g., SeqSLAM [14] or SeqConv [17], or continuous lines of high similarities with potentially varying slope, e.g., based on a flow network [70] or on a Hidden Markov Model [71]. Methods like SMART [72] additionally use available odometry information to find sequences with varying slope. On the other hand, a sequence of holistic image descriptors can be combined into a single vector. A **sequence descriptor** defined for a place thus accounts for the visual information observed in the preceding image frames. These sequence descriptors can be compared between the database and the query sets to obtain place match hypotheses. Existing methods include ABLE [73], MCN [74], delta descriptors [75], SeqNet [76], and different deep learning based approaches [77].
Besides leveraging descriptor similarities \(\mathbf{S}\)_between_ the database and query sets, **intra-database and intra-query similarities \(\mathbf{S}^{DB}\) and \(\mathbf{S}^{Q}\)**, i.e., descriptor similarities _within_ the database and query sets, can be used to improve performance. For example, in [17], the intra-set similarities \(\mathbf{S}^{DB}\) and \(\mathbf{S}^{Q}\) are used in combination with \(\mathbf{S}\) and sequence information to formulate a factor graph that can be optimized to refine the similarities in \(\mathbf{S}\). In this graph, the intra-set similarities are used to connect images within the database or query sets that are likely to show the same or different places due to a high or low intra-set similarity. For example, let us suppose that the \(l\)-th query image has high similarities \(s_{il}\) and \(s_{jl}\) to the \(i\)-th and \(j\)-th database images. Let us further suppose that the similarity \(s_{kl}\) to the \(k\)-th database image is low, although the \(i\)-th, \(j\)-th and \(k\)-th database images have high intra-database similarities \(s_{ij}^{DB}\), \(s_{ik}^{DB}\) and \(s_{jk}^{DB}\). The graph optimization then detects that the similarity \(s_{kl}\) between the \(k\)-th database image and the \(l\)-th query image is also likely to be high.
Methods such as experience maps [78] and co-occurrence maps [79] can be used in cases where the robot **frequently revisits the same places**. These methods continually observe each place and create a descriptor every time the appearance changes. During a comparison of a new query descriptor with this aggregated "map", only one descriptor of a similar condition needs to be matched to recognize the place, reducing the need for condition-invariant descriptors [80].
In the case of robot localization with known places (i.e., each visited query place is guaranteed to be in the database and no exploration beyond this mapped area is performed), VPR can benefit from **place-specific classifiers**, which can improve accuracy with reduced map storage or retrieval time [81, 82, 83]. A similar approach is to train a deep learning-based **place classifier** that directly outputs a place label for
a given image [11, 84], or to create environment-specific descriptors [85]. Another direction is to exploit known place types for place type matching to limit the number of potential matches between the database and query set [16]. For example, instead of searching through all database images, if the query image was taken in a forest, such semantic categorization constrains the database images to only those that were also taken in a forest.
## VII Conclusions
Visual Place Recognition (VPR) is a well established problem that has found widespread interest and use in both computer vision and robotics. In this tutorial, we have described the visual place recognition task, including its various problem categories and subtypes, their typical use cases, and how it is typically implemented and evaluated. Additionally, we discussed a number of methods that can be used to address common challenges in VPR.
There are a number of open challenges such as system integration, enriched reference maps, view synthesis, and the design of a "one-fits-all" solution that still need to be tackled by the community. While we do not discuss these challenges in this tutorial, we refer the interested reader to [1, 2].
| localization はモバイルロボットにとって必須な能力です。この分野で急速に成長している研究分野は、視覚的な場所認識（VPR）であり、これは、画像のみで以前見た場所を認識する能力です。この研究は、視覚的な場所認識に関する最初のチュートリアル論文です。VPR の terminologies を一括してまとめ、先人の研究に貢献しています。2つの重要な方向に: 1) 新しい研究者向けの体系的な導入を提供し、VPR の問題の構成方法、汎用アルゴリズムパイプライン、VPR 方法に対する評価方法論、および VPR の主要な課題とその解決方法について説明します。2) VPR 問題に精通した研究者にとっての貢献として、この論文は、入力、データ処理、出力に関する異なる VPR 問題の複雑さを調査します。このチュートリアルは、VPR アルゴリズムの評価の微妙な点についても議論します。たとえば |
2305.09344 | Dynamical modelling of ATLAS$^{\rm 3D}$ galaxies | Triaxial dynamical models of massive galaxies observed in the ATLAS3D project
can provide new insights into the complex evolutionary processes that shape
galaxies. The ATLAS3D survey is ideal as the sample comprises a good mix of
fast and slow rotators with vastly different mass assembly histories. We
present a detailed dynamical study with our triaxial modelling code DYNAMITE,
which models galaxies as a superposition of their stellar orbits. The models
allow us to constrain the intrinsic shape of the stellar component, the
distributions of the visible and invisible matter and the orbit distribution in
these nearby early-type galaxies and to relate it with different evolutionary
scenarios. Triaxial modelling is essential for these galaxies to understand
their complex kinematical features. | Sabine Thater, Prashin Jethwa, Edward J. Lilley, Alice Zocchi, Giulia Santucci, Glenn van de Ven | 2023-05-16T10:51:34 | http://arxiv.org/abs/2305.09344v1 | # Dynamical modes of ATLAS\({}^{\rm 3D}\) galaxies
###### Abstract
Triaxial dynamical models of massive galaxies observed in the ATLAS\({}^{\rm 3D}\) project can provide new insights into the complex evolutionary processes that shape galaxies. The ATLAS\({}^{\rm 3D}\) survey is ideal as the sample comprises a good mix of fast and slow rotators with vastly different mass assembly histories. We present a detailed dynamical study with our triaxial modelling code DYNAITE, which models galaxies as a superposition of their stellar orbits. The models allow us to constrain the intrinsic shape of the stellar component, the distributions of the visible and invisible matter and the orbit distribution in these nearby early-type galaxies and to relate it with different evolutionary scenarios. Triaxial modelling is essential for these galaxies to understand their complex kinematical features.
Galaxies: kinematics and dynamics, Galaxies: structure 0000-0002-4106-8000] Thater S. \({}^{1}\), Jethwa P.\({}^{1}\), Lilley E. J.\({}^{1}\), Zocchi A.\({}^{1}\), Santucci G.\({}^{2,3}\) and van de Ven G.\({}^{1}\)
## 1 Introduction
Early-type galaxies (ETGs) are distinguished into two classes based on their apparent angular momentum: fast rotators and slow rotators (e.g. Emsellem et al., 2007; Cappellari et al., 2007; Emsellem et al., 2011). While fast rotators are nearly axisymmetric and often have an oblate shape, slow rotators are weakly triaxial (but not far from isotropic). These two classes likely depict two different channels of galaxy formation. Slow rotators are thought to have assembled in the centres of massive halos and after an intense star formation period at high redshift have evolved from gas-poor major mergers. In contrast, fast rotators have likely formed out of star-forming discs and their evolution is dominated by gas accretion, bulge growth and quenching (Cappellari, 2016). In this study we investigate the evolutionary histories of ATLAS\({}^{\rm 3D}\) galaxies by searching for imprints in their dynamically inferred intrinsic shapes and stellar orbit distributions.
## 2 The ATLAS\({}^{\rm 3D}\) sample
ATLAS\({}^{\rm 3D}\)(Cappellari et al., 2011) is a multiwavelength survey that includes 260 ETGs with stellar masses \(M_{\rm s}>6\times 10^{9}\) M\({}_{\odot}\) within the local volume (42 Mpc). The sample was deduced from a parent sample which was carefully selected to be statistically representative of the nearby galaxy population. About 25% of the ATLAS\({}^{\rm 3D}\) ETGs were classified as elliptical galaxies and 75% as lenticular galaxies. The survey was carried out with the SAURON integral field unit on the William Herschel Telescope. From these observations, detailed 2-dimensional stellar kinematic maps of mean velocity, velocity dispersion, and \(h_{3}\) and \(h_{4}\) Gauss-Hermite polynomials with high S/N are available, which are ideal for our dynamical study. The kinematics (e.g. Krajnovic et al., 2011; Emsellem et al., 2011), star formation history (e.g. McDermid et al., 2015) and environment (e.g. Cappellari et al., 2011; Serra et al., 2012) of the ATLAS\({}^{\rm 3D}\) galaxy sample have been extensively studied in the past decade, allowing us to put our dynamical results in the context of galaxy assembly histories.
## 3 First dynamical results with DYNAITE
We obtained the data products from the ATLAS\({}^{\rm 3D}\) webpage1 and for the first time constructed triaxial Schwarzschild orbit-superposition models for these galaxies using DYNAITE1( van den Bosch et al., 2008; Jethwa et al., 2020; Thater et al., 2022). DYNAMITE allows us to recover the enclosed mass, the intrinsic shape of the stellar component, the dark matter fraction and the orbit distribution of the modelled galaxies. So far, we have finished modelling 63 galaxies, about a quarter of the full ATLAS\({}^{\rm 3D}\) sample. Figure 1 shows a comparison between the observed SAURON kinematics and the best-fit DYNAITE model of NGC 4365. NGC 4365 harbours a kinematically decoupled core (KDC) in its centre that is remarkably well recovered by our models. In the ATLAS\({}^{\rm 3D}\) survey, many galaxies are exhibiting complex kinematic features which require triaxial Schwarzschild modelling, and not only regularly rotating galaxies (that can be well modelled assuming axisymmetry).
Footnote 1: [http://www-astro.physics.ox.ac.uk/atlas3d/](http://www-astro.physics.ox.ac.uk/atlas3d/)
We modelled the ATLAS\({}^{\rm 3D}\) galaxies with the same modelling setup as described in Santucci et al. (2022) for SAMI galaxies. Our gravitational potential consists of a stellar component, a central black hole, and dark matter parametrised as a spherical halo with a Navarro-Frenk-White (NFW; Navarro et al., 1996) radial profile. In total, we consider five free parameters: a constant stellar mass-to-light ratio \(M_{*}/L\), intrinsic stellar axis-length ratios \(p\) (intermediate-to-long) and \(q\) (short-to-long), the stellar projected-to-intrinsic scale-length ratio \(u\) and the dark matter fraction \(f_{\rm DM}=M_{200}/M_{*}\) within the virial radius \(r_{200}\). The dark matter concentration \(c\) was fixed using the \(M_{200}-c\) relation by Dutton & Maccio (2014) to get a better handle on the degeneracy between \(M_{*}/L\) and dark matter. The ATLAS\({}^{\rm 3D}\) kinematics do not have the spatial resolution to constrain the central black hole mass (\(M_{\rm BH}\)), therefore we fixed black hole masses using the empirical \(M_{\rm BH}-\sigma_{\rm e}\) relation by van den Bosch (2016). From our modelling approach, we obtain dynamical masses that are consistent with those obtained by Cappellari et al. (2013) and Poci et al. (2017) using Jeans modelling.
In this study, we aim at constraining the intrinsic shape of the stellar component of massive ETGs. The triaxiality parameter at one effective radius (\(R_{\rm e}\)) is defined as \(T_{\rm Re}=(1-p_{\rm Re}^{2})/(1-q_{\rm Re}^{2})\). Galaxies with \(T_{\rm Re}=0\) are classified as oblate, galaxies with \(T_{\rm Re}=1\) as prolate and galaxies with values of \(T_{\rm Re}\) between 0.1 and 0.8 as triaxial. In Figure 2, we show the triaxiality parameter as a function of stellar mass; the ATLAS\({}^{\rm 3D}\) galaxies we modelled are shown as stars in the plot, with different colours indicating the kinematic classification
Figure 1: DYNAMITE makes it possible to dynamically model complex triaxial features such as kinematically decoupled components. We show here the surface brightness, mean velocity and velocity dispersion of the SAURON data (top) and of the Schwarzschild model (bottom) of the ATLAS\({}^{\rm 3D}\) galaxy NGC 4365.
by Krajnovic et al. (2011). While most of our modelled galaxies have triaxial intrinsic shapes, we find trends that depend on their kinematic classification. Most of the modelled galaxies are regularly fast-rotating galaxies (blue) and have a mildly triaxial shape (\(\overline{T_{\rm Re}}=0.25\)). However, the scatter is quite strong and some galaxies have oblate or strongly triaxial shapes. Featureless slow-rotators (\(\overline{T_{\rm Re}}=0.71\), red) on the other hand are strongly triaxial or even prolate (which is also seen in the kinematics). Interestingly, the modelled non-rotating galaxies (orange; only 3 galaxies) have a mildly triaxial shape (\(\overline{T_{\rm Re}}=0.23\)) similar to fast-rotators. The ATLAS\({}^{\rm 3D}\) survey contains about 20 galaxies with KDCs. For these galaxies we find on average a strongly triaxial intrinsic shape (\(\overline{T_{\rm Re}}=0.45\)) which is consistent with these galaxies having mainly formed through mergers. Surprisingly, we also find strong triaxiality for the three \(2\sigma\) galaxies (galaxies exhibiting two peaks in the velocity dispersion). As this feature is typically found in galaxies with a pair of extended, counter-rotating discs, we would have expected these galaxies to be more oblate. The triaxial shape might be a remnant signature of their merger history. More galaxies and a deeper analysis are essential to confirm our first results. In contrast to our modelled ATLAS\({}^{\rm 3D}\) galaxies, the majority of modelled galaxies in the SAMI (Santucci et al., 2022) and Fornax3D (Ding et al., 2023) surveys have oblate shapes, as shown in Fig. 2 (with grey dots and black diamonds, respectively).
DYNAMITE models also allow a detailed look at the distribution of the stellar orbits of the galaxies. In Fig 3, we show the orbit distribution of NGC 821, one of our fast-rotating ETGs. As expected for bulge-dominated galaxies, we see a high fraction of hot orbits with a circularity \(\lambda_{\rm z}\sim 0\) (consistent with a pressure-supported bulge). However, there is also a large fraction of warm and cold orbits, meaning that the ETG exhibits significant net angular momentum. We
Figure 2: Triaxiality parameter \(T_{\rm Re}\) as a function of stellar mass. We colour-code the ATLAS\({}^{\rm 3D}\) galaxies based on their kinematic classification from Krajnović et al. (2011): regular fast-rotating galaxies are represented here in blue, featureless slow-rotators in red, non-rotators in orange, KDCs in light purple, and \(2\sigma\) galaxies in dark purple. Typical uncertainties at \(1\sigma\) confidence level are shown in the top left corner of the plot. For comparison, we also show two other samples of galaxies that were modelled with DYNAMITE, from the SAMI (Santucci et al., 2022) and Fornax3D (Ding et al., 2023) surveys, represented here with grey dots and black diamonds, respectively.
will compare the orbit distribution for galaxies belonging to the various kinematic classes and search for signatures of accretion as in Zhu et al. (2020, 2022).
In the future, we will extend our dynamical study to the remaining galaxies of the ATLAS\({}^{3\mathrm{D}}\) survey and search for correlations between the intrinsic shapes/orbit distributions and kinematic properties, environment and star formation histories, in order to better understand the formation and evolutionary processes of early-type galaxies.
This research was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement NO 724857 (Consolidator Grand ArcheoDyn).
| 三軸的な動的モデルを用いた大質量星系を対象としたATLAS3Dプロジェクトの観測から、銀河の複雑な進化プロセスに関する新しい見通しを得ることが可能となる。ATLAS3Dサンプリングは、高速回転とゆっくり回転の星系が、 vastly different mass assembly historiesを持つため、理想的である。私たちはDYNAMITEと呼ばれる三軸的なモデル化コードを用いて、銀河を星々の軌道重ね合わせとしてモデル化する詳細な動的解析を行った。これらのモデルは、星々の固有形状、視覚的かつ非視覚的な物質の分布、そしてこれらの近傍の初期型星系における軌道分布を評価し、異なる進化シナリオと関連付けることができる。三軸的なモデル化は、これらの星系が複雑な運動特性を理解するために不可欠である。 |
2306.10345 | Do as I can, not as I get | This paper proposes a model called TMR to mine valuable information from
simulated data environments. We intend to complete the submission of this
paper. | Shangfei Zheng, Hongzhi Yin, Tong Chen, Quoc Viet Hung Nguyen, Wei Chen, Lei Zhao | 2023-06-17T13:23:22 | http://arxiv.org/abs/2306.10345v2 | # Do as I can, not as I get:
###### Abstract
Multi-modal knowledge graph (MKG) includes triplets that consist of entities and relations and multi-modal auxiliary data. In recent years, multi-hop multi-modal knowledge graph reasoning (MMKGR) based on reinforcement learning (RL) has received extensive attention because it addresses the intrinsic incompleteness of MKG in an interpretable manner. However, its performance is limited by empirically designed rewards and sparse relations. In addition, this method has been designed for the transductive setting where test entities have been seen during training, and it works poorly in the inductive setting where test entities do not appear in the training set. To overcome these issues, we propose **TMR** (Topology-aware **M**ulti-hop **R**easoning), which can conduct MKG reasoning under inductive and transductive settings. Specifically, TMR mainly consists of two components. (1) The topology-aware inductive representation captures information from the directed relations of unseen entities, and aggregates query-related topology features in an attentive manner to generate the fine-grained entity-independent features. (2) After completing multi-modal feature fusion, the relation-augment adaptive RL conducts multi-hop reasoning by eliminating manual rewards and dynamically adding actions. Finally, we construct new MKG datasets with different scales for inductive reasoning evaluation. Experimental results demonstrate that TMP outperforms state-of-the-art MKGR methods under both inductive and transductive settings.
Multi-hop reasoning, multi-modal knowledge graphs, inductive setting, adaptive reinforcement learning
## 1 Introduction
Knowledge graphs (KGs) store and manage huge amounts of data in reality and have been widely used in applications, including recommendation systems [43], information retrieval [28], and knowledge question answering [16]. A traditional KG consists of structural triplets that involve entities and relations, such as (_James Cameron_, \(Role\_create\), \(RoseBuckater\)). In recent years, as multi-modal data has received widespread attention in the field of data science and artificial intelligence, multi-modal knowledge graphs (MKGs) have emerged [27, 45]. As shown in Figure 1(a), an MKG contains extra multi-modal auxiliary data (images and text description) based on structural triplets, which provides diverse modalities of knowledge. However, the intrinsic incompleteness of MKGs severely limits knowledge applications [34]. To address this problem, the multi-modal knowledge graph reasoning (MKGR) technique is proposed to infer missing triplets in MKGs [53]. For instance, given a triple query (_James Cameron_, _Writer_,?), MKGR can utilize both structural and multi-modal auxiliary data to infer the missing entity _Titan_.
In the literature, existing MKGR methods can be categorized into two types: single-hop reasoning and multi-hop reasoning [53]. The former focuses on modeling score functions for one-step relations that contain relatively less information [34, 45], while the latter represents the latest work that interpretably infers missing elements by combining multi-hop relations and fusing the corresponding multi-modal features [53]. As shown in Figure 1(b), by connecting (_James Cameron_, _Role\_create_, _Jack Dawson_) and (_Jack Dawson_, _Hero_, _Titanic_), a missing triplet (_James Cameron_, _Writer_, _Titanic_) can be inferred. MMKGR [53] stands out as the unique multi-hop MKGR model in existing ones, garnering significant attention for its state-of-the-art (SOTA) performance and interpretability. To effectively utilize both structural features and corresponding multi-modal features, MMKGR first uses a unified gate-attention network to generate multi-modal complementary features with sufficient attention interactions and less noise. Then, these features are fed into a novel complementary feature-aware reinforcement learning (RL) framework. This framework selects a sequence of actions (i.e., multi-hop reasoning paths) to accumulate rewards on the basis of a manually designed 3D reward function. Finally, MMKGR aims to maximize reward values by successfully inferring missing entities and outputs interpretable reasoning paths.
Although MMKGR demonstrates impressive reasoning performance and interpretability on MKGs, there is still scope for enhancing its action and reward design. (1) _3D reward function is limited by manual design_. It comprises three manual sub-rewards, relying on the experience of domain experts and existing data distributions [12]. However, this necessitates time-consuming redesign when adapting to
new environments [2]. Moreover, the subjective nature of manual reward design can lead to variations among different designers [22]. (2) _MMKGR is sensitive to the sparsity of relations_. The selection of actions in MMKGR relies on the combination of multi-hop relations. The absence of any relation in this path causes the reasoning path to be unavailable, which limits the reasoning performance [50, 52]. For example, MMKGR infers the missing entity _Titanic_ through a two-hop reasoning path _James Cameron \(\overset{Role\_create}{\rightarrow}\) Jack Dawson \(\overset{Herg}{\rightarrow}\) Titanic_. If \(Role\_create\) or \(\overset{Herg}{\rightarrow}\) is unconnected, the aforementioned two-hop path does not exist, which results in the query (_James Cameron_, _Director_,?) cannot be inferred. Arguably, it is extremely challenging to design an adaptive reward without manual intervention and dynamically add actions to alleviate sparsity.
More importantly, MMKGR is difficult to apply to real scenarios since it primarily concentrates on the transductive setting while overlooking the importance of the inductive setting. As shown in Figure 1 (b), all entities are assumed to be seen during testing in the transductive setting [37]. However, knowledge is evolving and new entities are constantly emerging in the real world [29]. This observation is more in line with the inductive setting where the inferred entities in the test set do not appear in the training set [41]. These inferred entities, often referred to as unseen entities, lack knowable structural representations under the inductive setting. Intuitively, MMKGR can leverage multi-modal auxiliary data of unseen entities to infer the missing triples associated with them. This naturally raises an intriguing and fundamental question: How does MMKGR perform under the inductive setting? To answer this question, we first construct datasets of the induction setting where the entities in the test set and the train set are disjoint [33]. Then, we apply MMKGR to conduct multi-hop reasoning under the inductive setting. Experimental results reveal that MMKGR struggles to converge and has low reasoning performance under the inductive setting. Actually, the conclusion of inductive reasoning methods [33, 47] on traditional KGs is consistent with the above experimental findings: a transductive reasoning method that relies only on multi-modal auxiliary data and lacks generalizability to unseen entities is unsuitable for the inductive setting [11, 25, 41]. This prompts a subsequent goal: _How to develop the inductive capability of MMKGR to generalize unseen entities under an inductive setting?_
A technical challenge to achieving this goal lies in the lack of fine-grained entity-independent representations in existing MKGR methods. One of the key advantages of learning this representation is the development of the inductive capability to generalize unseen entities even in the absence of their specific structural features. [25, 37]. MMKGR lacking the inductive capability has no choice but to use multi-modal auxiliary features of unseen entities to understand these entities, which is highly dependent on the quality and quantity of multi-modal data and not suitable for unseen tasks [11, 41]. Additionally, existing entity-independent representation methods of inductive reasoning on traditional knowledge graph reasoning cannot be directly extended in MMKGR. This is because these methods struggle to aggregate the most relevant information based on specific query relations, resulting in the generation of coarse-grained representations for unseen entities. To make matters worse, the coarse-grained representation of each entity in the reasoning path iteratively disrupts decision-making abilities, which impairs the reasoning performance of MMKGR. Consequently, the fine-grained entity-independent representation is crucial to develop an inductive capability for MMKGR.
In light of the aforementioned challenges in MMKGR, we propose an extended method entitled **TRM (**Topology-aware **M**ulti-hop **R**easoning). The main difference between our method and existing ones is that TRM has not only a talent for exploiting multi-modal data, but also the inductive capability to generalize unseen entities. Thus, TRM is capable of conducting MKGR under both inductive and transductive settings. Specifically, TRM mainly contains _topology-aware inductive representation_ (TAIR) and _relation-augment adaptive reinforcement learning_ (RARL). To develop the inductive capability for MMKGR, **TAIR** learns fine-grained entity-independent representation from query-related topology knowledge. Its relation-aware entity initializer captures a coarse-grained entity-independent representation by leveraging type information of unseen entities from the connected directed relations. To further generate the fine-grained representation, an adaptive topology representation module introduces a query-aware graph neural network (GNN) to attentively capture the topological information. After completing multi-modal feature fusion, **RARL** infers the missing elements by multi-hop reasoning path on MKGs, aiming to further improve the comple
Fig. 1: MKGR under transductive and inductive settings. In the transductive setting, test entities have been seen in the training graph. In contrast, testing entities such as _Interstellar_ and _Christopher Nolan_ did not appear in the training graph under the inductive setting.
mentary feature-aware reinforcement learning framework of MMKGR. Technically, RARL not only dynamically adds relations as additional actions to eliminate relational sparsity but also adaptively generates rewards by imitating expert demonstrations with filtering low-contributing paths. In summary, as an extension of our conference paper [53], this work makes the following contributions:
* To the best of our knowledge, this is the first work to investigate _how to conduct MKGR under both inductive and transductive settings_.
* To resolve the above problem, we propose an RL-based MKGR model called TMR that mainly contains two components TAIR and RARL. Specifically, TAIR generates fine-grained entity-independent representations to generalize unseen entities. RARL conducts multi-hop reasoning by expanding action space and utilizing imitation learning to eliminate manually designed rewards.
* We construct MKG datasets under the inductive setting. To simulate unseen entities, we ensure that the entities in the test set and training set are disjoint.
* Extensive experiments are conducted under both the transductive and inductive settings. Experimental results demonstrate the superior performance of TMR surpasses MMKGR and various baselines.
The remaining sections are organized as follows. Preliminaries and definitions are presented in Section 2, followed by the overview of TMR. For different components in our proposed model, we introduce them in Sections 4, 5, and 6, respectively. Extensive experiments are shown in Section 7. Section 8 provides a review of the related literature. Finally, we conclude this work in Section 9.
## 2 Preliminaries and Definitions
A MKG is an extension of KG by adding multi-modal auxiliary data, it is denoted as \(\mathcal{G}_{m}=\{\mathcal{E}_{m},\mathcal{R},\mathcal{U}\}\), where \(\mathcal{R}\) is a set of semantic relations, and \(\mathcal{E}_{m}\) denotes a set of entities associated with related multi-modal auxiliary data. The features of an entity \(i\) are denoted \(\boldsymbol{f}_{i}=\boldsymbol{f}_{s}\circ\boldsymbol{f}_{m}\), where "\(\circ\)" represents a multi-modal fusion method, \(\boldsymbol{f}_{s}\) and \(\boldsymbol{f}_{m}\) denote structural features and multi-modal auxiliary features, respectively. \(\mathcal{U}=\{(e_{s},r,e_{d})\mid e_{s},e_{d}\in\mathcal{E}_{m},r\in\mathcal{ R}\}\) is a set of triplets, where \(e_{s}\), \(e_{d}\), and \(r\) denote a head entity, a tail entity, and the relation between these entities, respectively. MKGR typically refers to the link prediction task of the inferring triple query (\(e_{s}\), \(r_{q}\),?) and (?, \(r_{q}\), \(e_{d}\)), where \(r_{q}\) is a query relation. By adding inverse relation, each triplet (\(e_{s}\), \(r\), \(e_{d}\)) is equivalent to the triplet (\(e_{d}\), \(r^{-1}\), \(e_{s}\)). Without loss of generality, MKGR methods can predict missing head entities by converting (?, \(r_{q}\), \(e_{d}\)) to (\(e_{s}\),?,?).
**Definition 1**.: _MKGR under the transductive setting_. Given a MKG \(\mathcal{G}_{m}=\{\mathcal{E}_{m},\mathcal{R},\mathcal{U}\}\), MKGR under transductive setting aims to reason out a set of triple queries \(\mathcal{Q}\), \(\mathcal{Q}=\{(e_{s},r_{q},?)\mid(e_{s},r_{q},?)\notin\mathcal{U}\), \(e_{s}\), "\(\gamma^{\prime\prime}\in\mathcal{E}_{m}\), \(r_{q}\in\mathcal{R}\}\), where "?" is a missing entity, \(\mathcal{E}_{m}\in\mathcal{G}_{m}\) and \(\mathcal{R}\in\mathcal{G}_{m}\) represent entity and relation have been seen in the existing MKG \(\mathcal{G}_{m}\).
**Definition 2**.: _MKGR under the inductive setting_. Given two disconnected MKGs \(\mathcal{G}_{m}=\{\mathcal{E}_{m},\mathcal{R},\mathcal{U}\}\) and \(\mathcal{G}_{m}^{*}=\{\mathcal{G}_{m}^{*}\) = \(\{\mathcal{E}_{m}^{*},\mathcal{R}^{*},\mathcal{U}^{*}\}\), \(\mathcal{G}_{m}\) is often known as a training graph, while \(\mathcal{G}_{m}^{*}\) is considered as a testing graph composed of the triplets by the emerging entities \(\mathcal{E}_{m}^{*}\) and the relations \(\mathcal{R}^{*}\). MKGR under the inductive setting requires the model to learn inductive capability in the training graph to infer a set of queries \(\mathcal{Q}\) on the test graph, \(\mathcal{Q}=\{(e_{s},r_{q},\)?) \((e_{s},r_{q},?)\notin\{\mathcal{U}\cup\mathcal{U}^{*}\}\), \(e_{s}\), "?" \(\in\mathcal{E}_{m}^{*},r_{q}\in\mathcal{R}^{*}\}\), where \(\mathcal{E}_{m}\cap\mathcal{E}_{m}^{*}=\emptyset\) and \(\mathcal{R}\cup\mathcal{R}^{*}=\mathcal{R}\).
**Definition 3**.: _Multi-hop reasoning._ Multi-hop reasoning infers the missing element by the relational path shorter or equal \(L\) hops, where \(L\) is an integer not less than \(1\). A reasoning path is denoted as \(P\), which is obtained by summing all relations and entities in this path.
## 3 Overview of Trm
MMKGR, the version of our conference [53], is limited by manually designed reward functions and relation sparsity as well as poor performance under inductive settings, which motivates us to propose TMR in this paper. As shown in Figure 2, TMR mainly contains two components: **TAIR** and **RARL**. Specifically, inspired by the human inductive ability to generalize unseen tasks from existing relevant knowledge [19], **TAIR** generates fine-grained entity-independent features from the existing topological structure in an attentive manner to represent unseen entities. After employing the unified gate-attention network in MMKGR to complete multi-modal feature fusion, **RARL** conducts MKGR by dynamically adding actions and automatically generating rewards, which is mainly inspired by the fact that humans learn optimal policies by imitating demonstrations rather than predefined paradigms [18]. Notably, TMR is qualified for conducting MKGR under inductive and transductive settings. This is because TMR decouples representation and reasoning into independent components to ensure the flexibility of reasoning under different settings. When inductive settings are converted to transductive setting, TMR only needs to add additional structural representations of seen entities into the unified gate-attention network, while the reasoning module RARL continues to complete reasoning as a multi-modal perception interface without further changes.
## 4 Topology-aware Inductive Representation
Existing methods are powerless to capture fine-grained entity-independent representation, thereby restricting the inductive capability of MKGR models [11, 25, 41]. To address the problem, we propose a novel representation method called TAIR in this section. Notably, the technical difference from the existing method for representing unseen entities lies in the following two points: (1) Taking full advantage of type information derived from the connected directed relations of unseen entities. (2) Aggregating query-related neighbor relations in an attentive manner. Specifically, TAIR includes two modules, i.e., a relation-aware entity initializer and an adaptive topology representation. The former obtains coarse-grained representations of the unseen entity, and the latter aggregates topology information related to the query to generate ine-grained entity-independent representations.
### _Relation-aware Entity Initializer_
In general, entities with similar semantics have similar topological structures in MKGs, which are reflected in the connection patterns of their incoming and outgoing relations. By analyzing the connection patterns, we can obtain a coarse-grained representation that includes type information for unseen entities. For example, unseen entities \(James\_Cameron\) and \(Christopher\_Nolan\) both contain the outgoing edge \(Role\_create\), and these two entities have the same type-level representation, i.e., \(art\)\(creator\).
For an unseen entity \(e_{i}\), its initialized embedding \(\textbf{h}_{i}^{0}\in\mathbb{R}^{d}\) as follows:
\[\textbf{h}_{i}^{0}=\frac{\sum_{r\in I(i)}\textbf{W}_{i}\textbf{u}_{r}+\sum_{r \in O(i)}\textbf{W}_{o}\textbf{u}_{r}}{|I(i)|+|O(i)|} \tag{1}\]
where \(\textbf{W}_{i}\), \(\textbf{W}_{o}\in\mathbb{R}^{d\times d}\) are transformation matrices. \(I(i)\) and \(O(i)\) represent the set of incoming and outgoing relations of the entity \(e_{i}\), respectively. \(\textbf{u}_{r}\in\mathbb{R}^{d}\) is embedding of the relation \(r\).
Considering that the semantics of the same entity can be diverse under different query relations [39], we utilize the attention mechanism to filter out irrelevant neighbor relations. For example, the relations connected by unseen entity _Stephen Curry_ have different types, such as family and vocational relations. Given a triple query (_Stephen Curry_, _Father_, _?_), the vocational relations connected with the unseen entity indicate that _Stephen Curry_ is a professional basketball player, but this information is irrelevant to the query. Therefore, an attention mechanism that dynamically adjusts weights is employed to more accurately represent unseen entities in query tasks. The calculation process is as follows.
\[\alpha_{r}=softmax(\textbf{u}_{r},\textbf{u}_{r_{q}})=\frac{\exp(\textbf{u}_{ r}^{\top}\textbf{u}_{r_{q}})}{\sum_{r^{\prime}\in\mathcal{N}^{\prime}(i)} \exp(\textbf{u}_{r^{\prime}}^{\top}\textbf{u}_{r_{q}})} \tag{2}\]
where \(\textbf{u}_{r}\) and \(\textbf{u}_{r_{q}}\) are relation representations of neighbor relation \(r\) and query relation \(r_{q}\). \(\alpha_{r}\) denotes the correlation between \(r\) and \(r_{q}\).
After integrating \(\alpha_{r}\), Eq. (1) is updated as,
\[\textbf{h}_{i}^{0}=\frac{\sum_{r\in I(i)}\alpha_{r}\textbf{W}_{i}\textbf{u}_{ r}+\sum_{r\in O(i)}\alpha_{r}\textbf{W}_{o}\textbf{u}_{r}}{|I(i)|+|O(i)|} \tag{3}\]
### _Adaptive Topology Representation_
After obtaining the coarse-grained representation \(\textbf{h}^{0}\) of the unseen entity \(e_{i}\) by the initializer, we further capture the fine-grained semantic information from the topology of unseen entities. Inspired by the ability of GNNs to capture topology information in knowledge graphs [51], an adaptive topology representation module leverages GNNs to aggregate local structural information from multi-hop neighbors of entity \(e_{i}\). Specifically, we first concatenate the entities and their relations to obtain triplet information. Compared with individual entities or relations, triple information can provide sufficient topological information [51]. Then, we compute the correlation between the query relation \(r_{q}\) and these triplets that contain more contextual information, which effectively captures the fine-grained representation between topology and \(r_{q}\)[13, 54]. Next, we define updating process of the unseen entity \(e_{i}\) in a \(k\)-th layer as follows.
\[\textbf{h}_{i}^{k}=tanh(\textbf{W}_{self}^{k-1}\textbf{h}_{i}^{k-1 }+\sum_{(i^{\prime},r)\in\mathcal{N}_{i}(e_{i})}\alpha_{i,r}\textbf{W}_{in}^{k-1 }(\textbf{h}_{i}^{k-1}\circ\textbf{u}_{r}^{k-1})\] \[+\sum_{(r,i^{\prime})\in\mathcal{N}_{o}(e_{i})}\alpha_{i,r} \textbf{W}_{out}^{k-1}(\textbf{h}_{i}^{k-1}\circ\textbf{u}_{r}^{k-1})) \tag{4}\]
\[\alpha_{i,r}=\sigma(\textbf{W}_{2}\textbf{c}_{i,r}+\textbf{b}) \tag{5}\]
\[\textbf{c}_{i,r}=\sigma(\textbf{W}_{1}[\textbf{h}_{i}^{k-1}\oplus\textbf{h}_{ j}^{k-1}\oplus\textbf{u}_{r}^{k-1}\oplus\textbf{u}_{r}^{k-1}]) \tag{6}\]
where \(\mathcal{N}_{i}\) and \(\mathcal{N}_{o}\) are the incoming and outgoing neighbors of entity \(e_{i}\), respectively. \(\textbf{W}_{self}^{k-1}\), \(\textbf{W}_{in}^{k-1}\) and \(\textbf{W}_{out}^{k-1}\in\mathbb{R}^{d\times d}\) denote the transformation matrices, respectively. \(\circ\) is the element-wise product, and \(\sigma\) is the activation function \(sigmoid\). \(\alpha_{i,r}\) is the attention weight of the triplet (\(e_{i}\), \(r\), \(e_{j}\)). Based on this, we obtain the fine-grained entity-independent representation \(\textbf{h}_{i}^{k}\) of the unseen entity \(e_{i}\).
Fig. 2: TAIR first exploits query-related topological information to obtain fine-grained entity-independent features. Then, these features and multimodal auxiliary features are fed into the UGAN to generate multi-modal complementary features \(Z\). Next, the RL-based reasoner utilizes \(Z\) and the augmented actions to generate reasoning paths. The discriminator compares reasoning paths and demonstrations to output adaptive rewards for the reasoner. Finally, the reasoner updates the reasoning policies and interacts with MKG to complete the prediction.
Finally, to maintain the consistency of entities and relations within the embedding space, the embeddings of these relations are updated as follows:
\[\mathbf{u}_{r}^{k}=\mathbf{W}_{r}^{k-1}\mathbf{u}_{r}^{k-1} \tag{7}\]
where \(\mathbf{W}_{r}\in\mathbb{R}^{d}\) is a transformation matrix.
## 5 Unified Gate-attention Network
In this section, we employ the unified gate-attention network (UGAN) in MMKGR to conduct feature fusion of fine-grained entity-independent representation and multi-modal auxiliary representation. Specifically, the unified gate-attention network includes an attention-fusion module, and an irrelevance-filtration module. After extracting multi-modal auxiliary features, the attention-fusion module fuses these features and context features together, by attending them with a carefully designed fine-grained attention scheme. Then, the irrelevance-filtration module discards irrelevant or even misleading information and generates noise robust multi-modal complementary features. Based on this, the unified gate-attention network selects features of different modalities online and simultaneously completes intra-modal and inter-modal attention interactions with noise robustness.
### _Feature Extraction_
(1) Context features: The entity \(e_{l}\) at reasoning step \(l\) and query relation \(r_{q}\) are represented as the fine-grained entity-independent embedding \(\mathbf{h}_{l}\) and \(\mathbf{u}_{r_{q}}\), respectively. In addition, the history of the reasoning path that consists of the visited entities and relations is defined as \(b_{l}\) = (\(e_{s}\), \(r_{0}\), \(e_{1}\), \(r_{1}\),...,\(e_{l}\)). We leverage LSTM to integrate the vector of history information \(\mathbf{h}_{l}\) with \(d_{s}\) dimensions into context features. Given the query in our multi-hop reasoning process, we obtain the context features \(\mathbf{y}\) = \([\mathbf{b}_{l};\mathbf{h}_{l};\mathbf{u}_{r_{q}}]\) by concatenating these features. Following [53], a group of context features \(Y\) is calculated as \(Y=[\mathbf{y}_{1},\mathbf{y}_{2},...,\mathbf{y}_{m}]\), where \(Y\in\mathbb{R}^{m\times d_{y}}\), \(m\) and \(d_{y}\) are the number of entities and the dimension of the features, respectively. (2) Multi-modal auxiliary features: To initialize image features \(\mathbf{f}_{i}\), we extract a \(d_{i}\)-dimensional vector of the last fully-connected layer before the softmax in VGG model [4]. Textual features \(\mathbf{f}_{i}\) are initialized by the word2vec framework [30] and expressed as a \(d_{t}\)-dimensional vector. We concatenate the above two parts of features on rows to form the multi-modal auxiliary features \(\mathbf{x}=[\mathbf{f}_{i}W_{i};\mathbf{f}_{i}W_{i}]\). To flexibly add multi-modal auxiliary features, a group of \(\mathbf{x}\) is denoted as \(X=[\mathbf{x}_{1},\mathbf{x}_{2},...,\mathbf{x}_{m}]\), where \(W_{t}\in\mathbb{R}^{d_{t}\times d_{x}/2}\), \(W_{i}\in\mathbb{R}^{d_{t}\times d_{x}/2}\), and \(X\in\mathbb{R}^{m\times d_{x}}\) represents a group of multi-modal auxiliary features, \(d_{x}\) is the dimension of the feature.
### _Attention-fusion Module_
To obtain the complementary features with sufficient interactions and less noise, we need to fuse the context features \(Y\) and multi-modal auxiliary features \(X\) generated in feature extraction. However, redundant features tend to have a negative impact on the prediction during the multi-modal fusion [49]. Specifically, redundant features are either shifted versions of the features related to the triple query or very similar with little or no variations, which can amplify the negative effects of noise [23]. The redundant features add computational complexity and cause collinearity problems [36]. Consequently, we propose the attention-fusion module that fuses the context features and multi-modal auxiliary features effectively.
Specifically, we first utilize linear functions to generate the queries \(Q\), keys \(K\), and values \(V\) of the attention mechanism,
\[Q=XW_{q},K=YW_{k},V=YW_{v} \tag{8}\]
where \(W_{q}\in\mathbb{R}^{d_{x}\times d},W_{k},W_{v}\in\mathbb{R}^{d_{y}\times d}\), and \(Q,K,V\in\mathbb{R}^{m\times d}\) have the same shape. Then, the joint representation \(B^{l}\) of \(Q\) and \(K\) is learned based on the MLB pooling method [17], inspired by the recent successes of it in fine-grained multi-modal fusion,
\[B^{l}=KW_{k}^{l}\odot QW_{q}^{l} \tag{9}\]
Similarly, we can generate the joint representation \(B^{r}\) of \(V\) and \(Q\) with the following equation,
\[B^{r}=VW_{v}^{r}\odot QW_{q}^{r} \tag{10}\]
where \(W_{k}^{l},W_{q}^{l},W_{v}^{r},W_{q}^{r}\in\mathbb{R}^{d\times j}\) are embedding matrices, and \(\odot\) is Hadamard product.
Next, the filtration gate \(g_{t}\) applied to different feature vectors is defined as,
\[g_{t}=\sigma(B^{l}W_{m}) \tag{11}\]
where \(W_{m}\in\mathbb{R}^{j\times d}\) is an embedding matrix and \(\sigma\) denotes the sigmoid activation. Based on the filtration gate \(g_{t}\), we can filter out the redundant features generated during fusion and obtain a new representation with the following probability distributions,
\[G_{s}=softmax((g_{t}\odot K)((1-g_{t})\odot Q)) \tag{12}\]
where \(g_{t}\) and \(1-g_{t}\) are used to trade off how many context features and multi-modal auxiliary features are fused.
Finally, our attention-fusion module generates the attended features \(\hat{V}\)=\(\{\mathbf{v}_{i}\}_{i=1}^{m}\) by accumulating the enhanced bilinear values of context features and multi-modal auxiliary features,
\[\hat{V}=\sum\nolimits_{i=1}^{m}(G_{s}W_{g}^{l})B_{i}^{r} \tag{13}\]
where \(W_{g}^{l}\in\mathbb{R}^{d\times 1}\), and \(\mathbf{v}_{i}\in\mathbb{R}^{1\times j}\) denotes a row of the attended features \(\hat{V}\in\mathbb{R}^{m\times j}\), feature vector \(B_{i}^{r}\in\mathbb{R}^{1\times j}\) is a row of the embedding matrix \(B^{r}\).
By designing the attention-fusion module, we can complete the intra-modal and inter-modal feature interactions in a unified manner at the same time. This is because the inputs of this module are pairs from context features and multi-modal auxiliary features, where each vector of a pair may be learned from the same modality or different ones.
### _Irrelevance-filtration Module_
We use an irrelevance-filtration module to further improve the robustness of the model. The attended features \(\hat{V}\) obtained by the attention-fusion module may contain irrelevant features [14]. Specifically, irrelevant features are irrelevant to the triple query in the reasoning process. Since the attention mechanism assigns weights to all features,
these features tend to participate in model computation and mislead the reasoning policy [31]. This motivates our model to weight more on the most related complementary features and dynamically filter irrelevant ones. This is achieved by a well-designed irrelevance-filtration gate function. The output of this gate is a scalar, the value range of which is (0,1). The multi-modal complementary features \(Z\) are obtained as follows,
\[G_{f}=\sigma(B^{r}\odot\hat{V}) \tag{14}\]
\[Z=G_{f}(B^{r}\odot\hat{V}) \tag{15}\]
where \(\sigma\) and \(G_{f}\) denote the sigmoid activation function and irrelevance-filtration gate, respectively.
## 6 Relation-augment Adaptive reinforcement learning
The existing RL-based MKGR method is limited by manual rewards and sparse relations [1, 20]. To address this problem, we propose a novel RL-based framework entitled RARL in this section. Compared with MMKGR, the main technical difference of RARL lies in the following two points. (1) We effectively increase the additional actions to alleviate the negative impact of sparse relations on the RL-based model. (2) RARL utilizes generative adversarial imitating networks to adaptively learn rewards by imitating demonstrations, which can stabilize reasoning performance and eliminate manual intervention in reward design. This provides a new research perspective for RL-based reasoning methods for MKGR.
RARL consists of three modules namely RL-based reasoner, rule-based demonstration sampler, and modality-aware discriminator. Specifically, the reasoner leverages a rule-guided action augmentation method that dynamically adds additional actions and outputs diverse generated paths about missing elements. Then, the rule-based demonstration sampler filters out low-contributing paths as well as extracts trustworthy demonstrations from MKGs. Next, the modality-aware discriminator generates rewards to update the reasoner by evaluating the semantic similarity between demonstrations and reasoning paths. After sufficient adversarial training, the RL-based reasoner tries to deceive the discriminator to gain more adaptive reward values by imitating the demonstrations. We introduce the above three modules in subsections 6.1, 6.2, and 6.3, respectively.
### _RL-based Reasoner_
#### 6.1.1 Reinforcement Learning Formulation
RARL trains an agent to interact with the with MKGs by modeling by Markov decision process (MDP). The MDP consists of a 4-tuple, i.e., States, Actions, Transition, Rewards. The agent selects actions based on the current state and obtains rewards from the environment (MKGs) to update its behavior policy until it reaches a termination state or a predefined reasoning step.
**States**: The state of the agent at reasoning step \(l\) is denoted as \(s_{l}\)= (\(e_{l}\), (\(e_{s}\), \(r_{q}\)) \(\in\)\(\mathcal{S}\), where \(\mathcal{S}\) denotes a state space and \(e_{l}\) represents the entity at the current reasoning step \(l\). The source entity \(e_{s}\) and the query relation \(r_{q}\) are the global context shared throughout all steps.
**Actions**: For the given state \(s_{l}\), its original action space is the set of usable actions \(A_{l}^{o}\) at reasoning step \(l\) is expressed as \(A_{l}^{o}=\{(r_{l+1},\ e_{l+1})|\) (\(e_{l}\), \(r_{l+1}\), \(e_{l+1}\)) \(\in\)\(\mathcal{G}_{m}\}\). To alleviate relation sparsity, the rule-guided action augmentation module adds extra potential actions \(A_{l}^{o}\) into the original action space. Thus, the joint action space \(A_{l}^{o}\cup A_{l}^{a}=A_{l}\in\mathcal{A}\). In addition, we add \(STOP\) action to avoid infinitely unrolling in the reasoning process. The \(STOP\) action executes a self-loop when the reasoning step is unrolled to the maximum step \(L\).
**Transition**: \(\mathcal{P}_{r}\) is defined to facilitate the transition from the current state \(s_{t}\) to the next state \(s_{l+1}\). \(\mathcal{P}_{r}\): \(\mathcal{S}\)\(\times\)\(\mathcal{A}\)\(\rightarrow\)\(\mathcal{S}\) is defined as \(\mathcal{P}_{r}\) (\(s_{l}\), \(A_{l}\)) = \(\mathcal{P}_{r}\) (\(e_{l}\), (\(e_{s}\), \(r_{q}\)), \(A^{o}\), \(A^{a}\)).
**Rewards**: Different from existing manually-designed reward functions, we design an adaptive reward mechanism to eliminate manual intervention, which achieves high reasoning performance in complex and uncertain environments. The adaptive reward arises from path comparisons between the generator and expert demonstration, and it is defined in Eq. 29.
**Policy Network** The policy function \(\pi\) is used as a multi-modal perception interface to output the next action with the highest executable probability. For a given state, \(\pi\) selects the promising action with the maximum likelihood, which is defined as,
\[\pi_{\theta}(a_{l}|s_{l})=softmax(\textbf{A}_{l}(\textbf{W}\text{ReLu}(Z))) \tag{16}\]
where \(a_{l}\)\(\in\)\(A_{l}\), and \(A_{l}\) can be encoded to \(\textbf{A}_{t}\) by stacking the representations of existing available actions.
#### 6.1.2 Rule-guided Action Augmentation
The existing RL-based reasoning method assumes sufficient relation paths between entities, and regards these relations and connected tail entities as the next action. However, the intrinsic incompleteness of MKGs leads to sparse relations of an entity. Especially, emerging entities are sparsely connected to existing entities under the inductive setting. This sparsity limits the utilization of potential reasoning paths. Therefore, it is necessary to design an action space augmentation method to eliminate the sparsity of the action space. Although the idea of augmenting the action space is promising, a major challenge is how to _officially augment additional actions_.
Intuitively, enumerating all relations and entities to compose an additional action space can complement the existing action space. However, this combined search space is close to \(\mathcal{O}(|\mathcal{E}|\times|\mathcal{R}|)\), where \(|\mathcal{E}|\) and \(|\mathcal{R}|\) are the numbers of entities and relations in a MKG, respectively. For a large-scale MKG with millions of entities and thousands of relations, large search space becomes an intractable problem. To address the above problem, we propose a novel action augmentation method to efficiently augment additional actions. For a state \(s_{l}\), the candidate set of augmented action is denoted as \(C_{t}\) = \(\{(r^{\prime},e^{\prime})|r^{\prime}\in\mathcal{R}\wedge e^{\prime}\in \mathcal{E}_{m}\wedge(e_{l},r^{\prime},e^{\prime})\notin\mathcal{G}_{m}\}\). First, we calculate the probability for the candidate set \(C_{t}\):
\[p((r^{\prime},e^{\prime})|s_{l})=p(r^{\prime}|s_{l})p(e^{\prime}|r^{\prime},s_{l}) \tag{17}\]
To reduce the candidate action space and time-consuming, an approximate pruning strategy is used to filter out additional actions. The pruning strategy consists
of additional relation selection using \(p(r^{\prime}|s_{l})\) and entity generation using \(p(e^{\prime}|r^{\prime},s_{l})\).
Then, for the state \(s_{l}\), the attention score of the candidate relations is calculated as \(p(r|s_{l})\),
\[\textbf{w}=softmax(MLP(\textbf{s}_{l})\cdot[\textbf{u}_{r_{1}},...,\textbf{u}_ {r_{|R|}}]) \tag{18}\]
where **w** denotes the attention vector. We select top \(x\) relations with the largest attention values in **w** to obtain additional relation set \(\mathcal{R}_{add}=r^{1}\), \(r^{2}\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\\\\\\\_\
path features whose quality is higher than random noise into the reward.
\[R_{r}=\max(D(\mathbf{\nu_{P}})-D^{N}(\mathbf{\nu_{P}}),0) \tag{27}\]
\[R_{e}=\max(D(\mathbf{\kappa_{P}})-D^{N}(\mathbf{\kappa_{P}}),0) \tag{28}\]
where \(D^{N}(\mathbf{\nu_{P}})\) and \(D^{N}(\mathbf{\kappa_{P}})\) respectively denote noise embeddings from the entity and relation layers. These embeddings consist of random noise sampled from a continuous uniform distribution. \(R_{r}\) and \(R_{e}\) represent the adaptive rewards from relation layer and entity layer, respectively.
Then, we define the adaptive reward \(R(s_{l})\) at the state \(s_{l}\) as follows:
\[R(s_{l})=\alpha R_{e}+(1-\alpha)R_{r} \tag{29}\]
where \(\alpha\) is a balance factor. Note that, the agent learns adaptive rewards from demonstrations without manually designing and tuning, which reduces manual intervention and subjective bias [2]. In addition, the adaptive reward improves the generalizability of our proposed model and it is suitable for unseen tasks. This is because the adaptive reward mechanism automatically capturing common meta-knowledge by learning relation patterns and multi-modal features in demonstrations [32].
Next, we optimize the modality-aware discriminator by reducing training loss and expect it to possess expertise in distinguishing between \(P\) and \(\Omega\).
\[\mathcal{L}_{r}=D(\mathbf{\nu_{P}})-D(\mathbf{\nu_{\Omega}})+\lambda(\parallel \bigtriangledown_{\hat{p}}D(\hat{p})\parallel_{2}-1)^{2} \tag{30}\]
where \(\lambda\) is a penalty term and \(\hat{p}\) is sampled uniformly along straight lines between the generated path and the demonstration.
For the discriminator at the entity level, we define loss \(\mathcal{L}_{r}\) as follows:
\[\mathcal{L}_{e}=-(\log D(\mathbf{\kappa_{\Omega}})+\log(1-D(\mathbf{\kappa_{P}}))) \tag{31}\]
Finally, to maximize the accumulated rewards of adaptive reinforcement learning and obtain the optimal policy, the objective function is as follows,
\[\mathcal{J}(\theta)=\mathbb{E}_{(e_{s},r,e_{d})\sim\mathcal{G}_{m}}\mathbb{E} _{a_{1},\dots,a_{L}\sim\pi_{\theta}}[\mathbb{R}(s_{l}\mid e_{s},r)] \tag{32}\]
## 7 Experiment
### _Datasets_
Following MMKGR [34, 53], we use WN9-IMG-TXT and FB-IMG-TXT to verify the reasoning performance under the transductive setting. Each entity in these MKGs contains three modal information: structure, image, and text. Specifically, the relation triplets and textual descriptions are extracted from WordNet and Freebase. To extract the image features of the entities, 10 images and 100 images are crawled for each entity in WN9-IMG-TXT and FB-IMG-TXT, respectively [34]. Statistics are shown in Table I.
To perform inductive reasoning in MKGs, we construct new inductive benchmark datasets by extracting disjoint subgraphs from WN9-IMG-TXT and FB-IMG-TXT. In particular, each dataset contains a pair of graphs: _training graph_ and _ind_test graph_. Following [37], to generate the training graph, we first uniformly sample several entities as the root nodes, and then conduct the union of k-hop neighbor triplets around the roots. Next, we set the maximum number of samples at each hop to prevent the exponential growth of new neighbors. Finally, we remove the training graph from the whole graph and sample the test graph using the same procedure. In fact, the above division method destroys the link distribution in the original graph and reduces the number of triplets. Therefore, for a robust evaluation, we adjust the parameters of the above procedure to sample 5%, 10%, and 15% of the original graph (i.e., WN9-IMG-TXT and FB-IMG-TXT) to construct datasets with different sizes [41]. In summary, the model is trained on the training graph and tested on the ind_test graph in every version. Note that, (1) the two graphs have disjoint sets of entities, (2) training graphs contain all relations present in ind_test graphs.
### _Evaluation Protocol_
To evaluate the reasoning performance of TMR over inductive and transductive settings, we adopt the mean reciprocal rank (MRR) and Hits@N to report experimental results, which are common metrics for MKGR [34, 53].
### _Baselines and Implementation Details_
To investigate the performance of TMR, two categories of methods are compared. 1) Knowledge graph reasoning methods under the induction setting: DRUM [33], CoMPILE [29], Morse [6], RED-GNN [51]. 2) SOTA MKGR method: MMKGR [53]. Note that, the transductive reasoning methods (i.e., TransE [3] or RLPM [38]) on the traditional KG cannot be applied to MKGR. This is because these methods retrain the embedding from scratch whenever a new entity appears [41]. In our training stage, some core settings are as follows. The embedding dimension \(d_{s}\) of relation and history is set to 200, the embedding dimension \(d_{i}\) of the image feature is set to 128 and 4096 on FB-IMG-TXT and WN9-IMG-TXT respectively, and the embedding dimension \(d_{t}\) of textual feature is 1000. The maximum reasoning step \(L\) is 3. The number of additional relation \(I\) is 3 and 5 on different versions of WN9-IMG-TXT and FB-IMG-TXT, respectively. The size \(N\) is set to 5. We employ a 3-layer GNN to obtain the adaptive topology representation. \(\alpha\) in Eq.(29) is set to 0.4 and 0.2 on different versions of WN9-IMG-TXT and FB-IMG-TXT, respectively.
\begin{table}
\begin{tabular}{l l l l l l l} \hline Dataset & \#Ent & \#Rel & \#Train & \#Valid & \#Test \\ \hline WN9-IMG-TXT & 6,555 & 9 & 11,747 & 1,337 & 1,319 \\ FB-IMG-TXT & 11,757 & 1,231 & 285,850 & 29,580 & 34,863 \\ \hline \end{tabular}
\end{table} TABLE I: Statistics of the experimental datasets over the transductive setting.
\begin{table}
\begin{tabular}{l l l l l l l l} \hline & \multicolumn{3}{c}{WN9-IMG-TXT} & \multicolumn{3}{c}{FB-IMG-TXT} \\ \hline & & \#Ent & \#Rel & \#Triplets & \#Ent & \#Rel & \#Triplets \\ \hline V1 & Training & 420 & 8 & 760 & 2848 & 342 & 17561 \\ V1 & Ind\_test & 270 & 8 & 401 & 2002 & 263 & 3325 \\ \hline V2 & Training & 654 & 9 & 1465 & 3205 & 631 & 35184 \\ V2 & Ind\_test & 406 & 8 & 814 & 2111 & 343 & 6064 \\ \hline V3 & Training & 658 & 9 & 2180 & 3716 & 750 & 52544 \\ V3 & Ind\_test & 581 & 8 & 1442 & 2419 & 254 & 10623 \\ \hline \end{tabular}
\end{table} TABLE II: Statistics of the datasets over the inductive setting.
### _Inductive Reasoning on MKGs_
Reasoning performance over the inductive setting is reported in Table III (these scores are in percentage), where the competitive results of the baseline are marked by underlining and the highest performance results are highlighted in bold. Specifically, we have the following insightful analysis based on the experimental results in Table III. (1) The performance of our proposed TMR outperforms all baselines. This is because TMR not only generates fine-grained entity-independent representations for unseen entities, but also utilizes RARL to eliminate the negative impact of manual rewards and sparse relations on reasoning accuracy. (2) The experimental results of MMKGR are the lowest in the different datasets. This is because MMKGR without induction capability cannot obtain structured representations of unseen entities. Therefore, it is not a reasonable solution to only utilize multi-modal auxiliary information without learning inductive capabilities in the induction setting. (3) The rule-based SOTA model DRUM combines relations to infer unseen tasks, but this model is limited by the quality of the rules. (4) Similar to DRUM, CoMPILE, MorseE and RED-GNN are designed to infer unseen tasks in traditional knowledge graphs, but they adapt to conduct reasoning without utilizing multi-modal auxiliary features in the inductive setting. This is because the three models learn the inductive ability from the local graph structure information. In particular, the existing SOTA model RED-GNN remains competitive in all versions of MKGs. This is because RED-GNN recursively encodes multiple relational digraphs with shared edges and preserves the structural patterns at the same time [51]. In short, the key to performance improvement in the induction setting is to consider both induction capability and the ability to exploit multi-modal data. Note that, TMR is the first model to do this in the domain of MKGR.
### _Transductive Reasoning on MKGs_
In this section, we investigate whether TMR outperforms the SOTA model MMKGR under the transductive setting. To be consistent with MMKGR, we obtain the pre-trained embeddings of all entities by TransE. This is because unseen entities do not exist under the transductive setting. Specifically, the pre-trained representation \(\mathbf{e}_{i}\) of entity \(i\) is additionally added into the context features of TMR. Furthermore, a concern is that TMR is fed with pre-trained entity representations do not need entity-independent features generated by the TAIR component that contains topology information in the transductive setting. To eliminate the above concern, we added a variant of TMR-TAIR where the TAIR component is removed (i.e., the context features of TMR do not contain fine-grained entity-independent features).
The experimental results under the transductive setting are shown in Table IV. We have the following observations. (1) The reasoning performance of TMR surpasses that of the SOTA MMKGR on the original MKGs. This demonstrates the flexibility and generalizability of TMR under different settings. (2) The performance of TMR-TAIR is higher than that of MMKGR. This is because the RARL component in TMA-TAIR can eliminate the negative impact of manual rewards and sparse relations on reasoning accuracy. (3) The performance of TMR-TAIR is lower than that of TMR. A reason is that the entity-independent features generated by the TAIR component aggregate multi-hop topology information. The information can provide reasoning context to improve performance under both inductive and transductive settings.
### _Ablation Study_
The overall goal of the ablation study is to measure the contribution of different components by adding different variants. Figure 3 reports the experimental results of the TMR and its variants. (1) The variant version w/o TAIR only uses multi-modal features of unseen entities where the TAIR component is removed from the TMR. (2) w/o UGAN, in which the unified gate-attention network is removed and basic concatenation operation is added to fuse all features. (3) w/o RARL, a variant version where RARL is removed. To ensure the agent conducts the reasoning, we retain the RL-based reasoner with the original action space and basic
\begin{table}
\begin{tabular}{c|c c c|c c c} \hline \hline & \multicolumn{3}{c|}{WN-IMG-TXT} & \multicolumn{3}{c}{FB-IMG-TXT} \\ Model & MRR & Hist@1 & Hist@10 & MRR & Hist@1 & Hist@10 \\ \hline MMKGR & 80.2 & 73.6 & 92.8 & 71.3 & 65.8 & 82.6 \\ TMR-TAIR & 83.6 & 76.4 & 93.2 & 74.3 & 67.5 & 85.4 \\ TMR & **86.3** & **79.7** & **93.7** & **76.6** & **71.4** & **87.6** \\ \hline \hline \end{tabular}
\end{table} TABLE IV: Transductive link prediction results on different MKGs.
\begin{table}
\begin{tabular}{c|c c c c c c c c|c c c c c c c c c} \hline \hline & \multicolumn{8}{c|}{WN-IMG-TXT} & \multicolumn{8}{c}{FB-IMG-TXT} \\ & \multicolumn{3}{c|}{V1} & \multicolumn{3}{c|}{V2} & \multicolumn{3}{c|}{V3} & \multicolumn{3}{c}{V1} & \multicolumn{3}{c}{V2} & \multicolumn{3}{c}{V3} \\ & MRR & Hist@1 & Hist@10 & MRR & Hist@10 & MRR & Hist@1 & Hist@10 & MRR & Hist@1 & Hist@10 & MRR & Hist@1 & Hist@10 & MRR & Hist@1 & Hist@10 \\ \hline MMKGR & 27.0 & 25.1 & 30.2 & 27.2 & 25.5 & 30.6 & 29.3 & 26.9 & 33.3 & 21.4 & 19.3 & 25.3 & 23.3 & 21.7 & 26.5 & 26.0 & 23.9 & 29.1 \\ DRUM & 43.4 & 40.5 & 46.0 & 45.2 & 41.6 & 48.7 & 48.4 & 45.7 & 51.0 & 35.6 & 32.7 & 38.2 & 37.8 & 34.8 & 39.7 & 40.1 & 38.6 & 43.2 \\ CoMPILE & 45.2 & 41.7 & 47.3 & 47.0 & 43.9 & 50.8 & 49.1 & 47.6 & 53.2 & 36.9 & 34.5 & 39.3 & 39.8 & 35.9 & 40.9 & 42.3 & 39.9 & 44.4 \\ Morse & 48.2 & 45.8 & 52.4 & 50.3 & 48.6 & 53.1 & 54.2 & 52.1 & 56.7 & 38.3 & 36.6 & 41.2 & 40.7 & 38.3 & 43.1 & 44.2 & 42.1 & 46.2 \\ RED-GNN & 51.2 & 49.3 & 54.3 & 53.2 & 50.2 & 56.8 & 56.1 & 54.8 & 59.2 & **40.5** & 38.4 & **42.4** & 43.3 & 41.1 & 45.2 & 46.0 & **44.2** & 48.4 \\ TMR & **64.9** & **62.3** & **69.1** & **67.6** & **64.1** & **72.0** & **71.1** & **69.7** & **74.8** & **57.0** & **54.7** & **59.4** & **60.1** & **57.5** & **62.7** & **63.3** & **60.8** & **66.2** \\ \hline Improv. & 13.7\% & 13.0\% & 14.8\% & 14.3\% & 13.9\% & 15.2\% & 15.0\% & 14.9\% & 15.6\% & 16.5\% & 16.3\% & 17\% & 16.8\% & 16.4\% & 17.5\% & 17.3\% & 16.6\% & 17.8\% \\ \hline \hline \end{tabular}
\end{table} TABLE III: Inductive link prediction results for different versions of MKGs.
0/1 reward (i.e., the reward value is set to 1 if the target entity is the ground truth entity. Otherwise, the value is 0).
We have the following observations. (1) TMR has the best performance compared with both variant versions. This validates the effectiveness of all components. (2) The results of w/o TAIR are much lower than that of TMR, which verifies the importance of learning fine-grained entity-independent representation in the inductive setting. (3) Although the input of w/o UGAN includes multi-modal information, its reasoning performance still declines compared with TMR. This is because the concatenation operation cannot generate multi-modal complementary features. (4) After removing the rule-guided action augmentation method and adaptive reward mechanism, the performance of w/o RARA significantly degrades on different datasets. This demonstrates that RARL can eliminate the negative effects of sparse relations and manual rewards on reasoning performance.
### _Further Analysis_
#### 7.7.1 Convergence Analysis
To analyze the convergence rate and reasoning performance between our adaptive reward and manual rewards in MMKGR, we design a variant entitled TMR-3D. Specifically, this variant removes the _Rule-based Demonstration Sampler_ and _Modality-aware Discriminator_ in RARL. To ensure that the agent is rewarded, we add the 3D reward mechanism (i.e., the manual reward function in MMKGR). In addition, we design variant models TMR-R and TMR-E by removing the relation-level reward \(R_{r}\) and the entity-level reward \(R_{e}\) in TMR, respectively. This setting can investigate the contribution of different parts of the adaptive reward. Observed from Figure 4, (1) TMR adopted the adaptive reward has the fastest convergence and the most stable performance on MKGs. This is because the adaptive reward learned from both path semantic and multi-modal feature automatically eliminate the manual intervention and avoids decision bias on different MKG datasets [21]. (2) Although TMR-3D can converge slowly, its performance is still unstable. A reason is that the weak generalizability of manually-designed 3D rewards leads to unstable training on different datasets [10]. (3) The performance of TMR-R is slightly worse than that of TMR-E, which indicates that \(R_{r}\) has more contribution than \(R_{e}\) in the RARL.
#### 7.7.2 Effectiveness Analysis for Action Augmentation
To investigate the effectiveness of the action augmentation method for expanding the latent action space, we design a new variant model TMR-AA by removing the rule-guided action augmentation method in RARL. The experimental results are shown in Figure 5, and we have the following analysis: (1) The performance of TMR has declined to varying degrees in all datasets after removing the action augmentation method, which verifies the effectiveness of the proposed action augmentation method. (2) The performance improvement of the rule-guided action augmentation method on different versions of FB-IMG-TXT is more obvious than that on the different versions of WN9-IMG-TXT. This is because more relations on different versions of FB-IMG-TXT can be used to build more complex reasoning rules. (3) Compared with adaptive rewards, the performance improvement of the rule-guided action augmentation method is relatively small. One potential reason is that the reasoning processes completely depend on original actions, and the additional actions from this method mainly play an auxiliary role.
### _Key Parameter Analysis_
The balance factor \(\alpha\) is an important parameter for our proposed model TMR. As presented in Figure 6, 0.4 and 0.2 are the optimal values of \(\alpha\) on the different versions of WN9-IMG-TXT and FB-IMG-TXT, respectively. The reward \(R_{e}\) from entity level is assigned small weights, which demonstrates that the semantic correctness of relational paths
Fig. 4: The convergence rate of TMR and the variant model TMR-3D.
Fig. 5: Performance comparison between TMR and TMR-AA.
Fig. 3: Ablation on different components of the TMR.
Fig. 6: Performance of TMR w.r.t. varied \(\alpha\) on different datasets.
provides better reasoning clues than multi-modal entities in inductive reasoning tasks.
## 8 Related Work
### _Multi-modal Knowledge Graph_
A traditional KG is essentially a semantic graph that consists of entities (nodes) and relations (edges). At present, the actual internet data show multi-modal characteristics [15]. MKGs are developed to incorporate various types of data from different modalities into KGs [34]. A MKG typically includes structural triplets, and multi-modal data (i.e., texts and images) [26]. Common MKGs are IMGpedia [9], Ricpedia [40], and FB-Des [36]. However, the multi-modal auxiliary data of these MKGs is singular (i.e., image or text). To expand the auxiliary data with one modality, WN9-IMG-TXT and FB-IMG-TXT simultaneously add a number of textual descriptions and images to each entity, aiming to further enhance the data diversity of the MKGs [34, 48].
### _Multi-modal Knowledge Graph Reasoning_
Since MKGs inherently contain incomplete knowledge, MKGR technology that can synthesize the original knowledge and infer the missing knowledge is particularly important [34, 35]. Some studies employ the attention model or concatenation to fuse multi-modal auxiliary features and then adopt TransE to infer missing elements, such as IKRL [44] and TransAE [42], and MTRL [34]. However, these methods lack interpretability and are primarily suitable for single-hop reasoning containing limited information. To address this issue, MMKGR leverages the symbolic compositionality of the multi-step relational path (choices of actions) to infer the correct entity [5, 53]. MMKGR has been proven to be a SOTA model in the field of TKGR. Its multi-hop reasoning process is as intuitive as "going for a walk", which naturally forms an explainable provenance for MMKGR.
### _Inductive Reasoning on Knowledge Graph_
The inductive setting is receiving increasing attention since unseen entities are emerging in KGs. Therefore, completing knowledge reasoning in an inductive setting is a practical application. Several methods are proposed to solve this problem. Rule-based methods can leverage the logical rules of existing knowledge to infer new facts, because the rules are independent of specific entities [33]. In addition, GraIL [37] and CoMPILE [29] aim to generalize to unseen entities and improve reasoning performance by subgraph extraction, but the enclosing subgraphs cannot learn relational structures so as to weaken the inductive capability. Inspired by the powerful graph modeling capabilities of, SOTA models like MorsE [6] and RED-GNN [51] utilize GNNs to aggregate topological structures and mine existing neighbor information, which is a promising method for inductive reasoning. However, these methods still have limitations: (1) They do not extract fine-grained entity-independent features related to the query, which restricts their inductive capacity. (2) Lack of ability to utilize multi-modal auxiliary information in MKGR.
## 9 Conclusion
In this paper, we propose TMR as a solution to conduct TKGR in both inductive and transductive settings. Specifically, TMR mainly includes TAIR and RARL. TAIR learns fine-grained entity-independent representation from query-related topology knowledge to represent unseen entities. RARL eliminates the negative impact of sparse relations and artificial rewards on reasoning accuracy by introducing additional actions and adaptive rewards. To ensure that the entities in the training and testing sets are disjoint under the inductive setting, we construct six MKG datasets with varying scales. Experimental results demonstrate the superior performance of our proposed model compared to existing baselines across different settings.
| This paper proposes a model called TMR を使って、シミュレートされたデータ環境から貴重な情報を取り出します。この論文の提出を完了する予定です。 |
2307.07510 | NNLL Resummation for Projected Three-Point Energy Correlator | The projected energy correlator measures the energy deposited in multiple
detectors as a function of the largest angular distance $x_L = (1 -
\cos\chi_L)/2$ between detectors. The collinear limit $x_L\to 0$ of the
projected energy correlator is particularly interesting for understanding the
jet-substructures, while the large logarithms of $x_L$ could potentially spoil
the perturbation theory and must be resummed. As a necessary ingredient for its
resummation at next-to-next-to-leading logarithmic (NNLL) accuracy, we
calculate the two-loop jet functions for the projected three-point energy
correlator (E3C), using direct integration method and the parameter space
Integration-by-Part (IBP) method. We then present the NNLL resummation for
$e^+e^-$ annihilation and an approximate NNLL resummation for $pp\rightarrow
jj$ process, where the two-loop hard constant is estimated in the latter case.
The convergence is improved and the hadronization effect in the collinear limit
is suppressed when considering the ratio of E3C distribution to two-point
energy-energy correlator (EEC). Our results show potential in precision
determination of strong coupling constant using energy correlators from both
$e^+e^-$ data and $pp$ data. | Wen Chen, Jun Gao, Yibei Li, Zhen Xu, Xiaoyuan Zhang, Hua Xing Zhu | 2023-07-14T17:58:13 | http://arxiv.org/abs/2307.07510v1 | # NNLL Resummation for Projected Three-Point Energy Correlator
###### Abstract
The projected energy correlator measures the energy deposited in multiple detectors as a function of the largest angular distance \(x_{L}=(1-\cos\chi_{L})/2\) between detectors. The collinear limit \(x_{L}\to 0\) of the projected energy correlator is particularly interesting for understanding the jet-substructures, while the large logarithms of \(x_{L}\) could potentially spoil the perturbation theory and must be resummed. As a necessary ingredient for its resummation at next-to-next-to-leading logarithmic (NNLL) accuracy, we calculate the two-loop jet functions for the projected three-point energy correlator (E3C), using direct integration method and the parameter space Integration-by-Part (IBP) method. We then present the NNLL resummation for \(e^{+}e^{-}\) annihilation and an approximate NNLL resummation for \(pp\to jj\) process, where the two-loop hard constant is estimated in the latter case. The convergence is improved and the hadronization effect in the collinear limit is suppressed when considering the ratio of E3C distribution to two-point energy-energy correlator (EEC). Our results show potential in precision determination of strong coupling constant using energy correlators from both \(e^{+}e^{-}\) data and \(pp\) data.
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Footnote †: institutetext: \({}^{a}\) Department of Physics, University of California, Berkeley, CA 94720, USA
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Introduction
Energy correlators are a class of multi-particle angle correlation functions, weighted by the particle energy. Thanks to the energy weighting, they are infrared and collinear safe observables and can be calculated in perturbation theory. The simplest energy correlator is a two-point energy correlator, or Energy-Energy Correlation function (EEC). Proposed in 1970s [1; 2], EEC measures the correlation of energy deposited in two detectors as a function of the angle \(\chi\) between them. In perturbation theory, the definition of EEC reads
\[\frac{d\sigma^{[2]}}{d\cos\chi}\equiv\sum_{i,j}\int d\sigma\frac{E_{i}E_{j}}{Q ^{2}}\delta\left(\vec{n}_{i}\cdot\vec{n}_{j}-\cos\chi\right)\,, \tag{1}\]
where \(i\), \(j\) run over all the final state particles, \(\vec{n}_{i}\) and \(\vec{n}_{j}\) are unit three-vectors that define the directions of the particles, and \(Q\) is the total energy in the center-of-mass frame. Compared with other event shape variables studied at Large Electron-Positron Collider (LEP), one advantage of EEC is its simple analytic properties. As far as we are aware of, EEC is the only event shape that can be calculated analytically beyond leading order, e.g. it's now known analytically through to next-to-next-to-leading order (NNLO) [3; 4] in \(\mathcal{N}=4\) super Yang-Mills (SYM) theory and through to NLO in QCD [5; 6; 7].
In recent years, increasing attention has been paid to generalization of EEC to \(N\)-point energy correlators, which measure the energies of the outgoing particles with \(N\) detectors at colliders and turn out to be a function of \(N(N-1)/2\) angles among these detectors [8; 9; 10; 11; 12; 13; 14]. For example, the three-point energy correlator (EEEC) is defined as
\[\frac{d^{3}\sigma}{dx_{1}dx_{2}dx_{3}}\equiv \sum_{i,j,k}\int d\sigma\frac{E_{i}E_{j}E_{k}}{Q^{3}}\] \[\times\delta\left(x_{1}-\frac{1-\cos\theta_{jk}}{2}\right)\delta \left(x_{2}-\frac{1-\cos\theta_{ik}}{2}\right)\delta\left(x_{3}-\frac{1-\cos \theta_{ij}}{2}\right)\,, \tag{2}\]
which gives rise to rich functional dependence on the angles and can be used to probe various properties of perturbative QCD. The LO EEEC was first computed in the triple collinear limit in Ref. [9], later genelarized to arbitrary angle dependence in both \(\mathcal{N}=4\) SYM [15] and QCD [14]. To reduce the dimension of the kinematic space of the measured angles without losing too much useful information, one can project the kinematic dependence into a 1D subspace, which leads to the so-called _projected energy correlator_[16]. In momentum space, projected \(N\)-point energy correlator (ENC) is given by restricting the maximum angular distance to be \(x_{L}\):
\[\frac{d\sigma^{[N]}}{dx_{L}}\equiv\sum_{n}\sum_{1\leq i_{1},\cdots i_{N}\leq n }\int d\sigma\frac{\prod_{a=1}^{N}E_{i_{a}}}{Q^{N}}\delta(x_{L}-\max\{x_{i_{1 },i_{2}},x_{i_{1},i_{3}},\cdots x_{i_{N-1},i_{N}}\})\,, \tag{3}\]
and for example, EEEC is then reduced to the projected three-point correlator (E3C). In this work we are mainly interested in the small angle, or collinear limit of E3C, namely \(x_{L}\to 0\).
It is well-known in the boundary of phase space, incomplete cancellation of infrared divergences can lead to large logarithms that could possibly spoil the convergence of the perturbation theory and thus it is essential to resum these large logarithms to all orders. EEC is special as it exhibits both large logarithms in collinear limit and back-to-back limit. In this work we are interested in the large logarithms in the collinear limit, for which the most singular terms behave as \(\alpha_{s}^{n}\ln^{n}x_{L}\) at \(n\) loops. In the collinear region, EEC can be factorized into a hard function and a jet function, both of which live in the flavor space. The resummation of collinear EEC has been performed up to NNLL accuracy in both QCD [17] and \(\mathcal{N}=4\) SYM [17; 18; 19]. More interestingly, the collinear factorization can be easily generalized to three-point energy correlator [9] and even the projected \(N\)-point energy correlator [16]. Previously, LL and NLL resummation has been performed in [20; 16; 21]. To improve upon those results, it is necessary to compute the relevant jet and hard function to higher order. While the hard function is universal for them, the jet functions differ by the measurement function. One of the key new results in this paper is the calculation of two-loop jet function for projected three-point energy correlator, which is the last missing ingredient for NNLL resummation of projected three-point energy correlator in \(e^{+}e^{-}\) collider.
One of the main motivations for improving the theoretical accuracy of projected energy correlators comes from the possibility of determining the strong coupling constant \(\alpha_{s}\) by measuring the ratio of projected energy correlators [16]. Measurements of strong coupling constant using classical QCD event shape observable has been actively studied for a long time, e.g. [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. In recent years, there has been increasing attention to using jet substructure observables to extract \(\alpha_{s}\), such as soft-drop thrust and jet mass [41; 42], see also [43] for \(\alpha_{s}\) determination from jet substructure by demixing quark and gluon jets. Since we are mainly concerned with the collinear limit of projected energy correlators in this paper, our results naturally provide theory input for measuring projected energy correlator within a jet, treating it as a jet substructure observable. We will show that considering the ratio of E3C and EEC can significantly reduce scale uncertainties and hadronization corrections, which makes it a good candidate for precision determination of \(\alpha_{s}\) using jet substructure. We also note that energy correlators have the advantage that they can be defined and calculated using charged hadrons only [16; 44]. Using the track function formalism [45; 46], it is possible to perform precision calculation for projected energy correlators on tracks in the future.
The outline of this paper is as follows. In Sec. 2, we present the factorization theorem for ENC in the collinear limit and the RG evolution for both hard function and jet function. The desired orders required for all the ingredients to achieve NNLL resummation are briefly summarized there. In Sec. 2.4, we calculate the two-loop E3C jet function. Modern multiloop techniques like IBP and differential equation (DE) are applied for both finite and contact terms. Combining all together, we are able to extract the two-loop E3C jet constants, which is the last missing piece of the NNLL resummation for collinear E3C in \(e^{+}e^{-}\) collision. In Sec. 3, we present the matched NNLL results for both E3C and the ratio of E3C to EEC in \(e^{+}e^{-}\) collision. A qualitative analysis is performed to estimate the leading hadronization correction. The resummation procedure is extended to the case of \(pp\) collision, in particular, the \(pp\to\) dijet process in Sec. 4. We present the highest pertur
bative prediction given the available ingredients, the approximate NNLL, with the missing two-loop hard function constants estimated and included as an additional uncertainty. We summarize and conclude in Sec. 5.
## 2 Resummation formalism
### Factorization theorem
In this subsection, we summarize the factorization theorem for the projected \(N\)-correlator in the collinear limit and describe the necessary ingredients for NNLL resummation [16]. Similar to EEC, \(N\)-point energy correlator (ENC) in this limit is dominated by the logarithmic series of the largest angular distance \(x_{L}\)
\[\frac{d\sigma^{[N]}}{dx_{L}}=\sum_{L=1}^{\infty}\sum_{j=-1}^{L-1} \left(\frac{\alpha_{s}(\mu)}{4\pi}\right)^{L}c_{L,j}\mathcal{L}^{j}(x_{L})+ \ldots\,, \tag{1}\]
where \(\mathcal{L}^{-1}(x_{L})=\delta(x_{L})\) and \(\mathcal{L}^{j}(x_{L})=\left[\ln^{j}(x_{L})/x_{L}\right]_{+}\) for \(j\geq 0\), with standard plus distribution. We do the logarithm counting in the projected \(N\)-point energy correlator cumulant, defined as
\[\Sigma^{[N]}\left(x_{L},\ln\frac{Q^{2}}{\mu^{2}}\right)=\frac{1} {\sigma_{\rm tot}}\int_{0}^{x_{L}}dx_{L}^{\prime}\,\frac{d\sigma^{[N]}}{dx_{ L}^{\prime}}\left(x_{L}^{\prime},\ln\frac{Q^{2}}{\mu^{2}}\right)\,, \tag{2}\]
which maps \([\ln^{j}(x_{L})/x_{L}]_{+}\to 1/(j+1)\times\ln^{j+1}(x_{L})\) and \(\delta(x_{L})\to 1\). Then N\({}^{k}\)LL accuracy refers to the logarithmic series \(\sum_{i=0}^{\infty}\sum_{j=\max\{0,i-k\}}^{i}\left(\frac{\alpha_{s}(\mu)}{4\pi }\right)^{i}d_{i,j}\ln^{j}x_{L}\) in the cumulant \(\Sigma^{[N]}\).
At leading power, the \(e^{+}e^{-}\) cumulant \(\Sigma^{[N]}\) can be written in terms of a modified factorization formula in the collinear limit \(x_{L}\to 0\)[16]:
\[\Sigma^{[N]}_{ee}\left(x_{L},\ln\frac{Q^{2}}{\mu^{2}}\right)= \int_{0}^{1}dx\,x^{N}\vec{J}^{[N]}\left(\ln\frac{x_{L}x^{2}Q^{2}}{\mu^{2}} \right)\cdot\vec{H}_{ee}\left(x,\ln\frac{Q^{2}}{\mu^{2}}\right)\,, \tag{3}\]
where the hard function \(\vec{H}_{ee}^{[N]}\) encodes the production of a parent parton with energy fraction \(x\) with respect to the center of mass energy, and the jet function \(\vec{J}^{[N]}\) encodes the evolution of the parent parton into a number of collinear partons which contribute to the observable. Similar factorization formula for EEC was first obtained in [17], and checked explicitly with known NLO results in QCD [5; 6] and \(\mathcal{N}=4\) SYM [3; 4]. We note the explicit dependence on the variable \(x\) in both the jet function and the hard function. Ignoring the dependence on different quark flavor, both jet and hard functions are two-component vectors living in the flavor space, i.e. \(\vec{J}^{[N]}=\{J_{q}^{[N]},J_{g}^{[N]}\}\), \(\vec{H}_{ee}=\{H_{ee,q},H_{ee,g}\}\). We will describe their definition for both \(e^{+}e^{-}\) annihilation and \(pp\) collision in detail in the following subsections. We also emphasize that the factorization theorem holds for any \(N\) at leading power, though we only calculate the \(N=3\) case in this paper. Finally the energy weights in the distribution makes projected \(N\)-point energy correlator insensitive to the soft radiations and non-global logarithms.
In hadron colliders, the largest angular distance \(x_{L}\) is replaced by the rapidity-azimuth distance \(R_{L}=\max_{i,j\in X_{E}}\sqrt{\Delta\eta_{ij}^{2}+\Delta\phi_{ij}^{2}}\), where \(X_{E}\) is the set of particles that contributes
to the energy weight. When the projected energy correlators are measured within a jet, as is typical for jet substructure observable, the cumulant \(\Sigma_{\rm had}^{[N]}\) also depends on the jet radius \(R_{0}\) parameter. In the limit of \(R_{L}\ll R_{0}\), the modified factorization formula can be written as
\[\Sigma_{\rm had}^{[N]}\left(R_{0},R_{L},\ln\frac{p_{T}^{2}}{\mu^{2}}\right)= \int_{0}^{1}dx\,x^{N}\vec{J}^{[N]}\left(\ln\frac{R_{L}^{2}x^{2}p_{T}^{2}}{\mu^ {2}}\right)\cdot\vec{H}_{\rm had}\left(R_{0},x,\ln\frac{p_{T}^{2}}{\mu^{2}} \right)\,, \tag{4}\]
where \(p_{T}\) is the jet transverse momentum. Around \(R_{L}\sim R_{0}\), the jet function can also depend on \(R_{0}\). However, there is no large logarithms associated with \(R_{0}\), and its dependence can be obtained from fixed-order matching. For simplicity, we will ignore the \(R_{0}\) dependence in the jet function. In that case the jet function become universal between \(e^{+}e^{-}\) and \(pp\) collision. For \(pp\) collision, the hard function depends on the partonic scattering process, as well as parton distribution functions (PDFs).
### Hard functions
#### 2.2.1 \(e^{+}e^{-}\) annihilation
For \(e^{+}e^{-}\), the hard function is simply the semi-inclusive hadron fragmentation function [47], which depends on the parton flavor and parton energy fraction \(x=\frac{2p\cdot q}{Q^{2}}\), where \(q\) is the total momentum and \(p\) is the parton momentum. The leading order hard function follows from the born process \(e^{+}e^{-}\to q\bar{q}\), \(\vec{H}_{ee}^{(0)}(x)=\{2\delta(1-x),0\}\). At one-loop, we find
\[\frac{1}{2}H_{ee,q}^{(1)}(x) = \frac{\alpha_{s}}{4\pi}C_{F}\Bigg{[}\left(\frac{4\pi^{2}}{3}-9 \right)\delta(1-x)+4\left[\frac{\ln(1-x)}{1-x}\right]_{+}\] \[+\left(4\ln(x)-\frac{3}{2}\right)\left(2\frac{1}{\left[1-x\right] _{+}}-x-1\right)-\frac{9x}{2}-2(x+1)\ln(1-x)+\frac{7}{2}\Bigg{]}\,,\] \[H_{ee,g}^{(1)}(x) = \frac{\alpha_{s}}{4\pi}C_{F}\Bigg{[}\frac{4\left(x^{2}-2x+2 \right)\ln(1-x)}{x}+\frac{8\left(x^{2}-2x+2\right)\ln(x)}{x}\Bigg{]}\,. \tag{5}\]
The factor \(1/2\) in front of the quark channel indicates for identical contribution from anti-quark, since we do not distinguish quark and anti-quark flavor. At two-loop, the hard function can be found from the coefficient functions in [47]. Similar to the hadron fragmentation function, the renormalization group evolution (RGE) for the hard function \(\vec{H}\) is simply the DGLAP equation,
\[\frac{d\vec{H}(x,\ln\frac{Q^{2}}{\mu^{2}})}{d\ln\mu^{2}}=-\int_{x}^{1}\frac{dy }{y}\widehat{P}(y)\cdot\vec{H}\left(\frac{x}{y},\ln\frac{Q^{2}}{\mu^{2}}\right)\,, \tag{6}\]
with \(\widehat{P}(y)\) being the singlet timelike splitting matrix, which is now known to three loops [48; 49]. While it is very difficult to derive an analytic solution for DGLAP to all orders in \(\alpha_{s}\), as we will see below, our resummation only uses a \(\alpha_{s}\)-expanded solution (which turns out to be a very good approximation) and only requires certain moments of the hard function. Explicitly, we will only need the regular and logarithmic moments for the hard function
defined as the following [17],
\[\int_{0}^{1}dx\,x^{N}\,H_{q,g}(x,\mu=Q) = \sum_{L=0}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L}h_{L}^{q,g}(N)\,,\] \[\int_{0}^{1}dx\,x^{N}\,\ln x\,H_{q,g}(x,\mu=Q) = \sum_{L=1}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L}\dot{h} _{L}^{q,g}(N)\,,\] \[\int_{0}^{1}dx\,x^{N}\,\ln^{2}x\,H_{q,g}(x,\mu=Q) = \sum_{L=1}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L}\ddot{h }_{L}^{q,g}(N)\,. \tag{7}\]
Here we use \(x^{N}\ln x=\partial_{N}x^{N}\) and the dot on the RHS stands for the derivative. The expressions of needed hard function moments can be found in Appendix A.
#### 2.2.2 \(pp\) collision
In hadronic collisions, we mainly focus on the dijet production \(pp\to jj\), which has a relatively large cross section at the LHC. Different from \(e^{+}e^{-}\) collider, this hard function incorporates the partonic scattering cross sections, the contribution from parton distribution functions (PDFs) and the jet algorithms for clustering the particles. Currently, to the best of our knowledge, the hard function is not know at two-loop. However, important progress are being made to compute those hard functions, e.g. [50]. Similar to the \(e^{+}e^{-}\) case, our resummation will only need the hard function moments.
In this work we evaluate the needed moments of the hard function numerically in Madgraph5[51, 52]. To investigate the sensitivity of the result to the values of \(\alpha_{s}\), we used three different PDF sets: NNPDF31_nnlo_as_0112, NNPDF31_nnlo_as_0118 and NNPDF31_nnlo_as_0124 through Lhapdf[53]. Each PDF set fixes also the value of \(\alpha_{s}(m_{Z})\) and the corresponding evolution in Madgraph5. To address the fact that the hard function contains collinear divergence when resolving the energy fraction of the quarks and gluons, we use the one cut-off phase space slicing to regularize the collinear singularity, as implemented in [54]. With the collinear divergent contribution singled out and calculated analytically, the remaining contributions can be evaluated numerically. The detailed discussion can be found in Appendix A.
For \(pp\to jj\), we adopt the anti-\(k_{t}\) algorithm [55] for jet detection and use the following parameters in the calculation
\[R_{0}=0.4,\qquad p_{T}>15\,\text{GeV},\qquad|\eta|<1.5\,. \tag{8}\]
The two leading jets are further subject to the following cuts
\[|\Delta\phi(j_{1},j_{2})|>2,\qquad|p_{T}^{1}-p_{T}^{2}|/(p_{T}^{1}+p_{T}^{2})< 0.5\,, \tag{9}\]
and cast to the corresponding \(p_{t}\) bins for the analysis. The calculated moments need to be normalized with the cross section \(\sigma_{J}\) of jet production within specific \(p_{t}\) range. In particular, we expand \(H_{\text{had}}/\sigma_{J}\) to NLO in \(a_{s}\), and take the \(\mathcal{O}(a_{s}^{0})\) and \(\mathcal{O}(a_{s}^{1})\) as the leading and next-to-leading order results. For the purpose of phenomenological studies, we will focus on two different \(p_{t}\) ranges: \([300,350]\) GeV and \([500,550]\) GeV. The hard function moments needed for NNLL are also summarized in Appendix A.
### Jet functions
The E3C jet function, on the other hand, encodes the measurement information. From RG invariance of the modified factorization formula (3), the jet function satisfies a modified timelike DGLAP evolution equation
\[\frac{d\vec{J}^{[N]}(\ln\frac{x_{L}Q^{2}}{\mu^{2}})}{d\ln\mu^{2}}= \int_{0}^{1}dy\,y^{N}\vec{J}^{[N]}\left(\ln\frac{x_{L}y^{2}Q^{2}}{\mu^{2}} \right)\cdot\widehat{P}(y)\,. \tag{10}\]
In order to write down an operator description of the E3C jet function, we first recall the collinear EEEC jet function from [9]:
\[J_{q}(x_{1},x_{2},x_{3},Q,\mu^{2})=\] \[\int\frac{dl^{+}}{2\pi}\frac{1}{2N_{C}}\text{Tr}\int d^{4}xe^{il \cdot x}\langle 0|\frac{\not{n}}{2}\chi_{n}(x)\widehat{\mathcal{M}}_{\text{EEEC}} \ \delta(Q+\bar{n}\cdot\mathcal{P})\delta^{2}(\mathcal{P}_{\perp})\bar{\chi}_{n}( 0)|0\rangle\] \[J_{g}(x_{1},x_{2},x_{3},Q,\mu^{2})=\] \[\int\frac{dl^{+}}{2\pi}\frac{1}{2(N_{C}^{2}-1)}\text{Tr}\int d^{ 4}xe^{il\cdot x}\langle 0|\mathcal{B}^{a,\mu}_{n,\perp}(x)\widehat{\mathcal{M}}_{ \text{EEEC}}\ \delta(Q+\bar{n}\cdot\mathcal{P})\delta^{2}(\mathcal{P}_{\perp}) \mathcal{B}^{a,\mu}_{n,\perp}(0)|0\rangle\,, \tag{11}\]
where \(\chi_{n}\equiv W_{n}^{\dagger}\xi_{n}\) is the collinear quark and \(\mathcal{B}^{\mu}_{n,\perp}\equiv\frac{1}{g}\left[\frac{1}{\bar{n}\cdot P}W_{n }^{\dagger}[i\bar{n}\cdot D_{n},iD_{n\perp}^{\mu}|W_{n}\right]\) is the collinear gluon, and \(\mathcal{P}^{\mu}_{n\perp}\) form a complete set of collinear gauge invariant building blocks [56] in SCET [57; 58; 59; 60; 61]. The triple collinear measurement function \(\widehat{\mathcal{M}}_{\text{EEEC}}\) is defined as
\[\widehat{\mathcal{M}}_{\text{EEEC}}(x_{1},x_{2},x_{3})=\sum_{i,j, k}\frac{E_{i}E_{j}E_{k}}{Q^{3}}\delta\left(x_{1}-\frac{\theta_{ij}^{2}}{4} \right)\delta\left(x_{2}-\frac{\theta_{jk}^{2}}{4}\right)\delta\left(x_{3}- \frac{\theta_{ki}^{2}}{4}\right)\,, \tag{12}\]
with \(\theta_{ij}\) being the angle between parton \(i\) and \(j\). Then our E3C jet function has the same form as EEEC jet function, with a replacement of the measurement function:
\[\widehat{\mathcal{M}}_{\text{EEEC}}\Rightarrow\widehat{\mathcal{M }}_{\text{E3C}}(x_{L}) =\int_{0}^{x_{L}}dx_{L}^{\prime}\int_{K}dx_{1}dx_{2}dx_{3}\, \widehat{\mathcal{M}}_{\text{EEEC}}\,\delta\left(x_{L}^{\prime}-\max(x_{1},x_ {2},x_{3})\right)\] \[=\int_{K}dx_{1}dx_{2}dx_{3}\,\widehat{\mathcal{M}}_{\text{EEEC}} \,\theta\left(x_{L}-\max(x_{1},x_{2},x_{3})\right)\,. \tag{13}\]
There are two folds integration in the first line. The first one is performed in the allowed kinematic space \(\{x_{1},x_{2},x_{3}\}\in K\) that will be discussed below, projecting the shape-dependent EEEC jet function into a single-scale jet function. The second integration brings the differential measurement to the cumulant level. For \(N>3\), the measurement function takes a similar structure, with more \(\delta\) functions and integrations. Perturbatively, the E3C jet function can be written as \(J_{q,g}=\sum_{L}(\alpha_{s}/4\pi)^{L}J_{q,g}^{(L)}\), and we use the normalization condition \(2^{3}\cdot J_{q}^{(0)}=2^{3}\cdot J_{g}^{(0)}=1\) as in Ref. [16]. The one-loop correction can be calculated from the QCD \(1\to 2\) timelike splitting kernel and is given by
\[2^{3}J_{q}^{(1)}=\frac{9C_{F}}{2}\ln\frac{x_{L}Q^{2}}{\mu^{2}}- \frac{37C_{F}}{2}\,,\]
\[2^{3}J_{g}^{(1)}=\,\left(\frac{21C_{A}}{5}+\frac{3n_{f}}{10}\right)\ln\frac{x_{L} Q^{2}}{\mu^{2}}-\frac{449C_{A}}{25}-\frac{21n_{f}}{25}\,. \tag{14}\]
Note that the \(\mu\)-dependent terms are precisely captured by the jet RGE, while the remaining constants have to come from the fixed-order calculation. One of the main result in this paper is to calculate the two-loop constants described below.
### Two-loop calculation for the E3C jet function
In this subsection, we present the two-loop calculation of the E3C jet functions for both quark jets and gluon jets. Since they are universal in the small angle limit, they can be used in both \(e^{+}e^{-}\) collision and \(pp\) collision.
We start from recalling the definition of E3C at finite angle before taking the small angle limit. At two loops, E3C receives contributions from double-real (RR) and real-virtual (RV) as well as double-virtual (VV) corrections to \(q\to q\), from which the quark jet function can be extracted by matching to the factorization formula, (3). Similarly, the gluon jet function can be extracted from the NLO E3C distribution of Higgs gluonic decay \(H\to gg\). To organize the calculation, we rewrite the definition of E3C in Eq. (3) with the number of energy weight:
\[\frac{1}{\sigma_{0}}\frac{d\sigma^{[3]}}{dx_{L}} =\sum_{1\leq i_{1}\neq i_{2}\neq i_{3}\leq 4}\int\mathrm{d LIPS}_{4}\,|\mathcal{M}_{4}|^{2}\frac{E_{i_{1}}E_{i_{2}}E_{i_{3}}}{Q^{3}} \delta(x_{L}-\max\{x_{i_{1},i_{2}},x_{i_{1},i_{3}},x_{i_{2},i_{3}}\})\] \[+\sum_{n\in\{3,4\}}\sum_{1\leq i_{1}\neq i_{2}\leq n}\int\mathrm{ dLIPS}_{n}\,|\mathcal{M}_{n}|^{2}\frac{E_{i_{1}}^{2}E_{i_{2}}}{Q^{3}}\delta(x_{L}-x _{i_{1},i_{2}})\] \[+\sum_{n\in\{2,3,4\}}\sum_{1\leq i_{1}\leq n}\int\mathrm{dLIPS}_{ n}\,|\mathcal{M}_{n}|^{2}\frac{E_{i_{1}}^{3}}{Q^{3}}\delta(x_{L})\,, \tag{15}\]
where we normalize the distribution to the born cross-section in \(d\) dimension. The first line represents the contribution from nonidentical energy weights measurement and the other lines are called contact terms. If we define \(x_{1}=x_{L}z\bar{z}\), \(x_{2}=x_{L}(1-z)(1-\bar{z})\) and \(x_{3}=x_{L}\), then in the collinear limits, they are the contact terms for \(\delta(z\bar{z})\) that captures the strict squeeze limit and \(\delta(x_{L})\) that captures the strict triple collinear limit. The main goal of this section is to compute the collinear limit of Eq. (15) and extract the corresponding two-loop constants.
The lowest regular distribution of the E3C quark jet function comes from tree-level process \(\gamma^{*}\to 4\) partons in electron-positron annihilation, which under the triple collinear limit, factorizes into the born process \(\gamma^{*}\to q\bar{q}\) and the \(1\to 3\) splitting functions, and we will call it nonidentical energy weight term. Below we will introduce two different methods to compute this part. The traditional method is to calculate the EEEC jet function to order \(\mathcal{O}(\epsilon)\) and to integrate two angular distances \(x_{2},\,x_{3}\) numerically by the interpolation method. The OPE singularities (sometimes called squeezed singularities) of EEEC are subtracted and integrated in \(d\) dimension separately. The second approach benefits from the parameter space IBP method [62; 63; 64] developed very recently. Only 7 master integrals are needed to express EEEC, allowing the precise calculation of the remaining two-fold integral.
The other two parts contribute to the contact terms and cancel the infrared divergence, which is guaranteed by the Kinoshita-Lee-Nauenberg (KLN) theorem [65; 66]. Similar to EEC at NLO, the measurement function in the contact terms can be treated as a non-standard cut propagators, which allows for a generalized IBP reduction in Liltered[67; 68] and Fire6[69]. The master integrals then can be calculated in packages like Canonica[70] or Libra[71; 72] with the differential equation method implemented.
#### 2.4.1 Nonidentical energy-weight terms
We start by computing the nonidentical energy-weight contribution in the traditional approach. As discussed in Ref. [9], the inclusive jet function \(J_{i\overline{j}k}\) is related to the \(1\to 3\) splitting function \(P_{ijk}\)[73; 74; 75] through
\[J^{\rm nonid}\equiv J_{i\overline{j}k}=\int{\rm d}\Phi_{c}^{(3)}\left(\frac{ \mu^{2}e^{\gamma_{E}}}{4\pi}\right)^{2\epsilon}\frac{4g^{4}}{\delta_{123}^{2} }\sum_{i,j,k}P_{ijk}\widehat{\mathcal{M}}_{\rm EEEEC}\,, \tag{16}\]
where \({\rm d}\Phi_{c}^{(3)}\) is the triple collinear phase space [75; 76], and \(i,j,k\) run over all final-state particles. The fully differential distribution with respect to all angular distances \(\{x_{1},x_{2},x_{3}\}\) in \(d=4-2\epsilon\) dimension is then written as
\[\frac{dJ^{\rm nonid}}{dx_{L}d\text{Re}(z)d\text{Im}(z)}=\left( \frac{\mu^{2}}{Q^{2}}\right)^{2\epsilon}\frac{\alpha_{s}^{2}}{\pi^{3}}\frac{e ^{2\epsilon\gamma_{E}}}{\Gamma(1-2\epsilon)}\frac{1}{x_{L}^{1+2\epsilon}} \frac{1}{(2\text{Im}(z))^{2\epsilon}}\\ \times\left[G(z)+\epsilon F(z)+\epsilon^{2}H(z)+\mathcal{O}( \epsilon^{3})\right]\,, \tag{17}\]
where \(G(z),F(z),H(z),\cdots\) the shape function in \(\epsilon\) expansion. The order \(\mathcal{O}(1)\) part \(G(z)\) is computed analytically in [9] and following the same approach, we also calculate the complete result for \(F(z)\) and the \(z\to 1\) limit of \(H(z)\). We will see that these are all the needed ingredients for nonidentical part. Note that the \(x_{L}\) dependence is defined by plus distribution, where
\[\frac{1}{x_{L}^{1+2\epsilon}}=-\frac{\delta(x_{L})}{2\epsilon}+\left(\frac{1 }{x_{L}}\right)_{+}-2\epsilon\left(\frac{\ln x_{L}}{x_{L}}\right)_{+}+\cdots\,. \tag{18}\]
In order to perform the integral over \(z\), we need to figure out the integration region first. Compared with the first line in Eq. (15), it is straightforward to show that
\[\frac{dJ^{\rm nonid}}{dx_{L}}=\left(\frac{\mu^{2}}{Q^{2}}\right)^{2\epsilon} \frac{\alpha_{s}^{2}}{\pi^{3}}\frac{e^{2\epsilon\gamma_{E}}}{\Gamma(1-2 \epsilon)}\frac{6}{x_{L}^{1+2\epsilon}}\underbrace{\int_{\mathcal{S}}\frac{d \text{Re}(z)d\text{Im}(z)}{(2\text{Im}(z))^{2\epsilon}}\left[G(z)+\epsilon F( z)+\epsilon^{2}H(z)+\mathcal{O}(\epsilon^{3})\right]}_{\equiv A(\epsilon)}\,, \tag{19}\]
where the constant factor \(6\) comes from the \(S_{3}\) permutation symmetry and the integration region \(\mathcal{S}\) is given in Fig. 1. To calculate \(A(\epsilon)\) numerically, we also need to subtract the OPE singularities around \(z\to 1\) at the integrand level, and evaluate its \(z\) integration analytically in \(d\) dimension. The full asymptotic expansion of \(z\to 1\) is given in the appendix C. The most singular term is proportional to \(\frac{1}{(1-z)(1-\bar{z})}\), which gives rise to
\[\int_{0}^{\frac{\sqrt{3}}{2}}d\text{Im}(z)\int_{\frac{1}{2}}^{\sqrt{1 -(\text{Im}(z))^{2}}}d\text{Re}(z)\frac{1}{(2\text{Im}(z))^{2\epsilon}}\frac{1} {(1-z)(1-\bar{z})}\\ =-\frac{\pi}{4\epsilon}-\kappa+\epsilon\left(-\frac{167}{1080} \pi^{3}-\frac{1}{20}\pi\ln^{2}3+\kappa\ln 3+\frac{12}{5}\eta\right)+\mathcal{O}( \epsilon^{2})\,. \tag{20}\]
Here \(\kappa=\text{ImLi}_{2}e^{i\frac{\pi}{3}}\) is the Gieseking's constant living in the transcendentality-two family and \(\eta=\text{ImLi}_{3}\left(\frac{i}{\sqrt{3}}\right)\) is a parity-odd transcendentality-three constant. These constants are typical numbers in loop integrals, especially in trijet observable calculations.
With subtraction terms, the integral \(A\) in Eq. (19) up to order \(\mathcal{O}(\epsilon)\) is then written as
\[A=\int_{0}^{\frac{\sqrt{3}}{2}}d\text{Im}(z)\int_{\frac{1}{2}}^ {\sqrt{1-(\text{Im}(z))^{2}}}d\text{Re}(z)\frac{1}{(2\text{Im}(z))^{2\epsilon }}\left[G(z\to 1)+\epsilon F(z\to 1)+\epsilon^{2}H(z\to 1)\right]\\ +\int_{0}^{\frac{\sqrt{3}}{2}}d\text{Im}(z)\int_{\frac{1}{2}}^ {\sqrt{1-(\text{Im}(z))^{2}}}d\text{Re}(z)\frac{1}{(2\text{Im}(z))^{2\epsilon }}\left[(G(z)-G(z\to 1))+\epsilon\left(F(z)-F(z\to 1)\right)\right]. \tag{21}\]
The first term is proportional to Eq. (20) and it is straightforward to compute it to \(\mathcal{O}(\epsilon)\). For the second integral, we have to expand in \(\epsilon\) and evaluate it numerically. To implement the interpolation method, we first change the integration variables via \(v_{1}=\frac{2}{\sqrt{3}}\text{Im}(z)\) and \(v_{2}=\frac{\text{Re}(z)-\frac{1}{2}}{\sqrt{1-(\text{Im}(z))^{2}-\frac{1}{2}}}\), such that both \(v_{1,2}\) range from \(0\) to \(1\). Then we can build a 2D lattice by discretizing \(v_{1,2}\) and approximate our integrand with polynomials. This allows one to perform the two-fold numerical integral directly in Mathematica. To check the stability of the integration and estimate the statistical error, we vary the lattice size and the order of polynomials and see which significant figure remains unchanged. Eventually we obtain both
Figure 1: The integration region \(\mathcal{S}\) for E3C jet function. The \(S_{3}\) symmetry is applied to reduce the entire region of \(z\) into \(6\) times the blue region. The integration range for \(z\) then becomes \(\int_{0}^{\frac{\sqrt{3}}{2}}d\text{Im}(z)\int_{\frac{1}{2}}^{\sqrt{1-(\text{ Im}(z))^{2}}}d\text{Re}(z)\).
\(\delta(x_{L})\) contact term and \(\frac{1}{x_{L}}\) finite term for the nonidentical energy weight contribution. The explicit expression for both quark and gluon jet function can be found in Eq. (45)-(46) in the appendix.
Alternatively, benefiting from the recent development of the IBP method in the Feynman parameter space, we can simplify the whole jet function calculation with integral reduction. First of all, recall that Eq. (16) takes the form
\[J^{\rm nonid}\equiv\int{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}\frac{dJ^{R}}{dx_{1 }dx_{2}dx_{3}}\propto\int{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}{\rm d}\omega_{1}{ \rm d}\omega_{2}{\rm d}\omega_{3}\delta(1-\omega_{1}-\omega_{2}-\omega_{3}) \hat{P}_{ijk}\,. \tag{22}\]
Here \(\hat{P}\) is a homogeneous function of the energy fraction \(\omega_{i}\) of the final-state particles. Explicitly, it is of the form
\[\hat{P}_{ijk}=\frac{\omega_{1}^{\alpha_{1}}\omega_{2}^{\alpha_{2}}\omega_{3}^ {\alpha_{3}}}{f_{1}^{\beta_{1}}f_{2}^{\beta_{2}}}\,, \tag{23}\]
with \(f_{1}\) linear in \(\omega_{i}\), and \(f_{2}\) a polynomial of \(\omega_{i}\) of degree 2. Following the idea in Ref [9], the integral \(\frac{{\rm d}^{3}J^{R}}{{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}}\) in Eq. (22) can be related to a Feynman parameter integral through1
Footnote 1: In the special cases where \(\beta_{1}=0\) or \(f_{1}=U\), we don’t need to introduce the parameter \(\omega_{4}\).
\[\frac{d^{3}J^{\rm nonid}}{dx_{1}dx_{2}dx_{3}}= \frac{\Gamma(\beta_{1}+\beta_{2})}{\Gamma(\beta_{1})\Gamma(\beta _{2})}\int{\rm d}\omega_{1}{\rm d}\omega_{2}{\rm d}\omega_{3}{\rm d}\omega_{4 }\delta(1-\omega_{1}-\omega_{2}-\omega_{3})\frac{\omega_{1}^{\alpha_{1}}\omega _{2}^{\alpha_{2}}\omega_{3}^{\alpha_{3}}\omega_{4}^{\beta_{1}-1}}{(f_{2}+f_{1} \omega_{4})^{\beta_{1}+\beta_{2}}}\] \[= \frac{\Gamma(\beta_{1}+\beta_{2})}{\Gamma(\beta_{1})\Gamma(\beta _{2})}\int{\rm d}\omega_{1}{\rm d}\omega_{2}{\rm d}\omega_{3}{\rm d}\omega_{4 }\delta(1-\omega_{1}-\omega_{2}-\omega_{3})\frac{\omega_{1}^{\alpha_{1}}\omega _{2}^{\alpha_{2}}\omega_{3}^{\alpha_{3}}\omega_{4}^{\beta_{1}-1}}{(f_{2}+f_{1} \omega_{4})^{\beta_{1}+\beta_{2}}}\] \[= \frac{\Gamma(\beta_{1}+\beta_{2})}{\Gamma(\beta_{1})\Gamma(\beta _{2})}\int{\rm d}\omega_{1}{\rm d}\omega_{2}{\rm d}\omega_{3}{\rm d}\omega_{4 }\delta(1-U)\frac{\omega_{1}^{\alpha_{1}}\omega_{2}^{\alpha_{2}}\omega_{3}^{ \alpha_{3}}\omega_{4}^{\alpha_{4}}}{U^{\lambda_{1}}F^{\lambda_{2}}}\] \[\equiv \frac{\Gamma(\alpha_{1})\Gamma(\alpha_{2})\Gamma(\alpha_{3})}{ \Gamma(\beta_{1})\Gamma(\beta_{2})}I(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_ {3},\alpha_{4})\,, \tag{24}\]
where \(U=\omega_{1}+\omega_{2}+\omega_{3}\), \(F=f_{2}+f_{1}\omega_{4}\), \(\lambda_{1}=\alpha_{1}+\alpha_{2}+\alpha_{3}-\beta_{1}-2\beta_{2}+3\), \(\lambda_{2}=\beta_{1}+\beta_{2}\), and \(\alpha_{0}=-\beta_{1}-\beta_{2}\). The integral in the last line is a standard parametric Feynman integral, which can be reduced with IBP reduction [77; 78] in the parametric representation [62; 63; 64; 79]2. The master integrals are
Footnote 2: The algorithms described in ref. [63] to generate symbolic rules work only when all the indices are nonnegative. Thus, here we carry out the reduction by merely solving IBP identities using Kira[80; 81; 82; 83].
\[\mathcal{I}_{1} =I_{1}\left(\alpha_{0},-2\epsilon,1-2\epsilon,-2\epsilon\right), \mathcal{I}_{2} =I_{1}\left(\alpha_{0},1-2\epsilon,-2\epsilon,-2\epsilon\right),\] \[\mathcal{I}_{3} =I_{1}\left(\alpha_{0},-2\epsilon,-2\epsilon,1-2\epsilon\right), \mathcal{I}_{4} =I_{1}\left(\alpha_{0},-2\epsilon,-2\epsilon,-2\epsilon\right),\] \[\mathcal{I}_{5} =I_{2}\left(\alpha_{0},-2\epsilon,-2\epsilon,-2\epsilon,0\right), \mathcal{I}_{6} =I_{3}\left(\alpha_{0},-2\epsilon,-2\epsilon,-2\epsilon,0\right),\] \[\mathcal{I}_{7} =I_{4}\left(\alpha_{0},-2\epsilon,-2\epsilon,-2\epsilon,0\right)\,, \tag{25}\]
with the integrals \(I_{i}\) defined by the \(F\) polynomials
\[F_{1} =x_{1}\omega_{2}\omega_{3}+x_{2}\omega_{1}\omega_{3}+x_{3}\omega_{1 }\omega_{2}\,, F_{2} =F_{1}+(\omega_{1}+\omega_{2})\omega_{4}\,,\] \[F_{3} =F_{1}+(\omega_{1}+\omega_{3})\omega_{4}\,, F_{4} =F_{1}+(\omega_{2}+\omega_{3})\omega_{4}\,, \tag{26}\]
and \(\alpha_{0}=6\epsilon-2^{3}\). The master integrals can be evaluated using the differential equation technique [84; 85]. For simplicity, we set \(\mu=x_{3}=1\), and introduce \(u\) and \(v\) following \(z=u(1+iv)\). Then we construct the differential-equation system with respect to \(u\), and derive the canonical basis [86] using Libra[71; 72]
\[\mathcal{I}_{1}^{\prime}= 6u(v-1)\mathcal{I}_{4}+\frac{x_{1}(1-2\epsilon)}{\epsilon} \mathcal{I}_{1}\,,\] \[\mathcal{I}_{2}^{\prime}= 6(u-1)\mathcal{I}_{4}+\frac{x_{2}(1-2\epsilon)}{\epsilon} \mathcal{I}_{2}\,,\] \[\mathcal{I}_{3}^{\prime}= 6\left(uv+u-x_{1}\right)\mathcal{I}_{4}+\frac{x_{1}x_{2}(1-2 \epsilon)}{\epsilon}\mathcal{I}_{3}\,,\] \[\mathcal{I}_{4}^{\prime}= 6uv\mathcal{I}_{4}\,,\] \[\mathcal{I}_{5}^{\prime}= \left(x_{1}-x_{2}\right)\mathcal{I}_{5}\,,\] \[\mathcal{I}_{6}^{\prime}= \left(x_{3}-x_{1}\right)\mathcal{I}_{6}\,,\] \[\mathcal{I}_{7}^{\prime}= \left(x_{2}-x_{3}\right)\mathcal{I}_{7}\,, \tag{27}\]
with the corresponding alphabet \(\{u,\,2u-1,\,x_{2},\,x_{2}-1\}\). By solving the differential-equation system, we can express the master integrals via Goncharov polylogarithms (GPLs) [87; 88; 89]. The GPL is defined iteratively by
\[G(a_{1},\cdots a_{n};x)\equiv\int_{0}^{x}\frac{dt}{t-a_{1}}G(a_{2},\cdots a_{ n};t)\,, \tag{28}\]
with
\[G(;x)\equiv 1,\quad G(\vec{0}_{n};x)\equiv\frac{1}{n!}\ln^{n}(x)\,. \tag{29}\]
After finishing the simplified calculation of EEEC in the collinear limit, we still need to integrate two angular distances for the projected EEEC as the previous approach. By virtue of the \(S_{3}\) permutation symmetry, this amount to consider
\[\frac{dJ^{\rm nonid}}{dx_{L}}= 6\int{\rm d}x_{1}{\rm d}x_{2}\ \Theta(x_{1},x_{2})\frac{d^{3}J}{dx_{1}dx_{2}dx_{3}}\] \[= 24\int{\rm d}u{\rm d}v\ \Theta(x_{1},x_{2})u^{2}v\frac{d^{3}J}{dx _{1}dx_{2}dx_{3}}\] \[\equiv \int{\rm d}u{\rm d}v\ \Theta(x_{1},x_{2})\tilde{J}(u,v)\,, \tag{30}\]
where \(\Theta(x_{1},x_{2})\equiv\theta\left(1-\sqrt{x_{2}}\right)\theta\left(\sqrt{x _{2}}-\sqrt{x_{1}}\right)\theta\left(\sqrt{x_{2}}+\sqrt{x_{1}}-1\right)\). Now the OPE singularity corresponds to \(u\to 0\) limit, and similarly, we need to subtract the singular behavior and do the integration separately:
\[\frac{dJ^{\rm nonid}}{dx_{L}}=\int{\rm d}u{\rm d}v\ \Theta(x_{1},x_{2})\tilde{J}(u \to 0)+\int{\rm d}u{\rm d}v\ \Theta(x_{1},x_{2})\left[\tilde{J}(u,v)-\tilde{J}(u \to 0)\right]\,, \tag{31}\]
where again we can evaluate the first integral in \(d\) dimension and expand the integrand of the second one in \(\epsilon\).
To calculate the \(\tilde{J}(u\to 0)\), now we can directly extract the asymptotic expansion of the integral \(I\) in Eq. (24) from DE, in which we identify two expansion regions:
\[\text{hard region:}\quad\omega_{1}\sim\omega_{2}\sim\omega_{3} \sim 1\,,\] \[\text{small region:}\quad\omega_{2}\sim\omega_{3}\sim 1,\ \omega_{1} \sim u^{2}\,. \tag{32}\]
Eventually we only need to integrate the reduced master integrals in \(d\) dimension.
Regarding the second integral in Eq. (31), the \(u\) integral is straightforward since \(\tilde{J}(u,v)\) is expressed in terms of GPLs of the form \(G(\ldots,u)\). However, the \(v\) integral becomes unstable in two regions \(v\to 0\) and \(v\to\infty\). To resolve this problem, we decompose the \(v\in[0,\infty]\) integration into three parts: \([0,\ \frac{1}{C}]\), \([\frac{1}{C},\ C]\), and \([C,\ \infty]\), with a arbitrary cut parameter \(C>1\). In the region \((\frac{1}{C},\ C)\), we carry out the integration numerically, with the GPLs numerically using Handyg[90]. The other two regions require expanding the integrand in \(v\) (or \(\frac{1}{v}\)) to \(\mathcal{O}(v^{100})\) (or \(\mathcal{O}(v^{-100})\)) and performing the integration analytically. This expansion can easily be done by asymptotically solving the differential equations satisfied by the GPLs. Eventually, we find the same result as in Eq. (15)-(16).
#### 2.4.2 Contact terms
While it is convenient to calculate the nonidentical \(E_{i_{1}}E_{i_{2}}E_{i_{3}}\) part starting with the splitting functions, it is preferable to compute the full angular dependence on \(x_{L}\) for corresponding processes (namely \(e^{+}e^{-}\) annihilation and gluonic Higgs decay) with energy weights \(E_{i_{1}}^{2}E_{i_{2}}\) (\(i_{1}\neq i_{2}\)) and \(E_{i_{1}}^{3}\), and extract the contact term from the collinear limit \(x_{L}\to 0\). In other words, we will adopt the full matrix elements squared and compute the full phase space integral using modern multi-loop techniques, with which the collinear expansion gives \(\mathrm{d}\sigma^{[3]}_{\mathrm{E}^{2}\mathrm{EC}}(x_{L})/\mathrm{d}x_{L}\) (the \(E_{i_{1}}^{2}E_{i_{2}}\) (\(i_{1}\neq i_{2}\)) part) and \(\mathrm{d}\sigma^{[3]}_{\mathrm{E}^{3}\mathrm{C}}(x_{L})/\mathrm{d}x_{L}\) (the \(E_{i_{1}}^{3}\) part) in the \(x_{L}\to 0\) limit.
We start with the relevant processes in perturbation theory for two-loop jet functions,
\[\mathbf{e}^{+}\mathbf{e}^{-}\text{annihilation} \mathbf{Higgs decays}\] \[\gamma^{*}\to q\bar{q}+VV H\to gg+VV\] \[\gamma^{*}\to q\bar{q}g+V H\to ggg+V\] \[H\to q\bar{q}g+V\] \[\gamma^{*}\to q\bar{q}gg H\to gggg\] \[\gamma^{*}\to q\bar{q}q\bar{q} H\to q\bar{q}gg\] \[\gamma^{*}\to q\bar{q}q^{\prime}\bar{q} H\to q\bar{q}q\bar{q}\] \[H\to q\bar{q}\bar{q}^{\prime}\bar{q}^{\prime} \tag{33}\]
where \(V\) and \(VV\) denotes one-loop and two-loop correction respectively. In particular, in the \(x_{L}\to 0\) limit, \(1\to 2\) processes only contribute to \(\delta(x_{L})\)-terms (i.e., \(\mathrm{d}\sigma^{[3]}_{\mathrm{E}^{3}\mathrm{C}}(x_{L})/\mathrm{d}x_{L}\)).
The calculation setup of \(\mathrm{d}\sigma^{[3]}_{\mathrm{E}^{2}\mathrm{EC}}(x_{L},\epsilon)/\mathrm{d}x _{L}\) shares the same structure as the original EEC, which basically follows the approach described in Ref. [5] and more detail in [6]. Briefly speaking, using the Cutkosky rules [91, 92], we can replace the phase-space on-shell
delta functions with the cut propagators
\[\delta(p^{2})=\frac{1}{2\pi{\rm i}}\left(\frac{1}{p^{2}-{\rm i}0}-\frac{1}{p^{2}+{ \rm i}0}\right)\,, \tag{34}\]
and also the EEC measurement function \(\delta(x_{L}-x_{i,j})\) with
\[\delta\left(x_{L}-\frac{1-\cos\theta_{ij}}{2}\right)=\frac{(p_{i }\cdot p_{j})}{x_{L}}\delta\left[2x_{L}(p_{i}\cdot Q)(p_{j}\cdot Q)-p_{i}\cdot p _{j}\right]\] \[= \frac{1}{2\pi{\rm i}}\frac{(p_{i}\cdot p_{j})}{x_{L}}\left\{\frac{ 1}{[2x_{L}(p_{i}\cdot Q)(p_{j}\cdot Q)-p_{i}\cdot p_{j}]-{\rm i}0}-\frac{1}{[2x _{L}(p_{i}\cdot Q)(p_{j}\cdot Q)-p_{i}\cdot p_{j}]+{\rm i}0}\right\}\,, \tag{35}\]
where we set the center-of-mass energy \(Q=1\) for simplicity. After topology classification and identification as described in Ref. [6], the E\({}^{2}\)EC integral can be reduced to a set of master integrals \(\widetilde{\mathcal{I}}_{k}(x_{L},\epsilon)\) using IBP reduction and E\({}^{2}\)EC distribution can be written as a linear combination of the master integrals,
\[\frac{{\rm d}}{{\rm d}x_{L}}\sigma_{\text{E${}^{2}$EC}}^{[3]}(x_{L},\epsilon)= \sum_{k}\mathcal{C}_{k}(x_{L},\epsilon)\widetilde{\mathcal{I}}_{k}(x_{L}, \epsilon)\,. \tag{36}\]
Specifically, we generate the standard IBP equations using Litered[67, 68], add the missing one that is associated with the EEC measurement function by hand, and do the reduction in Fireg[69]. The master integrals turn out to be the same as in NLO EEC calculation for both \(e^{+}e^{-}\) annihilation and gluonic Higgs decays, which can be converted into the canonical basis using the DE package Canonica[70].
In order to obtain the collinear \({\rm d}\sigma_{\text{E${}^{2}$EC}}^{[3]}(x_{L},\epsilon)/{\rm d}x_{L}\), one could surely expand the differential equation asymptotically and derive the analytical expression of the master integrals in that limit. However, the fact that the most singular power of \(\mathcal{C}_{k}\)'s is \(x_{L}^{-8}\) requires us to compute the master integrals up to \(\mathcal{O}(x_{L}^{7})\) order, which turns out to be expensive and time-consuming. This becomes worse in the higher-point energy correlator since the singular power increases as well. One antidote is to reconstruct the coefficients from DE following an ansatz on the structure of asymptotic expansion. In fact, the pattern turns out to be \(x_{L}^{-\epsilon}U_{1}^{(1)}(x_{L},\epsilon)\) at \(\mathcal{O}(\alpha_{s})\) and \(x_{L}^{-\epsilon}U_{1}^{(2)}(x_{L},\epsilon)+x_{L}^{-2\epsilon}U_{2}^{(2)}(x_ {L},\epsilon)\) at \(\mathcal{O}(\alpha_{s}^{2})\), where \(U\) denotes a series in \(x_{L}\) with rational fractions of \(\epsilon\) as the coefficients.
Therefore, we perform the asymptotic expansion in the following way. First of all, we solve the canonical DE at \(0<x_{L}<1\) to transcendental-weight \(5\), which can be used to obtain the finite part of the contact term via Eq. (36). The result can be converted to Harmonic polylogarithms (HPLs) with the package Hpl[93] or even classical polylogarithms. Then we can extract the leading power \(x_{L}^{-1}\) and match it to a resummed ansatz
\[x_{L}^{-1-\epsilon}C_{1}(\epsilon)+x_{L}^{-1-2\epsilon}C_{2}(\epsilon)\,, \tag{37}\]
with unknown \(\epsilon\)-series \(C_{1}(\epsilon)\) and \(C_{2}(\epsilon)\). The matching between fixed order calculation and the resummed structure in \(\epsilon\) leads to the solution of \(C_{1}(\epsilon)\) and \(C_{2}(\epsilon)\) in \(\epsilon\) expansion. Since \(x_{L}^{-1-\epsilon}\) and \(x_{L}^{-1-2\epsilon}\) are defined with plus distribution similar to Eq. (18), now we obtain the correct \(\mathcal{O}(\epsilon^{0})\) formula for \({\rm d}\sigma_{\text{E${}^{2}$EC}}^{[3]}(x_{L},\epsilon)/{\rm d}x_{L}\) in the collinear limit.
The last remaining piece is \(\mathrm{d}\sigma^{[3]}_{\mathrm{E^{3}C}}(x_{L},\epsilon)/\mathrm{d}x_{L}\). The computation of the self-energy correlator is much easier since its dependence on \(x_{L}\) is factorized out by \(\delta(x_{L})\) and the integrals are simply standard cut integrals. The master integrals can be found in the literature, e.g. [94; 95]. Eventually adding \(\mathrm{d}\sigma^{[3]}_{\mathrm{E^{2}EC}}(x_{L})/\mathrm{d}x_{L}\) and \(\mathrm{d}\sigma^{[3]}_{\mathrm{E^{3}C}}(x_{L},\epsilon)/\mathrm{d}x_{L}\) together, we obtain the complete contact terms \(\mathrm{d}\sigma^{[3]}_{\mathrm{C}}(x_{L},\epsilon)/\mathrm{d}x_{L}\) for E3C distribution. The results are also summarized in Eq. (101)-(102). Combined with the nonidentical energy weight contributions, we find all \(\frac{1}{\epsilon}\) canceled and thus the infrared safety is guaranteed as expected.
#### 2.4.3 Results of two-loop jet function constants
With all individual contributions at hand, the full expressions of 2-loop E3Cs in the collinear limit can be written as
\[\frac{1}{\sigma_{0}}\frac{\mathrm{d}\sigma^{[3],2\text{-loop}}_{ \mathrm{q}}}{\mathrm{d}x_{L}}= \,2\,\frac{\mathrm{d}J^{\text{nonid,2-loop}}_{q}}{\mathrm{d}x_{L} }+\frac{1}{\sigma_{0}}\frac{\mathrm{d}\sigma^{[3],2\text{-loop}}_{\mathrm{C },\mathrm{q}}}{\mathrm{d}x_{L}}\quad(e^{+}e^{-}\,\text{annihilation})\,, \tag{38}\] \[\frac{1}{\sigma_{0}^{\prime}}\frac{\mathrm{d}\sigma^{[3],2\text{- loop}}_{\mathrm{d}x_{L}}}= \,2\,\frac{\mathrm{d}J^{\text{nonid,2-loop}}_{g}}{\mathrm{d}x_{L} }+\frac{1}{\sigma_{0}^{\prime}}\frac{\mathrm{d}\sigma^{[3],2\text{-loop}}_{ \mathrm{C},\mathrm{g}}}{\mathrm{d}x_{L}}\quad(\text{gluonic Higgs decay})\,. \tag{39}\]
Here a factor of 2 is added because we only consider a single jet in Sec. 2.4.1. Given the tree-level hard functions, \(\{H_{q}^{(0)},H_{g}^{(0)}\}=\{2\delta(1-x),0\}\) for \(e^{+}e^{-}\) annihilation and \(\{\tilde{H}_{q}^{(0)},\tilde{H}_{g}^{(0)}\}=\{0,2\delta(1-x)\}\) for the Higgs decay through the effective \(Hgg\) coupling, we can extract the two-loop jet constant directly from the \(\delta(x_{L})\) contribution from Eq. (38) and Eq. (39). We find that the \(\mu\) dependence are in full agreement with prediction from RG evolution, providing strong check to our calculation. The \(\mu\) independent part are the new results from this calculation. For the quark jet function, we get
\[j_{2}^{q,[3]}=12.3020\,C_{F}T_{F}n_{f}-26.2764\,C_{A}C_{F}+21.3943\,C_{F}^{2}\,, \tag{40}\]
and for gluon jet functions
\[j_{2}^{g,[3]}=17.5487\,C_{A}T_{F}n_{f}-2.05342\,C_{F}T_{F}n_{f}-5.97991\,C_{A}^ {2}+0.904693\,n_{f}^{2}T_{F}^{2}\,. \tag{41}\]
### Perturbative resummation
We start by defining the logarithmic order for our E3C resummation. The ingredients needed for our E3C resummation are summarized in Table 1. This includes the order of timelike splitting kernel \(\hat{P}(y)\), the boundary information (hard and jet constants), the \(\beta\) function for running coupling as well as the fixed-order matching.4 Due to the absent of analytic method to solve the RG equation exactly, we also truncate in the number of loops of the RGE solution to the desired logarithmic order [17].
Footnote 4: This is the same log counting as N\({}^{k}\)LL\({}^{\prime}\) in SCET, except that we omit all \({}^{\prime}\) for convenience.
We first review the LL resummation in \(e^{+}e^{-}\) annihilation. Based on our resummation setting, it is safe to set \(x=1\) in the argument of E3C jet function in Eq. (10), which only affects the higher-order terms beyond LL. This leads to
\[\frac{d\vec{J}_{\mathrm{LL}}^{[N]}(\ln\frac{x_{L}Q^{2}}{\mu^{2}})}{d\ln\mu^{2} }=\vec{J}_{\mathrm{LL}}^{[N]}(\ln\frac{x_{L}Q^{2}}{\mu^{2}})\cdot\frac{ \alpha_{s}}{4\pi}\int_{0}^{1}dy\,y^{N}\hat{P}^{(0)}(y)=-\vec{J}_{\mathrm{LL}}^ {[N]}(\ln\frac{x_{L}Q^{2}}{\mu^{2}})\cdot\frac{\alpha_{s}}{4\pi}\gamma_{T}^{(0 )}(N+1)\,. \tag{42}\]
Here, we introduce the anomalous dimension to be the moment of timelike splitting kernel
\[\gamma_{T}(N)\equiv-\int_{0}^{1}dy\,y^{N}\hat{P}(y)=\left(\frac{\alpha_{s}}{4\pi} \right)\gamma_{T}^{(0)}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\gamma_{T}^{(1 )}+\cdots\,. \tag{43}\]
Then given the boundary condition \(\vec{J}^{(0)}=\{2^{-N},2^{-N}\}\), we can directly write down the solution to LL jet function:
\[\vec{J}^{[N]}_{\rm LL}=2^{-N}(1,1)\cdot\exp\left[-\frac{\gamma_{T}^{(0)}}{ \beta_{0}}\ln\frac{\alpha_{s}\left(\sqrt{x_{L}}Q\right)}{\alpha_{s}(\mu)} \right]\,. \tag{44}\]
Plugging both jet and hard functions into the factorization for the cumulant \(\Sigma^{[N]}\) and differentiating it with respect to \(x_{L}\), we obtain the LL resummed physical spectrum for E3C.
Beyond LL, the \(x=1\) approximation is no longer valid, and instead we have to solve the jet RGE directly. While it is difficult to obtain a close-form solution for this modified DGLAP equation, we find that a truncated solution in \(\alpha_{s}\) is already in good convergence. Explicitly, we assume the jet function takes the form
\[\vec{J}^{[N]}=\underbrace{\sum_{i=1}^{\infty}\alpha_{s}^{i}L^{i}\vec{c}_{i,i} }_{\rm LL}+\underbrace{\sum_{i=1}^{\infty}\alpha_{s}^{i}L^{i-1}\vec{c}_{i,i- 1}}_{\rm NLL}+\underbrace{\sum_{i=1}^{\infty}\alpha_{s}^{i}L^{i-2}\vec{c}_{i, i-2}}_{\rm NNLL}+\cdots\,, \tag{45}\]
with \(L\equiv\ln\frac{x_{L}Q^{2}}{\mu^{2}}\) and \(c_{i,j}\) unknown constants, and solve both the jet RGE and \(\beta\) RGE order by order in \(\alpha_{s}\) (which is referred as expanded solution). In practice, we evaluate it numerically up to \(\mathcal{O}(\alpha_{s}^{50})\). Another advantage of using expanded solution is that we only need certain moments of the hard functions. For example, consider one term from the jet function, \(\vec{J}^{[N]}\supset\alpha_{s}^{2}\vec{c}_{2,2}L^{2}\), and plug into Eq. (3), we find
\[\Sigma^{[N]}\supset\alpha_{s}^{2}\vec{c}_{2,2}\cdot\int_{0}^{1}dx \,x^{N}\ln^{2}\left(\frac{x_{L}x^{2}Q^{2}}{\mu^{2}}\right)\cdot\vec{H}_{ee} \left(x,\ln\frac{Q^{2}}{\mu^{2}}\right)\] \[=\alpha_{s}^{2}\vec{c}_{2,2}\cdot\left[\ln^{2}\left(\frac{x_{L}Q^ {2}}{\mu^{2}}\right)^{2}\int_{0}^{1}dx\,x^{N}\vec{H}_{ee}\left(x,\ln\frac{Q^{ 2}}{\mu^{2}}\right)\right.\] \[+2\ln\left(\frac{x_{L}Q^{2}}{\mu^{2}}\right)\int_{0}^{1}dx\,\ln x ^{2}x^{N}\vec{H}_{ee}\left(x,\ln\frac{Q^{2}}{\mu^{2}}\right)+\int_{0}^{1}dx\, \ln^{2}x^{2}x^{N}\vec{H}_{ee}\left(x,\ln\frac{Q^{2}}{\mu^{2}}\right)\bigg{]}\] \[=\alpha_{s}^{2}\vec{c}_{2,2}\cdot\left[\ln^{2}\left(\frac{x_{L}Q^ {2}}{\mu^{2}}\right)+2\ln\left(\frac{x_{L}Q^{2}}{\mu^{2}}\right)\partial_{N} +4\partial_{N}^{2}\right]\int_{0}^{1}x^{N}\vec{H}_{ee}\left(x,\ln\frac{Q^{2}}{ \mu^{2}}\right)\,, \tag{46}\]
\begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline resummation order & \(\hat{P}(y)\) & \(\vec{H}\), \(\vec{J}\) constants & \(\beta[\alpha_{s}]\) & fixed-order matching \\ \hline LL & tree & tree & 1-loop & LO \\ \hline NLL & 1-loop & 1-loop & 2-loop & NLO \\ \hline NNLL & 2-loop & 2-loop & 3-loop & NNLO \\ \hline \end{tabular}
\end{table}
Table 1: Definition of the resummation order and their corresponding fixed-order matching.
where the three terms correspond to the standard moment, the single logarithmic moment and the double logarithmic moment of the E3C hard function. To derive the last line, we also use the following relation
\[\int_{0}^{1}\ln^{k}x^{2}x^{N}\vec{H}_{ee}\left(x,\ln\frac{Q^{2}}{\mu^{2}}\right) =2^{k}\partial_{N}^{k}\int_{0}^{1}x^{N}\vec{H}_{ee}\left(x,\ln\frac{Q^{2}}{\mu^ {2}}\right)\,. \tag{47}\]
In the Appendix A, we provide all the hard moments with \(N=2,3\) that are required for NNLL resummation.
In this paper, we present results for the NNLL resummation of E3C for \(e^{+}e^{-}\) annihilation, and approximate NNLL resummation for jets from the hadronic collision process \(pp\to jj\). For \(e^{+}e^{-}\) annihilation, we have all ingredients needed for NNLL resummation. And since there is no accurate fixed-order data for E3C at NNLO, we will instead match the NNLL result to NLO. Regarding the dijet production, due to the absence of the two-loop hard constant, we will present the approximate NNLL resummation (which we refer as NNLL\({}_{\rm approx}\)), with an additional uncertainty coming from the missing two-loop hard constant. Resummation with the accurate two-loop hard function as well as the matching with fixed-order result are left as future improvements.
## 3 NNLL resummation in \(e^{+}e^{-}\) annihilation
With all the ingredients at hand, now we can present the NNLL resummation prediction. In this section, we first consider \(e^{+}e^{-}\) collision at two different energies: 250 GeV and 1 TeV. In the resummation calculation, we will use \(\alpha(m_{Z})=0.118\).
### Resummation results
Following the discussion in Sec. 2.5, our resummation is performed by perturbatively solving the jet function RG equation to order \(\mathcal{O}(\alpha_{s}^{50})\), plugging back to the cumulant factorization and finally truncating the logarithms \(\ln\frac{x_{L}Q^{2}}{\mu^{2}}\) to the desired order. In the resummation formula, we set canonical jet scale \(\mu_{j}=\mu_{h}\sqrt{x_{L}}\) in the factorization, leaving a single hard scale \(\mu_{h}=\mu\) in the resummed expression. We vary the scale \(\mu\) to estimate the uncertainty from higher order corrections. Regarding the observables, below we consider three cases: \(N=2\), \(N=3\) and their ratio.
The \(N=2\) case is precisely the EEC observable, where we directly use the result from Ref. [17], and the singular expansion has been verified against the NLO EEC fixed-order calculation. For \(N=3\) case, this is the main result of this paper. In Fig. 2, we first check our \(\mathcal{O}(\alpha_{s}^{2})\) expansion with the Monte Carlo program Event2. In the collinear limit, we find excellent agreement between theory and numeric result, while in the meantime, this also suggests the non-singular contribution from fixed-order calculation is negligible in this limit.
Nevertheless, the matching formula can be written as
\[\frac{d\sigma^{\rm match}}{dx_{L}}=\frac{d\sigma^{\rm resum}}{dx_{L}}-\frac{ d\sigma^{\rm sing}}{dx_{L}}+\frac{d\sigma^{\rm FO}}{dx_{L}}\,. \tag{48}\]
Here each term is a function of \(\alpha_{s}(\mu)\) evaluated at the hard scale \(\mu_{h}=\mu\). In Fig. 3, we present the E3C resummation up to NNLL, matched to fixed-order. As explained above, due to the absence of NNLO data, we only match NNLL to NLO. The hard scale is chosen to be half of the center-of-mass energy \(\mu=Q_{\rm jet}\equiv Q/2\), the typical energy for each quark jet, and the scale uncertainty is obtained by varying the hard scale by a factor of 2. In both energies, the uncertainty band width goes down as we increase the resummation order, while at 1 TeV, we have a tighter band because the coupling \(\alpha_{s}\) runs slower at high energy. At NNLL, we find a relative 4% hard uncertainty for \(Q=250\) GeV and 2% for \(Q=1\) TeV. We find large corrections as we go from LL to NNLL, as was also observed previously in [17], which emphasize the importance of higher order corrections. For higher center-of-mass energy, the convergence between different orders is improved.
To improve the convergence, we also introduce the ratio of different point energy correlators, namely [16]
\[\Delta_{m,n}(x_{L},\mu,\mu^{\prime})\equiv\frac{d\sigma^{[m]}/dx_{L}}{d\sigma^ {[n]}/dx_{L}},\quad m,n\geq 2\,, \tag{3.2}\]
where \(\mu\) and \(\mu^{\prime}\) are the hard scale in \(\frac{d\sigma^{[m]}}{dx_{L}}\) and \(\frac{d\sigma^{[n]}}{dx_{L}}\) respectively. In particular, we focus on the ratio between fully matched E3C and EEC, i.e. \(\Delta_{3,2}(x_{L})\). In Fig. 4, we show the NNLL resummed \(\Delta_{3,2}(x_{L})\) at again \(Q=250\) GeV and \(Q=1\) TeV, and find good convergence. This implies that the ratio can be used as precision observable. For hard scale uncertainty, we use the seven-point scale variation, which amounts to varying the scales in both numerator and denominator independently by a factor of 2, to a combination of
\[\left(\frac{\mu}{Q_{\rm jet}},\frac{\mu^{\prime}}{Q_{\rm jet}}\right)\in \left\{\left(\frac{1}{2},\frac{1}{2}\right)\text{, }\left(2,2\right)\text{, }\left(1,2\right)\text{, }\left(1,1\right)\text{, }\left(2,1\right)\text{, }\left(1,\frac{1}{2}\right)\text{, }\left(\frac{1}{2},1\right)\right\}, \tag{3.3}\]
and take the envelope as the uncertainty estimation. The convergence also indicates that ENC shares similar non-perturbative behavior in the collinear limit and taking the ratio strongly suppresses the power corrections.
Figure 2: The comparison of fixed-order result and the singular expansion from resummation. The left panel shows good agreement between LO expression and the \(\mathcal{O}(\alpha_{s})\) singular expansion of E3C resummed result. The difference between them is the non-singular result. The right panel gives the corresponding contributions at NLO, with the numerical full NLO prediction from Event2.
### Hadronization corrections
In this subsection, we consider the power-suppressed hadronization corrections in the collinear limit. At present hadronization corrections cannot be computed from first principle. For
Figure 4: The ratio between resummed E3C and EEC distribution \(\Delta_{3,2}(x_{L})\) up to NNLL+NLO for \(e^{+}e^{-}\) collision at 250 GeV (left top panel) and 1 TeV (right top panel). Uncertainty bands are obtained in the seven-point scale variation scheme, namely varying the scales \(\mu\) and \(\mu^{\prime}\) in the two factors independently by a factor of 2 around the central value. The lower panels show the relative scale uncertainty of the NNLL+NLO distribution around the central value.
Figure 3: The resummed E3C distribution up to NNLL+NLO, multiplied by a factor \(x_{L}(1-x_{L})\), for \(e^{+}e^{-}\) collision at 250 GeV (left top panel) and 1 TeV (right top panel). Uncertainty bands are obtained by varying the hard scale \(\mu\) around the nominal value \(Q_{\rm jet}=Q/2\) by a factor of 2. The lower panels show the relative scale uncertainty of the NNLL+NLO distribution around the central value.
simplicity, we use a phenomenological form for the leading non-perturbative power correction as suggested in [96], and fit the unknown parameters from a Monte Carlo program. This provides some insights on how to model the hadronization effect for a global fit in the future.
In general, the non-perturbative corrections in infrared-collinear safe observables are (at least) suppressed as \(\Lambda_{\rm QCD}/Q\) to some power, where \(Q\) is the hard scale of the process. Following from the LL result in Eq. (44), we observe that in the collinear limit, there exists a lower scale \(\sqrt{x_{L}}Q\) in the coupling, and the most important non-perturbative correction that could potentially appear is linear in \(\Lambda_{\rm QCD}\) and takes the form \(\Lambda_{\rm QCD}/(\sqrt{x_{L}}Q)\), multiplied with an extra kinematic factor \(1/x_{L}\). The sub-leading non-perturbative corrections with additional powers of \(\Lambda_{\rm QCD}/(\sqrt{x_{L}}Q)\) will become necessary down to small \(x_{L}\sim\Lambda_{\rm QCD}^{2}/Q^{2}\), where the perturbation theory also breaks down. For the leading non-perturbative correction we are considering, such structure is in fact recovered for the EEC in the fragmentation modeling of non-perturbative radiations [1] and and analysis using renormalon or dispersive techniques [96; 97; 98].
As a qualitative analysis, we use the following parametrization of the leading non-perturbative correction,
\[\frac{d\sigma^{\rm NP-soft}}{dx_{L}}=\frac{1}{x_{L}}\cdot\left(\,\frac{ \tilde{\Lambda}}{\sqrt{x_{L}}\,Q}\,\right)^{1+\gamma}\qquad(\,\text{soft fragmentation}\,), \tag{45}\]
we verify the scaling behaviour of the non-perturbative correction in the collinear limit for both EEC and E3C distributions with Pythia8[99], and extract the non-perturbative parameters by fitting from the difference of the hadron level and parton level predictions. Note that the issues of extracting the non-perturbative power corrections from Monte Carlo generators have been pointed out in Ref. [36]. In particular, the corrections from the hadronization modeling in the Monte Carlo programs in fact unfaithfully absorb partial subleading-log contributions, as the hadronization modeling has been tuned to reproduce some collider data with limited perturbative accuracy. Therefore, in this paper we only use Monte Carlo to illustrate the impact of power correction for individual EEC and E3C distribution as well as their ratio.
For our case, we stay in the default settings of Pythia8 and obtain the following fit at the 95% confidence level. At \(Q=250\) GeV, we find for EEC and E3C:
\[\tilde{\Lambda}_{2} =(0.956\pm 0.031)\,\text{GeV}\,,\quad\gamma_{2}=0.462\pm 0.017\,,\] \[\tilde{\Lambda}_{3} =(0.500\pm 0.040)\,\text{GeV}\,,\quad\gamma_{3}=0.335\pm 0.031\,. \tag{46}\]
And in the case with \(Q=1\) TeV, we have
\[\tilde{\Lambda}_{2} =(0.775\pm 0.013)\,\text{GeV}\,,\quad\gamma_{2}=0.383\pm 0.008\,,\] \[\tilde{\Lambda}_{3} =(0.435\pm 0.015)\,\text{GeV}\,,\quad\gamma_{3}=0.325\pm 0.012\,. \tag{47}\]
We emphasis that for too small \(x_{L}\) value, the leading order non-perturbative approximation itself becomes invalidated. The enhancement of the non-perturbative corrections in the collinear limit must be turned off before entering the fully non-perturbative phase,
where the degrees of freedom become freely interacting hadrons and a nice scaling behavior follows [20]. In this qualitative analysis, we choose the lower bound of the fit range by finding the extreme point of the distributions from hadron level prediction in Pythia8. Multiplying the extreme point by a factor of 2 gives a good estimate of the lower bound for the range where the non-perturbative correction follows the described scaling behavior. In Fig. 5, we show the relative hadronization correction from both Pythia8 and our two-parameter fit. Except the shaded region, our parameterization agrees with the Monte Carlo result and it is sufficient for understanding their structure.
In Fig. 6, we include the non-perturbative correction in the matched E3C resummation, which strongly enhances the extreme collinear limit. At \(Q=1\) TeV, the non-perturbative correction changes our NNLL+NLO prediction by only a few percent at \(x_{L}\sim 0.1\), while this modification reaches 50% at \(x_{L}\sim 10^{-4}\). This shows that the non-perturbative corrections for energy correlators, though being power suppressed at high energies, can become sizable even at the energy level of future \(e^{+}e^{-}\) colliders. However, since EEC and E3C share a close power law in the leading power correction, the enhancement is significantly canceled when considering their ratio \(\Delta_{3,2}(x_{L})\). As shown in Fig. 7, the leading non-perturbative correction only gives rise to roughly 4% effect at \(Q=250\) GeV and 2% at \(Q=1\) TeV for matched NNLL. This confirms that \(\Delta_{3,2}(x_{L})\) is insensitive to the hadronization and indeed
Figure 5: Comparison of Pythia8 result and the fitting using Eq. (14). The blue curves are the difference of hadron-level and parton-level distribution for EEC and E3C, at both 250 GeV and 1 TeV. The red curves are our fitting result with parameters from Eq. (15) and (16). The shaded region stands for the range where the parametrization of the leading non-perturbative correction is no longer valid and should be excluded from the fit range.
a good candidate for precise \(\alpha_{s}\) measurement.
We also investigate the impact on the final resummation results caused by the uncertainties from the two-parameter fit. The statistical error for both \(\tilde{\Lambda}\) and \(\gamma\) are given in Eq. (3.5) and (3.6). Fig. 8 shows the final uncertainty in the matched NNLL distribution from varying these two NP parameters. In both \(Q=250\) GeV and \(Q=1\) TeV, excluding
Figure 6: The E3C distribution to NNLL+NLO including the non-perturbative (NP) hadronization corrections estimated with Pythia8 data, with different plot ranges for collision energies at 250 GeV (left panel) and 1 TeV (right panel).
Figure 7: The E3C/EEC ratio to NNLL+NLO with non-perturbative (NP) hadronization corrections. The non-perturbative hadronization corrections are estimated as above by \(\tilde{\Lambda}/(x_{L}^{3/2}Q)\) for both the EEC and E3C distributions with the coefficients fitted from Pythia8.
the shaded region, the NP uncertainty is much smaller than the hard uncertainty estimated by seven-point variation. In particular, at \(Q=1\) TeV, the NP uncertainty is reduced to 1% in the potential fit region. Despite that, we admit that the effect of non-perturbative corrections turns to increase for such small \(x_{L}\) region, and more accurate understanding of the non-perturbative corrections will be required to further improve the precision.
### Anticipation of \(\alpha_{s}\) determination
In this subsection, we discuss the potential of extracting the strong coupling constant \(\alpha_{s}\) from measuring the resumed E3C/EEC ratio \(\Delta_{3,2}(x_{L})\). In literature [100; 101], the back-to-back limit of EEC is resummed to NNLL+NLO and has been use for \(\alpha_{s}\) measurement from \(e^{+}e^{-}\) data. Similar to other event shapes, the non-perturbative correction is significantly large in this region and require careful modeling. And how we profile the resummation and power correction has a sizable effect on the final theory uncertainty.
Alternatively, we can also do the \(\alpha_{s}\) measurement _only_ in the collinear limit. First of all, as we discussed in Sec. 3.1, the non-singular contribution is almost zero in this limit, and thus it is safe to ignore the higher fixed-order contribution. Secondly, by considering the ratio distribution, \(\Delta_{m,n}(x_{L})\), the suppressed power corrections will lead to a smaller theory uncertainty and thus more precise \(\alpha_{s}\) determination. As illustration, we first investigate the sensitivity of \(\Delta_{3,2}(x_{L})\) when slightly changing the value of \(\alpha_{s}\). In particular, we vary the value of strong coupling at \(Z\)-pole \(\alpha_{s}(m_{Z})\) by a factor of 5%, namely \(\alpha_{s}(m_{Z})=\{0.112,0.118,0.124\}\) and compare the effect on matched resummation result.
We first consider the NNLL+NLO \(\Delta_{3,2}(x_{L})\) at \(Q=91.2\) GeV with all three values of \(\alpha_{s}(m_{Z})\). As observed in Fig. 9, the slope become sensitive to the \(\alpha_{s}\) in the collinear region \(x_{L}=10^{-3}\sim 10^{-4}\), while the relative difference with respect to \(\alpha_{s}(m_{Z})=0.118\) ranges from
Figure 8: The uncertainty from varying the non-perturbative parameters \(\tilde{\Lambda}\) and \(\gamma\) in the resummed ratio distribution \(\Delta_{3,2}(x_{L})\). The bottom panels stand for the relative difference of NNLL+NLO+NP with respect to its central value.
10% to 20%. The slope sensitivity and the cancellation of hadronization correction have made the ratio of E3C and EEC \(\Delta_{3,2}(x_{L})\) an advantageous observable for extracting the \(\alpha_{s}\) from \(e^{+}e^{-}\) annihilation. Similar behaviors also exist at other energies and for completeness, we present the comparison at \(Q=250\) GeV and \(Q=1\) TeV in Fig. 10.
The fact that the resummed E3C/EEC ratio has larger sensitivity to \(\alpha_{s}\) and reduced non-perturbative corrections in the collinear limit makes it a promising candidate for the \(\alpha_{s}\) determination. To further improve the \(\alpha_{s}\) determination requires improving the resummation accuracy, matching with NNLO fixed-order correction, as well as the non-perturbative modeling.
## 4 Approximate NNLL resummation in \(pp\) collisions
In this section, we consider the dijet production \(pp\to jj\) at the LHC. There are several motivations to study energy correlators in \(pp\) collisions. First of all, LHC provides unique opportunities to study energy flows correlation in QCD at extremely high energy. While the LEP or future CEPC provides a very clean environment for precise measurements, \(pp\) collisions at the LHC can produce multiple jets with very high energies (\(p_{T}\gtrsim 500\) GeV), and high angular resolution can be achieved to probe the underlying dynamics for their formation and evolution. Secondly, as we have observed in the \(e^{+}e^{-}\) collisions, the non-perturbative corrections for ENC have a relatively simple form compared to other event shape observables (at least in leading power), which might be easier to study non-perturbative QCD. At the same time, with multiple scales involved, \(pp\) collision can provide robust data from high energy to low energy, which is beneficial for understanding non-perturbative effects.
In this section, we still focus on improving the perturbative predictions for ENC. As in Sec. 2, the jet functions are universal across different hard processes and the new ingredients are the moments of \(pp\) hard function, both regular and logarithmic. The main complication
Figure 9: Left panel is the matched NNLL ratio distribution \(\Delta_{3,2}(x_{L})\) with different strong coupling constants: \(\alpha_{s}(m_{Z})=\{0.112,0.118,0.124\}\) at \(Q=91.2\) GeV. The uncertainty is the hard scale variation. Right panel shows the relative deviation of all three bands with respect to the central prediction of \(\alpha_{s}(m_{Z})=0.118\).
for \(pp\) collision is that the hard function now involves convolution with PDFs and algorithmic definition of jet, allowing only numeric calculation of the hard function.
For the numerical calculation of the hard function, we adopt the anti-\(k_{t}\) jet algorithm and choose the jet radius to be \(R_{0}=0.4\). The complete kinematic cuts are summarized in Eqs. (8)-(9). The \(\mu\) independent part of the NLO hard function are presented in Appendix. A. We observes large corrections going from LO to NLO. The \(\mu\) dependent part of the NNLO hard function can be derived using the RG equation in (6). The \(\mu\) independent part requires a genuine two-loop corrections and are beyond the scope of this work. Instead we make a simple estimate of the two-loop constant terms, and dubbed the resulting prediction approximate NNLL resummation (NNLL\({}_{\rm approx}\)). Specifically, we use a modified Pade approximation to estimate the two-loop hard function constants in both quark channel and gluon channel:
\[a_{s}^{2}h_{0}^{(2)}\approx\kappa\frac{(a_{s}h_{0}^{(1)})^{2}}{h_{0}^{(0)}}\,, \tag{10}\]
where we vary \(\kappa\) in the range \([0,1/2]\) as a naive way to estimate our theory uncertainties on the missing two-loop constants. For the splitting function, \(\beta\) function, as well as the jet
Figure 10: Left panels are the matched NNLL ratio distribution \(\Delta_{3,2}(x_{L})\) with different strong coupling constants: \(\alpha_{s}(m_{Z})=\{0.112,0.118,0.124\}\) at \(Q=250\) GeV and \(Q=1\) TeV. Right panels are the relative deviation of all three bands with respect to the central prediction of \(\alpha_{s}(m_{Z})=0.118\).
functions, we used the ones required by NNLL accuracy as shown in Table 1.
In Fig. 11, we show the E3C/EEC ratio \(\Delta_{3,2}(R_{L})\) up to NNLL\({}_{\rm approx}\), with the hard uncertainty estimated by seven-point variation. Due to the lack of knowledge of the genuine two-loop hard function moment, we have chosen to normalize the E3C/EEC distribution in the range of \(R_{L}\in[0.01,0.4]\) to reduce the impact from not knowing the full two-loop hard function. We find good convergence for both \(p_{t}\) ranges: \([300,350]\) GeV and \([500,550]\) GeV. In the future, it would be interesting the compute the two-loop hard function, as well as match the resummed results to fixed order to improve the prediction around \(R_{L}\sim R_{0}\).
### Anticipation of \(\alpha_{s}\) determination
Similar to \(e^{+}e^{-}\) annihilation, in this subsection we discuss the potential of extracting the strong coupling constant \(\alpha_{s}\) from the resummed \(\Delta_{3,2}(R_{L})\) distribution in \(pp\to jj\). In particular, we also investigate the slope sensitivity of the distribution with respect to different values of \(\alpha_{s}\). For hadron colliders, we need to change the PDFs as we vary the strong coupling among \(\alpha_{s}(m_{Z})=0.118\pm 0.06\). For this purpose, we use three PDF sets: NNPDF31_nnlo_as_0112, NNPDF31_nnlo_as_0118 and NNPDF31_nnlo_as_0124 when calculating the hard function using the method in [54].
As shown in Fig. 12, for each \(p_{t}\) range, the uncertainty is significantly reduced from NLL to NNLL\({}_{\rm approx}\), leading to distinguishable slopes with respect to different \(\alpha_{s}\). This suggests that ratios of energy correlators are good candidate for extracting \(\alpha_{s}\). We note that there is larger slope variation for lower \(p_{t}\) of the jet, in agreement with the expectation that the measurement at lower energy is more sensitive to \(\alpha_{s}\) due to asymptotic free nature of QCD.
## 5 Conclusion
In this paper we have performed a systematic study of resummation of projected three-point energy correlator E3C [16], and its ratio to EEC, in both \(e^{+}e^{-}\) collider and \(pp\) collider. We
Figure 11: Normalized E3C/EEC ratio \(\Delta_{3,2}(R_{L})\) for \(pp\to jj\) with \(\alpha_{s}(m_{Z})=0.118\), and jet \(p_{t}\) ranges \([300,350]\) GeV (left) and \([500,550]\) GeV (right). Uncertainty bands are obtained in the 7-point scale variation scheme, with additional uncertainty from varying the estimation of NNLO hard function constant for NNLL\({}_{\rm approx}\).
have achieved the first NNLL accuracy for the \(e^{+}e^{-}\) case, and NNLL\({}_{\rm approx}\) accuracy for the \(pp\) case. Our results show that good perturbative convergence can be achieved for the ratios of projected energy correlators. The current theoretical uncertainties are at a level of a few percent, and can be further improved in the future when the higher order ingredients become available. We have also shown that the ratio observable is sensitive to variation of \(\alpha_{s}\), therefore provides a good candidate for precision \(\alpha_{s}\) determination using jet substructure.
To achieve the above theory accuracy, one of the main new ingredients is the two-loop E3C jet function computed in this work. The calculation includes three pieces: double-real, real-virtual and double-virtual. The last two contributions only involve a single \(\delta\) measure
Figure 12: Normalized NLL (upper) and NNLL\({}_{\rm approx}\) (lower) resummation result for E3C/EEC ratio at \(pp\to jj\) with \(\alpha_{s}(m_{Z})=0.118\pm 0.06\), i.e., varied by about 5%, for two different jet \(p_{t}\) ranges [300,350] GeV and [500,550] GeV. Lower panels show the relative difference from the result at the central scale with \(\alpha_{s}(m_{Z})=0.118\).
ment function in the phase space integral and share a similar form as the analytic EEC calculation at NLO [5]. Regarding the double-real emissions, which amounts to integrating the fully-differential EEEC distribution within the collinear kinematic space, we used two different approaches and find the same results. The first method is to subtract the infrared divergence in the collinear EEEC jet function, integrate it separately with \(d\)-dimension kinematic space, and expand the finite terms in \(\epsilon\). The second approach benefits from the recently developed parametric IBP, where we can also simplify the integrand with IBP reduction and calculate the integrals via differential equations.
Regarding the ENC resummation, for \(e^{+}e^{-}\) annihilation, we solve the E3C jet RGE (which is a modified DGLAP equation) order by order in \(\alpha_{s}\) with the two-loop boundary, and push the resummation up to NNLL. For \(pp\) collisions, we calculate the combined hard function moments using the method in [54] for dijet production. We present the complete NLL and the approximate NNLL resummation result, where the approximation is due to the missing of genuine two-loop hard function constant. The uncertainty is reduced compared with the previous results [16; 20; 21]. For the fixed-order matching, we notice that the singular contribution dominates the collinear limit and the non-singular contribution from matching has only small effects in the \(e^{+}e^{-}\) case. Nevertheless, we perform the matching for \(e^{+}e^{-}\) given the fixed-order result is already available, but leave the matching with fixed-order in the \(pp\) case for the future study.
For a complete phenomenological analysis and precise \(\alpha_{s}\) extraction at hadron collider, there are still several ingredients needed in the future. Perturbatively, we need to compute both two-loop hard function and the NLO non-singular distribution for \(pp\to jj\), in order to achieve a full NNLL story. More over, it would be interesting to solve the RG equation exactly following [102], and compare the results with the truncation method. At the same time, for both \(e^{+}e^{-}\) and \(pp\), it would be interesting to better understand the hadronization power corrections to help further reduce theoretical uncertainties. We hope that all these efforts can lead to a precision determination of \(\alpha_{s}\) from jet substructure in the future.
The authors thank Hao Chen, Kyle Lee, Meng Xiao, Tong-Zhi Yang, Yulei Ye for useful discussions. XYZ also thanks the MIT CTP for its hospitality while part of this work was performed. The work of WC, YL, ZX, and HXZ was supported by the National Natural Science Foundation of China under the Grant No. 11975200. The work of JG was sponsored by the National Natural Science Foundation of China under the Grant No.12275173 and No.11835005.
## Appendix A Hard and jet functions
### \(e^{+}e^{-}\) Hard function
The ENC hard function for \(e^{+}e^{-}\) can be obtained from the semi-inclusive hadron fragmentation function. At NNLL, following our resummation procedure, we need the regular up
to two-loop, single logarithmic up to one-loop and the double logarithmic moments at tree level with respect to the energy fraction \(x\):
\[\int_{0}^{1}dx\,x^{N}\,H_{q,g}(x,\mu=Q) = \sum_{L=0}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L}h_{L}^{q,g}(N)\,,\] \[\int_{0}^{1}dx\,x^{N}\,\ln x\,H_{q,g}(x,\mu=Q) = \sum_{L=1}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L}\dot{h} _{L}^{q,g}(N)\,,\] \[\int_{0}^{1}dx\,x^{N}\,\ln^{2}x\,H_{q,g}(x,\mu=Q) = \sum_{L=1}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L}\ddot{h }_{L}^{q,g}(N)\,. \tag{100}\]
For EEC (\(N=2\)), we have
\[h_{0}^{q} =2\,,\qquad h_{0}^{g}=0\,,\qquad\qquad h_{1}^{q}=\frac{131}{4}\,C _{F}\,,\qquad h_{1}^{g}=-\frac{71}{12}\,C_{F}\,,\] \[h_{2}^{q} =\left(64\zeta_{4}-\frac{1172}{3}\zeta_{3}-166\zeta_{2}+\frac{2 386397}{2592}\right)C_{A}C_{F}\] \[+\left(-128\zeta_{4}+\frac{1016}{3}\zeta_{3}+\frac{1751}{18} \zeta_{2}-\frac{1105289}{5184}\right)C_{F}^{2}+\left(32\zeta_{3}+\frac{118}{15 }\zeta_{2}-\frac{8530817}{54000}\right)C_{F}T_{F}n_{f}\,,\] \[h_{2}^{g} =\left(-\frac{76}{3}\zeta_{3}+\frac{188}{45}\zeta_{2}-\frac{2980 2739}{324000}\right)C_{A}C_{F}+\left(\frac{124}{3}\zeta_{3}+\frac{523}{18} \zeta_{2}-\frac{674045}{5184}\right)C_{F}^{2}\,,\] \[\dot{h}_{0}^{q} =0\,,\qquad\dot{h}_{1}^{q}=\left(40\zeta_{3}+\frac{61}{3}\zeta_{ 2}-\frac{5303}{72}\right)C_{F}\,,\qquad\dot{h}_{0}^{g}=0\,,\qquad\dot{h}_{1}^{ g}=\left(-\frac{7}{3}\zeta_{2}+\frac{31}{4}\right)C_{F}\,,\] \[\ddot{h}_{0}^{q} =0\,,\qquad\ddot{h}_{0}^{g}=0\,. \tag{101}\]
Note that the EEC hard moments are also summarized in the appendix of Ref. [17]). However, the normalization condition in [17] is different from ours, due to the scaled energy \(E_{i}/(Q/2)\) there in contrast with \(E_{i}/Q\) here in the definition of the jet function. For E3C (\(N=3\)), we find
\[h_{0}^{q} = 2,\qquad h_{0}^{g}=0,\qquad h_{1}^{q}=\frac{11909}{300}C_{F}, \qquad h_{1}^{g}=-\frac{547}{150}C_{F}\,,\] \[h_{2}^{q} = \left(-\frac{942}{5}\zeta_{3}-\frac{17}{45}\zeta_{2}+\frac{17147 309}{32400}\right)C_{A}C_{F}+\left(32\zeta_{3}+\frac{322}{25}\zeta_{2}-\frac{ 6169957}{30000}\right)C_{F}n_{f}T_{F}\] \[+\left(-\frac{2012}{15}\zeta_{3}-\frac{8987}{30}\zeta_{2}+\frac{ 3256506739}{3240000}\right)C_{F}^{2}\,,\] \[h_{2}^{g} = \left(\frac{52}{5}\zeta_{3}+\frac{4396}{225}\zeta_{2}-\frac{1017 63773}{810000}\right)C_{A}C_{F}+\left(\frac{392}{15}\zeta_{3}+\frac{397}{15} \zeta_{2}-\frac{163115357}{1620000}\right)C_{F}^{2}\,,\] \[\dot{h}_{0}^{q} = 0\,,\qquad\dot{h}_{1}^{q}=\left(40\zeta_{3}+\frac{337}{15}\zeta_ {2}-\frac{709693}{9000}\right)C_{F}\,,\] \[\dot{h}_{0}^{g} = 0\,,\qquad\dot{h}_{1}^{g}=\left(-\frac{22}{15}\zeta_{2}+\frac{167 39}{4500}\right)C_{F}\,,\] \[\ddot{h}_{0}^{q} = 0\,,\qquad\ddot{h}_{0}^{g}=0\,. \tag{102}\]
For completeness, we also provide the E3C (\(N=3\)) hard moments for the gluonic Higgs decay, which is needed for extracting the two-loop gluon jet constants. Here we use
\(\hat{h}\) to distinguish from the \(e^{+}e^{-}\) case.
\[\tilde{h}_{0}^{q} = 0,\qquad\tilde{h}_{0}^{g}=2,\qquad\tilde{h}_{1}^{q}=-\frac{2461}{45 0}n_{f}T_{F},\qquad\tilde{h}_{1}^{g}=\frac{11491}{150}C_{A}-\frac{494}{45}n_{f}T _{F}\,,\] \[\tilde{h}_{2}^{q} = n_{f}T_{F}\left[C_{A}\left(\frac{88}{3}\zeta_{3}+\frac{3428}{75} \zeta_{2}-\frac{219509243}{810000}\right)+\left(\frac{1727}{225}\zeta_{2}-\frac {187858397}{1620000}\right)C_{F}\right]\] \[+\left(-\frac{352}{45}\zeta_{2}+\frac{7224}{125}\right)n_{f}^{2} T_{F}^{2}\,,\] \[\tilde{h}_{2}^{g} = n_{f}T_{F}\left[C_{A}\left(-\frac{208}{3}\zeta_{3}+\frac{1264}{ 15}\zeta_{2}-\frac{38190113}{40500}\right)+C_{F}\left(96\zeta_{3}-\frac{242}{2 25}\zeta_{2}-\frac{113165189}{810000}\right)\right]\] \[+C_{A}^{2}\left(-388\zeta_{3}-\frac{31684}{75}\zeta_{2}+\frac{83 7482633}{270000}\right)+n_{f}^{2}T_{F}^{2}\left(-\frac{64}{9}\zeta_{2}+\frac{ 44252}{675}\right)\,,\] \[\dot{\tilde{h}}_{0}^{q} = 0\,,\qquad\dot{\tilde{h}}_{1}^{q}=n_{f}T_{F}\left(-\frac{22}{15} \zeta_{2}+\frac{404}{125}\right)\,,\] \[\dot{\tilde{h}}_{0}^{g} = 0\,,\qquad\dot{\tilde{h}}_{1}^{g}=C_{A}\left(40\zeta_{3}+\frac{346 }{15}\zeta_{2}-\frac{2134817}{27000}\right)+\left(-\frac{8}{3}\zeta_{2}+\frac {5369}{1350}\right)n_{f}T_{F}\,,\] \[\ddot{\tilde{h}}_{0}^{q} = 0\,,\qquad\ddot{\tilde{h}}_{0}^{g}=0\,. \tag{100}\]
\(pp\to jj\) **Hard function**
The following table gives the hard function moments for \(pp\to jj\) calculated in Madgraph5 in two different \(p_{t}\) ranges: \([300,350]\) GeV and \([500,550]\) GeV, needed for the resummation of both EEC (\(N=2\)) and E3C (\(N=3\)).
As one of the important checks of our calculation, we show in Fig. 13 the independence of the slicing parameter \(\delta_{\rm cut}\) when evaluating the hard function moments using the method in [54]. The values of the moments are in agreement within the numeric uncertainty for three values of \(\delta_{\rm cut}\) across two orders of magnitude, namely \(\delta_{\rm cut}\in\{0.003,0.03,0.3\}\).
\begin{table}
\begin{tabular}{||c c c c c c c||} \hline \multicolumn{6}{||c||}{\(pp\to jj\) at 13 TeV, with NNPDF31\_nnlo\_as\_0118} \\ \hline \hline (300,350) GeV & \(h_{0}^{q}\) & \(h_{0}^{g}\) & \(a_{s}\,h_{1}^{q}\) & \(a_{s}\,h_{1}^{g}\) & \(a_{s}\,\dot{h}_{1}^{q}\) & \(a_{s}\,\dot{h}_{1}^{g}\) \\ \hline \(N=2\) & 0.3571 & 0.6429 & 0.1003 & 0.3304 & 0.0546 & 0.2149 \\ \hline \(N=3\) & 0.3571 & 0.6429 & 0.1463 & 0.4996 & 0.0393 & 0.1379 \\ \hline \hline (500,550) GeV & \(h_{0}^{q}\) & \(h_{0}^{g}\) & \(a_{s}\,h_{1}^{q}\) & \(a_{s}\,h_{1}^{g}\) & \(a_{s}\,\dot{h}_{1}^{q}\) & \(a_{s}\,\dot{h}_{1}^{g}\) \\ \hline \(N=2\) & 0.4417 & 0.5583 & 0.1337 & 0.2473 & 0.0568 & 0.1816 \\ \hline \(N=3\) & 0.4417 & 0.5583 & 0.1820 & 0.3894 & 0.0417 & 0.1150 \\ \hline \end{tabular}
\end{table}
Table 2: Values for hard function moments in \(pp\) collision for different \(p_{t}\) ranges. The NLO corrections turn out to be significant.
Figure 13: NLO hard function moments for \(N=2\) (left), and \(N=3\) (right), with \(p_{t}\in[300,350]\) GeV and \(\delta_{\rm cut}\in\{0.003,\,0.03,\,0.3\}\). The lower panels show the relative variation of the moments compared with the average value for three \(\delta_{\rm cut}\). The error bars represent the Monte-Carlo numeric uncertainty given by Madgraph5.
#### Jet function
For ENC, solving the jet function RGE requires the regular anomalous dimensions and their derivatives, and at NNLL, similar to hard function, we need the regular terms up to two-loop, the first derivative up to one-loop as well as the second derivative at tree-level.
The QCD timelike splitting function is expanded in \(\frac{\alpha_{s}}{4\pi}\)
\[P_{ij}(x)=\sum_{L=0}^{\infty}\left(\frac{\alpha_{s}}{4\pi}\right)^{L+1}P_{ij}^{ (L)}(x)\,, \tag{100}\]
and the anomalous dimension for ENC is defined to be the (N+1) Mellin moment of the splitting function. Explicitly,
\[\gamma^{(L)}_{T,ij} \equiv-\int_{0}^{1}\mathrm{d}x\,x^{N}\,P_{ij}^{(L)}(x)\,,\] \[\dot{\gamma}^{(L)}_{T,ij} \equiv-\int_{0}^{1}\mathrm{d}x\,\ln x\,x^{N}P_{ij}^{(L)}(x)\,,\] \[\ddot{\gamma}^{(L)}_{T,ij} \equiv-\int_{0}^{1}\mathrm{d}x\,\ln^{2}x\,x^{N}P_{ij}^{(L)}(x)\,. \tag{101}\]
Here the dot also represents the derivative with respect to \(N\). Note that \(\{i,j\}=\{q,g\}\) and the anomalous dimension is a \(2\times 2\) matrix.
The results for EEC (\(N=2\)) are derived and summarized in the appendix of Ref. [17], so here we provide the expressions for E3C (\(N=3\)).
At LO, we find
\[\gamma^{(0)}_{T,qq} =\frac{157}{30}\,C_{F}\,,\qquad\gamma^{(0)}_{T,gq}=-\frac{11}{15} \,C_{F}\,,\qquad\gamma^{(0)}_{T,qg}=\frac{11}{30}\,n_{f}\,,\qquad\gamma^{(0)}_{ T,gg}=\frac{21}{5}\,C_{A}+\frac{2}{3}\,n_{f}\,,\] \[\dot{\gamma}^{(0)}_{T,qq} =\left(4\zeta_{2}-\frac{10169}{1800}\right)C_{F}\,,\quad\dot{ \gamma}^{(0)}_{T,gq}=\frac{247}{900}\,C_{F}\,,\quad\dot{\gamma}^{(0)}_{T,qg}= \frac{137}{1800}\,n_{f}\,,\quad\dot{\gamma}^{(0)}_{T,gg}=\left(4\zeta_{2}- \frac{2453}{450}\right)C_{A}\,,\] \[\ddot{\gamma}^{(0)}_{T,qq} =\left(-8\zeta_{3}+\frac{507103}{54000}\right)C_{F}\,,\quad\ddot{ \gamma}^{(0)}_{T,gq}=-\frac{5489}{27000}\,C_{F}\,,\quad\ddot{\gamma}^{(0)}_{ T,qg}=-\frac{1919}{54000}\,n_{f}\,,\] \[\ddot{\gamma}^{(0)}_{T,gg} =\left(-8\zeta_{3}+\frac{124511}{13500}\right)C_{A}\,, \tag{102}\]
and at NLO, we have
\[\gamma^{(1)}_{T,qq} =\left(-\frac{628}{15}+\frac{2905763}{54000}\right)C_{F}^{2}+ \frac{16157}{675}C_{A}C_{F}-\frac{13427}{3000}\,C_{F}n_{f}\,,\] \[\gamma^{(1)}_{T,gq} =\left(\frac{88}{15}\zeta_{2}-\frac{104389}{27000}\right)C_{F}^{2} -\frac{142591}{13500}C_{A}C_{F}\,,\] \[\gamma^{(1)}_{T,qg} =\left(\frac{44}{15}\zeta_{2}-\frac{60391}{27000}\right)C_{A}n_{f} -\frac{166729}{54000}\,C_{F}n_{f}-\frac{6}{25}\,n_{f}^{2}\,,\] \[\gamma^{(1)}_{T,gg} =\left(-\frac{168}{5}\zeta_{2}+\frac{90047}{1500}\right)C_{A}^{2} +\left(-\frac{16}{3}\zeta_{2}+\frac{2273}{1350}\right)C_{A}n_{f}+\frac{57287}{2 7000}\,C_{F}n_{f}\,,\] \[\dot{\gamma}^{(1)}_{T,qq} =\left(-120\zeta_{4}+\frac{422}{3}\zeta_{3}+\frac{10169}{150} \zeta_{2}-\frac{162656941}{1080000}\right)C_{F}^{2}\]
\[\gamma^{(2)}_{T,gg}=\left(840\zeta_{4}-\frac{3752}{25}\zeta_{3}-\frac{ 342578}{375}\zeta_{2}+\frac{1069405919}{1350000}\right)C_{A}^{3}\] \[\qquad+\left(\frac{400}{3}\zeta_{4}-\frac{29534}{225}\zeta_{3}- \frac{30316}{675}\zeta_{2}+\frac{129284923}{2430000}\right)C_{A}^{2}n_{f}\]
\[+\left(\frac{2744}{45}\zeta_{3}-\frac{2158}{125}\zeta_{2}-\frac{18828 3293}{6075000}\right)C_{A}C_{F}n_{f}+\left(-\frac{352}{225}\zeta_{3}+\frac{4037} {3375}\zeta_{2}+\frac{27742123}{24300000}\right)C_{F}^{2}n_{f}\] \[+\left(-\frac{64}{9}\zeta_{3}+\frac{160}{27}\zeta_{2}-\frac{71341 }{27000}\right)C_{A}n_{f}^{2}+\left(-\frac{484}{675}\zeta_{2}-\frac{165553}{270 000}\right)C_{F}n_{f}^{2}\,.\] (A.9)
## Appendix B \(\beta\)-function RGE and running coupling
The well-known QCD \(\beta\)-function is written as
\[\frac{\mathrm{d}\alpha_{s}(\mu)}{\mathrm{d}\ln\mu}=\beta(\alpha_{s}(\mu)), \quad\beta(\alpha)=-2\alpha\,\left[\left(\frac{\alpha}{4\pi}\right)\beta_{0}+ \left(\frac{\alpha}{4\pi}\right)^{2}\beta_{1}+\left(\frac{\alpha}{4\pi}\right) ^{3}\beta_{2}+\cdots\right]\,,\] (B.1)
where the coefficient up to three loops are given by [103; 104; 105; 106]
\[\beta_{0} =\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f}\,,\] (B.2) \[\beta_{1} =\frac{34}{3}C_{A}^{2}-\frac{20}{3}C_{A}T_{F}n_{f}-4C_{F}T_{F}n_{ f}\,,\] \[\beta_{2} =n_{f}^{2}T_{F}^{2}\left(\frac{158}{27}C_{A}+\frac{44}{9}C_{F} \right)+n_{f}T_{F}\left(2C_{F}^{2}-\frac{205}{9}C_{F}C_{A}-\frac{1415}{27}C_{A }^{2}\right)+\frac{2857}{54}C_{A}^{3}\,,\] \[\beta_{3} =\frac{1093}{729}n_{f}^{3}+\left(\frac{50065}{162}+\frac{6472}{8 1}\zeta_{3}\right)n_{f}^{2}+\left(-\frac{1078361}{162}-\frac{6508}{27}\zeta_{ 3}\right)n_{f}+3564\zeta_{3}+\frac{149753}{6}\,.\]
At one-loop, the \(\beta\)-RGE can be solved exactly. At two-loop and beyond, there are different solutions. In terms of \(L\equiv\ln\frac{\mu^{2}}{\Lambda_{\text{QCD}}^{2}}\), a expanded solution can be written as:
\[\alpha_{s}(\mu)=\frac{4\pi}{\beta_{0}}\left[\frac{1}{L}-\frac{ \beta_{1}}{\beta_{0}^{2}L^{2}}\ln L+\frac{\beta_{1}^{2}}{\beta_{0}^{4}L^{3}}( \ln^{2}L-\ln L-1)+\frac{\beta_{2}}{\beta_{0}^{3}L^{3}}\right.\\ \left.+\frac{\beta_{1}^{3}}{\beta_{0}^{6}L^{4}}\left(-\ln^{3}L+ \frac{5}{2}\ln^{2}L+2\ln L-\frac{1}{2}\right)-3\frac{\beta_{1}\beta_{2}}{ \beta_{0}^{5}L^{4}}\ln L+\frac{\beta_{3}}{2\beta_{0}^{4}L^{4}}\right]\,.\] (B.3)
Here we can obtain the two-loop running coupling for NLL resumation by setting \(\beta_{2}=\beta_{3}=0\) and three-loop running coupling for NNLL by only \(\beta_{3}=0\).
Alternatively, one can iteratively solve the RGE order by order in a formal expansion parameter \(\epsilon\sim\frac{\beta_{a}}{\beta_{0}}\), with \(n\geq 1\). For NLL, the two-loop running coupling is written as
\[\alpha_{s}(\mu)=\alpha_{s}(Q)\left[X+\alpha_{s}(Q)\frac{\beta_{1}}{4\pi\beta_ {0}}\ln X\right]^{-1},\quad X\equiv 1+\frac{\alpha_{s}(Q)}{2\pi}\beta_{0}\ln \frac{\mu}{Q}\,,\] (B.4)
and at three loops for NNLL
\[\alpha_{s}(\mu)=\alpha_{s}(Q)\left\{X+\alpha_{s}(Q)\frac{\beta_{1}}{4\pi\beta _{0}}\ln X+\frac{\alpha_{s}^{2}(Q)}{16\pi^{2}}\left[\frac{\beta_{2}}{\beta_{0}} \left(1-\frac{1}{X}\right)+\frac{\beta_{1}^{2}}{\beta_{0}^{2}}\left(\frac{1}{X }-1+\frac{\ln X}{X}\right)\right]\right\}^{-1}\,.\] (B.5)
For the resummation in this paper, we use the iterative solution (the latter one) and set the coupling at \(Q=91.2\) GeV to be the world average value \(\alpha_{s}(m_{Z})=0.118\).
Squeeze limit of EEEC jet functions
In this section, we provide the perturbative data for the squeeze limit of the EEEC jet function in Eq. (17), which is needed for E3C jet function calculation. Given the conformal parameterization,
\[x_{1}=x_{L}z\bar{z},\quad x_{2}=x_{L}(1-z)(1-\bar{z}),\quad x_{3}=x_{L}\,, \tag{114}\]
the squeeze limits correspond to \(z\to 0,1,\infty\), related by a \(\mathbb{S}_{3}\) symmetry. Without loss of generality, we provide the \(z\to 1\) limit for the shapes function up to \(\mathcal{O}(\epsilon^{2})\). In the quark jet, we find for \(G(z)\)
\[G_{q}(z) \stackrel{{ z\to 1}}{{\approx}}C_{F}T_{F}n_{f}\bigg{(} \frac{13}{4800(1-z)(1-\bar{z})}+\frac{z-2}{1440(1-\bar{z})^{2}}+\frac{\bar{z}}{ 1440(1-z)^{2}}-\frac{39z+1}{28800(1-z)^{2}}\] \[+\frac{13}{9600(1-\bar{z})}\bigg{)}+C_{F}C_{A}\bigg{(}\frac{91}{4 800(1-z)(1-\bar{z})}+\frac{2-z}{2880(1-\bar{z})^{2}}-\frac{\bar{z}}{2880(1-z)^ {2}}\] \[-\frac{273z-293}{28800(1-z)^{2}}+\frac{91}{9600(1-\bar{z})} \bigg{)}+C_{F}^{2}\bigg{(}\frac{1}{20(1-z)(\bar{z}-1)}-\frac{z+\bar{z}-2}{40(z -1)(1-\bar{z})}\bigg{)}\,, \tag{115}\]
and for \(F(z)\):
\[F_{q}(z) \stackrel{{ z\to 1}}{{\approx}}C_{F}T_{F}n_{f}\bigg{(} \frac{649}{28800(1-z)(1-\bar{z})}-\frac{259}{43200(1-z)^{2}}-\frac{259}{43200(1 -\bar{z})^{2}}\bigg{)}\] \[+C_{F}C_{A}\bigg{(}\frac{561}{3200(1-z)(1-\bar{z})}+\frac{229}{86 400(1-z)^{2}}+\frac{229}{86400(1-\bar{z})^{2}}\bigg{)}\] \[+C_{F}^{2}\bigg{(}\frac{3307}{7200(1-z)(1-\bar{z})}\bigg{)}\,, \tag{116}\]
as well as the \(H(z)\):
\[H_{q}(z) \stackrel{{ z\to 1}}{{\approx}}C_{F}T_{F}n_{f}\bigg{(} \frac{664193-23400\pi^{2}}{4320000(1-z)(1-\bar{z})}+\frac{1800\pi^{2}-53191}{ 1296000(1-z)^{2}}+\frac{1800\pi^{2}-53191}{1296000(1-\bar{z})^{2}}\bigg{)}\] \[+C_{F}C_{A}\bigg{(}\frac{1805867-54600\pi^{2}}{1440000(1-z)(1-\bar {z})}+\frac{45421-1800\pi^{2}}{2592000(1-\bar{z})^{2}}-\frac{1800\pi^{2}-45421 }{2592000(1-z)^{2}}\bigg{)}\] \[+C_{F}^{2}\bigg{(}\frac{352451-10800\pi^{2}}{108000(1-z)(1-\bar{ z})}\bigg{)}\,. \tag{117}\]
Here the red stands for the most singular term, which contributes to \(\frac{1}{\epsilon}\) divergence in the E3C jet function calculation. For the gluon jet, we also find
\[G_{g}(z) \stackrel{{ z\to 1}}{{\approx}}C_{F}T_{F}n_{f}\bigg{(} \frac{3}{320(1-z)(1-\bar{z})}+\frac{3}{640(1-z)}+\frac{3}{640(1-\bar{z})}\bigg{)}\] \[+C_{A}T_{F}n_{f}\bigg{(}\frac{7}{800(1-z)(1-\bar{z})}+\frac{z-2}{1 440(1-\bar{z})^{2}}+\frac{\bar{z}}{1440(1-z)^{2}}-\frac{63z-43}{14400(1-z)^{2}}\] \[+\frac{7}{1600(1-\bar{z})}\bigg{)}+C_{A}^{2}\bigg{(}\frac{49}{800( 1-z)(1-\bar{z})}+\frac{2-z}{2880(1-\bar{z})^{2}}-\frac{\bar{z}}{2880(1-z)^{2}}\]
\[-\frac{441z-451}{14400(1-z)^{2}}+\frac{49}{1600(1-\bar{z})}\Bigg{)}\,, \tag{109}\]
\[F_{g}(z) \stackrel{{ z\to 1}}{{\approx}}C_{F}T_{F}n_{f}\bigg{(} \frac{241}{3200(1-z)(1-\bar{z})}\bigg{)}+C_{A}T_{F}n_{f}\bigg{(}\frac{343}{4800 (1-z)(1-\bar{z})}-\frac{259}{43200(1-z)^{2}}\] \[-\frac{259}{43200(1-\bar{z})^{2}}\bigg{)}+C_{A}^{2}\bigg{(}\frac{ 557}{960(1-z)(1-\bar{z})}+\frac{229}{86400(1-z)^{2}}+\frac{229}{86400(1-\bar{z })^{2}}\bigg{)}\,,\]
\[H_{g}(z) \stackrel{{ z\to 1}}{{\approx}}C_{F}T_{F}n_{f} \bigg{(}\frac{434309-16200\pi^{2}}{864000(1-z)(1-\bar{z})}+C_{A}T_{F}n_{f} \bigg{(}\frac{1033981-37800\pi^{2}}{2160000(1-z)(1-\bar{z})}\] \[+\frac{1800\pi^{2}-53191}{1296000(1-z)^{2}}+\frac{1800\pi^{2}-531 91}{1296000(1-\bar{z})^{2}}\bigg{)}+C_{A}^{2}\bigg{(}\frac{2999389-88200\pi^{2} }{720000(1-z)(1-\bar{z})}\] \[-\frac{1800\pi^{2}-45421}{2592000(1-z)^{2}}-\frac{1800\pi^{2}-454 21}{2592000(1-\bar{z})^{2}}\bigg{)}\,. \tag{110}\]
## Appendix D Result of two-loop E3C jet function calculation
We list the individual results for the two-loop jet function calculation in Sec. 2.4. As we discussed above, the calculation is reorganized as nonidentical energy weight contribution and contact terms. For the nonidentical energy weight in Sec. 2.4.1, we find for the quark jet
\[\frac{dJ_{q}^{\rm nonid}}{dx_{L}} =\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{2}\left\{\delta(x_{L})f_{ q}(\mu,Q,\epsilon)+\frac{1}{x_{L}}\bigg{[}C_{F}T_{F}n_{f}\bigg{(}-\frac{13}{200 \epsilon}+\frac{13}{100}\ln\bigg{(}\frac{Q^{2}x_{L}}{\mu^{2}}\bigg{)}\] \[-0.44158(3)\bigg{)}+C_{F}^{2}\left(-\frac{6}{5\epsilon}+\frac{12} {5}\ln\bigg{(}\frac{Q^{2}x_{L}}{\mu^{2}}\bigg{)}-10.963(1)\right)\] \[+C_{F}C_{A}\left(-\frac{91}{200\epsilon}+\frac{91}{100}\ln\bigg{(} \frac{Q^{2}x_{L}}{\mu^{2}}\bigg{)}-4.3743(7)\bigg{)}\,\bigg{]}\right\}, \tag{111}\]
with the coefficient of the \(\delta(x_{L})\) being
\[f_{q}(\mu,Q,\epsilon) =C_{F}T_{F}n_{f}\bigg{[}\frac{13}{400\epsilon^{2}}+\frac{1}{ \epsilon}\left(\frac{13}{200}\ln\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}+0.22079( 2)\right)+0.44158(3)\ln\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}\] \[+\frac{13}{200}\ln^{2}\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}+0.544 1(8)\bigg{]}+C_{F}C_{A}\bigg{[}\frac{91}{400\epsilon^{2}}+\frac{1}{\epsilon} \left(\frac{91}{200}\ln\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}+2.1871(8)\right)\] \[+4.3743(7)\ln\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}+\frac{91}{200} \ln^{2}\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}+10.483(2)\bigg{]}+C_{F}^{2}\bigg{[} 24.60(4)+\frac{3}{5\epsilon^{2}}\] \[+\frac{1}{\epsilon}\left(\frac{6}{5}\ln\bigg{(}\frac{\mu^{2}}{Q^{2 }}\bigg{)}+5.4815(3)\right)+10.963(1)\ln\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}+ \frac{6}{5}\ln^{2}\bigg{(}\frac{\mu^{2}}{Q^{2}}\bigg{)}\,\bigg{]}\,. \tag{112}\]
The \(\ln\Big{(}\frac{Q^{2}x_{L}}{\mu^{2}}\Big{)}\) term is verified by the jet RGE. For a gluon jet, the \(\mathcal{O}(\alpha_{s}^{2})\) contribution is
\[\frac{dJ_{g}^{\rm nonid}}{dx_{L}}=\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{2} \left\{\delta(x_{L})f_{g}(\mu,Q,\epsilon)+\frac{1}{x_{L}}\bigg{[}C_{F}T_{F}n_{f} \left(-\frac{9}{40\epsilon}+\frac{9}{20}\ln\bigg{(}\frac{Q^{2}x_{L}}{\mu^{2}} \right)\]
\[-1.8862(6))+C_{A}T_{F}n_{f}\left(-\frac{21}{100\epsilon}+\frac{21}{5 0}\ln\left(\frac{Q^{2}x_{L}}{\mu^{2}}\right)-1.5376(9)\right)\] \[+C_{A}^{2}\left(-\frac{147}{100\epsilon}+\frac{147}{50}\ln\left( \frac{Q^{2}x_{L}}{\mu^{2}}\right)-14.031(3)\right)\bigg{]}\bigg{\}}\,, \tag{103}\]
with the corresponding coefficient
\[f_{g}(\mu,Q,\epsilon) =C_{A}T_{F}n_{f}\bigg{[}\frac{21}{200\epsilon^{2}}+\frac{1}{ \epsilon}\left(\frac{21}{100}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+0.7688(5) \right)+1.5376(9)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)\] \[+\frac{21}{100}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)+2.350(8 )\bigg{]}+C_{F}T_{F}n_{f}\bigg{[}\frac{9}{80\epsilon^{2}}+\frac{1}{\epsilon} \left(\frac{9}{40}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+0.9431(3)\right)\] \[+1.886(3)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{9}{40}\ln^{ 2}\left(\frac{\mu^{2}}{Q^{2}}\right)+3.757(1)\bigg{]}+C_{A}^{2}\bigg{[}33.188( 4)+\frac{147}{200\epsilon^{2}}\] \[+\frac{1}{\epsilon}\left(\frac{147}{100}\ln\left(\frac{\mu^{2}}{Q ^{2}}\right)+7.01569(5)\right)+14.031(3)\ln\left(\frac{\mu^{2}}{Q^{2}}\right) +\frac{147}{100}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\bigg{]}\,. \tag{104}\]
Regarding the contact term in Sec. 2.4.2, for \(e^{+}e^{-}\) annihilation, we have the sum of E\({}^{2}\)EC and E\({}^{3}\)C
\[\frac{1}{\sigma_{0}}\frac{\mathrm{d}_{\mathrm{C},q}^{[3],2\text{- loop}}(x_{L},\epsilon)}{\mathrm{d}x_{L}}= \left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Bigg{\{}\delta(x_{L})r_{ q}(\mu,Q,\epsilon)+\left[\frac{1}{x_{L}}\right]_{+}\left[C_{A}C_{F}\bigg{(}\frac{91}{100 \epsilon}+\frac{1189}{200}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)\right.\] \[-6\zeta_{3}+\frac{25\pi^{2}}{6}-\frac{52307}{18000}\bigg{)}+C_{F }n_{f}T_{F}\left(\frac{13}{100\epsilon}-\frac{31}{25}\ln\left(\frac{\mu^{2}}{Q ^{2}}\right)-\frac{14809}{2000}\right)\] \[+C_{F}^{2}\left(\frac{12}{5\epsilon}+\frac{24}{5}\ln\left(\frac{ \mu^{2}}{Q^{2}}\right)+12\zeta_{3}-\frac{43\pi^{2}}{6}+\frac{274081}{3600} \right)\bigg{]}\] \[+\left[\frac{\ln(x_{L})}{x_{L}}\right]_{+}\left(-\frac{1343}{200} C_{A}C_{F}+\frac{113}{100}C_{F}n_{f}T_{F}+\frac{87}{80}C_{F}^{2}\right)\Bigg{\}}\,, \tag{105}\]
with the singular part \(r_{q}(\mu,Q,\epsilon)\)
\[r_{q}(\mu,Q,\epsilon) =C_{A}C_{F}\Bigg{[}-\frac{91}{200\epsilon^{2}}+\frac{1}{\epsilon} \Bigg{(}-\frac{91}{100}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+3\zeta_{3}-\frac {25\pi^{2}}{12}+\frac{452921}{36000}\Bigg{)}-\frac{91}{100}\ln^{2}\left(\frac{ \mu^{2}}{Q^{2}}\right)\] \[+\left(6\zeta_{3}+\frac{890167}{36000}-\frac{25\pi^{2}}{6} \right)\ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{347\zeta_{3}}{2}+\frac{7 \pi^{4}}{20}-\frac{6697\pi^{2}}{1800}+\frac{47220317}{270000}\Bigg{]}\] \[+C_{F}n_{f}T_{F}\Bigg{[}-\frac{13}{200\epsilon^{2}}+\frac{1}{ \epsilon}\Bigg{(}-\frac{13}{100}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{ 5299}{12000}\Bigg{)}-\frac{13}{100}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\] \[-\frac{4349}{6000}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+4\zeta_{3} +\frac{137\pi^{2}}{400}-\frac{1413979}{720000}\Bigg{]}+C_{F}^{2}\Bigg{[}- \frac{6}{5\epsilon^{2}}\] \[+\frac{1}{\epsilon}\Bigg{(}-\frac{12}{5}\ln\left(\frac{\mu^{2}}{Q ^{2}}\right)-6\zeta_{3}+\frac{43\pi^{2}}{12}-\frac{281641}{7200}\Bigg{)}- \frac{12}{5}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)\] \[+\left(-12\zeta_{3}-\frac{281641}{3600}+\frac{43\pi^{2}}{6}\right) \ln\left(\frac{\mu^{2}}{Q^{2}}\right)+293\zeta_{3}-\frac{7\pi^{4}}{10}+\frac{15 371\pi^{2}}{1440}-\frac{380074411}{864000}\Bigg{]}\,. \tag{106}\]
Similarly, in the gluonic Higgs decay, we get
\[\frac{1}{\sigma_{0}^{\prime}}\frac{\text{d}\sigma_{\text{C,g}}^{[3] \text{,2-loop}}(x_{L},\epsilon)}{\text{d}x_{L}}= \lambda(\mu)\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Bigg{\{} \delta(x_{L})r_{g}(\mu,Q,\epsilon)+\left[\frac{1}{x_{L}}\right]_{+}\left\{n_ {f}^{2}T_{F}^{2}\bigg{(}-\frac{3}{5}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)- \frac{131}{60}\bigg{)}\right.\] \[+n_{f}T_{F}\bigg{[}C_{A}\left(\frac{21}{50\epsilon}-\frac{171}{100 }\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{7\pi^{2}}{15}-\frac{140917}{9000} \right)\] \[+C_{F}\left(\frac{9}{10}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)+ \frac{9}{20\epsilon}+\frac{1579}{400}\right)\bigg{]}\] \[+C_{A}^{2}\bigg{(}\frac{147}{50\epsilon}+\frac{1743}{100}\ln \left(\frac{\mu^{2}}{Q^{2}}\right)+6\zeta_{3}-\frac{97\pi^{2}}{30}+\frac{21182 9}{2250}\bigg{)}\bigg{\}}\] \[+\left[\frac{\ln(x_{L})}{x_{L}}\right]_{+}\left[n_{f}T_{F}\left( \frac{51}{25}C_{A}-\frac{69}{40}C_{F}\right)-\frac{133}{25}C_{A}^{2}+\frac{2} {5}n_{f}^{2}T_{F}^{2}\right]\right\}, \tag{100}\]
with the gluonic singular term \(r_{g}(\mu,Q,\epsilon)\)
\[r_{g}(\mu,Q,\epsilon) =C_{A}T_{F}n_{f}\bigg{[}-\frac{21}{100\epsilon^{2}}+\frac{1}{ \epsilon}\bigg{(}-\frac{21}{50}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{7 \pi^{2}}{30}+\frac{6887}{9000}\bigg{)}-\frac{1163}{150}\ln^{2}\left(\frac{\mu^ {2}}{Q^{2}}\right)\] \[+\left(-\frac{948847}{18000}-\frac{7\pi^{2}}{15}\right)\ln\left( \frac{\mu^{2}}{Q^{2}}\right)-\frac{211\zeta_{3}}{10}+\frac{3037\pi^{2}}{1800}- \frac{5585159}{67500}\bigg{]}+C_{F}T_{F}n_{f}\] \[\left[-\frac{9}{40\epsilon^{2}}+\frac{1}{\epsilon}\bigg{(}-\frac {9}{20}\ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{1509}{800}\bigg{)}-\frac{9} {20}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{3109}{400}\ln\left(\frac{ \mu^{2}}{Q^{2}}\right)+15\zeta_{3}\right.\] \[+\frac{5\pi^{2}}{8}-\frac{230393}{6000}\bigg{]}+C_{A}^{2}\bigg{\{} -\frac{147}{100\epsilon^{2}}+\frac{1}{\epsilon}\bigg{[}-\frac{147}{50}\ln \left(\frac{\mu^{2}}{Q^{2}}\right)-3\zeta_{3}+\frac{97\pi^{2}}{60}-\frac{47485 7}{18000}\bigg{]}\] \[+\frac{2143}{300}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)+\left( -6\zeta_{3}+\frac{261281}{18000}+\frac{97\pi^{2}}{30}\right)\ln\left(\frac{\mu ^{2}}{Q^{2}}\right)+\frac{1133\zeta_{3}}{10}-\frac{7\pi^{4}}{20}\] \[+\frac{373\pi^{2}}{100}-\frac{12512789}{90000}\bigg{\}}+n_{f}^{2} T_{F}^{2}\bigg{[}\frac{4}{3}\ln^{2}\left(\frac{\mu^{2}}{Q^{2}}\right)+\frac{2971}{300} \ln\left(\frac{\mu^{2}}{Q^{2}}\right)-\frac{23\pi^{2}}{45}+\frac{579043}{27000} \bigg{]}\,, \tag{101}\]
where \(\lambda\) is the effective \(Hgg\) coupling5[107]. These results are then used to extract the two-loop jet constants.
Footnote 5: For the case of gluonic Higgs decays, we normalize the E3C into the form where the LO E3C is \(\frac{1}{\sigma_{0}^{\prime}}\frac{\text{d}\sigma_{\text{C,g}}^{[3]}}{\text{d}x_{L }}=\lambda(\mu)\left(\frac{1}{4}\delta(x_{L})+\frac{3}{4}\delta(1-x_{L})\right)\) in \(d=4-2\epsilon\) dimensions.
## Appendix E Fixed-order expansion
In this section, we provide the singular expansion of projected energy correlator up to NNLO \(\mathcal{O}(\alpha_{s}^{3})\) in \(e^{+}e^{-}\) annihilation. This can be achieved by expanding our resummed distribution with canonical scale \(\mu=Q\). For EEC, we find
\[\frac{1}{\sigma_{0}}\frac{d\sigma^{[2]}}{dx_{L}}=\left(\frac{\alpha_{s}}{4\pi} \right)C_{F}\frac{3}{2x_{L}}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}C_{F} \bigg{\{}\bigg{[}\frac{53}{30}n_{f}T_{F}+\frac{25}{4}C_{F}-\frac{107}{15}C_{A} \bigg{]}\frac{\ln x_{L}}{x_{L}}\]
\[+\bigg{[}-\frac{4913}{450}n_{f}T_{F}+\bigg{(}-\frac{8263}{216}+\frac{4 3}{9}\pi^{2}-8\zeta_{3}\bigg{)}C_{F}+\bigg{(}\frac{35336}{675}-\frac{25}{9}\pi^{ 2}+4\zeta_{3}\bigg{)}C_{A}\bigg{]}\frac{1}{x_{L}}\bigg{\}}\] \[+\bigg{(}\frac{\alpha_{s}}{4\pi}\bigg{)}^{3}\,C_{F}\bigg{\{}\bigg{[} \frac{8059}{300}C_{A}^{2}-\frac{340}{9}C_{F}C_{A}+\frac{625}{48}C_{F}^{2}- \frac{16259}{900}C_{A}T_{F}n_{f}+\frac{4619}{360}C_{F}T_{F}n_{f}\] \[+\frac{92}{45}n_{f}^{2}T_{F}^{2}\bigg{]}\frac{\ln^{2}x_{L}}{x_{L} }+\bigg{[}-\frac{17734}{675}n_{f}^{2}T_{F}^{2}+\bigg{(}-\frac{64\zeta_{3}}{3}- \frac{6760183}{32400}+\frac{416\pi^{2}}{27}\bigg{)}C_{F}T_{F}n_{F}\] \[+\bigg{(}\frac{32\zeta_{3}}{3}+\frac{6644267}{27000}-\frac{36 \pi^{2}}{5}\bigg{)}C_{A}T_{F}n_{f}+\bigg{(}-\frac{172\zeta_{3}}{3}-\frac{7235 33}{2592}+\frac{1849\pi^{2}}{54}\bigg{)}C_{F}^{2}\] \[+\bigg{(}-\frac{74\zeta_{3}}{3}-\frac{2916859}{6750}+\frac{503 \pi^{2}}{30}\bigg{)}C_{A}^{2}+\bigg{(}\frac{262\zeta_{3}}{3}+\frac{105425}{14 4}-\frac{550\pi^{2}}{9}\bigg{)}C_{F}C_{A}\bigg{]}\frac{\ln x_{L}}{x_{L}}\] \[+\bigg{[}\bigg{(}\frac{88031}{1125}+\frac{4\pi^{2}}{5}\bigg{)}n_{ f}^{2}T_{F}^{2}+\bigg{(}-\frac{15988\zeta_{3}}{45}+\frac{236\pi^{4}}{135}- \frac{15161\pi^{2}}{360}+\frac{164829499}{243000}\bigg{)}C_{F}T_{F}n_{F}\] \[+\bigg{(}\frac{3679\zeta_{3}}{15}-\frac{118\pi^{4}}{135}+\frac{3 79579\pi^{2}}{16200}-\frac{1025118113}{1080000}\bigg{)}C_{A}T_{F}n_{F}\] \[+\bigg{(}8\pi^{2}\zeta_{3}+52\zeta_{3}+208\zeta_{5}-\frac{167\pi^ {4}}{27}-\frac{18805\pi^{2}}{1296}+\frac{742433}{1944}\bigg{)}C_{F}^{2}\] \[+\bigg{(}4\pi^{2}\zeta_{3}-\frac{47483\zeta_{3}}{90}+56\zeta_{5} -\frac{481\pi^{4}}{540}-\frac{906257\pi^{2}}{16200}+\frac{964892417}{540000} \bigg{)}C_{A}^{2}\] \[+\bigg{(}-12\pi^{2}\zeta_{3}+\frac{10604\zeta_{3}}{15}-216\zeta_ {5}+\frac{847\pi^{4}}{180}+\frac{137305\pi^{2}}{1296}-\frac{105395741}{51840 }\bigg{)}C_{F}C_{A}\bigg{]}\frac{1}{x_{L}}\bigg{\}}\,. \tag{100}\]
Similarly, for E3C, we have
\[\frac{1}{\sigma_{0}}\frac{d^{\sigma[3]}}{dx_{L}} =\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}\,C_{F}\frac{9}{8x_{L}}+ \Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{2}\,C_{F}\bigg{\{}\bigg{[}\frac{139}{100 }n_{f}T_{F}+\frac{471}{80}C_{F}-\frac{979}{200}C_{A}\bigg{]}\frac{\ln x_{L}}{x _{L}}\] \[+\bigg{[}-\frac{24863}{3000}n_{f}T_{F}-\frac{21}{10}C_{F}+\frac{ 66769}{3000}C_{A}\bigg{]}\frac{1}{x_{L}}\bigg{\}}\] \[+\bigg{(}\frac{\alpha_{s}}{4\pi}\bigg{)}^{3}\,C_{F}\bigg{\{}\bigg{[} \frac{17743}{1000}C_{A}^{2}-\frac{412753}{12000}C_{F}C_{A}+\frac{24649}{1600} C_{F}^{2}-\frac{19019}{1500}C_{A}T_{F}n_{f}\] \[+\frac{35369}{3000}C_{F}T_{F}n_{f}+\frac{128}{75}n_{f}^{2}T_{F}^{ 2}\bigg{]}\frac{\ln^{2}x_{L}}{x_{L}}+\bigg{[}-\frac{4559891}{22500}C_{A}- \frac{814823}{48000}C_{F}^{2}\] \[+\bigg{(}\frac{34399441}{120000}-\frac{11\pi^{2}}{2}\bigg{)}C_{F}C _{A}+\bigg{(}2\pi^{2}-\frac{1026851}{10000}\bigg{)}C_{F}T_{F}n_{f}+\frac{305590 7}{22500}C_{A}T_{F}n_{f}\] \[-\frac{23494}{1125}n_{f}^{2}T_{F}^{2}\bigg{]}\frac{\ln x_{L}}{x_{L }}+\bigg{[}j_{2}^{q,[3]}\bigg{(}\frac{157}{15}-\frac{44C_{A}}{3C_{F}}+\frac{1 6n_{f}T_{F}}{3C_{F}}\bigg{)}-\frac{22}{15}j_{2}^{g,[3]}\] \[+\bigg{(}\frac{106027}{54000}-\frac{22\pi^{2}}{225}\bigg{)}n_{f}^{ 2}T_{F}^{2}+\bigg{(}\frac{1827\zeta_{3}}{25}-\frac{3877\pi^{2}}{3000}-\frac{32 39027203}{10800000}\bigg{)}C_{F}T_{F}n_{f}\] \[+\bigg{(}-\frac{1037\zeta_{3}}{50}-\frac{2167\pi^{2}}{4500}-\frac{2 4958553}{3600000}\bigg{)}C_{A}T_{F}n_{f}\] \[+\bigg{(}\frac{3267\zeta_{3}}{20}-\frac{111313\pi^{2}}{14400}-\frac{6 031520921}{17280000}\bigg{)}C_{F}^{2}+\bigg{(}-\frac{829\zeta_{3}}{100}+\frac{4 4333\pi^{2}}{2250}+\frac{363491521}{5400000}\bigg{)}C_{A}^{2}\]
\[+\bigg{(}-\frac{42321\zeta_{3}}{200}+\frac{284797\pi^{2}}{36000}+ \frac{4941457181}{7200000}\bigg{)}C_{F}C_{A}\bigg{]}\frac{1}{x_{L}}\bigg{\}}\,, \tag{114}\]
with the two-loop jet constant \(j_{2}^{q/g,[3]}\) from Eq. (4.4)-(2.41).
| projected energy correlatorは、複数の検出器におけるエネルギーの積算を、検出器間の最大角距離 $x_L = (1 -\cos\chi_L)/2$ と関数の形式で測定する。 $x_L$ は、検出器間の角度距離の最大値で、 collinear limit $x_L\to 0$ は、ジェット構造を理解するための興味深い点である。一方、$x_L$ の大きな対数はおそらくPerturbation theoryを破綻させ、補正する必要がある。この補正のために、次世代の次世代の次世代の精度で補正する必要がある、次の世代の精度で補正する必要がある、次の世代の精度で補正する必要がある。 |
2306.07639 | Hidden Lagrangian coherence and memory effects in the statistics of
Hamiltonian motions | This paper is focused on the coherent effects that appear in tracer
statistics in two-dimensional incompressible turbulence in the presence of an
average velocity. We show that this determines strong modifications of the
transport and trajectory statistics, which are essentially caused by hidden
coherent components of the motion. | Madalina Vlad, Dragos Iustin Palade, Florin Spineanu | 2023-06-13T09:18:03 | http://arxiv.org/abs/2306.07639v1 | # Hidden Lagrangian coherence and memory effects
###### Abstract
Turbulence is a complex nonlinear process, which appears in many domains as fluid mechanics, plasma physics, astrophysics, atmosphere and ocean sciences, chemistry [1]-[3]. One of the main difficulty in understanding the dynamics of turbulence is the complicated combination of stochastic and quasi-coherent aspects that is typical for the strongly nonlinear regimes.
Quasi-coherence or order appears at the basic level of tracer trajectories in smooth stochastic velocity fields with finite correlation lengths \(\lambda\) and times \(\tau_{c}.\) Trajectory coherence (or Lagrangian coherence) is usually a transitory process that lasts during the time of flight over \(\lambda\) with the amplitude \(V\) of the stochastic velocity, \(\tau_{fl}=\lambda/V.\) It exists only for slow time variation of the velocity with \(\tau_{c}>\tau_{fl}\). In the case of two-dimensional incompressible velocity fields, a much stronger Lagrangian coherence appears that is due to trajectory eddying or trapping. It generates vortical structures of trajectories, which produce non-Gaussian statistics of tracer displacements and strongly modified transport [4]-[7]. The order of the tracer trajectories determines much more complicated effects on turbulence evolution, which essentially amplifies the degree of quasi-coherence. The turbulence that is dominantly two-dimensional has a self-organizing character [8]-[11], which consists of the generation of quasi-coherent large scale structures [12].
This paper is focused on the coherent effects that appear in tracer statistics in two-dimensional incompressible turbulence in the presence of an average velocity \({\bf V}_{d}\). We show that \({\bf V}_{d}\) determines strong modifications of the transport and trajectory statistics, which are essentially caused by hidden coherent components of the motion.
The results are based on the numerical simulation of the trajectories and consist of a conditional statistical analysis adapted to the special properties of the two-dimensional incompressible velocity fields.
The formulation of the problem and the simulation method are presented in Section 2. The motion is of Hamiltonian type with the Hamiltonian function \(\phi_{t}\) composed of the stochastic and average potentials. The trajectories evolve on the contour lines of \(\phi_{t}({\bf x})\) in static (frozen) potentials (\(\tau_{c}\rightarrow\infty\)), and they remain strongly correlated to these lines for slow time variation (large \(\tau_{c}\)). We consider first (Sections 3-5) frozen potentials. We discuss the configuration of the contour lines of the potential and the main features of tracer advection in Section 3. The space of trajectories is organized in two categories: trapped and free. The statistics of the Lagrangian velocity and of the trajectories are examined for each category, and their specific contributions to the global statistics (on the whole set of trajectories) are identified (Section 4). We show that quasi-coherent Lagrangian velocities parallel to \({\bf V}_{d}\) are generated for both categories. They are hidden in the global statistics, in the sense that their contributions compensate each other. However, they provide explanations for the nonstandard transport produced in these conditions. A deeper examination of the coherence induced by the average velocity is presented in Section 5, where we determine the Lagrangian statistics for the trajectories of each category that evolve on contour lines of the potential with the same value of \(\phi_{t}.\) These results reveal other (hidden) coherent elements of the motion, and provide important properties that are used for the understanding of the effects of the average velocity on the statistics of trajectories and transport. Sections 6 and 7 deal with time dependent potentials. The Lagrangian statistics conditioned by the value of the potential shows that the order found in frozen potentials does not decay due to the random time variation of the potential, as expected. On the contrary, important quasi-coherent elements significantly increase (Section 6). Explanations are provided in Section 7. They are essentially related to the constraint of invariance of the potential, which are approximately valid at large \(\tau_{c},\) and to the slow transition of the trajectories between the two categories. Long memory effects are identified and their effects is discussed. A short summary of the results and the conclusions are presented in Section 8.
## II 2. The problem and the simulation method
Tracer trajectories in two-dimensional stochastic velocity fields are obtained from
\[\frac{d{\bf x}}{dt}={\bf v}({\bf x},\!t)=\widetilde{\bf v}({\bf x},\!t)+V_{d}{ \bf e}_{2}, \tag{1}\]
where \({\bf e}_{1}\), \({\bf e}_{2}\) are the unit vectors in the plane of the motion \({\bf x}=(x_{1},x_{2})\), \({\bf e}_{3}\) is perpendicular on this plane. The velocity \({\bf v}({\bf x},\!t)\) has a stochastic component \(\widetilde{\bf v}({\bf x},\!t)\) superposed on a constant average velocity \({\bf V}_{d}=V_{d}\;{\bf e}_{2}.\) The incompressibility condition \(\nabla\cdot\widetilde{\bf v}({\bf x},\!t)=0\) of the velocity field is equivalent with the representation of \(\widetilde{\bf v}({\bf x},\!t)\) by a stochastic potential (or stream function) \(\phi({\bf x},\!t)\)
\[\widetilde{\bf v}({\bf x},\!t)=-\nabla\phi({\bf x},\!t)\times{\bf e}_{3}. \tag{2}\]
The equation of motion is of Hamiltonian type, with \(x_{1}\) and \(x_{2}\) the conjugate variables and \(\phi_{t}({\bf x},\!t)=\phi({\bf x},\!t)+x_{1}V_{d}\) the Hamiltonian function.
Dimensionless quantities are used in Eq. (1) with the potential normalized by its amplitude \(\Phi\), the distances by \(\lambda_{0}\) that is of the order of the correlation lengths, the velocities (including \(V_{d}\)) by \(V_{0}=\Phi/\lambda_{0}\) and the time by \(\tau_{0}=\lambda_{0}/V_{0}.\)
The potential is represented by a homogeneous and stationary Gaussian stochastic field. Its Eulerian correlation (EC) \(E({\bf x},\!t)\equiv\langle\phi({\bf x}_{0},\!t_{0})\ \phi({\bf x}_{0}+{\bf x},\!t_{0}+t)\rangle\) in dimensionless quantities is modeled in the simulations presented here by
\[E({\bf x},\!t)\equiv\exp\left(-\frac{x_{1}^{2}}{2\lambda_{1}^{2}}-\frac{x_{2 }^{2}}{2\lambda_{2}^{2}}-\frac{t^{2}}{2\tau_{c}^{2}}\right), \tag{3}\]
where \(\lambda_{i}\) are the correlation lengths of the 2-dimensional potential and \(\tau_{c}\) is the correlation time. The EC's of the velocity components \(E_{ii}({\bf x},\!t)\equiv\langle v_{i}({\bf x}_{0},\!t_{0})\ v_{i}({\bf x}_{0}+ {\bf x},\!t_{0}+t)\rangle\) are
\[E_{11}({\bf x},\!t)=-\partial_{2}\partial_{2}E({\bf x},\!t),\ E_{22}({\bf x},\!t )=-\partial_{1}\partial_{1}E({\bf x},\!t), \tag{4}\]
which determine the normalized amplitudes of the velocity fluctuations \(V_{1}=\sqrt{E_{11}({\bf 0},\!0)}=1/\lambda_{2}\), \(V_{2}=1/\lambda_{1}\).
The statistical properties of the trajectories obtained from Eq. (1) are numerically analyzed. More precisely, we determine the statistics of the trajectories and of the Lagrangian velocity, and a class of conditional Lagrangian correlations that reveal the quasi-coherent components of the motion and their properties.
We use statistical averages, which consists of generating a large number of realizations (\(r\)) of the stochastic Gaussian potential and of determining the trajectory with the initial condition \({\bf x}(0)=0\) in each \(r\) by effectively computing the velocity on the trajectory at each time step, \({\bf v}({\bf x}(t_{i}),t_{i}).\) However, the analysis of the results is connected to the equivalent space averaging procedure. This corresponds to the statistical ensemble of trajectories obtained in a single typical realization of the potential by different initial conditions \({\bf x}(0)={\bf x}_{0}^{r},\) where the points \({\bf x}_{0}^{r}\) are uniformly distributed in a very large domain.
We use the simulation code presented in [13], which is based on a fast generator of Gaussian fields with prescribed spectra. In the present work, we have implemented the so called FRD representation
\[\phi({\bf X})=\sum_{i=1}^{N_{c}}\sqrt{S({\bf K}_{i})}\sin\left({\bf K}_{i}{ \bf X}+\frac{\pi}{4}\zeta_{i}\right), \tag{5}\]
where \({\bf X}\equiv({\bf x},\!t)\) is the three-dimensional space-time and \({\bf K}_{i}\equiv({\bf k}_{\perp}^{i},\omega^{i})\) are the \(N_{c}\) discrete values of the wave numbers \({\bf k}_{\perp}^{i}\) and frequencies \(\omega^{i}.\)\(S({\bf K})\) is the spectrum of the stochastic potential, the Fourier transform of the EC (3). This representation is different of the usual discrete Fourier decomposition by the set of the values of \({\bf K}_{i}\) that are not the fixed points of a three-dimensional mesh, but random values with uniform distribution. Also, the random phases \(\zeta_{i}\) have not continuous distributions, but discrete values \(\pm 1\) (with equal probabilities).
Each set of the \(N_{c}\) random values of \({\bf K}_{i}\) and \(\zeta_{i}\) determines a realization \(r\) of the potential and a trajectory (solution of Eq. (1) with initial condition \({\bf x}(0)={\bf 0}\)). The statistical ensemble \(R\) consists of a number \(M\) of these sets.
The representation (5) provides a fast convergence of the Eulerian properties of the stochastic potential. We have shown that reasonable errors in the EC and in the probability of the potential are obtained at much smaller values of \(N_{c}\) and \(M\) than in the usual fast Fourier representation (FFR). This leads to the decrease of the computing times by roughly one order of magnitude compared to the usual FFR method in two-dimensional potentials [13]. Most of the simulations analyzed here are performed with \(N_{c}=500\) and \(M=10^{5}.\)
## III Main features of tracer advection
The incompressibility of the two-dimensional velocity field (\(\nabla\cdot\mathbf{v}(\mathbf{x}\),\(t)=0\)) determines two invariance laws of the solutions of Eq. (1). It leads to equations of motion of Hamiltonian type, with \(x_{1}\) and \(x_{2}\) conjugate variables and \(\phi_{t}\) the Hamiltonian function. In the case of time independent (frozen) potentials \(\phi\left(\mathbf{x}\right)\), the trajectories are linked to the contour line of the potential \(\phi_{t}(\mathbf{x})\), which means that the Lagrangian potential \(\phi_{t}(\mathbf{x}(t))\) is invariant along each trajectory. The other invariant law is statistical and applies to the motion in both frozen and time dependent potentials \(\phi_{t}(\mathbf{x}(t),t)\) for any value of the correlation time \(\tau_{c}\). It concerns the distribution of the Lagrangian velocity \(\mathbf{v}(\mathbf{x}(t),t)\), that is shown to be time independent, and thus identical with the distribution of the Eulerian velocity \(\mathbf{v}(\mathbf{x},t)\). The Lagrangian potential is statistically invariant too. This property is trivial in frozen potentials where \(\phi_{t}(\mathbf{x}(t))=\phi_{t}(\mathbf{x}(0))\) and it is similar to the case of the velocity in time dependent potentials where \(\phi_{t}(\mathbf{x}(t),t)\) changes in time.
An example of configuration of the contour lines of the potential can be seen in Fig. 1, where a typical realization of \(\phi_{t}(\mathbf{x})\) is presented for \(V_{d}=0\) (left panel) and for \(V_{d}=0.3\) (right panel). The contour lines at \(V_{d}=0\) are nested closed curves with multi-scale sizes that have the dimensions \(r_{\max}\) from \(r_{\max}\ll\lambda_{i}\) to \(r_{\max}\to\infty.\) The average velocity \(\mathbf{V}_{d}\) completely changes the field lines by breaking the large size contour lines and generating (winding) open paths along its direction. Islands of closed contour lines remain between the network of open paths, but their average size decreases as \(V_{d}\) increases, and, for \(V_{d}\) much larger than the amplitude of the stochastic velocity, all the lines are open. The average velocity also determines the limitation of the excursion of the contour lines perpendicular to \(\mathbf{V}_{d}\).
This configuration of the contour lines of \(\phi_{t}(\mathbf{x})\) determines solutions of Eq. (1) that are, in the presence of an average velocity, a mixture of localized periodic (or trapped) trajectories that are closed, and of free trajectories that have unlimited displacements along \(\mathbf{V}_{d}\). The space of trajectories \(R\) is organized in two disjointed subensembles: \(tr\) for the trapped trajectories and \(fr\) for the free ones (\(R=tr\cup fr\), \(tr\cap fr=\varnothing\)). The classification criterion is the periodicity of the trajectories. A trajectory \(r\) with period \(T_{r}\) belongs to \(tr\) if \(T_{r}\) is finite and to \(fr\) otherwise. \(T_{r}\) is defined as the time of the first return in the initial point \(\mathbf{x}(0)=\mathbf{0}\), and is determined as the first solution of \(r(t)=0\), where \(r(t)=\sqrt{x_{1}^{2}(t)+x_{2}^{2}(t)}.\) Practically, a trajectory belongs to the subensemble \(tr\) when its period is smaller than the time of integration \(t_{\max}\). The size of each trajectory \(r_{\max}=Max(r(t))\) is also calculated.
For \(V_{d}=0\), all trajectories \(\mathbf{x}(t)\) are closed, periodic functions of time when \(t\to\infty.\) At finite time \(t\), open trajectories are found, which correspond to large periods \(T_{r}>t\) (and to large size contour lines). As \(t\) increases the fraction of free trajectories decreases, and, in the limit \(t\to\infty\), all trajectories are trapped (\(tr=R\) and \(fr=\varnothing\)). The probability of trajectory sizes \(P(r_{\max},t)\) is represented in Fig. 2 at two time moments, \(t=60\) (dashed line) and \(t=120\) (solid line). One can see that the time evolution of \(P(r_{\max},t)\) affects only the large distances, while the small \(r_{\max}\) domain has invariant probability. The contributions of the closed and open trajectories to \(P(r_{\max},t)\) are also represented
Figure 1: Typical realization of the potential \(\phi_{t}(\mathbf{x})\) for \(V_{d}=0\) (left panel) and for \(V_{d}=0.3\) (right panel).
in the figure. The closed trajectories (red points) determine the invariant part of \(P(r_{\max},t).\) The open trajectories (green points) have large sizes and they determine the time variation of \(P(r_{\max},t).\) Their contribution move toward larger \(r_{\max}\) as time increases and it decays, such that \(P(r_{\max},t)\) is determined only by closed trajectories in the limit \(t\rightarrow\infty\). It is a decaying function of \(r_{\max}\) that scales as \(P(r_{\max},t)\sim r_{\max}^{-1.3}\) at large \(t.\) The slow algebraic decay of the asymptotic probability shows that the sizes of the trajectories cover multiple scales from \(r_{\max}\ll\lambda_{i}\) to \(r_{\max}\rightarrow\infty.\)
Thus, the invariance of the Lagrangian potential determines a process of trajectory trapping manifested by eddying in the structure of \(\phi\left(\mathbf{x}\right).\)
The average velocity \(V_{d}\) that strongly modifies the structure of the field lines of \(\phi_{t}\left(\mathbf{x}\right)\) determines a significant change of the probability of trajectory sizes. Two categories of trajectories coexist for \(V_{d}\lesssim V:\) periodic, closed trajectories situated on the islands of closed contour lines of \(\phi_{t}\left(\mathbf{x}\right)\) and non-periodic trajectories along the open paths generated by the average potential \(xV_{d}.\) The latter are free trajectories that make large displacements along \(\mathbf{V}_{d}.\) The probability \(P(r_{\max},t)\) can be written as the sum of the contributions of these two types of trajectories
\[P(r_{\max},t)=n_{tr}(r_{\max},t)+n_{fr}(r_{\max},t), \tag{6}\]
where \(n_{tr},\)\(n_{fr}\) are determined in the subensembles \(tr\) and \(fr\) at time \(t.\)\(P(r_{\max},t),\) shown in Fig. 3 (left panel) has a second maximum. It appears at a large value of \(r_{\max}\) that increases with the increase of \(V_{d}.\) Also, the amplitude and the width of this peak increase with \(V_{d}.\) It is determined by the free trajectories. The narrow peak \(n_{tr}(r_{\max},t)\) at small \(r_{\max}\) is the contribution of the trapped, periodic trajectories. It is represented in the right panel of Fig. 3, which shows that both the maximum size and the amplitude of the trapped trajectories decrease as \(V_{d}\) increases. The average velocity hinders and eventually eliminates the trapping process. The contribution \(n_{tr}(r_{\max},t)\) in Eq. (6) decreases with \(V_{d}\) and become negligible at \(V_{d}\gg 1.\) The contribution of the free trajectories is negligible in this range of small sizes, at any \(V_{d},\) as shown in Fig. 3 (right panel) where the black points for \(P(r_{\max},t)\) are superposed on the red curves for \(n_{tr}(r_{\max},t).\) The two contributions in Eq. (6) separates at large time.
The probability of the periods of the closed trajectories \(P(T,t)\) calculated from the trajectories \(\mathbf{x}(t)\) at \(t=60\) is shown in Fig. 4. One can see that, at small \(V_{d},\) this probability extends to large values of \(T\lesssim 100\) and it has a weak decay. As \(V_{d}\) increases, the width of \(P(T,t)\) decreases and its decay is steeper. This behavior is in agreement with the decay of the trajectory sizes at large \(V_{d}.\) An average velocity can be defined for the trapped trajectories as the maximum displacement over the period, \(v^{eff}=r_{\max}/T.\) Its probability is weakly dependent on the average velocity.
The fraction of trajectories that are not closed at time \(t,\)\(n_{fr}(t,V_{d})\) is obtained from the probability of the periods of the closed trajectories (calculated at the time of integration, \(t_{\max}\))
\[n_{fr}(t,V_{d})=1-n_{tr}(t,V_{d}),\text{ \ \ }n_{tr}(t,V_{d})=\int_{0}^{t}P(T,t_{ \max})\text{ }dT. \tag{7}\]
This function decreases in time from \(n_{fr}(0,V_{d})=1,\) as seen in Fig. 5 (left panel). In the case \(V_{d}\neq 0,\)\(n_{fr}(t,V_{d})\) saturates at a value \(n_{fr}(V_{d})\) in a time that becomes shorter at larger \(V_{d}\). In the case of \(V_{d}=0,\) the decay is not limited, and it scales as \(n_{fr}(t,0)\sim t^{-0.6}.\)
Figure 2: The probability of trajectory sizes \(P(r_{max},t)\) for \(V_{d}=0\) at \(t=60\) (dashed black line) and \(t=120\) (solid black line). Also shown are the contributions of the trapped (red points) and free (green points) trajectories at \(t=120.\)
The results obtained for the asymptotic fraction of free trajectories \(n_{fr}(V_{d})\equiv\lim\limits_{t\to\infty}n_{fr}(t,V_{d})\), presented in Fig. 5 (right panel), are well approximated by
\[n_{fr}(V_{d})=\left[1-\exp\left(-V_{d}^{2}\right)\right]^{1/4}. \tag{8}\]
The fraction of trapped trajectories is \(n_{tr}(t,V_{d})=1-n_{fr}(t,V_{d})\) at any time, with the asymptotic value \(n_{tr}(V_{d})=1-n_{fr}(V_{d})\) that is also represented in Fig. 5 (right panel).
## IV 4. Lagrangian statistics in static potentials
Thus, the trajectories obtained in the stochastic potential \(\phi_{t}(\mathbf{x})\) were divided into two categories: trapped and free. They have different topologies and different sizes, which suggests that their contributions to the global statistical properties of the trajectories are qualitatively different.
We analyze here the statistics of the Lagrangian velocity and of the displacements of each category of trajectories. For any Lagrangian quantity \(A(\mathbf{x}(t))\), we determine \(\left\langle A(\mathbf{x}(t))\right\rangle_{tr}\) and \(\left\langle A(\mathbf{x}(t))\right\rangle_{fr}\) that are conditional averages restricted to the trapped and free trajectories, respectively. These are statistical averages calculated over the subspaces \(tr\) and \(fr.\) The contribution of each subensemble to the global average (over \(R\)) is the product of the probability that a trajectory belongs to the subensemble multiplied by the statistical average over the subensemble, \(n_{c}(t,V_{d})\left\langle A(\mathbf{x}(t))\right\rangle_{c},\) where \(c=tr,\;fr.\) It yields
Figure 4: The probability of the periods of the trapped trajectories \(P(T,t)\) as functions of \(T\) at \(t=60\) and at the values of \(V_{d}\) that label the curves.
Figure 3: The probability \(P(r_{max},t)\) for several average velocities \(V_{d}\) that label the curves, at \(t=60\) as function of \(r_{max}\) (the curves in the left panel and black points in the right panel) and the contribution of the trapped trajectories \(n_{tr}(r_{max},t)\) (right panel, red lines).
\[\left\langle A({\bf x}(t))\right\rangle=n_{tr}(t,V_{d})\ \left\langle A({\bf x}(t)) \right\rangle_{tr}+n_{fr}(t,V_{d})\ \left\langle A({\bf x}(t))\right\rangle_{fr}. \tag{9}\]
The separation of the trajectories in these categories is performed at a large time such that \(n_{fr}(t,V_{d})\) is saturated (see Fig. 5, left panel).
### 4.1 Statistics of the Lagrangian velocity
The statistical parameters of the Lagrangian velocity \({\bf v}\left({\bf x}(t)\right)\equiv{\bf v}(t)\) are shown in Fig. 6 for a stochastic potential with \(\lambda_{1}=1\), \(\lambda_{2}=2\) and \(V_{d}=0.2.\) The average Eulerian velocity and fluctuation amplitudes are in this case \(\left\langle v_{1}\right\rangle=0,\ \left\langle v_{2}\right\rangle=V_{d}\), \(V_{1}=0.5\) and \(V_{2}=1\), where \(V_{i}=\sqrt{\left\langle\widehat{v}_{i}^{2}\right\rangle}\) are obtained from Eq. (4).
The Lagrangian quantities maintain the Eulerian values at any time, as stated by Lumley theorem. Besides this, the conditional average velocity and fluctuation amplitudes are time invariant, as seen in Fig. 6, but their values depend on the category.
It is interesting to note that the average velocity is determined only by the free trajectories, while the trapped trajectories do not contribute (\(\left\langle v_{2}(t)\right\rangle_{tr}=0\) at any time). The average velocity of the free trajectories is larger than \(V_{d}\), and it can be approximated with
\[\left\langle v_{2}(t)\right\rangle_{fr}=\frac{V_{d}}{n_{fr}}>V_{d} \tag{10}\]
Figure 5: Left panel: the fractions of free trajectories as function of time for the values of \(V_{d}\) that label the curves. Right panel: the asymptotic values \(n_{fr}\) and \(n_{tr}\) and the average velocity of the free trajectories (see next Section) as functions of \(V_{d}\).
Figure 6: The average Lagrangian velocities (left panel) and the fluctuations of the Lagrangian velocities (right panel) as functions of time. The dashed lines are for the \(v_{1}\) and the solid lines for the \(v_{2}\). The green lines are for the free trajectories and the red lines for the trapped trajectories, while the black are averages on the whole statistical ensemble \(R\). \(V_{d}=0.2\).
It is \(\left\langle v_{2}(t)\right\rangle_{fr}=0.45\) for the example presented in Fig. 6, left panel, obtained for \(V_{d}=0.2\). The conditional average velocity \(\left\langle v_{2}(t)\right\rangle_{fr}\) is also shown in Fig. 5 (right panel) as function of \(V_{d}.\) One can see that this average velocity is significantly larger than \(V_{d}\) only in the presence of trajectory trapping (for \(V_{d}\lesssim 1\)).
This result shows that a supplementary ordered component of the Lagrangian velocity appears for the free trajectories that exactly compensates the missing contribution of the trapped particles, such that \(\left\langle v_{2}(t)\right\rangle=n_{fr}\left\langle v_{2}(t)\right\rangle_{ fr}=V_{d}.\) It seems to be a trivial consequence of \(\left\langle v_{2}(t)\right\rangle_{tr}=0,\) but the underlying physical process is rather complex. It essentially consists of generation of ordered motion from the stochastic velocity \(\widetilde{\mathbf{v}}(\mathbf{x},\)\(t)\) for both types of trajectories
\[\left\langle\widetilde{v}_{2}(t)\right\rangle_{tr}=-V_{d},\ \ \left\langle \widetilde{v}_{2}(t)\right\rangle_{fr}=V_{d}\frac{n_{tr}}{n_{fr}}. \tag{11}\]
The supplementary average velocity of the trapped trajectories is opposite to \(\mathbf{V}_{d}\) and exactly compensates it. The supplementary average velocity of the free trajectories is along \(\mathbf{V}_{d}\) and it contributes to the increase of the Lagrangian over the Eulerian velocity.
Equations (11) are valid at any time, including \(t=0.\) It can be interpreted as the condition for the separation of the trajectories in the free and trapped categories. The trapped trajectories start from the geometric locus for which \(\left\langle\widetilde{v}_{2}(\mathbf{x})\right\rangle=-V_{d}\) and they remain in this domain, while the free trajectories are confined in the complement of this domain. These ordered components of the motion are hidden, in the sense that they are not "seen" in the average velocity calculated on the whole ensemble \(R\) (\(\left\langle v_{2}(t)\right\rangle=V_{d}).\) However, as shown below, they have strong effects on the transport along \(\boldsymbol{V}_{d}\) through the modification of the correlation of the Lagrangian velocity.
Figure 8: The correlations of the Lagrangian velocity \(v_{1}(t)\) (left panel) and \(v_{2}(t)\) (right panel). The correlations on the whole statistical ensemble \(L_{i}(t)\) (black lines) are compared to the subensemble correlations \(L_{i}^{tr}(t)\) (red lines) and \(L_{i}^{fr}(t)\) (green lines). \(V_{d}=0.2\).
Figure 7: The amplitudes of fluctuations of the Lagrangian velocities as functions of \(V_{d}\). The dashed lines are for the \(v_{1}\) and the solid lines for the \(v_{2}\). The green lines are for the free trajectories and the red lines for the trapped trajectories, while the black are averages on the whole statistical ensemble \(R.\)
The amplitudes of velocity fluctuations around the average velocity are shown in Fig. 6 (right panel). They are different for the two types of trajectories. It is interesting to underline that the supplementary order that characterizes trapped and free trajectories appears in the fluctuations of the velocity in \(R\). The average of the square velocity decomposed on \(tr\) and \(fr\) subensembles according to Eq. (9) for large time
\[\left\langle v_{i}^{2}(t)\right\rangle=n_{tr}\left\langle v_{i}^{2}(t)\right\rangle _{tr}+n_{fr}\left\langle v_{i}^{2}(t)\right\rangle_{fr}, \tag{12}\]
leads to
\[V_{2}^{2}=n_{tr}(V_{2}^{tr})^{2}+n_{fr}(V_{2}^{fr})^{2}+\frac{n_{tr}}{n_{fr}}V_ {d}^{2}, \tag{13}\]
where
\[(V_{i}^{c})^{2}\equiv\left\langle\left(v_{i}(t)-\left\langle v_{i}(t)\right \rangle_{c}\right)^{2}\right\rangle_{c} \tag{14}\]
are the amplitudes of the fluctuations of the velocity \(\delta v_{i}(t)\equiv v_{i}(t)-\left\langle v_{i}(t)\right\rangle_{c},\)\(i=1,2,\) conditioned by the category of trajectories \(c=tr,\)\(fr\) (on the subensembles \(tr\) and \(fr\)). Thus, a contribution produced by the ordered motion appears (the last term of Eq. (13)) besides the direct contributions of the conditional fluctuations. It is determined by the ordered motion (11) generated by \(V_{d}\) in the presence of trapping (for \(V_{d}\lesssim 1).\) The results presented in Fig. 6 (right panel) show values \(V_{2}^{tr}<V_{2}\) and \(V_{2}^{fr}<V_{2},\) which reproduce Eq. (13).
The conditioned amplitudes of the velocity fluctuations \(V_{i}^{c}\) depend on the average velocity \(V_{d}.\) As seen in Fig. 7, the amplitudes of both components of the trapped trajectory velocity (red curves) are continuously decreasing functions of \(V_{d}\). This is the effect of the decrease of the size of the islands of closed contour lines of the potential, which, as \(V_{d}\) increases, shrink around the maxima and minima of \(\phi(\mathbf{x})\) where the gradients are small. In the case of free trajectories (green lines), the amplitudes of the Lagrangian velocities are different of \(V_{i}\) only in the range of \(V_{d}\) that corresponds to the existence of islands of closed contour lines of the potential. The perpendicular amplitude is increased (\(V_{1}^{fr}>V_{1}),\) while the parallel amplitude is decreased (\(V_{2}^{fr}<V_{2}\)) such that the supplementary parallel velocity is compensated (Eq. (13)).
One can deduce from these results that the EC defined on the geometric locus of the free trajectories is different of the EC (3). As shown below (Section 6), the amplitude of the stochastic potential of the free trajectories \(\Delta\) is smaller than in the whole space (\(\Delta<\Phi\)). The correlation lengths are evaluated using the amplitudes of velocity fluctuations of the free trajectories
\[\lambda_{1}^{fr}\sim\frac{\Delta}{V_{2}^{fr}}=\lambda_{1}\frac{\Delta}{\Phi} \frac{V_{2}}{V_{2}^{fr}},\;\lambda_{2}^{fr}\sim\frac{\Delta}{V_{1}^{fr}}= \lambda_{2}\frac{\Delta}{\Phi}\frac{V_{1}}{V_{1}^{fr}}, \tag{15}\]
where \(\lambda_{1}=\Phi/V_{2}\) and \(\lambda_{2}=\Phi/V_{1}.\) Thus, the correlation lengths on the domain of free trajectories decrease with the factor \(\Delta/\Phi\) on both directions and are modified by the velocity amplitudes (decreased along \(\mathbf{V}_{d}\) and increased across \(\mathbf{V}_{d}\)).
Figure 9: Histograms of the Lagrangian velocities \(v_{1}\) (left panel) and \(v_{2}\) (right panel) represented by the black curves and the contributions determined by the free (green curves) and trapped (red curves) trajectories. \(V_{d}=0.4.\)
The correlations of the Lagrangian velocity are shown in Fig. 8, where the notations are
\[L_{i}(t)=\left\langle\delta v_{i}(0)\ \delta v_{i}(t)\right\rangle,\ L_{i}^{c}(t)= \left\langle\delta v_{i}(0)\ \delta v_{i}(t)\right\rangle_{c},\ \ c=tr,fr. \tag{16}\]
One can see that all the conditional correlations (for both categories and both components of the velocity) decay to zero at large \(t.\) However, the correlation of the velocity along \({\bf V}_{d}\) calculated on all trajectories, \(L_{2}(t),\) has a finite asymptotic value. It is determined by the ordered components of motion produced in subensembles \(tr\) and \(fr.\) An equation similar to (13) can be obtained from (9) written for \(A=v_{i}(0)\ v_{i}(t)\)
\[L_{2}(t)=n_{tr}L_{2}^{tr}(t)+n_{fr}L_{2}^{fr}(t)+\frac{n_{tr}}{n_{fr}}V_{d}^{2}, \tag{17}\]
which shows that \(L_{2}(t)\) has a finite asymptotic tail in spite of the decay to zero of \(L_{2}^{tr}(t)\) and \(L_{2}^{fr}(t).\) It is determined by the presence of trapped trajectories at small average velocity \(V_{d}.\)
The histograms for the Lagrangian velocity components are time invariant for all statistical ensembles \(R,\)\(tr\) and \(fr.\) The histogram for all trajectories (in \(R\)) is shown in Fig. 9 together with the contributions of the trapped and free trajectories (that include the fractions of trajectories). One can see that the distribution is Gaussian in \(R,\) while significant departures from Gaussianity appear in the subensembles \(tr\) and \(fr,\) especially for the velocity component \(v_{2}\) (right panel). The domain of large positive velocities is dominated by the free trajectories, while the trapped trajectories have the main contribution for the large negative \(v_{2}.\) The most probable value of \(v_{2}\) on \(tr\) (that is slightly negative) is compensated by a longer tail at positive \(v_{2}.\)
The non-Gaussian distribution of the Lagrangian velocity of the trapped trajectories provides additional information on the geometric locus of this category of trajectories. It shows that the space average of the parallel velocity \(v_{2}({\bf x})\) on this locus (that is zero for any \(V_{d}\)) results from the elimination of the regions with large, positive \(v_{2}({\bf x}).\) In other words, the regions where the stochastic velocity is oriented parallel to \({\bf V}_{d}\) belong to the geometric locus \(fr.\)
### 4.2 Transport and statistics of trajectories
The statistics of the displacements is strongly non-Gaussian, in spite of the Gaussian Lagrangian velocity. Moreover, the average and mean square displacements (calculated for all trajectories and in the subensembles \(tr\) and \(fr\)) have asymptotic regimes that can be linear, quadratic or saturated functions of time, which shows that the transport has anomalous aspects.
The average displacements are in agreement with the average Lagrangian velocities
\[\left\langle x_{1}(t)\right\rangle = \left\langle x_{1}(t)\right\rangle_{tr}=\left\langle x_{1}(t) \right\rangle_{fr}=0, \tag{18}\] \[\left\langle x_{2}(t)\right\rangle = V_{d}t,\ \left\langle x_{2}(t)\right\rangle_{tr}=0,\ \left\langle x_{2}(t)\right\rangle_{fr}=\frac{V_{d}}{n_{fr}}t.\]
The dispersion \(\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle,\) where \(\delta x_{i}(t)=x_{i}(t)-\left\langle x_{i}(t)\right\rangle\) are shown in Fig. 10 for \(V_{d}=0\) (left panel) and for \(V_{d}=0.2\) (right panel), as functions of time.
In the absence of the average velocity (\(V_{d}=0\)), the dispersions are similar along the two directions. The curves in the left panel of Fig. 10 are only translated due to the different amplitudes of the stochastic velocities \(V_{1}=0.5,\ V_{2}=1.\) The dispersions are sub-diffusive, with time increase that is slower than linear \(\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle\sim t^{0.68}.\) The reason is the progressive saturation of the contributions of the trajectories with small periods. At a time \(t,\) all the trajectories with \(T<t\) have saturated dispersion and only the free trajectories (that are still not closed) determine the time variation of \(\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle.\) The latter results from two factors with opposite effects: the fraction of free trajectories at time \(t\) and their average size. As seen in Fig. 5, left panel, \(n_{fr}(t,0)\) is a decreasing function of time, \(n_{fr}(t,0)\sim t^{-0.6}.\) The average size of the closed trajectories is an increasing function of \(t,\) because it is an increasing function of the average period.
The average velocity \(V_{d}\) makes trajectory dispersion strongly non-isotropic, as seen in Fig. 10, right panel. The dispersion \(\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle\) for all trajectories (black lines) are compared to the results obtained for the trapped \(\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle_{tr}\) (red lines) and free \(\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle_{fr}\) (green lines) trajectories. The dispersions across \({\bf V}_{d}\) (of \(x_{1}(t)\)) for the whole set of trajectories and for the subensembles \(tr\) and \(fr\) are all saturated (the dashed curves in Fig. 10, right panel), which
corresponds to the minimum sub-diffusive transport. This means that the average velocity completely hinders the perpendicular transport in the case of static stochastic potentials. The contrary happens to the transport parallel to \(\mathbf{V}_{d}\): the dispersion of the trajectories has a very fast time-increase, \(\left\langle\left(\delta x_{2}(t)\right)^{2}\right\rangle\sim t^{2}\), which correspond to the maximum super-diffusive transport that is of the ballistic type. It appears in spite of the much weaker transport of the trapped and free trajectories \(\left(\left\langle\left(\delta x_{2}(t)\right)^{2}\right\rangle_{tr}\right.\) saturates and \(\left\langle\left(\delta x_{2}(t)\right)^{2}\right\rangle_{fr}\sim t\) is diffusive).
This super-diffusive parallel transport is the effect of the coherent parallel motion generated by \(V_{d}\), as demonstrated using Eq. (9) for \(A=x_{i}^{2}(t).\) The relations between the dispersion of all trajectories (in \(R\)) and the subensemble \(tr\) and \(fr\) dispersions are
\[\left\langle\left(\delta x_{1}(t)\right)^{2}\right\rangle=n_{tr}\left\langle \left(\delta x_{1}(t)\right)^{2}\right\rangle_{tr}+n_{fr}\left\langle\left( \delta x_{1}(t)\right)^{2}\right\rangle_{fr}, \tag{19}\]
\[\left\langle\left(\delta x_{2}(t)\right)^{2}\right\rangle=n_{tr}\left\langle \left(\delta x_{2}(t)\right)^{2}\right\rangle_{tr}+n_{fr}\left\langle\left( \delta x_{2}(t)\right)^{2}\right\rangle_{fr}+\frac{n_{tr}}{n_{fr}}V_{d}^{2}\ t^{2}. \tag{20}\]
The last term in Eq. (20) is dominant at large time and it makes the asymptotic regime superdiffusive of ballistic type. This term is determined by the supplementary average velocity generated from the stochastic components for the free and trapped trajectories, Eq. (11). It leads to the "concentration" of the average velocity along the free trajectories Eq. (10). Thus, the super-diffusive parallel transport is determined by the average velocity (\(V_{d}\neq 0\)) only in the presence of the islands of trapped trajectories (\(n_{tr}\neq 0\)), which corresponds to \(V_{d}\lesssim 1\).
The dispersions of the trajectories (Fig. 10) are connected to the correlations of the Lagrangian velocity (Fig. 8) and to the time dependent diffusion coefficients, defined by \(2D_{i}(t)=d\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle/dt,\) by Taylor formulas [14]
\[D_{i}(t)=\int_{0}^{t}L_{i}(\tau)\ d\tau,\ \ \left\langle\left(\delta x_{i}(t) \right)^{2}\right\rangle=2\int_{0}^{t}\left(t-\tau\right)\ L_{i}(\tau)\ d\tau. \tag{21}\]
Similar equations can be written for each category of trajectories (trapped or free). Figure 11 presents the time dependent diffusion coefficients compared to their restrictions to the trapped and free trajectories for the \(x_{1}\) (left panel) and \(x_{2}\) (right panel) directions. This confirms that the perpendicular diffusion is completely hindered (even for the free trajectories). The time integral of \(L_{1}(t)\) vanishes for all categories at a finite time. The parallel transport is ballistic \(D_{2}(t)\sim t\), in spite of the normal diffusion of the free trajectories and of the total confinement of the trapped ones. It is the result of the ordered parallel motion, as seen by performing the time derivative in Eq. (20).
The probability of the displacements \(P(\mathbf{x},t)\) is strongly non-Gaussian, as seen in in Fig. 12 (black curves). The contributions of the two categories of trajectories are completely different: the trapped trajectories determine the steep peak in \(\mathbf{x}=\mathbf{0}\), and the free ones have a large Gaussian distribution with the average parallel displacement \(\left\langle x_{2}(t)\right\rangle_{fr}=V_{d}t/n_{fr}.\) We note that the transport, which is essentially produced by the free trajectories, results from a Gaussian distribution.
Figure 10: The dispersions of the trajectories on the whole statistical ensemble (black line) for \(V_{d}=0\) (left panel) and \(V_{d}=0.2\) (right panel) as functions of time. The dashed lines are for \(x_{1}(t)\) and the solid lines for \(x_{2}(t)\). The conditional dispersions for trapped (red) and free (green) trajectories are also shown in the right panel.
## V Coherence induced by an average velocity
The Hamiltonian structure of equation (1) is the origin of the order that characterizes the two-dimensional incompressible turbulence. It determines the strong connection between trajectories and the contour lines of the potential, which are paths of the motion. The order (quasi-coherence) of the motion is essentially represented by the existence of correlations between the potential and the trajectories. They are represented by nonzero average displacements or velocities conditioned by the (initial) potential.
Significant quasi-coherent characteristics of the transport process can be found by analyzing statistical Lagrangian quantities restricted on the contour lines with given potential \(\phi^{0}.\) The trajectories that belong to this class correspond to solutions of Eq. (1) that start (in \(\mathbf{x}(0)=\mathbf{0}\)) from points with \(\phi(\mathbf{0})=\phi^{0}.\) The invariance of the total potential in this class gives
\[\phi_{t}(\mathbf{x}(t))=\phi(\mathbf{x}(t))+x_{1}(t)V_{d}=\phi^{0}. \tag{22}\]
The fractions of trajectories that evolve on the \(\phi^{0}\) potential lines, the average and the amplitude of fluctuations of their displacements and Lagrangian velocities are determined below for each type of trajectories using conditional averages.
The analysis starts from the representation (9) and introduces a supplementary condition for the trajectories, namely that the initial potential is \(\phi^{0}\) [\(\phi(\mathbf{x}(0))=\phi^{0}\)]. Defining the fraction of these trajectories by \(n(\phi^{0})\) and the
Figure 11: The time dependent diffusion coeficients in the direction perpendicular (left panel) and parallel (right panel) to the average velocity for the whole statistical ensemble \(R\) (black lines) and restricted to the \(tr\) (red lines) and \(fr\) (green lines) subensembles. \(V_{d}=0.2.\)
Figure 12: The probabilities of the \(x_{1}\) (left panel) and \(x_{2}\) (right panel) for the whole set of realizations (black lines) compared to the contributions of the free (green lines) and trapped (red lines) trajectories for \(V_{d}=0.2\) and \(t=97.\)
corresponding conditional average by \(\langle\rangle_{\phi^{0}}\), the average \(\langle A({\bf x}(t))\rangle\) is the sum of the contributions from each value \(\phi^{0}\)
\[\langle A({\bf x}(t))\rangle=\int_{-\infty}^{\infty}\left\langle A({\bf x}(t)) \right\rangle_{\phi^{0}}\ P(\phi^{0})\ d\phi^{0}, \tag{23}\]
where \(P\left(\phi^{0}\right)\) is the Gaussian distribution of the (normalized) potential
\[P\left(\phi^{0}\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\left(\phi^{0} \right)^{2}}{2}\right). \tag{24}\]
Similar equations can be written for the contributions of the free and trapped trajectories
\[n_{c}\ \left\langle A({\bf x}(t))\right\rangle_{c}=\int_{-\infty}^{\infty} \left\langle A({\bf x}(t))\right\rangle_{\phi^{0},c}\ n^{c}(\phi^{0})\ d\phi^{0}, \tag{25}\]
where \(n^{c}(\phi^{0})\) is the fraction of trajectories that evolve on contour lines \(\phi^{0}\) and are in the category \(c=tr,\ fr,\) and \(\langle\rangle_{\phi^{0},c}\) is the conditional average taken on the subensemble of these trajectories. \(n^{c}(\phi^{0})\) is related to \(n_{tr},\)\(n_{fr}\) (defined in Section 3)
\[n_{c}=\int_{-\infty}^{\infty}n^{c}(\phi^{0})\ d\phi^{0}. \tag{26}\]
One obtains using Eq. (9)
\[\left\langle A({\bf x}(t))\right\rangle_{\phi^{0}}P(\phi^{0})=\left\langle A( {\bf x}(t))\right\rangle_{\phi^{0},tr}\ n^{tr}(\phi^{0})+\ \left\langle A({\bf x}(t))\right\rangle_{\phi^{0},fr}\ n^{fr}(\phi^{0}), \tag{27}\]
which connects the contribution of all trajectories \(\left\langle A\right\rangle_{\phi^{0}}P(\phi^{0})\) to the contributions of each category \(\left\langle A\right\rangle_{\phi^{0},c}n^{c}(\phi^{0}).\)
The fractions of trajectories fulfil the equation
\[P(\phi^{0})=n^{tr}(\phi^{0})+n^{fr}(\phi^{0}), \tag{28}\]
which is obtained from Eq. (27) for \(A=1.\) The numerical results obtained for \(n^{fr}(\phi^{0})\) and \(n^{tr}(\phi^{0})\) (represented by points), are compared to analytical approximations (solid lines) in Fig. 13. One can see that the fraction of trajectories that evolve on \(\phi^{0}\) contour lines (black line and points in Fig. 13) reproduces Eq. (24). The fraction of the free trajectories is narrower, but it is still Gaussian. We have found, as seen in Fig. 13 (green curve), a good approximation of the data by
\[n^{fr}(\phi^{0})=n_{fr}\ G(\phi^{0};\Delta), \tag{29}\]
Figure 13: The fraction of the trajectories that evolve on the contour lines with potential \(\phi^{0}\) for the free (green points), trapped (red points) and for all trajectories (black points) obtained from the numerical simulations compared to \(P(\phi^{0})\) (solid black line), with Eq. (29) (solid green line) and Eq. (31) (solid red line).
where \(G(\phi^{0};\Delta)\) is the Gaussian distribution
\[G(\phi^{0};\Delta)=\frac{1}{\sqrt{2\pi}\Delta}\exp\left(-\frac{\left(\phi^{0} \right)^{2}}{2\Delta^{2}}\right) \tag{30}\]
with a width \(\Delta\) that depends on the average velocity \(V_{d}.\) The fraction of the trapped trajectories (red curve in Fig. 13), which according to Eq. (28) is
\[n^{tr}(\phi^{0})=P(\phi^{0})-n_{fr}\ G(\phi^{0};\Delta), \tag{31}\]
provides a good representation of the numerical results (red points).
The width \(\Delta\) as function of the average velocity \(V_{d}\) is shown in Fig. 14 together with the fraction of free trajectories \(n_{fr}.\) The numerical results for \(\Delta(V_{d})\) (diamonds) are well approximated by
\[\Delta(V_{d})=\left[1-\exp\left(-V_{d}^{2}\right)\right]^{0.17}. \tag{32}\]
Both functions saturate at large \(V_{d}\) (\(V_{d}>V_{1},V_{2}\)), and they have power law dependence for small \(V_{d}\)
\[n_{fr}\sim V_{d}^{0.5},\ \Delta\sim V_{d}^{0.34}. \tag{33}\]
The asymptotic value \(n_{fr}\to 1\) for \(V_{d}\rightarrow\infty\) corresponds to the complete elimination of the islands of close contour lines of the potential (\(n_{tr}=0\)). The limit \(\Delta\to 1\) confirms that all trajectories are free, because \(G(\phi^{0};1)=P(\phi^{0})\), where \(P(\phi^{0})\) is the probability of the Eulerian potential.
Thus, the free trajectories are localized on the contour lines with small values of \(\left|\phi^{0}\right|\lesssim\Delta.\) The potential on the geometrical locus of free trajectories is Gaussian with an amplitude that is smaller than in the whole space.
The trapped (periodic) trajectories mainly have large \(\left|\phi^{0}\right|\) : they completely occupy the range of large potential \(\left|\phi^{0}\right|\gg\Delta\), but also have significant presence at small potential \(\left|\phi^{0}\right|\lesssim\Delta\) that correspond to free trajectories.
The average displacements conditioned by the value of the initial potential and by the category of the trajectories are shown in Fig. 15 for free (green), trapped (red) and all (black) trajectories. The perpendicular (left panel) and the parallel (right panel) displacements are shown at a large time \(t=97\), larger than the saturation time of \(n_{fr}(t,V_{d})\) (seen in Fig. 5, left panel). These represent quasi-coherent components of the motion, and appear only in the presence of an average velocity. One can see that the average conditional displacements are small for the trapped trajectories (red points), and that significant values appear for the free trajectories in both directions (green points). As shown in Fig. 15, these quantities can be approximated by
\[\left\langle x_{1}(t)\right\rangle_{\phi^{0},fr}=\frac{\phi^{0}}{V_{d}},\ \ \left\langle x_{1}(t)\right\rangle_{\phi^{0},tr}\cong 0, \tag{34}\]
Figure 14: The width \(\Delta\) of the initial potential of the free trajectories (diamonds) and the fraction of free trajectories (circles) as functions of the average velocity \(V_{d}\). The numerical results are interpolated by Eqs. (32) and (8)
\[\left\langle x_{2}(t)\right\rangle_{\phi^{0},fr}=\frac{V_{d}t}{n_{fr}},\ \ \left\langle x_{2}(t)\right\rangle_{\phi^{0},tr}=0, \tag{35}\]
which are represented by red and green lines respectively. The black lines have the equations
\[\left\langle x_{1}(t)\right\rangle_{\phi^{0}}=\frac{\phi^{0}}{V_{d}}F(\phi^{0} ),\ \ \left\langle x_{2}(t)\right\rangle_{\phi^{0}}=\frac{V_{d}t}{n_{fr}}F(\phi^{0}), \tag{36}\]
where
\[F(\phi^{0})=\frac{n^{fr}(\phi^{0})}{P(\phi^{0})}=\frac{n_{fr}}{\Delta}\exp \left(-\frac{\left(\phi^{0}\right)^{2}}{2}\frac{1-\Delta^{2}}{\Delta^{2}}\right). \tag{37}\]
They result from Eq. (27) with \(A=x_{i}(t)\) using (34) and (35), and provide, as seen in Fig. 15, good approximations of the data for \(\left\langle x_{i}(t)\right\rangle_{\phi^{0}}\) (black points).
The contributions to the average displacements, obtained by multiplying the conditional averages with the corresponding fractions of trajectories (\(P(\phi^{0})\), \(n^{fr}(\phi^{0})\) or \(n^{tr}(\phi^{0})\)), are shown in Fig. 16. It appears more clearly that the coherent displacements are produced only by the free trajectories. The black points for all trajectories are practically superposed on the green points and they are well approximated by the green lines that represent the contributions of
Figure 16: The contributions of the trapped (red points) and free (green points) trajectories, to the average displacements (black points) along \(x_{1}\) (left panel) and \(x_{2}\) (right panel) directions as functions of \(\phi^{0}\) for the values of \(V_{d}\) label the curves. The approximations for the free trajectories are represented by the green lines.
Figure 15: The conditional average displacements along \(x_{1}\) axis (left panel) and along \(x_{2}\) axis (right panel) as functions of \(\phi^{0}\) for the trapped (red points), free (green points) and all (black points) trajectories, compared to the approximations (34)-(35) (green lines) and (36) (black lines). \(V_{d}=0.3\).
the free trajectories (\(\phi^{0}/V_{d}\;n^{fr}(\phi^{0})\) in the left panel and \(V_{d}\;t/n_{fr}\;n^{fr}(\phi^{0})\) in the right panel). The dependence on \(\phi^{0}\) is different across and along the average velocity \(\mathbf{V}_{d}.\) In the first case, it is an anti-symmetrical function of \(\phi^{0}\) that saturates in time, and, in the second case, it is a symmetrical Gaussian function that increases linearly in time.
The parallel average displacement on the contour lines with initial potential \(\phi^{0},\) which increases linearly with \(t\) (35), leads to an average Lagrangian velocity
\[\left\langle v_{2}(t)\right\rangle_{\phi^{0},fr}=\frac{V_{d}}{n_{fr}}. \tag{38}\]
It is important to note that this velocity does not depend on \(\phi^{0},\) and it equals the average velocity (10). The contribution of the conditional average velocity is determined only by the free trajectories since \(\left\langle v_{2}(t)\right\rangle_{\phi^{0},tr}=0.\) The perpendicular average displacement of the free trajectories also determines an average velocity, but it is transitory since \(\left\langle x_{1}(t)\right\rangle_{\phi^{0},fr}\) saturates in time.
The dispersion of the trajectories conditioned by the value of the initial potential and by the category are shown in Fig. 17 for free (green), trapped (red) and all (black) trajectories, in the perpendicular (left panel) and parallel (right panel) directions. One can see that the dispersion of the free trajectories along both directions are not dependent on the initial potential \(\phi^{0},\) and can be approximated by
\[\left\langle\delta x_{1}^{2}\right\rangle_{\phi^{0},fr}=\frac{\Delta^{2}}{V_{ d}^{2}},\;\;\left\langle\delta x_{2}^{2}\right\rangle_{\phi^{0},fr}=2\;D_{2}^{fr}t \tag{39}\]
represented by the green lines. On the contrary, the trapped trajectories have dispersions that decay with the increase of \(\phi^{0}\) (the red points). The dispersion of the trajectories conditioned by the potential is obtained using Eq. (27) for \(A=x_{i}^{2}(t)\)
\[\left\langle\delta x_{i}^{2}\right\rangle_{\phi^{0}}=\left\langle\delta x_{1} ^{2}\right\rangle_{\phi^{0},fr}F+\left\langle\delta x_{1}^{2}\right\rangle_{ \phi^{0},tr}(1-F)+\left\langle x_{i}(t)\right\rangle_{\phi^{0},fr}^{2}F(1-F), \tag{40}\]
which depends on \(\phi^{0}\) as seen in Fig. 17 (black points).
Thus, the analysis of the Lagrangian statistics conditioned by the initial potential reveals a coherent component of motion perpendicular to \(\mathbf{V}_{d}\). It consists of average displacements that have the sign correlated with the sign of \(\phi^{0}.\) They appear for the free trajectories and are hidden in the sense that the contributions of all contour lines (with all values of \(\phi^{0}\)) mix to zero. The amplitude of the ordered motion is defined by the displacements conditioned by the sign of \(\phi^{0},\)\(\left\langle x_{1}(t)\right\rangle_{+}\) obtained by integration over \(\phi^{0}\) on the interval \([0,\infty)\) and \(\left\langle x_{1}(t)\right\rangle_{-}\) that is the integral over \((-\infty,0].\) These are symmetrical quantities since \(\left\langle x_{1}(t)\right\rangle=\left\langle x_{1}(t)\right\rangle_{+}+ \left\langle x_{1}(t)\right\rangle_{-}=0.\) The time derivatives of these functions determine a pair of average velocities with opposite directions that exactly compensate each other (the hidden drifts, HDs) that are oriented across \(\mathbf{V}_{d}\). The HDs were first found in [15] using an approximate theoretical approach, the decorrelation trajectory method [16]. In the presence of components of the motion that introduce a
Figure 17: The conditional trajectory dispersion along \(x_{1}\) axis (left panel) and along \(x_{2}\) axis (right panel) as functions of \(\phi^{0}\) for the trapped (red points), free (green points) and all (black points) trajectories, compared to the approximations (39) (green lines). \(V_{d}=0.3\) and \(t=97.\)
small compressibility, an average velocity can be generated by breaking the equilibrium of the HDs. Such effects were found in magnetically confined plasmas [17]-[19]. The HDs are transitory in frozen potentials because the average displacements saturate \(\left\langle x_{1}(t)\right\rangle_{\phi^{0},fr}\rightarrow\phi^{0}/V_{d}.\)
The analysis also shows that the parallel motion is similar on the contour lines with different \(\phi^{0},\) and that it depends only on the category of trajectories. The asymptotic value \(\left\langle x_{1}(t)\right\rangle_{\phi^{0},fr}\rightarrow\phi^{0}/V_{d}\) represents the centrum of the space domain that contains the trajectories, which start from points on the line \(x_{1}=0\) where the potential is \(\phi^{0}.\) It is limited by the lines \(x_{1}^{-}=(\phi^{0}-\Delta)/V_{d},\)\(x_{1}^{+}=(\phi^{0}+\Delta)/V_{d},\) and has infinite dimension along \({\bf V}_{d}.\) Reported at this centrum, the free trajectories are statistically identical for all values of the initial potential \(\phi^{0}.\)
## VI 6. Lagrangian statistics in time dependent potentials
The general conclusion of the analysis of trajectory statistics in frozen potentials is the existence of a high degree of coherence, which reflects the structure of the contour lines of \(\phi_{t}({\bf x})\) on which the trajectories are bounded.
The time-dependence of the potential determines the variation of the Lagrangian potential and the decorrelation from its initial value \(\phi^{0}\). It is expected to strengthen the random aspects of the trajectories and to cause the elimination of the Lagrangian coherence in a time of the order of the decorrelation time \(\tau_{c}\). More precisely, the random time-variation of the potential should vanish the averages and correlations conditioned by \(\phi^{0},\) which show the existence of hidden order. It is thus expected that the order found in static potential is in this case only a transitory processes with life-time \(\tau_{c}.\)
The trajectories are more complex than in static potentials. Closed periodic trajectories do not exist in time dependent potentials, but trapping events represented by almost closed eddying segments appear on all trajectories when the decorrelation time \(\tau_{c}\) is large compared to the time of flight \(\tau_{fl}=\lambda/V,\)\((\lambda=(\lambda_{1}^{2}+\lambda_{2}^{2})^{1/2}\) and \(V=(V_{1}^{2}+V_{2}^{2})^{1/2}),\) and the integration time is much longer than \(\tau_{c}.\) The trapping events are separated by long jumps, which are similar with the free trajectories.
The separation of the trajectories in the categories \(c=tr,\)\(fr\) has no meaning in time-dependent potentials. However one can define related quantities that are not properties of the trajectories but of the contour lines of the potential. The latter are geometric objects. The fraction of free/trapped trajectories can be defined using the number of trajectories that stay on open/closed contour lines of the potential at time \(t.\) These fractions do not depend on time for stationary stochastic potentials, because the amplitude, the space correlation and the structure of the contour lines are statistically time-invariant. They equal the asymptotic values of the time dependent fractions \(n_{c}(t,V_{d})\) obtained in static potentials from the trajectories (Sections 3)
\[n_{c}(V_{d})=\ \lim_{t\rightarrow\infty}\ n_{c}(t,V_{d}), \tag{41}\]
for any \(\tau_{c}\) and \(c=tr,fr.\) In a similar way, the fraction of trajectories that stay at time \(t\) on contour lines of category \(c\) with the potential \(\phi^{t}=\phi({\bf x}(t))\) is a time-independent function of \(\phi^{t}\) and \(c,\) which is the asymptotic value \(n^{c}(\phi^{t})\) of the fractions obtained in static potential (\(\tau_{c}\rightarrow\infty\)) in Eqs. (29-31). The physical meaning of these quantities will be clarified after analyzing the significant modifications of the statistics of trajectories produced by the time-variation of the potential.
We underline that, in time-dependent potentials, one can define the statistics conditioned by the initial potential, but not by the category.
Our aim is to see if the special order determined by the average velocity survives at finite \(\tau_{c}.\) We analyze here the statistics on the whole set of trajectories (in \(R\)), while, in the next section, the statistics conditioned by the initial potential will be used for understanding the direct effects of time variation on the coherent elements found here.
The time variation of \(\phi\) represents a decorrelation mechanism, because it determines the stochastic change of the Lagrangian velocity, which vanishes the Lagrangian correlations \(L_{i}(t)\) at times \(t\gg\tau_{c}.\) Usually, this produces the saturation of the time dependent diffusion coefficients \(D_{i}(t)\to D^{i}\) and diffusive transport with \(\left\langle\delta x_{i}^{2}(t)\right\rangle\to 2D^{i}t\) (as obtained from Eq. (21)). The memory of the initial velocity is lost in a time \(\tau_{c},\) which means that the displacements at large time \(t\gg\tau_{c}\) are sequences of non-correlated random steps that yield a Gaussian distribution.
We show that this general behavior is not at all observed in the presence of \({\bf V}_{d}\) at large correlation times (weak time variation). A strong non-standard influence appears both on the transport and on the probability of the displacements.
The Lagrangian velocity is, as expected, Gaussian at any time and for any \(\tau_{c},\) as in the static case. The time variation influences only its correlation.
The dispersions of the trajectories \(\left<\delta x_{i}^{2}(t)\right>\) and the probabilities \(P(x_{i},t)\) for a typical case that illustrates the effects of weka time variation of the potential are shown in Figs. 18 and 19, by the black lines for \(V_{d}=0.3\) and \(\tau_{c}=33\). We also present, for comparison, two examples with \(V_{d}=0.3\) (for the static case \(\tau_{c}=\infty\) (red lines) and for fast time variation \(\tau_{c}=3.3\) (blue)), and two examples with \(V_{d}=0\) (for \(\tau_{c}=\infty\) (green) and \(\tau_{c}=33\) (cyan)).
When \(V_{d}=0\), the subdiffusive transport at \(\tau_{c}=\infty\) with \(\left<\delta x_{i}^{2}(t)\right>\sim t^{0.68}\) is transformed into normal transport (\(\left<\delta x_{i}^{2}(t)\right>\to 2D^{i}t\)) at large \(\tau_{c}\) (Fig. 18, green and cyan curves). The process appears for all finite values of \(\tau_{c}\), which only influence the diffusion coefficients \(D^{i}.\) However, the probabilities of displacements are Gaussian only for fast time variation. In the static case, \(P(x_{i},t)\) has a steep peak in \(x_{i}=0,\) which corresponds to trapped trajectories, superposed on a large Gaussian component, which yields from the trajectories that are not closed at time \(t.\) The steep peak is flattened by time variation and the probabilities have extended exponential shapes at large \(\tau_{c}.\) When \(\tau_{c}\) decreases, \(P(x_{i},t)\) evolves to Gaussian distribution, which is attained when \(\tau_{c}<\tau_{fl}.\)
The average velocity makes the transport strongly anisotropic. In frozen potential \(\tau_{c}=\infty,\) the transport is ballistic in the parallel direction and saturated perpendicular to \(\mathbf{V}_{d},\) as discusses in Section 4.2 and also shown in Figs. 18 and 19 (red curves). The normal transport and the Gaussian probability are reached only for fast time variation of the potential (\(\tau_{c}<\tau_{fl}\)), as seen in the example for \(\tau_{c}=3.3\) (blue curves). In these conditions, the motion along the contour lines of \(\phi(\mathbf{x},t)\) is completely hindered, which means that the quasi-coherent components are eliminated.
Compared to these cases, the trajectory statistics at slow time variation in the presence of \(V_{d}\) (the black curves) is strongly anomalous with complex behavior. Trajectory dispersion at large time has nonstandard time-dependence in both directions
\[\left<\delta x_{1}^{2}(t)\right>\sim t^{\alpha_{1}},\ \ \left<\delta x_{2}^{2}(t) \right>\sim t^{\alpha_{2}},\]
where \(\alpha_{1}<1\) and \(\alpha_{2}>1,\) which corresponds to subdiffusive perpendicular transport (but not saturated as in the static case) and superdiffusive parallel transport (but not of ballistic type). These powers are functions of \(\tau_{c}\) and \(V_{d},\)\(\alpha_{i}(\tau_{c},V_{d}).\) When \(\tau_{c}\) decreases, \(\alpha_{1}\) increases and saturates \(\alpha_{1}\to 1,\) while \(\alpha_{2}\) decreases and saturates \(\alpha_{2}\to 1.\) A similar effect is determined by the increase of \(V_{d},\) which leads to normal transport at \(V_{d}\gtrsim 1.\) In the example presented in Fig. 18 (black curves), \(\alpha_{1}(33,0.3)=0.57,\)\(\alpha_{2}(33,0.3)=1.35.\) The probabilities are very large, especially in the parallel direction, and the peak in \(\mathbf{x}=0\) persists for very long time (\(t=300=6\tau_{c}\) in Fig. 19).
Thus, the transport and the statistics of displacements are non-standard when \(V_{d}\lesssim 1\) and \(\tau_{c}>\tau_{fl}.\) In these conditions the structure of the contour lines of the potential shows island of closed lines in a network of open lines. Also, the trajectories approximately follow the contour lines for distances of the order of the correlation length before they are removed by the time variation of the potential.
Figure 18: Comparision of the dispersions of the trajectories in time-dependent potential with the static cases along \(x_{1}\) (dashed lines) and \(x_{2}\) (solid lines) directions. The values of \(V_{d}\) and \(\tau_{c}\) label the curves.
## VII 7. Enhanced Coherence and Long Memory Effects
### 7.1 Hidden ordered motion
Ordered motion conditioned by the initial potential \(\phi^{0}\) was found in the presence of an average velocity \(V_{d}\) for the free trajectories. It is represented by the average displacements \(\left\langle x_{i}(t)\right\rangle_{\phi^{0},fr}\) that are conditioned by the initial potential and by the category \(c=fr\) [Eq. (36)]. These quantities obtained in a time-dependent potential (with \(\tau_{c}=33\)) are shown in Fig. (20) for \(x_{1}\) and in Fig. (21) for \(x_{2}\), compared to the static case. Significant differences appear for both directions.
One can see in Fig. (20, left panel) that the perpendicular displacements \(\left\langle x_{1}(t)\right\rangle_{\phi^{0}}\) are larger in time-depending potential, although the calculations are at a very large time, \(t=6\tau_{c}\) (where the EC of the potential (3) is negligible, \(E(t)=10^{-8}\)). The main contribution comes from large values of the potential \(\left|\phi^{0}\right|,\) which is negligible in the static potential. The amplitude of the ordered motion is represented by the average displacements conditioned by the sign of \(\phi^{0},\)\(\left\langle x_{1}(t)\right\rangle_{+}\) and \(\left\langle x_{1}(t)\right\rangle_{-},\) which determine by time derivative the HDs. Surprisingly, the time evolution of these quantities shows a continuous increase in time-dependent potential, while it saturates in the static case, as seen in Fig. (20, right panel) for \(\left\langle x_{1}(t)\right\rangle_{+}\). This means that the hidden drifts are transitory in static potentials, but they are long-life statistical quantities in time dependent potential. Their amplitude decays on a long time scale, much longer than the decorrelation time of the potential.
The time variation of the potential modifies the parallel displacements \(\left\langle x_{2}(t)\right\rangle_{\phi^{0}}\) by determining the extension of
Figure 19: The probabilities of the displacement along \(x_{1}\) (left panel) and \(x_{2}\) (right panel) directions, show the effect of the time variation of the potential for typical cases with the values of \(V_{d}\) and \(\tau_{c}\) that label the curves.
Figure 20: Ordered perpendicular displacements \(\left\langle x_{1}(t)\right\rangle_{\phi^{0}}\) as functions of \(\phi^{0}\) at \(t=200\) (left panel) and \(\left\langle x_{1}(t)\right\rangle_{+}\) as function of time (right panel) for a time-dependent potential with \(\tau_{c}=33\) (black points) compared to the static case (dashed blue lines)
the contribution on the whole range of \(\phi^{0}\) and the dependence on these quantities of the initial potential (Fig. (21, left panel). It is peaked on \(\phi^{0}=0\) and has a weak (algebraic) decay at large \(\left|\phi^{0}\right|,\) instead of the concentration on the domain of free trajectories with uniform average displacement Eq. (35). However, the average Lagrangian velocity is uniform on the whole range of \(\phi^{0}\) and it has the Eulerian value \(V_{d},\) as seen in Fig. (21, right panel). The process of concentration of the Lagrangian average velocity on the domain of free trajectories found in static potentials is eliminated by the time-variation.
The fluctuations of the trajectories \(\left\langle\delta x_{i}^{2}(t)\right\rangle_{\phi^{0}}\) and the transport \(\left\langle\delta x_{i}(t)\delta v_{i}(t)\right\rangle_{\phi^{0}}\) conditioned by the initial potential are all asymptotically uniform on the whole domain of \(\phi^{0}\). They reach this stage in a long time compared to the correlation time of the potential (\(t\gg\tau_{c}\)) staring from the values corresponding to static potential that are maintained at small time \(t<\tau_{c}.\)
These results show that the trajectories are statistically identical with the exception of the perpendicular average displacement \(\left\langle x_{i}(t)\right\rangle_{\phi^{0}},\) which depend on \(\phi^{0}.\)
### 7.2 Long memory
The persistent Lagrangian order and the non-standard characteristics of the trajectories in the time dependent case can be understood by analyzing the statistics of the Lagrangian potential \(\phi(t)\equiv\phi(\mathbf{x}(t),t).\)
The distribution of the Lagrangian potential has the same invariance property as the Lagrangian velocity. It has the Gaussian probability of the Eulerian potential at any time \(t,\) for both the static and the time-dependent cases, at any value of the average velocity \(V_{d}.\) However, significant differences appear between these cases concerning the correlation and the average conditioned by the initial potential, as seen in Figs. (22) and 23.
The correlation of the Lagrangian potential \(L_{\phi}(t)=\left\langle\phi(0)\right\rangle\)\(\phi(t)\) is far from the Eulerian time-correlation, \(E(\mathbf{0},t).\) Starting from the trivial case of static potential with \(V_{d}=0,\) where the invariance of the potential implies \(L_{\phi}(t)=E(\mathbf{0})=1,\) in all the other cases shown in Fig. (22), the Lagrangian correlation is stronger than the Eulerian one, as it has a long tail with much slower decay. It demonstrates that the Lagrangian potential has a long-time memory.
The memory effect is strongest in almost static potentials (very large \(\tau_{c}\)) with average velocity \(V_{d}=0.\) The Lagrangian correlation decreases much slower than the Eulerian one, and it is larger than than \(E(\mathbf{0},t)\) at any time (Fig. 22, the curve for \(V_{d}=0,\ \tau_{c}=33\)). In this example, at \(t=200\cong 6\tau_{c},\)\(L_{\phi}\) decreases only at \(0.4,\) while \(E(\mathbf{0},t)=1.5\)\(10^{-8}.\)
The average velocity (\(V_{d}\neq 0\)) determines a faster decrease of \(L_{\phi}(t)\) at small time that leads to smaller values compared to the case \(V_{d}=0\) (Fig. 22, the curve for \(V_{d}=0.3,\ \tau_{c}=33\)). The decorrelation takes place on two time-scales. There is a fast decay at small time that is followed by a slab decrease of \(L_{\phi}(t).\) The fast decay is the same for \(\tau_{c}=\infty\) and \(\tau_{c}=33\) at \(V_{d}=0.3,\) which shows that this process is not a consequence of the potential time variation, but rather of the presence of \(V_{d}.\)
In the static case, the memory of the Lagrangian potential is infinite (\(L_{\phi}(t)\) saturates). The asymptotic value is positive and it is a decreasing function of \(V_{d}\). The time-dependence of \(L_{\phi}(t)\) is the result of a selective decorrelation mechanism determined by the average velocity. This process can be understood by analyzing the correlation of \(\phi(t)\)
conditioned by the category \(c=tr,\;fr\) in the static potential (\(\tau_{c}=\infty\)). As seen in Fig. 23, left panel, \(\left\langle\phi(0)\;\phi(t)\right\rangle_{c}\) decays to zero for the free trajectories, while it saturates for the trapped trajectories at a value that is comparable to the conditioned amplitude \(\left\langle\phi^{2}(0)\right\rangle_{tr}\). This demonstrates that in static potential the decorrelation affects only the free trajectories and that the memory effect is determined by the trapped trajectories, which approximately maintain the initial potential \(\phi(0)\). The asymptotic value of the average Lagrangian potential is thus
\[\left\langle\phi(0)\;\phi(t)\right\rangle=\left\langle\phi(0)\;\phi(t)\right \rangle_{tr}n_{tr}=\phi^{0}n_{tr}. \tag{42}\]
It is interesting to note that at finite \(\tau_{c}\), the significant decrease of the correction appears at any time, although it seems that the process determined by \(V_{d}\) is transitory in static potentials. This is caused by the interaction of the effects of \(V_{d}\) with the influence produced by the time variation, which determine the two time-scale evolution of \(L_{\phi}(t)\). It is clearly evidenced by the average Lagrangian potential conditioned by the initial value \(\phi^{0}\) normalized by this value, \(\left\langle\phi(t)\right\rangle_{\phi^{0}}/\phi^{0}\) represented in Fig. 23, right panel for the static case (lines) and for \(\tau_{c}=33\) (points) for several values of \(\phi_{0}\).
An important property of the Lagrangian potential in slow time-dependent potential is that its correlation and the average conditioned by \(\phi^{0}\) have the same time-decay (as seen in Fig. 22 for the case \(V_{d}=0.3\), \(\tau_{c}=33\) and in the right panel of Fig. 23, all the curves have the same behavior \(\exp\left(-t/88\right)\)).
The long memory of the potential and the increase of the average displacements \(\left\langle x_{1}(t)\right\rangle_{\phi^{0}}\) (Fig. 20) are the result of the same process. It consists of the liberation of the trapped trajectories with large \(\left|\phi^{0}\right|\) followed by repeated
Figure 23: Caracterization of the memory of the Lagrangian potential. Left: The correlations conditioned by the category \(<\phi(0)\phi(t)>_{c}\) for \(V_{d}=0.1\) (dotted), \(V_{d}=0.2\) (dashed), \(V_{d}=0.3\) (continuous). Right: The normalized average potential conditioned by the initial potential for \(V_{d}=0.3\), for \(\tau_{c}=\infty\) (continuous), \(\tau_{c}=33\) (dotted), and the values of \(\phi_{0}\) that label the curves.
Figure 22: Correlation of the Lagrangian potential for \(V_{d}=0\) (dashed lines) and \(V_{d}=0.3\) (continuous lines) for static (\(\tau_{c}=\infty\)), slow (\(\tau_{c}=33\)) and fast time-variation (\(\tau_{c}=3.3\)), compared to the Eulerian correlations (dotted blue lines). Long tails with exponential decay appear in time-dependent potentials with large \(\tau_{c}\).
stochastic events of capture and release combined with the constraint of the total potential invariance (22) that approximately holds for small time intervals. Considering the case of the peaks of the potential, the liberation of the trapped trajectories with large \(\phi^{0}\) is produced when the time variation determines the decrease of the potential to \(\Delta\). The contour lines of the potential that are open have the average perpendicular displacement \(\Delta/V_{d}\) and the average potential along them equal to zero (as imposed by Eq. (22)). The stochastic recapture is uniformly distributed over the potential and has the average perpendicular location \(\Delta/V_{d}\). This cancels asymptotically the average of the potentials on the trapping events and leads to the average \(\Delta/V_{d}\) of the positions of the trapping events. This happens on a time scale that is much larger than \(\tau_{c}\). These released trajectories with large \(\phi^{0}\) determine the slow decay of their initial average potential and the increase of their average displacement from zero to the largest possible value \(\Delta/V_{d}\). Thus, the memory of the Lagrangian potential and the strengthening of the coherence of the trajectories are both determined by the slow evolution toward uniform distribution of the trapping events on the trajectories caused by the time-variation of the potential.
## VIII Summary and conclusions
A detailed study of the Lagrangian coherence of the trajectories in 2-dimensional incompressible velocity fields is presented. The strong order that appear in this kind of velocity fields is determined by the Hamiltonian structure of the advection equations (1), (2) and by the space correlation of the stochastic potential. The trajectories follow the contour lines of the potential in the static case, and, for slowly varying potentials, they remain close to them for long time intervals. This study is focused on the identification and understanding of the order generated by an average velocity \(V_{d}\) superposed on the stochastic field. It determines an average potential, which strongly modifies the structure of contour lines of the total potential \(\phi_{t}({\bf x},t)=\phi({\bf x},t)+x_{1}Vd\) by generation of a network of open lines between islands of closed lines. As a result, two categories of trajectories are found in static (frozen) potential: trapped (closed, periodic) and free (with unlimited displacement in the parallel direction to \({\bf V}_{d}\)).
The results presented here are based on the numerical simulation of the trajectories and on a complex statistical analysis that includes conditional averages and correlations connected to the topology of the contour lines of the potential \(\phi_{t}.\) The statistics of displacements and of the Lagrangian velocity are determined for the whole ensemble \(R,\) for the categories trapped (\(tr\)) and free (\(fr\)), and also on sets of contour lines of the potential conditioned by the value \(\phi^{0}\) at the starting point of the trajectories. This analysis reveals the origin of coherence and provides explanations for the nonstandard statistics, transport and memory of this motion.
In the case of frozen potentials, we have found that the statistical properties determined for the two categories \(tr,\)\(fr\) are completely different compared to those obtained in the whole space \(R\). The average velocity \(V_{d}\) generates coherence in the Lagrangian velocity, which acquires average components from he stochastic ones for both categories. The supplementary coherent velocity cancels the average velocity \(V_{d}\) of the trapped trajectories, and it determines larger velocity for the free trajectories that compensate the missing contribution of the trapped ones (\(\left<v_{2}(t)\right>_{fr}=V_{d}/n_{fr}\)). Thus, the statistical invariance of the Lagrangian velocity (Lumley theorem) is ensured in a rather non-trivial manner that involves hidden coherent parallel velocity.
The statistical analysis conditioned by the initial potential \(\phi^{0}\) reveals additional important aspects. The free trajectories have Gaussian distribution of \(\phi^{0}\) with a width \(\Delta\) that is smaller than the dispersion of the Eulerian potential. It shows the existence of ordered perpendicular motion (average displacements across \({\bf V}_{d}\) (34)) that appear for the free trajectories and are proportional with \(\phi^{0}.\) These averages \(\left<x_{1}(t)\right>_{\phi^{0},fr}\) increase in time from zero and saturate at \(\phi^{0}/V_{d}.\) They generate the hidden drifts, a pair of transitory average velocities perpendicular on \({\bf V}_{d}\) conditioned by the sign of \(\phi^{0},\) which have opposite directions and exactly compensate each other. We have also found that the Lagrangian statistics of the free trajectories conditioned by \(\phi^{0}\) depends on the value \(\phi^{0}\) only through the average perpendicular displacement. This means that the trajectories in the category \(fr\) for different values of \(\phi^{0}\) are statistically identical, and are organized in strips with limited perpendicular extensions.
The probability of the displacements is non-Gaussian with separated contributions of the categories: a steep peak for \(tr\) and Gaussian distribution that move along \({\bf V}_{d},\) but with larger velocity \(V_{d}/n_{fr}\) for the \(fr\) subensemble. The time-invariant Gaussian distribution of the Lagrangian velocity is the sum of the non-Gaussian contributions of the two categories, which are both non-Gaussian, but invariant.
The transport is produced only by the free trajectories. A paradoxical behavior was found: the statistics of the trajectories is strongly non-Gaussian, but the transport is produced by Gaussian trajectories, which in fact yield from the non-Gaussian velocity distribution of the free trajectories. The transport is anomalous, subdiffusive across \({\bf V}_{d}\) and superdiffusive of ballistic type along \({\bf V}_{d}.\) The latter results from the ordered parallel motion. category.
The free trajectories can be associated to a geometrical locus on the two-dimensional space \((x_{1},x_{2}),\) in the sense
that each point is the initial condition for such trajectory and all trajectories are confined in this domain \(fr\). The complement of \(fr\) is the geometric locus of the trapped trajectories \(tr\), which is composed of the islands of closed contour lines of the potentials. The (Eulerian) statistical characteristics of each geometrical locus were identified.
The time-dependence of the stochastic potential produces anomalous increase of the Lagrangian coherence, instead of the expected decay after the decorrelation time. In particular, the perpendicular average displacements conditioned by \(\phi^{0}\) significantly increase and the transitory hidden drifts become long-life structure that survive at \(t\gg\tau_{c}.\) The enhanced coherence is found to be associated to a long memory of the Lagrangian potential. Also, the trajectories conditioned by the initial potential become statistically identical for all values of \(\phi^{0},\) not only on the domain of small potential with width \(\Delta,\) as in frozen potentials. These effects are caused by the stochastic liberation by the time-variation of the potential of the trajectories that initially are trapped, followed by repeated stochastic captures that are constraint by of the approximate invariance of the total potential.
| この論文では、2次元非 compressible 乱流における速度の平均存在下でトラッカー統計における連続的な効果を重点的に分析しています。輸送統計と軌跡統計に強い修正をもたらす、これらの効果の基礎は、動きの隠れた連続的な成分によって引き起こされることが明らかになります。 |
2307.08623 | HYTREL: Hypergraph-enhanced Tabular Data Representation Learning | Language models pretrained on large collections of tabular data have
demonstrated their effectiveness in several downstream tasks. However, many of
these models do not take into account the row/column permutation invariances,
hierarchical structure, etc. that exist in tabular data. To alleviate these
limitations, we propose HYTREL, a tabular language model, that captures the
permutation invariances and three more structural properties of tabular data by
using hypergraphs - where the table cells make up the nodes and the cells
occurring jointly together in each row, column, and the entire table are used
to form three different types of hyperedges. We show that HYTREL is maximally
invariant under certain conditions for tabular data, i.e., two tables obtain
the same representations via HYTREL iff the two tables are identical up to
permutations. Our empirical results demonstrate that HYTREL consistently
outperforms other competitive baselines on four downstream tasks with minimal
pretraining, illustrating the advantages of incorporating the inductive biases
associated with tabular data into the representations. Finally, our qualitative
analyses showcase that HYTREL can assimilate the table structures to generate
robust representations for the cells, rows, columns, and the entire table. | Pei Chen, Soumajyoti Sarkar, Leonard Lausen, Balasubramaniam Srinivasan, Sheng Zha, Ruihong Huang, George Karypis | 2023-07-14T05:41:22 | http://arxiv.org/abs/2307.08623v2 | # HyTrel: Hypergraph-enhanced
###### Abstract
Language models pretrained on large collections of tabular data have demonstrated their effectiveness in several downstream tasks. However, many of these models do not take into account the row/column permutation invariances, hierarchical structure, etc. that exist in tabular data. To alleviate these limitations, we propose **HyTrel**, a tabular language model, that captures the permutation invariances and three more _structural properties_ of tabular data by using hypergraphs-where the table cells make up the nodes and the cells occurring jointly together in each row, column, and the entire table are used to form three different types of hyperedges. We show that HyTrel is maximally invariant under certain conditions for tabular data, i.e., two tables obtain the same representations via HyTrel_iff_ the two tables are identical up to permutations. Our empirical results demonstrate that HyTrel**consistently** outperforms other competitive baselines on four downstream tasks with minimal pretraining, illustrating the advantages of incorporating the inductive biases associated with tabular data into the representations. Finally, our qualitative analyses showcase that HyTrel can assimilate the table structures to generate robust representations for the cells, rows, columns, and the entire table. 1
Footnote 1: Code, data, and checkpoints will be available soon.
## 1 Introduction
Tabular data that is organized in bi-dimensional matrices are widespread in webpages, documents, and databases. Understanding tables can benefit many tasks such as table type classification, table similarity matching, and knowledge extraction from tables (e.g., column annotations) among others. Inspired by the success of pretrained language models in natural language tasks, recent studies (Yin et al., 2020; Yang et al., 2022) proposed Tabular Language Models (TaLMs) that perform pretraining on tables via self-supervision to generate expressive representations of tables for downstream tasks.
Among the TaLMs, many works (Herzig et al., 2020; Yin et al., 2020; Deng et al., 2020; Iida et al., 2021) serialize tables to a sequence of tokens for leveraging existing pretrained language model checkpoints and textual self-supervised objectives like the Masked Language Modeling. However, due to the linearization of tables to strings, these models do not explicitly incorporate the structural properties of a table, e.g., the invariances to arbitrary permutations of rows and columns (independently). Our work focuses on obtaining representations of tables that take table structures into account. We hypothesize that incorporating such properties into the table representations will benefit many downstream table understanding tasks.
**Motivation:** Tabular data is structurally different in comparison to other data modalities such as images, audio, and plain texts. We summarize four _structural properties_ present in the tables below:
* Most tables are invariant to row/column permutations. This means, in general, if we arbitrarily (and independently) permute the rows or columns of a table, it is still an equivalent table. For other tables with an explicit ordering of rows or columns, we can make them permutation invariant by appropriately adding a ranking index as a new column or row.
* Data from a single column are structurally similar-for example, they oftentimes have the same semantic types. Similarly, the cells within a single row are not independent of each other, and they usually describe the different aspects of one object.
* The interactions within cells/rows/columns are necessarily not pairwise, i.e., the cells within the same row/column, and rows/columns from the same table can have high-order multilateral relations.
* Information in tables is generally organized in a hierarchical fashion where the information at the table-level can be aggregated from the column/row-level, and further from the cell-level.
However, the linearization-based approaches are not designed to explicitly capture most of the above properties. We aim to address the limitation by modeling all the aforementioned structural properties as inductive biases while learning the table representations.
**Our Approach:** In line with recent studies (Deng et al., 2020; Yang et al., 2022; Wang et al., 2021) which have elucidated upon the importance of the structure of a table, we propose the HyTrel that uses hypergraphs to model the tabular data. We propose a modeling paradigm that aims _capture all of the four properties_ directly. Figure 1 provides an example of how a hypergraph is constructed from a table. As observed, converting a table into a hypergraph allows us to incorporate the first two properties inherent to the nature of hypergraphs. Hypergraphs seamlessly allow the model to incorporate row/column permutation invariances, as well as interactions among the cells within the same column or row. Moreover, the proposed hypergraph structure can capture the high-order (not just pairwise) interactions for the cells in a column or a row, as well as from the whole table, and an aggregation of hyperedges can also help preserve the hierarchical structure of a table.
**Contributions:** Our theoretical analysis and empirical results demonstrate the advantages of modeling the four structural properties. We first show that HyTrel is maximally invariant when modeling tabular data (under certain conditions), i.e. if two tables get the same representations via the hypergraph table learning function, then the tables differ only by row/column permutation (independently) actions and vice versa. Empirically, we pretrain HyTrel on publicly available tables using two self-supervised objectives: a table content based ELECTRA\({}^{2}\) objective (Clark et al., 2020; Iida et al., 2021) and a table structure dependent contrastive objective (Wei et al., 2022). The evaluation of the pretrained HyTrel model on four downstream tasks (two knowledge extraction tasks, a table type detection task, and a table similarity prediction task) shows that HyTrel can achieve state-of-the-art performance.
We also provide an extensive qualitative analysis of HyTrel-including visualizations that showcase that (a) HyTrel representations are robust to arbitrary permutations of rows and columns (independently), (b) HyTrel can incorporate the hierarchical table structure into the representations, (c) HyTrel can achieve close to state-of-the-art performance even without pretraining, and the model is
Figure 1: An example of modeling a table as a hypergraph. Cells make up the nodes and the cells in each row, column, and the entire table form hyperedges. The table caption and the header names are used for the names of the table and column hyperedges. The hypergraph keeps the four structural properties of tables, e.g., the invariance property of the table as the row/column permutations result in the same hypergraph.
extremely efficient with respect to the number epochs for pretraining in comparison to prior works, further demonstrating the advantages of HyTel in modeling the structural properties of tabular data. In Appendix B, we provide additional analysis that demonstrates HyTrel's ability to handle input tables of arbitrary size and underscore the importance of the independent row/column permutations.
## 2 HyTrel Model
Formally, a table in our work is represented as \(\mathcal{T}=[M,H,R]\), where \(M\) is the caption, \(H=[h_{1},h_{2},h_{3},...,h_{m}]\) are the \(m\) column headers, \(R\) represents the \(n\) rows \([R_{1},R_{2},R_{3},...,R_{n}]\). Each row \(R_{i}\) has \(m\) cells \([c_{i1},c_{i2},c_{i3},...,c_{im}]\). The caption, header, and cell values can be regarded as sentences that contain several words. We note that each cell \(c_{ij}\) or header \(h_{i}\) also belongs to the corresponding column \(C_{j}\). We use \(C=[C_{1},C_{2},C_{3},...,C_{m}]\) to represent all the columns that include the headers, so a table can also be defined as \(\mathcal{T}=[M,C]\).
### Formatter & Embedding Layer
The formatter transforms a table into a hypergraph. As shown in Figure 1, given a table \(\mathcal{T}\), we construct a corresponding hypergraph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}\), \(\mathcal{E}\) denote the set of nodes and hyperedges respectively. We treat each cell \(c_{ij}\) as a node \(v_{ij}\in\mathcal{V}\), and each row \(R_{i}\), each column \(C_{j}\), and the entire table \(\mathcal{T}\) as hyperedges \(e^{e}_{i},e^{r}_{j},e^{t}\in\mathcal{E}_{\{1\leq i\leq n,1\leq j\leq m\}}\), respectively. As a part of our hypergraph construction, each cell node \(v_{ij}\) is connected to 3 hyperedges: its column hyperedge \(e^{c}_{i}\), row hyperedge \(e^{r}_{j}\), and the table hyperedge \(e^{t}\). The hypergraph can be conveniently be represented as an incidence matrix \(\mathbf{B}\in\{0,1\}^{mn\times(m+n+1)}\), where \(\mathbf{B}_{ij}=1\) when node \(i\) belong to hyperedge \(j\) and \(\mathbf{B}_{ij}=0\) otherwise.
An embedding layer is then employed over the nodes and hyperedges. Each node \(v_{ij}\) corresponds to a cell \(c_{ij}\) that has several tokens, and we obtain the feature vector \(\mathbf{X}_{v_{ij},:}\in\mathbb{R}^{F}\) for a given node by feeding its constituent cell tokens into the embedding layer and then averaging the embeddings over the tokens. After obtaining node embeddings \(\mathbf{X}\in\mathbb{R}^{nm\times F}\) for all nodes, a similar transformation is applied over hyperedges. For different hyperedge types, we use different table content for their initialization : for a column hyperedge \(e^{c}_{i}\), we use all the tokens from the corresponding header \(h_{i}\). For the table hyperedge \(e^{t}\), we use the the entire caption associated with \(M\). For the row hyperedge
Figure 2: The overall framework of HyTrel. We first turn a table \(\mathcal{T}\) into a hypergraph \(\mathcal{G}\) and then initialize the embeddings of the nodes \(\mathcal{V}\) and hyperedges \(\mathcal{E}\). After that, we encode the hypergraph using stacked multiple-layer hypergraph-structure-aware transformers (HyperTrans). Each HyperTrans layer has two attention blocks that work on hypergraph (HyperAtt) and one hyperedge fusion block. Lastly, we use the node and hyperedge representations from the final layer for pretraining and fine-tuning.
when no semantic information is available, we randomly initialize them as \(\mathbf{S}_{e_{:}^{r},:}\in\mathbb{R}^{F}\). Performing the above operations yields an initialization for all the hyperedge embeddings \(\mathbf{S}\in\mathbb{R}^{(m+n+1)\times F}\).
### Hypergraph Encoder
After embedding, we propose to use a structure-aware transformer module (HyperTrans) to encode the hypergraphs. HyperTrans encoder can encode the table content, structure, and relations among the table elements (including cells, headers, captions, etc.). As shown in Figure 2, one layer of the HyperTrans module is composed of two hypergraph attention blocks (HyperAtt, \(f\)) [12] that interact with the node and hyperedge representations, and one Hyperedge Fusion block. The first HyperAtt block is the Node2Hyperedge attention block as defined below:
\[\mathbf{\tilde{S}}_{e,:}^{(t+1)}=f_{\mathcal{V}\rightarrow\mathcal{E}}\left(K_ {e,\mathbf{X}^{(t)}}\right) \tag{1}\]
Where \(f_{\mathcal{V}\rightarrow\mathcal{E}}\) is a hypergraph attention function defined from nodes to hyperedges. \(K_{e,\mathbf{X}}=\{\mathbf{X}_{v,:}\ v\in e\}\) denotes the sets of hidden node representations included in the hyperedge \(e\). The Node2Hyperedge block will aggregate information to hyperedge \(e\) from its constituent nodes \(v\in e\).
We then use a Hyperedge Fusion module (a Multilayer Perceptron Network, \(\mathrm{MLP}\)) to propagate the hyperedge information from the last step, as defined below:
\[\mathbf{S}_{e,:}^{(t+1)}=\mathrm{MLP}\left(\mathbf{S}_{e,:}^{(t)},\mathbf{ \tilde{S}}_{e,:}^{(t+1)}\right) \tag{2}\]
A second HyperAtt block Hyperedge2Node then aggregates information from a hyperedge to its constituent nodes as follows:
\[\mathbf{X}_{v,:}^{(t+1)}=f_{\mathcal{E}\rightarrow\mathcal{V}}\left(L_{v, \mathbf{S}^{(t+1)}}\right) \tag{3}\]
Where \(f_{\mathcal{E}\rightarrow\mathcal{V}}\) is another hypergraph attention function defined from hyperedges to nodes. \(L_{v,\mathbf{S}}=\{\mathbf{S}_{e,:}\ v\in e\}\) is defined as the sets of hidden representations of hyperedges that contain the node \(v\).
As for the HyperAtt block \(f\), similar to transformer [25], it is composed of one multi-head attention, one Position-wise Feed-Forward Network (FFN), two-layer normalization (\(\mathrm{LN}\)) [11] and two skip connections [11], as in Figure 2. However, we do not use the self-attention [25] mechanism from the transformer model because it is not designed to keep the invariance structure of tables or hypergraphs. Inspired by the deep set models [10, 12], we use a set attention mechanism that can keep the permutation invariance of a table. We define HyperAtt \(f\) as follows:
\[f_{\mathcal{V}\rightarrow\mathcal{E}\ \mathtt{or}\ \mathcal{E} \rightarrow\mathcal{V}}(\mathbf{I}):=\mathrm{LN}(\mathbf{Y}+\mathrm{FFN}( \mathbf{Y})) \tag{4}\]
Where \(\mathbf{I}\) is the input node or hyperedge representations. The intermediate representations \(\mathbf{Y}\) is obtained by:
\[\mathbf{Y}=\mathrm{LN}\left(\mathbf{\omega}+\mathrm{SetMH}(\mathbf{\omega}, \mathbf{I},\mathbf{I})\right) \tag{5}\]
Where \(\mathrm{SetMH}\) is the multi-head set attention mechanism defined as:
\[\mathrm{SetMH}(\mathbf{\omega},\mathbf{I},\mathbf{I})=\|_{i=1}^{h}\mathbf{O}_ {i} \tag{6}\]
and
\[\mathbf{O}_{i}=\mathrm{Softmax}\left(\mathbf{\omega}_{i}\left(\mathbf{I} \mathbf{W}_{i}^{K}\right)^{T}\right)\left(\mathbf{I}\mathbf{W}_{i}^{V}\right) \tag{7}\]
Where \(\mathbf{\omega}\) is a learnable weight vector as the query and \(\mathbf{\omega}:=\|_{i=1}^{h}\mathbf{\omega}_{i}\), \(\mathbf{W}_{i}^{K}\) and \(\mathbf{W}_{i}^{V}\) are the weights for the key and value projections, \(\|\) means concatenation.
So the HyperTrans module will update node and hyperedge representations alternatively. This mechanism enforces the table cells to interact with the columns, rows, and the table itself. Similar to BERT\({}_{base}\)[13] and TaBERT\({}_{base}\)[25], we stack \(12\) layers of HyperTrans.
### Invariances of the HyTrel Model
Let \(\phi:\mathcal{T}\mapsto\mathbf{z}\in\mathbb{R}^{d}\) be our target function which captures the desired row/column permutation invariances of tables (say for tables of size \(n\times m\)). Rather than working on the table \(\mathcal{T}\) directly, the proposed HyTrel model works on a hypergraph (via Eqns (1-5)) that has an incidence matrix \(\mathbf{B}\) of size \(mn\times(m+n+1)\). Correspondingly, we shall refer to HyTrel as a function \(g:\mathbf{B}\mapsto\mathbf{y}\in\mathbb{R}^{k}\).
In this section we will make the connections between the properties of the two functions \(\phi\) and \(g\), demonstrating a maximal invariance between the two-as a result of which we prove that our HyTrel can also preserve the permutation invariances of the tables. First, we list our assumptions and resultant properties of tabular data. Subsequently, we present the maximal invariance property of \(\phi\) and our hypergraph-based learning framework \(g\). As a part of our notation, we use \([n]\) to denote \(\{1,2,\ldots,n\}\). Preliminaries and all detailed proofs are presented in the Appendix A.1 and A.2 respectively.
**Assumption 2.1**.: For any table \((\mathcal{T}_{ij})_{i\in[n],j\in[m]}\) (where \(i,j\) are indexes of the rows, columns), an arbitrary group action \(a\in\mathbb{S}_{n}\times\mathbb{S}_{m}\) acting appropriately on the rows and columns leaves the target random variables associated with tasks on the entire table unchanged.
This assumption is valid in most real-world tables-as reordering the columns and the rows in the table oftentimes doesn't alter the properties associated with the entire table (e.g. name of the table, etc). As noted earlier, for tables with an explicit ordering of rows or columns, we can make them permutation invariant by adding a ranking index as a new column or row appropriately. To model this assumption, we state a property required for functions acting on tables next.
_Property 1_.: A function \(\phi:\mathcal{T}\mapsto\mathbf{z}\in R^{d}\) which satisfies Assumption 2.1 and defined over tabular data must be invariant to actions from the (direct) product group \(\mathbb{S}_{n}\times\mathbb{S}_{m}\) acting appropriately on the table i.e. \(\phi(a\cdot\mathcal{T})=\phi(\mathcal{T})\ \ \forall a\in\mathbb{S}_{n}\times \mathbb{S}_{m}\).
However, HyTrel (or the function \(g\) via hypergraph modeling) through Eqns (1-5)) models invariances of the associated incidence matrix to the product group \(\mathbb{S}_{mn}\times\mathbb{S}_{m+n+1}\) (proof presented in the appendix). To make the connection between the two, we present the maximal invariance property of our proposed HyTrel model.
**Theorem 2.2**.: _A continuous function \(\phi:\mathcal{T}\mapsto\mathbf{z}\in\mathbb{R}^{d}\) over tables is maximally invariant when modeled as a function \(g:\mathbf{B}\mapsto\mathbf{y}\in\mathbb{R}^{k}\) over the incidence matrix of a hypergraph \(\mathcal{G}\) constructed per Section 2.1 (Where \(g\) is defined via Eqns (1-5)) if \(\exists\) a bijective map between the space of tables and incidence matrices (defined over appropriate sizes of tables, incidence matrices). That is, \(\phi(\mathcal{T}_{1})=\phi(\mathcal{T}_{2})\) iff \(\mathcal{T}_{2}\) is some combination of row and/or column permutation of \(\mathcal{T}_{1}\) and \(g(\mathbf{B}_{1})=g(\mathbf{B}_{2})\) where \(\mathbf{B}_{1},\mathbf{B}_{2}\) are the corresponding (hypergraph) incidence matrices of tables \(\mathcal{T}_{1},\mathcal{T}_{2}\)._
_Proof Sketch_: Detailed proof is provided in Appendix A.2. The above theorem uses Lemma 1 from (Tyshkevich and Zverovich, 1996) and applies the Weisfeiler-Lehman test of isomorphism over the star expansion graphs of the hypergraphs toward proving the same.
As a consequence of Theorem 2.2, two tables identical to permutations will obtain the same representation, which has been shown to improve generalization performance (Lyle et al., 2020).
### Pretraining Heads
**ELECTRA Head**: In the ELECTRA pretraining setting, we first corrupt a partial of cells and headers from a table and then predict whether a given cell or header has been corrupted or not Iida et al. (2021). Cross-entropy loss is used to train the binary classification head.
**Contrastive Head**: In the contrastive pretraining setting, we randomly corrupt a table-transformed hypergraph by masking a portion of the connections between nodes and hyperedges, as inspired by the hypergraph contrastive learning (Wei et al., 2022). For each hypergraph, we corrupt two augmented views and use them as the positive pair, and use the remaining in-batch pairs as negative pairs. Following this, we contrast the table and column representations from the corresponding hyperedges. The InfoNCE (van den Oord et al., 2018) objective is used for optimization as in 8.
\[loss=-\log\frac{\exp\left(\mathbf{q}\cdot\mathbf{k}_{+}/\tau\right)}{\sum_{i=0}^{K} \exp\left(\mathbf{q}\cdot\mathbf{k}_{i}/\tau\right)} \tag{8}\]
where \((\mathbf{q},\mathbf{k}_{+})\) is the positive pair, and \(\tau\) is a temperature hyperparameter.
Experiments
### Pre-training
**Data** In line with previous TaLMs (Yin et al., 2020; Iida et al., 2021), we use tables from Wikipedia and Common Crawl for pretraining. We utilize preprocessing tools provided by Yin et al. (2020) and collect a total of 27 million tables (1% are sampled and used for validation).3 During pretraining, we truncate large tables and retain a maximum of 30 rows and 20 columns for each table, with a maximum of 64 tokens for captions, column names, and cell values. It is important to note that the truncation is solely for efficiency purposes and it does not affect HyTrel's ability to deal with large tables, as elaborated in appendix B.1.
Footnote 3: As the version of Wikipedia used by (Yin et al., 2020) is not available now, we use an updated version so we collect slightly more tables than previous TaLMs.
**Settings** With the ELECTRA pretraining objective, we randomly replace 15% of the cells or headers of an input table with values that are sampled from all the pretraining tables based on their frequency, as recommended by Iida et al. (2021). With the contrastive pretraining objective, we corrupted 30% of the connections between nodes and hyperedges for each table to create one augmented view. The temperature \(\tau\) is set as 0.007. For both objectives, we pretrain the HyTrel models for 5 epochs. More details can be found the Appendix C.1.
### Fine-tuning4
Footnote 4: More details about experimental settings, the datasets, and the baselines can be found the Appendix C.2
After pretraining, we use the HyTrel model as a table encoder to fine-tune downstream table-related tasks. In order to demonstrate that our model does not heavily rely on pretraining or on previous pretrained language models, we also fine-tune the randomly initialized HyTrel model for comparison. In this section, we introduce the evaluation tasks and the datasets. We choose the following four tasks that rely solely on the table representations since we want to test the task-agnostic representation power of our model and avoid training separate encoders for texts (e.g., questions in table QA tasks) or decoders for generations. As mentioned, our encoder can be used in all these scenarios and we leave its evaluation in other table-related tasks as future work.
**Column Type Annotation** (CTA) task aims to annotate the semantic types of a column and is an important task in table understanding which can help many knowledge discovery tasks such as entity recognition and entity linking. We use the column representations from the final layer of HyTrel with their corresponding hyperedge representations for making predictions. We evaluate HyTrel on the TURL-CTA dataset constructed by Deng et al. (2020).
**Column Property Annotation** (CPA) task aims to map column pairs from a table to relations in knowledge graphs. It is an important task aimed at extracting structured knowledge from tables. We use the dataset TURL-CPA constructed by Deng et al. (2020) for evaluation.
**Table Type Detection** (TTD) task aims to annotate the semantic type of a table based on its content. We construct a dataset using a subset from the public WDC Schema.org Table Corpus.
**Table Similarity Prediction** (TSP) task aims at predicting the semantic similarity between tables and then classifying a table pair as similar or dissimilar. We use the PMC dataset proposed by Habibi et al. (2020) for evaluation.
### Baselines
**TaBERT**(Yin et al., 2020) is a representative TaLM that flattens the tables into sequences and jointly learns representations for sentences and tables by pretraining the model from the BERT checkpoints. _K=1_ and _K=3_ are the two variants based on the number of rows used.
**TURL**(Deng et al., 2020) is another representative TaLM that also flattens the tables into sequences and pretrains from TinyBERT (Jiao et al., 2020) checkpoints. It introduces a vision matrix to incorporate table structure into the representations.
**Doduo**(Suhara et al., 2022) is a state-of-the-art column annotation system that fine-tunes the BERT and uses table serialization to incorporate table content.
### Main Results
The results are presented in Tables 1 and 2. Overall, HyTrel can consistently outperform the baselines and achieve superior performance. A salient observation is that our model (even without pretraining) can achieve close to state-of-the-art performance. In comparison, we notice that the performance slumps significantly for TaBERT without pretraining. This phenomenon empirically demonstrates the advantages of modeling the table structures as hypergraphs over the other methods that we compare.
Additionally, we observe that the two pretraining objectives help different tasks in different ways. For the CTA, CPA, and TTD tasks, the two objectives can help HyTrel further improve its performance. In general, the ELECTRA objective performs better than the contrastive objective. These results are also in line with the representation analysis in Section 4.2 where we observe that the ELECTRA objective tends to learn table structure better than the contrastive objective. However, for the TSP task, we observe that the contrastive objective can help the HyTrel model while the ELECTRA objective fails to bring any improvement. One possible reason for the ineffectiveness of the ELECTRA objective could be its inability to transfer well across domains. HyTrel pretrained with tables from Wikipedia and Common Crawl could not transfer well to the medical domain PMC dataset. As for the improvement observed from the contrastive objective, the reason could be that contrastive learning that uses similarity metrics in the objective function can naturally help the similarity prediction task.
**Scalability:** As stated in Section 3, we have truncated large tables during pretraining. However, this truncation does not hinder the ability of HyTrel to handle large table inputs in downstream tasks. In Appendix B.1, we present additional experiments demonstrating that: (a) HyTrel can effectively
\begin{table}
\begin{tabular}{l|c|c} \hline \hline Systems & Column Type Annotation & Column Property Annotation \\ \hline Sherlock & 88.40 / 70.55 / 78.47 & - \\ BERT\({}_{base}\) & - & 91.18 / 90.69 / 90.94 \\ TURL + metadata & 92.75 / 92.63 / 92.69 & 92.90 / 93.80 / 93.35 \\ Doduo + metadata & 93.25 / 92.34 / 92.79 & 91.20 / 94.50 / 92.82 \\ \hline TaBERT\({}_{base}\)_(K=1)_ & 91.40\({}_{\pm 0.06}\) / 89.49\({}_{\pm 0.21}\) / 90.43\({}_{\pm 0.11}\) & 92.31\({}_{\pm 0.24}\) / 90.42\({}_{\pm 0.53}\) / 91.36\({}_{\pm 0.30}\) \\ _w/o_ Pretrain & 90.00\({}_{\pm 0.14}\) / 85.50\({}_{\pm 0.09}\) / 87.70\({}_{\pm 0.10}\) & 89.74\({}_{\pm 0.40}\) / 68.74\({}_{\pm 0.93}\) / 77.84\({}_{\pm 0.64}\) \\ TaBERT\({}_{base}\)_(K=3)_ & 91.63\({}_{\pm 0.21}\) / 91.12\({}_{\pm 0.25}\) / 91.37\({}_{\pm 0.08}\) & 92.49\({}_{\pm 0.18}\) / 92.49\({}_{\pm 0.22}\) / 92.49\({}_{\pm 0.10}\) \\ _w/o_ Pretrain & 90.77\({}_{\pm 0.11}\) / 87.23\({}_{\pm 0.22}\) / 88.97\({}_{\pm 0.12}\) & 90.10\({}_{\pm 0.17}\) / 84.83\({}_{\pm 0.89}\) / 87.38\({}_{\pm 0.48}\) \\ \hline HyTrel _w/o_ Pretrain & **92.92\({}_{\pm 0.11}\)** / 92.50\({}_{\pm 0.10}\) / 92.71\({}_{\pm 0.08}\) & 92.85\({}_{\pm 0.35}\) / 91.50\({}_{\pm 0.54}\) / 92.17\({}_{\pm 0.38}\) \\ HyTrel _w/_ ELECTRA & 92.85\({}_{\pm 0.21}\) / **94.21\({}_{\pm 0.09}\)** / **93.53\({}_{\pm 0.10}\)** & 92.88\({}_{\pm 0.24}\) / **94.07\({}_{\pm 0.27}\)** / **93.48\({}_{\pm 0.12}\)** \\ HyTrel _w/_ Contrastive & 92.71\({}_{\pm 0.20}\) / 93.24\({}_{\pm 0.08}\) / 92.97\({}_{\pm 0.13}\) & **93.01\({}_{\pm 0.40}\)** / 93.16\({}_{\pm 0.40}\) / 93.09\({}_{\pm 0.17}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Test results on the CTA and CPA tasks (Precision/Recall/F1 Scores,%). The results of TaBERT and HyTrel are from the average of 5 system runs with different random seeds. For fair comparisons, we use the results of TURL and Doduo with metadata, i.e., captions and headers.
\begin{table}
\begin{tabular}{l|c|c c} \hline \hline Systems & Table Type Detection & Table Similarity Prediction \\ \cline{2-4} & Accuracy & Precision/Recall/F1 & Accuracy \\ \hline TFIDF+Glove+MLP & - & 87.36 / 83.81 / 84.47 & 85.06 \\ TabSim & - & 88.65 / 85.45 / 86.13 & 87.05 \\ \hline TaBERT\({}_{base}\)_(K=1)_ & 93.11\({}_{\pm 0.31}\) & 87.04\({}_{\pm 0.64}\) / 85.34\({}_{\pm 0.93}\) / 86.18\({}_{\pm 1.13}\) & 87.35\({}_{\pm 1.42}\) \\ _w/o_ Pretrain & 85.04\({}_{\pm 0.41}\) & 33.61\({}_{\pm 12.70}\) / 50.31\({}_{\pm 12.75}\) / 40.30\({}_{\pm 12.03}\) & 63.45\({}_{\pm 1.011}\) \\ TaBERT\({}_{base}\)_(K=3)_ & 95.15\({}_{\pm 0.14}\) & 87.76\({}_{\pm 0.64}\) / 86.97\({}_{\pm 0.59}\) / 87.36\({}_{\pm 0.95}\) & 88.29\({}_{\pm 0.98}\) \\ _w/o_ Pretrain & 89.88\({}_{\pm 0.26}\) & 82.96\({}_{\pm 1.84}\) / 81.16\({}_{\pm 1.45}\) / 82.05\({}_{\pm 1.02}\) & 82.57\({}_{\pm 1.20}\) \\ \hline HyTrel w/o_ Pretrain & 93.84\({}_{\pm 0.17}\) & 88.94\({}_{\pm 1.83}\) / 85.72\({}_{\pm 1.52}\) / 87.30\({}_{\pm 1.02}\) & 88.38\({}_{\pm 1.43}\) \\ HyTrel w/ ELECTRA & **95.81\({}_{\pm 0.19}\)** & 87.35\({}_{\pm 0.42}\) / 87.29\({}_{\pm 0.48}\) / 87.32\({}_{\pm 0.50}\) & 88.29\({}_{\pm 0.49}\) \\ HyTrel w/ Contrastive & 94.52\({}_{\pm 0.30}\) & **89.41\({}_{\pm 0.58}\)** / **89.10\({}_{\pm 0.90}\)** / **89.26\({}_{\pm 0.53}\)** & **90.12\({}_{\pm 0.49}\)** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Test results on the TTD (Accuracy Score,%) and TSP (Precision/Recall/F1 Scores,%) tasks. The results of TaBERT and HyTrel are from the average of 5 system runs with different random seeds.
process tables of any size as inputs, and (b) down-sampling can be a favorable strategy when working with large input tables, significantly improving efficiency without compromising performance.
## 4 Qualitative Analysis
### HyTrel Learns Permutation Robust Representations
We sample 5,000 tables from the validation data for analysis. We analyze the impact of applying different permutations to a table, including permuting rows, columns, and both rows/columns independently.
Toward our analysis, we measure the Euclidean distance (L2 Norm) of the representations (cells, rows, columns and tables). As shown in Figure 3, the distance is almost always 0 for the HyTrel model because of its explicit invariance-preserving property. On the other hand, for the TaBERT model, the distance is not trivial. We observe that when more rows (K=3) are enabled, the value of the L2 norm increases as we introduce different permutations. Moreover, we also observe that permuting the columns has a greater impact on the L2 norm than the row permutations. A combination of rows and columns permutations has the largest impact among all three actions. Note that when K=1 with just one row, the effect of row permutation is disabled for TaBERT.
### HyTrel Learns the Underlying Hierarchical Table Structure
Next, we demonstrate that the HyTrel model has learned the hierarchical table structure into its representations, as we target at. We use t-SNE (Van der Maaten and Hinton, 2008) for the visualization of different table elements from the same 5,000 validation tables, as shown in Figure 4.
We observe that with random initializations, different table elements cannot be distinguished properly. After the encoding of the randomly initialized HyTrel model, we start to observe noticeable differences for different table elements in the visualization space. Notably, the individual cell representations start to concentrate together and can be distinguished from high-level table elements (tables, columns, and rows) which occupy their separate places in the space of representations. We also notice that, by pretraining the HyTrel with the ELECTRA objective, all table elements can be well separated, showing that it incorporates the hierarchical table structure into its latent representations. As for the contrastive pretraining, we see that it can distinguish columns from rows as
Figure 4: t-SNE visualization of the representations learned by HyTrel. ‘tab’, ‘col’, ‘row’, and ‘cell’ are the representations for different table elements: tables, columns, rows, and cells.
Figure 3: Average distance between table element representations before and after permutations. The HyTrel is immune to the permutations while the TaBERT is sensitive to them.
compared with randomly initialized HyTrel, but could not to well separate the table representations in comparison with the ELECTRA pretraining. This also partially explains the better performance of the ELECTRA pretraining in the CTA, CPA and TTD tasks in contrast to the contrastive pretraining.
### HyTrel Demonstrates Effective Pretraining by Capturing the Table Structures
Our evaluation shows that the HyTrel model can perform well without pretraining, which demonstrates its training efficiency by modeling table structures. Here we further analyze how much pretraining is required for HyTrel to further enhance its performance, as compared with the baseline model. We plot the validation performance of the tasks evaluated at different pretraining checkpoints in Figure 5.
Overall, we can observe that the performance of HyTrel improves drastically during the first several pretraining epochs, and then saturates at about 5 epochs. With the minimal pretraining, HyTrel can outperform the fully pretrained TaBERT models. This demonstrates that our model does not require extensive pretraining to further improve its performance in contrast with previous TaLMs (e.g., TURL for 100 epochs, TaBERT for 10 epochs). Besides, we also observe from the curves that the ELECTRA objective consistently outperforms the contrastive objective for the CTA, CPA, and TTD tasks, but under-performs on the TSP task. Also, the ELECTRA objective has a negative impact on the TSP task when pretrained for longer duration, which is in line with our previous findings.
## 5 Related Work
There are two avenues of research that have studied tabular data representation learning. The first group of studies focus on predicting labels (essentially one row) for classification and regression problems, using row information and column schema as inputHuang et al. (2020); Arik and Pfister (2021); Sompalli et al. (2021); Gorishniy et al. (2021); Grinsztajn et al. (2022); Wang and Sun (2022); Du et al. (2022); Wydmanski et al. (2023). These studies use gradient descent-based end-to-end learning and aim to outperform tree-based models through task-specific model pretraining and fine-tuning.
The second group of studies proposes TaLMs to retrieve task-agnostic tabular data representations for different downstream table understanding tasks. Drawing inspiration from textual Language Models like BERT Devlin et al. (2019), many works Herzig et al. (2020); Yin et al. (2020); Deng et al. (2020); Iida et al. (2021) serialize tables to a sequence of tokens, leveraging existing checkpoints and textual self-supervised objectives. However, the representations of the tables can not only be learned from table content and by utilizing table structures, similar to other forms of semi-structured data like code and HTML. Some contemporary works have noticed the importance of the table structures and introduce many techniques to learn a certain aspect of them, such as masking Deng et al. (2020), coordinates Wang et al. (2021); Dash et al. (2022), and attention bias Yang et al. (2022). Our work belongs to this second group of studies and we propose to use hypergraphs to comprehensively model the rich table structures, and this is close to previous graph-based neural networks Mueller et al. (2019); Wang et al. (2021); Wang et al. (2021) where tables have been structured as graphs to incorporate row and column order information.
Table representation learning that focuses on joint text and table understanding is a separate field of research that partially overlaps with our work. Among them, some work Herzig et al. (2020); Shi
Figure 5: Performance with different pretraining checkpoints on the validation set of the four tasks. For the TaBERT models, we can only access the randomly initialized and fully pretrained (10 epochs) checkpoints. All results are from 5 system runs with different random seeds.
et al., 2022; Herzig et al., 2021; Glass et al., 2021; Yang et al., 2022) specialize in question-answering (QA) on tables and they jointly train a model that takes the question and the table structure as input together, allowing the pretraining to attend to the interactions of both texts and tables and boosting the table-based QA tasks. Another branch of joint text and table understanding work focuses on text generation from tables(Parikh et al., 2020; Yoran et al., 2021; Wang et al., 2022; Andrejczuk et al., 2022), relying on an encoder-decoder model like T5(Raffel et al., 2020) that can encode tables and decode free-form texts. In contrast to these studies, our work centers on the importance of structures in tables for table representation only, without extra text encoding or decoding.
Learning on hypergraphs has gone through a series of evolution (Agarwal et al., 2006; Yadati et al., 2019; Arya et al., 2020) in the way the hypergraph structure is modeled using neural networks layers. However, many of them collapse the hypergraph into a fully connected graph by clique expansion and cannot preserve the high-order interaction among the nodes. The recent development of permutation invariant networks (Zaheer et al., 2017; Lee et al., 2019) has enabled high-order interactions on the hypergraphs (Chien et al., 2022) that uses parameterized multi-set functions to model dual relations from node to hyperedges and vice versa. Closely related to the latest advancement, our HyTrel model adopts a similar neural message passing on hypergraphs to preserve the invariance property and high-order interactions of tables.
## 6 Limitations
The proposed HyTrel is a table encoder, and by itself cannot handle joint text and table understanding tasks like table QA and table-to-text generation. While it's possible to add text encoders or decoders for these tasks, it can potentially introduce additional factors that may complicate evaluating our hypothesis about the usefulness of modeling structural table properties. Moreover, the current model structure is designed for tables with simple column structures, like prior TaLMs, and cannot handle tables with complex hierarchical column structures. Additionally, HyTrel does not consider cross-table relations. Although we believe the hypergraph can generalize to model complicated column structures and cross-table interactions, we leave these aspects for future research.
## 7 Conclusion
In this work, we propose a tabular language model HyTrel that models tables as hypergraphs. It can incorporate the permutation invariances and table structures into the table representations. The evaluation on four table-related tasks demonstrates the advantages of learning these table properties and show that it can consistently achieve superior performance over the competing baselines. Our theoretical and qualitative analyses also support the effectiveness of learning the structural properties.
| Large amounts of tabular data have been used to train language models, and these models have demonstrated effectiveness in various downstream tasks. However, most of these models do not take into account the row/column permutation invariances, hierarchical structure, etc. inherent in tabular data. To address these limitations, we propose HYTREL, a tabular language model that captures permutation invariances and three additional structural properties of tabular data using hypergraphs. Table cells form the nodes in this model, and the cells within each row, column, and the entire table are used to form three different types of hyperedges. We demonstrate that HYTREL is maximally invariant under certain conditions for tabular data. Specifically, two tables obtain the same representations through HYTREL if and only if the two tables are identical up to permutations. Our empirical results demonstrate that HYTREL consistently outperforms other competitive baselines on four downstream tasks with minimal pretraining, illustrating the advantages of incorporating the inductive biases associated with tabular data |
2305.06077 | Relightify: Relightable 3D Faces from a Single Image via Diffusion
Models | Following the remarkable success of diffusion models on image generation,
recent works have also demonstrated their impressive ability to address a
number of inverse problems in an unsupervised way, by properly constraining the
sampling process based on a conditioning input. Motivated by this, in this
paper, we present the first approach to use diffusion models as a prior for
highly accurate 3D facial BRDF reconstruction from a single image. We start by
leveraging a high-quality UV dataset of facial reflectance (diffuse and
specular albedo and normals), which we render under varying illumination
settings to simulate natural RGB textures and, then, train an unconditional
diffusion model on concatenated pairs of rendered textures and reflectance
components. At test time, we fit a 3D morphable model to the given image and
unwrap the face in a partial UV texture. By sampling from the diffusion model,
while retaining the observed texture part intact, the model inpaints not only
the self-occluded areas but also the unknown reflectance components, in a
single sequence of denoising steps. In contrast to existing methods, we
directly acquire the observed texture from the input image, thus, resulting in
more faithful and consistent reflectance estimation. Through a series of
qualitative and quantitative comparisons, we demonstrate superior performance
in both texture completion as well as reflectance reconstruction tasks. | Foivos Paraperas Papantoniou, Alexandros Lattas, Stylianos Moschoglou, Stefanos Zafeiriou | 2023-05-10T11:57:49 | http://arxiv.org/abs/2305.06077v2 | # Relightify: Reliable 3D Faces from a Single Image via Diffusion Models
###### Abstract
Following the remarkable success of diffusion models on image generation, recent works have also demonstrated their impressive ability to address a number of inverse problems in an unsupervised way, by properly constraining the sampling process based on a conditioning input. Motivated by this, in this paper, we present the first approach to use diffusion models as a prior for highly accurate 3D facial BRDF reconstruction from a single image. We start by leveraging a high-quality UV dataset of facial reflectance (diffuse and specular albedo and normals), which we render under varying illumination settings to simulate natural RGB textures and, then, train an unconditional diffusion model on concatenated pairs of rendered textures and reflectance components. At test time, we fit a 3D morphable model to the given image and unwrap the face in a partial UV texture. By sampling from the diffusion model, while retaining the observed texture part intact, the model inpaints not only the self-occluded areas but also the unknown reflectance components, in a single sequence of denoising steps. In contrast to existing methods, we directly acquire the observed texture from the input image, thus, resulting in more faithful and consistent reflectance estimation. Through a series of qualitative and quantitative comparisons, we demonstrate superior performance in both texture completion as well as reflectance reconstruction tasks.
## 1 Introduction
Creating digital avatars of real people is of paramount importance for a range of applications, including VR, AR or the film industry. Human faces have been studied extensively over the years, attracting attention at the intersection of Computer Vision, Graphics and Machine Learning research. Although vast literature exists around the estimation of the 3D shape and reflectance of a face from unconstrained inputs such as "in-the-wild" RGB images, it still remains a challenging problem in the field. In particular, the recent breakthrough in image synthesis using diffusion generative models creates a new perspective towards photo-realistic 3D face reconstruction, which has not been explored so far and stems from the state-of-the-art performance of these models in solving inverse problems without supervised training.
Facial reflectance capture typically requires a controllable illumination system equipped with multiple cameras, first introduced as a Light Stage [13]. Polarized illumination and gradient patterns can be employed for diffuse-specular separation [49, 27], using which, spatially varying facial reflectance maps can be acquired, that describe BRDF parameters, including the diffuse and specular albedo and normals. Although recent works attempt to simplify the capturing apparatus and process using inverse rendering [29, 56] or commodity devices [39], such methods still require a laborious capturing process and expensive equipment.
Since their introduction by Blanz and Vetter [4], 3D Mor
phable Models (3DMMs) [55, 12, 45, 6, 5] have been established as a robust methodology for monocular 3D face reconstruction [19, 70] by regularizing the otherwise ill-posed optimization problem towards a known statistical prior of the facial geometry, which is usually defined by the linear space of a PCA model. In addition to the coarse geometry estimation, 3DMMs have been used in conjunction with powerful CNN-based texture models, leading to impressively detailed avatar reconstructions even from low-resolution images [58, 24, 25]. Furthermore, another line of research [7, 33, 69, 3, 18, 17, 40, 42] revolves around the reconstruction of rendering assets such as reflectance components (diffuse and specular albedo) and high-frequency normals of the facial surface. As a result, the recovered 3D faces can be realistically rendered in arbitrary illumination environments. However, prior work either contains scene illumination inhibiting relighting [14, 22, 24] or is restricted by the models' generalization, lowering the identity similarity [24, 40, 48]. Our work shares the same objective in that we couple a 3DMM with high-quality UV reflectance maps, but attempts to solve both of these issues, by preserving the observed texture details from the input image and jointly inferring the facial reflectance.
In fact, the visible pixels of the facial texture by the given camera pose are directly recoverable from the input image via inverse rasterization of the fitted 3D mesh. Therefore, we cast the 3D face reconstruction problem as an image inpainting task in the UV space; _i.e_. the goal is to fill in the missing pixels in a consistent manner with respect to some statistical prior. In particular, we propose to use a diffusion model as the generative backbone of our method. Diffusion models [62] are naturally associated with guided image synthesis since they treat image generation as a sequence of denoising steps in the form of a learnable Markov process. This allows to directly interfere with the sampling process, given that samples at each part of the chain are distorted versions of real images with known noise variances. Thus, by properly modifying the sampling process, a single unconditional diffusion model can be used for different inverse problems, such as image editing [51], inpainting [47, 11], restoration [37] or super-resolution [10, 9], without problem-specific training.
In this paper, we build a high-quality statistical model of facial texture and reflectance by means of a diffusion model and adopt an inpainting approach to complete the partially reconstructed UV texture produced by a 3DMM fitting step. We further extend the sampling process to recover the missing reflectance components by enforcing consistency with the input texture. As a result, our method, dubbed _Relightfy_, generates accurate and render-ready 3D faces from unconstrained images, as shown in Fig. 1.
In summary, we make the following contributions:
* We present the first, to the best of our knowledge, diffusion-based approach for relightable 3D face reconstruction from images. By training on a pseudo-ground-truth dataset of facial reflectance, while directly recovering texture parts from the input, we achieve high-quality rendering assets that preserve important details of the input face (_e.g_. wrinkles, moles).
* We propose an efficient way of predicting different modalities in a consistent way by learning a generative model on concatenated reflectance maps and casting the reconstruction as an inpainting problem, spatially, but also channel-wise.
* We qualitatively and quantitatively demonstrate the superiority of our approach against previous methods regarding both the completed textures as well as the recovered reflectance maps.
## 2 Related Work
### Diffusion Models for Inverse Problems
Diffusion models [62] are latent variable generative models which artificially corrupt the data distribution by adding noise and attempt to approximate the reverse process. They have lately emerged as a powerful image synthesis model [31, 16, 64] outperforming previous state-of-the-art approaches in both conditional and unconditional tasks. While they achieve excellent image quality and are robust to multi-modal distributions, they are computationally demanding to sample from, since they require a large sequence of denoising steps (_e.g_. 1000), each of which operates in the high dimensional image space. To alleviate this, a number of works [63, 38, 59] have proposed alternative strategies to accelerate sampling by reducing the steps of the reverse process. Another line of research [68, 60] proposes to train an encoding model and learn a diffusion model on its lower-dimensional latent space. Recently, Rombach _et al_. [57] have further explored the use of a VQGAN [20] as the auto-encoding model, showing that a mild compression is enough to reduce the training/sampling time without sacrificing sample quality. The latter approach is our method of choice for this work, as we elaborate on a high-resolution UV image space, which would otherwise significantly increase the computational overhead.
One of the most interesting aspects of diffusion models is that they can be used as unsupervised solvers for different inverse problems, where the goal is to reconstruct a sample from some distorted observation, _i.e_. conditioning input. Song _et al_. [64] propose a conditioning mechanism during inference that allows applications such as class-conditional generation, inpainting and colorization. Similarly, [9] uses a low-pass filtered version of the conditioning image to guide the denoising process at each step and SDEdit [51] addresses image translation and editing using a diffused ver
sion of the input image to initialize sampling from an intermediate timestep. RePaint [47] achieves state-of-the-art results on image inpainting by repeating multiple forward and backward diffusion steps to enforce harmonization. Despite its improved performance, this resampling strategy significantly increases the computational time. In contrast, CCDF [10] and DDRM [37] propose efficient techniques for reducing the length of the reverse process while retaining image quality at a high level. More recently, MCG [11] introduced a novel manifold constraint step, which combined with the standard reverse diffusion outperforms the aforementioned methods on a number of inverse tasks, including inpainting. We adopt this approach in our work to accurately fill in the missing pixels of both texture and reflectance maps of a face from a given image via diffusion-based inpainting, while fully preserving the observed ones. Note also that this approach does not assume any specific distribution of visibility masks, as it is trained unconditionally on complete textures.
### Facial Reconstruction
3DMMs [4] are the typical models for facial reconstruction from "in-the-wild" images, using a linear model for the identity, and additional linear models for expression or color. Current facial 3DMMs include the Basel Face Model (BFM) [55] and the Large Scale Facial Model (LSFM) [5]. Egger _et al_. [19] provide a thorough review on the subject. AlbedoMM [61] first created a 3DMM of facial reflectance, which can be relighted, but is restricted to a linear and per-vertex color model. Dib _et al_. [17, 18] improved on prior works' simplistic shading models and used inverse ray tracing to acquire photorealistic facial reflectance. Recently, GANFit [24, 25] introduced a potent method for fitting 3DMMs with a GAN-based [28] facial texture generator, achieving high-fidelity facial avatars, but lacking relighting capabilities due to baked illumination in the textures. AvatarMe++ [40, 42] overcame this issue by translating the reconstructed textures to facial reflectance using a conditional GAN, while adding extra processing steps. While we use AvatarMe++ to augment our training data, our method significantly outperforms them by using a powerful diffusion model and inferring only the occluded facial texture.
TBGAN [23] first introduced a deep generative network for facial reflectance, based on ProgressiveGAN [34] and [44] introduced a more powerful model, based on StyleGAN [35]. However, both works did not showcase fitting capabilities. An extension of the latter [48], introduced a set of multiple networks, with a StyleGAN2 [36] base, that can be used to generate shape and albedo from images with arbitrary illumination and expression. While close to our work, our method uses a single and more powerful diffusion model, inferring not only the diffuse albedo, but also the specular albedo and normals. Moreover, our work inpaints only the occluded facial areas, preserving the visible part of the texture and achieves higher reconstruction fidelity.
Although our method is applied to facial reconstruction, we simultaneously solve a facial texture inpainting problem in UV space. Initially explored in 2D facial images [46] and expanded to UV completion using deep encoder-decoder architectures (UV-GAN [14]), such works recover the facial texture from partial and masked facial images. Recently, OSTeC [22], used a pre-trained StyleGAN in 2D to recover multiple poses of the input subject so as to create a complete UV facial texture. While prior works achieve impressive results, all are restricted facial textures with baked illumination. In contrast, we jointly recover the facial reflectance, making the reconstruction relightable in standard rendering engines.
## 3 Method
We propose a diffusion-based inpainting approach to estimate both the UV texture with existing baked illumination and the actual reflectance of a face in a single process. At the core of our approach lies an unconditional diffusion generative model trained on pairs of textures and their accompanying reflectance. This coupled texture-reflectance modeling along with the sequential denoising process of diffusion models allows us to reconstruct the reflectance from a partial texture of the input face, as shown in Fig. 2. Our method, thus, generates high-quality 3D face avatars from 'in-the-wild' images, which can be realistically relighted.
In the following sections, we first analyze the training of our diffusion model, and then explain the 3D shape reconstruction and texture inpainting strategies in further detail.
### Diffusion Models: Background
Given a distribution of real images \(x\), diffusion models [62] define a forward diffusion process which gradually adds Gaussian noise to the input image in \(T\) consecutive steps. This corresponds to a fixed Markov Chain, where starting from a clean image \(x_{0}\), the noisy samples \(x_{t}\) at each timestep \(t\) are drawn from the following distributions (with timestep-depending variances \(\beta_{t}\)) conditioned on the previous samples:
\[q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}\mathbf{ I}) \tag{1}\]
This is equivalent to directly sampling \(x_{t}\) conditioned on the clean image \(x_{0}\) via:
\[q(x_{t}|x_{0})=\mathcal{N}(x_{t};\sqrt{\bar{\alpha}_{t}}x_{0},(1-\bar{\alpha}_ {t})\mathbf{I}) \tag{2}\]
where \(\alpha_{t}\coloneqq 1-\beta_{t}\) and \(\bar{\alpha}_{t}\coloneqq\prod_{s=1}^{t}\alpha_{s}\). Given large enough \(T\), this process leads to normally distributed noise \(x_{T}\). Then, the goal is to learn the reverse Markov process:
\[p_{\theta}(x_{t-1}|x_{t})=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\Sigma_{ \theta}(x_{t},t)) \tag{3}\]
which gradually denoises the random noise \(x_{T}\) towards a realistic image, by minimizing the variational lower bound of the negative log likelihood [31, 16]. Following the reparameterization proposed in [31], the model consists of time-conditioned denoising autoencoders \(\epsilon_{\theta}(x_{t},t);t\in\{1,2,\dots,T\}\), which are trained to predict the noise \(\epsilon\sim\mathcal{N}(\mathbf{0},\mathbf{I})\) that was added to the input image \(x_{0}\) to account for the noisy version \(x_{t}\):
\[L=E_{x_{0},\epsilon,t}\left[||\epsilon-\epsilon_{\theta}(x_{t},t)||^{2}\right] \tag{4}\]
Once trained, we can generate images by starting from random noise \(x_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})\) and sequentially drawing denoised images around the mean:
\[\mu_{\theta}(x_{t},t)=\frac{1}{\sqrt{\alpha_{t}}}\left(x_{t}-\frac{\beta_{t}} {\sqrt{1-\hat{\alpha}_{t}}}\epsilon_{\theta}(x_{t},t)\right) \tag{5}\]
### Training of our Diffusion Model
In this work, we harness the power of diffusion models to learn a strong generative prior over the domain of facial texture/reflectance. In particular, we adopt a physically-based perspective by separating the facial reflectance into different UV maps, namely diffuse albedo (\(\mathbf{A}_{d}\)), specular albedo (\(\mathbf{A}_{s}\)) and surface normals (\(\mathbf{N}\)) with high-frequency details. This allows realistic rendering under different illumination conditions. We learn our prior using a high-quality dataset consisting of _complete_ pairs of facial reflectance, and a corresponding rendered texture \(\mathbf{T}\) under arbitrary illumination. More details on the data we use are provided in section 4.1. We train an unconditional diffusion model (as described in section 3.1) on the quadruples:
\[x=[\mathbf{T},\mathbf{A}_{d},\mathbf{A}_{s},\mathbf{N}]\in\mathbb{R}^{512\times 5 12\times 12} \tag{6}\]
where we concatenate the components of Eq. 6 across channels (each of the 4 UV images measures \(512\times 512\times 3\) pixels). By sampling from this model, we can synthesize pairs of shaded RGB textures (\(\mathbf{T}\)) and reflectance components (\(\mathbf{A}_{d},\mathbf{A}_{s},\mathbf{N}\)) which are in correspondence, meaning that the texture is a rendered version of the UV reflectance under some illumination environment.
In practice, to reduce the computational requirements to a reasonable level, we follow the paradigm of **latent diffusion models** proposed by Rombach _et al_. [57], where the images are first compressed to a latent space \(z=\mathcal{E}(x)\in\mathbb{R}^{h\times w\times c}\) by training a perceptual auto-encoder, consisting of an encoder \(\mathcal{E}\) and a decoder \(\mathcal{D}\). Using perceptual and adversarial losses similar to VQGAN [20], the autoencoder achieves an excellent quality of reconstructed samples \(\tilde{x}=\mathcal{D}(\mathcal{E}(x))\), while allowing to efficiently train the diffusion model on the lower dimensional pixel-space of the learned embeddings. In our case, we train four similar auto-encoders, one for each of \(\mathbf{T},\mathbf{A}_{d},\mathbf{A}_{s}\) and \(\mathbf{N}\), all of them reducing the input resolution to latent dimensions of \(h=w=64\), \(c=3\). Therefore, our latent diffusion model [57] is trained on the concatenation of the 4 embeddings:
\[z=[z_{\mathbf{T}},z_{\mathbf{A}_{d}},z_{\mathbf{A}_{s}},z_{\mathbf{N}}]\in \mathbb{R}^{64\times 64\times 12} \tag{7}\]
Samples from our diffusion model (after being decoded through each \(\mathcal{D}\)) can be seen in the left part of Fig. 1.
### Inference
We use the aforementioned trained diffusion model to perform inpainting on both the texture and reflectance UV
Figure 2: Overview of our method during inference. Please note that we use a latent diffusion model [57], yet we illustrate the denoising process in the original image space for visualization purposes. We perform standard 3DMM fitting to get a partial UV texture via image-to-uv rasterization. Then, starting from random noise, we utilize the known texture to guide the sampling process of a texture/reflectance diffusion model towards completing the unobserved pixels. Each denoising step, from \(z_{t}\) to \(z_{t-1}\) (\(t\in\{1,\dots,T\}\)), follows an inpainting approach similar to MCG [11] (see Eq. 9): 1). The reflectance maps and unobserved texture pixels are updated based on reverse diffusion sampling and manifold constraints, while 2) the known pixels are directly sampled from the input texture via forward diffusion (\(\odot\) and \(\oplus\) denote the Hadamard product and addition respectively). Note that masking is only applied to the texture, while the reflectance maps (diffuse/specular albedo, normals) are entirely predicted from random noise. At the end of the process, we acquire high-quality rendering assets, making our 3D avatar realistically renderable.
maps based on a partial UV texture obtained by 3DMM fitting. We provide a detailed description below.
3DMM Fitting and Texture Initialization.We rely on 3DMMs to recover a rough 3D shape of the face from a 2D image as a mesh \(\mathbf{S}\in\mathbb{R}^{n\times 3}\) with \(n\) vertices. Specifically, we employ a linear 3DMM:
\[\mathbf{S}(\mathbf{p}_{s},\mathbf{p}_{e})=\mathbf{m}+\mathbf{U}_{s}\mathbf{p}_ {s}+\mathbf{U}_{e}\mathbf{p}_{e} \tag{8}\]
consisting of the LSFM [5] shape eigenbasis \(\mathbf{U}_{s}\in\mathbb{R}^{3n\times 158}\) and the expression eigenbasis \(\mathbf{U}_{e}\in\mathbb{R}^{3n\times 29}\) from the 4DFAB database [8]. We fit the 3DMM to the input image by optimizing the shape coefficients \(\mathbf{p}_{s}\), expression coefficients \(\mathbf{p}_{e}\) and camera parameters \(\mathbf{p}_{c}\) utilizing an off-the-shelf framework [1].
We use a standard UV topology for texturing the 3D mesh, where each vertex is assigned to a fixed 2D coordinate on the UV plane. By rasterizing the fitted 3D mesh and using barycentric interpolation, we can reverse the rendering process and unfold the face in UV, hence reconstructing the visible parts of the texture directly from the input image. This initial texture is accompanied by a UV visibility mask, with 1 for pixels that are observed from the input image, and 0 for those that are occluded and, thus, need to be inpainted by our model.
Texture Completion and Reflectance Prediction.Starting from the partially completed UV texture \(\mathbf{T}_{0}\) of the face and a binary visibility mask \(m\) produced by the previous step, our goal is to inpaint the remaining pixels along with the pixels of the 3 reflectance maps. We use the latent representation \(z_{\mathbf{T}_{0}}=\mathcal{E}(\mathbf{T}_{0})\in\mathbb{R}^{h\times\nu\times c}\) of this texture image to constrain the reverse diffusion process. Note that the mask \(m\) is downsampled to the same resolution \(h=w=64\) of the latent space for the next steps. Our inpainting algorithm starts with a random noise image \(z_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})\) and uses the denoising procedure of MCG [11], consisting of the following repeated steps:
\[z_{t-1}^{\text{unknown}} \sim\mathcal{N}(\mu_{\theta}(z_{t},t),\Sigma_{\theta}(z_{t},t)) \tag{9a}\] \[z_{\mathbf{T}_{t-1}}^{\text{known}} \sim\mathcal{N}(\sqrt{\bar{\alpha}_{t-1}}z_{\mathbf{T}_{0}},(1- \bar{\alpha}_{t-1})\mathbf{I})\] (9b) \[\dot{z}_{0} =\left(z_{t}-\sqrt{1-\bar{\alpha}_{t}}\epsilon_{\theta}(z_{t},t) \right)/\sqrt{\bar{\alpha}_{t}}\] (9c) \[\mathcal{L} =\left\|\left(z_{\mathbf{T}_{0}}-\hat{z}_{\mathbf{T}_{0}}\right) \odot m\right\|_{2}^{2}\] (9d) \[z_{\mathbf{T}_{t-1}} =m\odot z_{\mathbf{T}_{t-1}}^{\text{known}}+(1-m)\odot\left(z_{ \mathbf{T}_{t-1}}^{\text{unknown}}-\alpha\frac{\mathcal{L}}{\partial z_{ \mathbf{T}_{t}}}\right)\] (9e) \[z_{k_{t-1}} =z_{k_{t-1}}^{\text{unknown}}-\alpha\frac{\partial\mathcal{L}}{ \partial z_{k_{t}}},\quad k=\{\mathbf{A}_{d},\mathbf{A}_{s},\mathbf{N}\} \tag{9f}\]
Given a sample \(z_{t}\) at timestep \(t\), we first sample the next denoised sample \(z_{t-1}\) using the original reverse diffusion step (Eq. 9a). We term this as \(z_{t-1}^{\text{unknown}}\) (borrowing the notation from [47]) as it does not take into account the known parts of the observed texture. To exploit the known texture, we sample a noisy version of it \(z_{\mathbf{T}_{t-1}}^{\text{known}}\) at timestep \(t-1\) via a forward diffusion step (Eq. 9b). Then, we directly impose this known noisy texture \(m\odot z_{\mathbf{T}_{t-1}}^{\text{known}}\) (\(\odot\) denotes the Hadamard product) as in the first half of Eq. 9e. Finally, for the unknown pixels, we add the manifold constraint introduced in MCG [11]; _i.e_. we make a prediction of the clean sample \(\hat{z}_{0}\) (Eq. 9c) based on the previous timestep \(z_{t}\), compare this (\(\ell_{2}\) loss) with the ground truth in the known regions (Eq. 9d), and use the gradient of this loss to update the unknown pixels of \(z_{t-1}\) (Eq. 9e and 9f) so as to minimize this distance.
Note on inpainting algorithm.We have chosen to adopt the recently proposed MCG [11] inpainting algorithm, which outperforms related state-of-the-art diffusion-based methods (_e.g_. RePaint [47], DDRM [37]), as we empirically found it to produce excellent results. Motivated by the original algorithm, which aims at inpainting standard RGB images, we expand it to account for different input domains:
Figure 3: Examples of 3D reconstructions by our method, rendered using different environment maps in a commercial renderer [50].
by treating our images as concatenated texture/reflectance maps, we challenge the model to perform not only spatial inpainting, but also 'channel-wise inpainting', thus predicting accurate reflectance maps from just a partial illuminated version of them.
## 4 Experiments
### Dataset and Implementation Details
We create a high-quality dataset that consists of facial textures and their corresponding reflectance. Each item includes a texture \(\mathbf{T}\), shaded in some illumination, diffuse albedo \(\mathbf{A}_{d}\), specular albedo \(\mathbf{A}_{s}\) and normals \(\mathbf{N}\). To achieve this, firstly, we acquire the public MimicMe dataset [52], which contains \(\mathbf{\tilde{T}}=\{\mathbf{T}_{0},\dots,\mathbf{T}_{n_{T}}\},n_{T}=4,700\) diverse facial textures, whose statistics are reported in [52]. However, such textures contain the illumination of the scanning apparatus and are not relightable. Hence, we then train an image-to-image translation network based on AvatarMe++ model using the available dataset [42], which translates the textures \(\mathbf{\tilde{T}}\) to facial reflectance: \(\alpha(\mathbf{\tilde{T}})\rightarrow\{\mathbf{A}_{D},\mathbf{A}_{S},\mathbf{ N}\}\). Moreover, we augment the skin-tone diversity, using histogram matching albedo augmentation following [41]. Given the memory requirement of our network, all textures have a resolution of \(512\times 512\). Finally, to enable the diffusion model to perform well in "in-the-wild" images, we use the shapes \(\mathbf{S}\) of MimicMe and the acquired reflectance, to re-render the textures under arbitrary realistic environments, directly on the UV space: \(\rho(\mathbf{A}_{D},\mathbf{A}_{S},\mathbf{N},\mathbf{S})\rightarrow\mathbf{T}\). Although AvatarMe++ uses a similar method to augment training data, we do not require this process to be differentiable and use a ray-tracing renderer [50] (_Baker_ algorithm) to achieve more realistic textures.
To train our model, we use a KL-regularized latent diffusion model with the default hyper-parameters proposed by the authors of [57]. Specifically, we use a downsampling factor of \(f=8\) for the perceptual auto-encoder and a diffusion length of \(T=1000\) for the denoising model. We train our model once and use it for texture and reflectance reconstruction from "in-the-wild" images. Below we provide comprehensive qualitative and quantitative evaluations.
spect to the input thanks to our carefully designed inpainting approach. We also show an extensive qualitative comparison with related 3D reconstruction methods in Fig. 5 (most of which can only recover the texture), where similar observations can be made. Finally, we test our method on images from the Digital Emily [2] and show the results in Fig. 8 together with related works [18, 42]. We yield similar results regardless of the lighting, thanks to our coupled texture/reflectance modeling that combines reflectance with randomly rendered textures during training.
### Texture Completion
Following [22, 14], we evaluate our method on the task of texture completion using the Multi-PIE [30] subset of the UVDB dataset [14]. This consists of complete UV textures for 337 different identities, and corresponding 2D images of the faces from various camera poses. In accordance with [22, 14], we use the last 137 subjects for evaluation (as the first 200 were used as training data in prior works). We perform texture completion with our diffusion-based approach for each different viewing angle and compare it with existing texture completion methods, namely CE [54], UV-GAN [14] and OSTeC [22]. We use the widely adopted Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) metrics to compare the completed textures with the ground truth and report the results in Tab. 1. As can be seen, _Relighify_ outperforms the related methods in almost all settings, especially for challenging angles. A visual comparison with [22, 14] is provided in Fig. 6. Note that in contrast to CE [54] and UV-GAN [14], our model was not trained on the Multi-PIE dataset.
### Identity Preservation
We perform quantitative evaluations of our method's ability to preserve the subject's identity, by comparing the distribution of identity scores between the input image and rendered reconstruction, on the LFW dataset [32], against prior work [24, 25, 26, 67]. Following the existing benchmark [25], we evaluate our results using VGG-Face [53]. We present our analysis in Fig. 7, measuring the distance between the input image and reconstruction for all subjects. Our method shows a significant improvement in similarity, while also producing not just a facial texture, but a set of relightable reflectance textures.
and specular albedos, while the normals closely match that of [42]. This demonstrates our method's ability to better capture subject-specific details by directly leveraging texture information from the input image.
### Experimentation with Inpainting Algorithms
Although we adopt the MCG [11] approach for our texture/reflectance diffusion model, we have experimented with different inpainting algorithms. We compare four of them in Fig. 9 and Tab. 4. We also provide the runtime for each algorithm in Tab. 3. The baseline method of Score-SDE [64], which can be interpreted as Eq. 9 without the gradient term, produces sub-optimal results, the occluded areas are often inpainted in an inconsistent way with the observed ones, which is especially apparent in the texture (Fig. 9) and albedos (Tab. 4). RePaint [47] also produces unsatisfactory textures while at the same time increasing the reverse diffusion steps by a factor of \(n\) (we use \(n=10\) as suggested by the authors of [47]), which significantly affects the computational time. In contrast, MCG [11] preserves the original sampling length (\(T=1000\) timesteps), hence being much more efficient. However, it is still slower than Score-SDE [64] since it requires the computation of a gradient for the manifold constraint at each step. In general, we found MCG [11] to perform better in most cases. To further strengthen the efficiency of our method, we have additionally incorporated the DDIM [63] acceleration technique in the MCG algorithm, which allows reducing the denoising steps to \(N<T\) (we use \(N=200\)) without a significant drop in quality. In such case, our method can generate high-quality texture and reflectance assets from a partial UV texture in roughly 12 seconds, which is significantly faster than competing texture completion algorithms (OSTeC [22] requires around 10 minutes). | Diffusionモデルの画像生成分野での顕著な成功を踏まえ、最近の研究では、条件付き入力に基づいてサンプリングプロセスを適切に制限することで、多くの逆問題を非 supervised で解決できる能力を démontrer, これは、この論文では、Diffusionモデルを3D顔 BRDF 再構成の事前として初めて使用したアプローチを提案しています。このアプローチでは、顔の反射率（ diffuse と specular アルベドと法線）を備えた高品質の UV データセットを利用し、多様な照明設定でレンダリングすることで、自然な RGB テクスチャをシミュレートします。そして、連想されたテクスチャと反射率の組を組み合わせたデータセットを用いて、条件付き拡散モデルを訓練します。テスト時に、3D モルパブルモデルを画像にフィットさせ、その画像の顔部分で UV テクスチャを展開します。サンプリングを行うことで、観察された |
2303.14925 | Stratifications of abelian categories | This paper studies abelian categories that can be decomposed into smaller
abelian categories via iterated recollements - such a decomposition we call a
stratification. Examples include the categories of (equivariant) perverse
sheaves and epsilon-stratified categories (in particular highest weight
categories) in the sense of Brundan-Stroppel (2018).
We give necessary and sufficient conditions for an abelian category with a
stratification to be equivalent to a category of finite dimensional modules of
a finite dimensional algebra - this generalizes the main result of
Cipriani-Woolf (2022). Furthermore, we give necessary and sufficient conditions
for such a category to be epsilon-stratified - this generalizes the
characterisation of highest weight categories given by Krause (2017). | Giulian Wiggins | 2023-03-27T05:48:34 | http://arxiv.org/abs/2303.14925v1 | # Stratifications of abelian categories
###### Abstract.
This paper is a study of abelian categories that can be decomposed into smaller abelian categories via iterated recollements - such a decomposition we call a _stratification_. Examples include the categories of (equivariant) perverse sheaves and \(\varepsilon\)-stratified categories (in particular highest weight categories) in the sense of Brundan-Stroppel [1].
We give necessary and sufficient conditions for an abelian category with a stratification to be equivalent to a category of finite dimensional modules of a finite dimensional algebra - this generalizes the main result of Cipriani-Wolf [1]. Furthermore, we give necessary and sufficient conditions for such a category to be \(\varepsilon\)-stratified - this generalizes the characterisation of highest weight categories given by Krause [1].
###### Contents
* 1 Introduction
* 1.1 Outline and explanation of main results
* 2 Preliminaries
* 3 The intermediate-extension functor
* 4 Recollements with enough projectives/injectives
* 4.1 Projective covers
* 4.2 Ext-finiteness
* 4.3 Main results
* 5 Standard and costandard objects
* 6 Brundan and Stroppel's \(\varepsilon\)-stratified categories
* 6.1 Highest weight categories
## 1. Introduction
A _recollement_ of abelian categories is a short exact sequence of abelian categories
in which \(\mathcal{A}_{Z}\) is a Serre subcategory of \(\mathcal{A}\) (with Serre quotient \(\mathcal{A}_{U}\)), \(i_{*}\) has both a left and right adjoint, and \(j^{*}\) has fully-faithful left and right adjoints. We usually denote such a recollement of abelian categories by the diagram
where \((i^{*},i_{*},i^{!})\) and \((j_{!},j^{*},j_{*})\) are adjoint triples.
This definition is motivated by recollements of triangulated categories as defined by Beilinson, Bernstein and Deligne [1] to generalize Grothendieck's six functors relating the constructible derived category, \(\mathcal{D}(X,\Bbbk)\), of sheaves on a variety \(X\) (with coefficients in a field \(\Bbbk\)) and the constructible derived categories, \(\mathcal{D}(Z,\Bbbk)\) and \(\mathcal{D}(U,\Bbbk)\), of sheaves on a closed subvariety \(Z\subset X\) and open complement \(U:=X\backslash Z\) i.e. the situation:
Here, \(i:Z\hookrightarrow X\) is the closed embedding and \(j:U\hookrightarrow X\) is the complimentary open embedding. Note that if \(\mathcal{S}h(X,\Bbbk)\) is the category of sheaves on \(X\), and \(\mathcal{H}^{i}:\mathcal{D}(X,\Bbbk)\to\mathcal{S}h(X,\Bbbk)\) is the \(i\)-th cohomology functor, then the following is an example of a recollement of abelian categories.
More generally, given a recollement of triangulated categories with \(t\)-structure, there is a canonical recollement of abelian categories on the hearts of the \(t\)-structure [1, Proposition 1.4.16] defined using the zero-th cohomology functor arising from the \(t\)-structure.
In representation theory, recollements of abelian categories arise from modules of algebras \(A\) with an idempotent \(e\). In this situation there is a recollement
where \(i_{*}:A/AeA\text{-Mod}\to A\text{-Mod}\) is the restriction functor, and \(j^{*}:A\text{-Mod}\to eAe\text{-Mod}\) is the functor sending an \(A\)-module \(M\) to the \(eAe\)-module \(eM\).
In applications, recollements often arise as part of an iterated collection of recollements, giving a kind of filtration of a larger category by smaller categories. Consider, for example, the category, \(P_{\Lambda}(X,\Bbbk)\), of perverse sheaves that are constructible with respect to a stratification, \(X:=\bigcup_{\lambda\in\Lambda}X_{\lambda}\), of a variety \(X\). For each strata \(X_{\lambda}\), there is a recollement
where \(\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)\) is the category of local systems on \(X_{\lambda}\) of finite type. More generally, if \(\Lambda^{\prime}\subset\Lambda\) is such that \(X_{\Lambda^{\prime}}:=\bigcup_{\lambda\in\Lambda^{\prime}}X_{\lambda}\) is a closed subvariety of \(X\), and \(X_{\lambda}\) is open in \(X_{\Lambda^{\prime}}\), then the following is a recollement of abelian categories.
Similar iterations of recollement occur in highest weight categories (as defined in [1]). Indeed, let \(\mathcal{A}\) be a highest weight category with respect to a poset \(\Lambda\), and let \(\{\Delta_{\lambda}\mid\lambda\in\Lambda\}\) be the collection of standard objects in \(\mathcal{A}\). For each lower1 subposet \(\Lambda^{\prime}\subset\Lambda\), let \(\mathcal{A}_{\Lambda^{\prime}}\) be the Serre subcategory generated by objects
\(\{\Delta_{\lambda}\mid\lambda\in\Lambda^{\prime}\}\). For each lower subpost \(\Lambda^{\prime}\subset\Lambda\) and maximal \(\lambda\in\Lambda^{\prime}\) there is a recollement of abelian categories
where \(j^{*}:=\operatorname{Hom}(\Delta_{\lambda},-):\mathcal{A}_{\Lambda^{\prime}} \rightarrow\operatorname{End}_{\mathcal{A}}(\Delta_{\lambda})^{op}\text{-Mod}\).
These examples are unified by the following definition.
**Definition 2.4**.: A _stratification_ of an abelian category \(\mathcal{A}\) by a non-empty poset \(\Lambda\) consists of the following data
1. For each lower subposet \(\Lambda^{\prime}\subset\Lambda\), define a Serre subcategory, \(\mathcal{A}_{\Lambda^{\prime}}\), of \(\mathcal{A}\).
2. For each \(\lambda\in\Lambda\), define an abelian category \(\mathcal{A}_{\lambda}\). These we call _strata categories_.
This data must satisfy the conditions
1. \(\mathcal{A}_{\emptyset}=0\) and \(\mathcal{A}_{\Lambda}=\mathcal{A}\).
2. For each pair of lower subposets \(\Lambda^{\prime}_{1}\subset\Lambda^{\prime}_{2}\subset\Lambda\), there are inclusions of Serre categories \(\mathcal{A}_{\Lambda^{\prime}_{1}}\hookrightarrow\mathcal{A}_{\Lambda^{\prime }_{2}}\).
3. For each lower subposet \(\Lambda^{\prime}\subset\Lambda\), and maximal \(\lambda\in\Lambda^{\prime}\) there is a recollement
For example, the category \(P_{\Lambda}(X,\Bbbk)\) has a stratification by the poset \(\Lambda\) with closure order: \(\lambda\leq\mu\) if \(X_{\lambda}\subset\overline{X_{\mu}}\). Further examples include equivariant perverse sheaves, and \(\varepsilon\)-stratified categories in the sense of Brundan and Stroppel [1].
This paper is inspired by a result of Cipriani and Woolf [15, Corollary 5.2] that says that a category of perverse sheaves (with coefficients in a field) on a space stratified by finitely many strata is finite2 if and only if the same is true for each category of finite type local systems on each stratum. We extend this result by showing that an abelian category with a stratification by a finite poset is finite if and only if the same if true of all strata categories (Corollary 4.10). Moreover, we give new necessary and sufficient conditions for a finite abelian category to be \(\varepsilon\)-stratified (Theorem 6.4). This result specializes to give necessary and sufficient conditions for a finite abelian category to be standardly stratified in the sense of Cline, Parshall, Scott [11], and further specialises to recover a characterisation of highest weight categories due to Krause [16].
Footnote 2: For any field \(\Bbbk\), a \(\Bbbk\)-linear abelian category \(\mathcal{A}\) is _finite_ (over \(\Bbbk\)) if \(\mathcal{A}\) is equivalent to a category of finite dimensional modules of a finite dimensional \(\Bbbk\)-algebra.
This article revises and extends part of the author's PhD thesis [20]. This paper has benefited from discussions with Oded Yacobi, Kevin Coulembier, and feedback from a referee of [20]. This work was supported by Australian Research Council grant DP190102432.
### Outline and explanation of main results
Let \(\mathcal{A}\) be an abelian category with a stratification by a poset \(\Lambda\). For each \(\lambda\in\Lambda\), define the Serre quotient functor \(j^{\lambda}:\mathcal{A}_{\{\mu\in\Lambda\mid\mu\leq\lambda\}}\rightarrow \mathcal{A}_{\lambda}\), and let \(j^{\lambda}_{!}:\mathcal{A}_{\lambda}\rightarrow\mathcal{A}_{\{\mu\in\Lambda \mid\mu\leq\lambda\}}\) and \(j^{\lambda}_{*}:\mathcal{A}_{\lambda}\rightarrow\mathcal{A}_{\{\mu\in\Lambda \mid\mu\leq\lambda\}}\) be the left and right adjoints of \(j^{\lambda}\). By a slight abuse of notation, write \(j^{\lambda}_{!}:\mathcal{A}_{\lambda}\rightarrow\mathcal{A}\) and \(j^{\lambda}_{*}:\mathcal{A}_{\lambda}\rightarrow\mathcal{A}\) for the functors obtained by postcomposing with the inclusion functor \(\mathcal{A}_{\{\mu\in\Lambda\mid\mu\leq\lambda\}}\hookrightarrow\mathcal{A}\).
In Section 2 we recall some basic features of recollements and stratifications, and give examples of stratifications of abelian categories.
In Section 3 we define, for each \(\lambda\in\Lambda\), the _intermediate-extension functor_\(j^{\lambda}_{!*}:\mathcal{A}_{\lambda}\to\mathcal{A}\):
\[j^{\lambda}_{!*}X:=\operatorname{im}(\overline{1_{X}}:j^{\lambda}_{!}X\to j^{ \lambda}_{*}X),\]
where \(\overline{1_{X}}\) is the morphism corresponding to the identity \(1_{X}\) under the natural isomorphism
\[\operatorname{Hom}_{\mathcal{A}}(j^{\lambda}_{!}X,j^{\lambda}_{*}X)\simeq \operatorname{Hom}_{\mathcal{A}_{\lambda}}(X,j^{\lambda}j^{\lambda}_{*}X)\simeq \operatorname{Hom}_{\mathcal{A}_{\lambda}}(X,X).\]
Proposition 3.4 says that every simple object \(L\in\mathcal{A}\) is of the form \(j^{\lambda}_{!*}L_{\lambda}\), for a unique (up to isomorphism) simple object \(L_{\lambda}\in\mathcal{A}_{\lambda}\) and unique \(\lambda\in\Lambda\). Proposition 3.6 says that if \(\mathcal{A}\) is an abelian category with a stratification by a finite poset, then every object in \(\mathcal{A}\) has a finite filtration by simple objects if and only if the same is true of all the strata categories. The proofs here are almost identical to the standard proofs of these results in the theory of perverse sheaves (see e.g. [1, Chapter 3]).
In Section 4 we prove the following condition for a category to have enough projectives.
**Theorem 4.9**.: _Consider a recollement:_
_Suppose \(\mathcal{A}_{U}\) and \(\mathcal{A}_{Z}\) have finitely many simple objects and every object has a finite filtration by simple objects. Suppose moreover that for any simple objects \(A,B\) in \(\mathcal{A}\),_
\[\dim_{\operatorname{End}_{\mathcal{A}}(B)}\operatorname{Ext}^{1}_{\mathcal{A }}(A,B)<\infty.\]
_Then \(\mathcal{A}\) has enough projectives if and only if both \(\mathcal{A}_{U}\) and \(\mathcal{A}_{Z}\) have enough projectives.3_
Footnote 3: Since \(\operatorname{End}_{\mathcal{A}}(B)\) is a division ring, any \(\operatorname{End}_{\mathcal{A}}(B)\)-module is free. For an \(\operatorname{End}_{\mathcal{A}}(B)\)-module \(M\), \(\dim_{\operatorname{End}_{\mathcal{A}}(B)}M\) is the rank of \(M\) as a free \(\operatorname{End}_{\mathcal{A}}(B)\)-module.
Theorem 4.9 has the following important consequence.
**Corollary 4.11**.: _For any field \(\Bbbk\), a \(\Bbbk\)-linear abelian category with a stratification by a finite poset is finite if and only if the same is true for all strata categories._
To state the results in Sections 5 and 6, let \(\mathcal{A}\) have enough projectives and injectives, finitely many simple objects, and suppose every object in \(\mathcal{A}\) has finite length. Let \(B\) be a set indexing the simple objects in \(\mathcal{A}\) (up to isomorphism) and write \(L(b)\) (respectively \(P(b)\), \(I(b)\)) for the simple (respectively projective indecomposable, injective indecomposable) object corresponding to \(b\in B\).
Define the _stratification function_
\[\rho:B\to\Lambda\]
that maps each \(b\in B\) to the corresponding \(\lambda\in\Lambda\) in which \(L(b)=j^{\lambda}_{!*}L_{\lambda}(b)\) for some simple object \(L_{\lambda}(b)\in\mathcal{A}_{\lambda}\). Write \(P_{\lambda}(b)\) and \(I_{\lambda}(b)\) for the projective cover and injective envelope of \(L_{\lambda}(b)\) in \(\mathcal{A}_{\lambda}\).
For \(b\in B\) and \(\lambda=\rho(b)\), define the _standard_ and _costandard_ objects
\[\Delta(b):=j^{\lambda}_{!}P_{\lambda}(b),\qquad\nabla(b):=j^{\lambda}_{*}I_{ \lambda}(b).\]
Porism 5.1 says that every projective indecomposable, \(P(b)\), in \(\mathcal{A}\) has a filtration by quotients of standard objects, \(\Delta(b^{\prime})\), in which \(\rho(b^{\prime})\geq\rho(b)\). Dually, every
injective indecomposable, \(I(b)\), of \(\mathcal{A}\) has a filtration by subobjects of costandard objects, \(\nabla(b^{\prime})\), in which \(\rho(b^{\prime})\geq\rho(b)\).
Standard and costandard objects play a crucial role in representation theoretic applications of stratifications of abelian categories, where one often requires that projective and/or injective indecomposable objects have filtrations by standard and/or costandard objects. For example, both of these conditions are required in Cline-Parshall-Scott's definition of highest weight category [1].
Categories whose projective indecomposables have filtrations by standard objects have been widely studied - beginning with the work of Dlab [10] and Cline, Parshall and Scott [1]. Categories in which both projective objects have a filtration by standard objects and injective objects have a filtration by costandard objects have been studied by various authors (see e.g. [11], [12], [13]). Brundan and Stroppel [1] (based on the work of [1]) define a general framework called an \(\varepsilon\)_-stratified category_ that includes these situations as special cases. We recall this definition now.
For a _sign function_\(\varepsilon:\Lambda\to\{+,-\}\), define the \(\varepsilon\)_-standard_ and \(\varepsilon\)_-costandard objects_
\[\Delta_{\varepsilon}(b):=\left\{\begin{array}{ll}j_{\uparrow}^{\lambda}P_{ \lambda}(b)&\text{if $\varepsilon(\lambda)=+$}\\ j_{\downarrow}^{\lambda}L_{\lambda}(b)&\text{if $\varepsilon(\lambda)=-$} \end{array}\right.,\quad\nabla_{\varepsilon}(b):=\left\{\begin{array}{ll}j_ {*}^{\lambda}L_{\lambda}(b)&\text{if $\varepsilon(\lambda)=+$}\\ j_{*}^{\lambda}I_{\lambda}(b)&\text{if $\varepsilon(\lambda)=-$}\end{array}\right.,\]
where \(\lambda=\rho(b)\).
Brundan and Stroppel [1] say that \(\mathcal{A}\) is \(\varepsilon\)_-stratified_ if the following equivalent conditions hold:
1. Every projective indecomposable, \(P(b)\), has a filtration by \(\varepsilon\)-standard objects, \(\Delta_{\varepsilon}(b^{\prime})\), in which \(\rho(b^{\prime})\geq\rho(b)\).
2. Every injective indecomposable, \(I(b)\), has a filtration by \(\varepsilon\)-costandard objects, \(\nabla_{\varepsilon}(b^{\prime})\), in which \(\rho(b^{\prime})\geq\rho(b)\).
Note that \(\mathcal{A}\) is a _highest weight category_ if and only if each \(\mathcal{A}_{\lambda}\) is semisimple and \(\mathcal{A}\) is \(\varepsilon\)-stratified for any (and all) \(\varepsilon:\Lambda\to\{+,-\}\).
The following result gives a new criterion for a finite abelian category to be \(\varepsilon\)-stratified.
**Theorem 6.4**.: _A finite abelian category \(\mathcal{A}\) with a stratification by a finite poset \(\Lambda\) is \(\varepsilon\)-stratified (for a function \(\varepsilon:\Lambda\to\{+,-\}\)) if and only if the following conditions hold:_
1. _For each inclusion of Serre subcategories_ \(i_{*}:\mathcal{A}_{\Lambda^{\prime}}\to\mathcal{A}_{\Lambda}\)_, and objects_ \(X,Y\in\mathcal{A}_{\Lambda^{\prime}}\)_, there is a natural isomorphism_ \(\operatorname{Ext}^{2}_{\mathcal{A}_{\Lambda^{\prime}}}(X,Y)\simeq \operatorname{Ext}^{2}_{\mathcal{A}}(i_{*}X,i_{*}Y)\)_._
2. _For each_ \(\lambda\in\Lambda\)_,_ 1. _If_ \(\varepsilon(\lambda)=+\) _then_ \(j_{*}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) _is exact._ 2. _If_ \(\varepsilon(\lambda)=-\) _then_ \(j_{\uparrow}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) _is exact._
One application of Theorem 6.4 is a criterion for a category of (equivariant) perverse sheaves to be \(\varepsilon\)-stratified - although we remark that checking these conditions (particularly the first condition) is difficult in general.
## 2. Preliminaries
We begin with an axiomatic definition of recollement. The notation used in this definition will be used throughout the paper.
**Definition 2.1**.: A _recollement of abelian categories_ consists of three abelian categories \(\mathcal{A}_{Z}\), \(\mathcal{A}\) and \(\mathcal{A}_{U}\) and functors:
(2.1)
satisfying the conditions:
1. \((i^{*},i_{*},i^{!})\) and \((j_{!},j^{*},j_{*})\) are adjoint triples.
2. The functors \(i_{*}\), \(j_{!}\), \(j_{*}\) are fully-faithful. Equivalently the adjunction maps \(i^{*}i_{*}\to\operatorname{Id}\to i^{!}i_{*}\) and \(j^{*}j_{*}\to\operatorname{Id}\to j^{*}j_{!}\) are isomorphisms.
3. The functors satisfy \(j^{*}i_{*}=0\) (and so by adjunction \(i^{*}j_{!}=0=i^{!}j_{*}\)).
4. The adjunction maps produce exact sequences for each object \(X\in\mathcal{A}\): (2.2) \[j_{!}j^{*}X\to X\to i_{*}i^{*}X\to 0\] (2.3) \[0\to i_{*}i^{!}X\to X\to j_{*}j^{*}X\]
Alternatively, condition (R4) can be replaced by the condition
1. For any object \(X\in\mathcal{A}\), if \(j^{*}X=0\) then X is in the essential image of \(i_{*}\).
A _recollement of triangulated categories_ is defined in the same way as a recollement of abelian categories except that condition (R4) is replaced by the existence of the triangles:
\[j_{!}j^{*}X\to X\to i_{*}i^{*}X\to \tag{2.5}\] \[i_{*}i^{!}X\to X\to j_{*}j^{*}X\to \tag{2.4}\]
for each object \(X\).
**Remark 2.2**.: The interchangibility of (R4) and (R4') follows from the following argument. If \(j^{*}X=0\) then (R4) implies that \(i_{*}i^{!}X\simeq X\simeq i_{*}i^{*}X\) and so \(X\) is in the essential image of \(i_{*}\). Conversely let \(\mu:j_{!}j^{*}\to\operatorname{Id}\) and \(\eta:\operatorname{Id}\to i_{*}i^{*}\) be the adjunction natural transformations. Then there is a commutative diagram
in which the rows are exact. By applying \(j^{*}\) to the top row we see that \(j^{*}\operatorname{cok}\mu_{X}=0\) and so (R4') implies that \(\operatorname{cok}\mu_{X}\simeq i_{*}i^{*}(\operatorname{cok}\mu_{X})\simeq i _{*}i^{*}X\). Equation (2.3) holds by a similar argument.
Write \(\mathcal{A}^{Z}\) for the essential image of \(i_{*}\). To reconcile Definition 2.1 with the definition of recollement in Section 1, note that by (R2), \(\mathcal{A}^{Z}\simeq\mathcal{A}_{Z}\), and by (R4'), \(\mathcal{A}^{Z}\) is the kernel of the exact functor \(j^{*}\) and is hence a Serre subcategory of \(\mathcal{A}\).
It will be useful to note that if we extend the sequences (2.2) and (2.3) to exact sequences
\[0\to K\to j_{!}j^{*}X\to X\to i_{*}i^{*}X\to 0 \tag{2.7}\] \[0\to i_{*}i^{!}X\to X\to j_{*}j^{*}X\to K^{\prime}\to 0 \tag{2.6}\]
then \(K\) and \(K^{\prime}\) are in \(\mathcal{A}^{Z}\). Indeed, by applying the exact functor \(j^{*}\) to (2.6) we get that \(j^{*}K=0\) and so \(i_{*}i^{!}K\simeq K\simeq i_{*}i^{*}K\). Likewise by applying \(j^{*}\) to (2.7) we get that \(K^{\prime}\in\mathcal{A}^{Z}\).
Given a recollement of abelian or triangulated categories with objects and morphisms as in (2.1), the opposite categories form the following recollement
which we call the _opposite recollement_.
The following proposition describes a useful way to characterise the functors \(i^{*}\) and \(i^{!}\) in any recollement.
**Proposition 2.3**.: _Let \(\mathcal{A}\) be an abelian category with a recollement as in (2.1). Then for any object \(X\in\mathcal{A}\):_
1. \(i_{*}i^{*}X\) _is the largest quotient object of_ \(X\) _in_ \(\mathcal{A}^{Z}\)_._
2. \(i_{*}i^{!}X\) _is the largest subobject of_ \(X\) _in_ \(\mathcal{A}^{Z}\)_._
Proof.: By the adjunction \((i_{*},i^{*})\) and since \(i_{*}\) is fully-faithful we have natural isomorphisms for \(X\in\mathcal{A}\), \(Y\in\mathcal{A}_{Z}\):
\[\operatorname{Hom}_{\mathcal{A}}(i_{*}i^{*}X,i_{*}Y)\simeq\operatorname{Hom} _{\mathcal{A}_{Z}}(i^{*}X,Y)\simeq\operatorname{Hom}_{\mathcal{A}}(X,i_{*}Y)\]
sending \(f\) to \(f\circ\eta\) where \(\eta:X\to i_{*}i^{*}X\) is the adjunction unit. In particular any morphism \(X\to i_{*}Y\) factors through \(i_{*}i^{*}X\). Statement (i) follows. Statement (ii) follows by taking the opposite recollement.
Say that a subset \(\Lambda^{\prime}\) of a poset \(\Lambda\) is _lower_ if for any \(\lambda\in\Lambda^{\prime}\), if \(\mu\leq\lambda\) then \(\mu\in\Lambda^{\prime}\).
**Definition 2.4**.: A _stratification_ of an abelian/triangulated category \(\mathcal{A}\) by a non-empty poset \(\Lambda\) consists of the following data:
1. An assignment of an abelian/triangulated category \(\mathcal{A}_{\Lambda^{\prime}}\) for every lower \(\Lambda^{\prime}\subset\Lambda\), and for lower subsets \(\Lambda^{\prime\prime}\subset\Lambda^{\prime}\subset\Lambda\) an embedding \(i_{\Lambda^{\prime\prime},\Lambda^{\prime*}}:\mathcal{A}_{\Lambda^{\prime \prime}}\hookrightarrow\mathcal{A}_{\Lambda^{\prime}}\).
2. For each \(\lambda\in\Lambda\) an abelian/triangulated category \(\mathcal{A}_{\lambda}\). We call these _strata categories_.
This data must satisfy the following conditions
1. \(\mathcal{A}_{\emptyset}=0\) and \(\mathcal{A}_{\Lambda}=\mathcal{A}\).
2. For each \(\lambda\in\Lambda\) and lower subset \(\Lambda^{\prime}\subset\Lambda\) in which \(\lambda\in\Lambda^{\prime}\) is maximal, the functor \(i_{*}=i_{\Lambda^{\prime}\setminus\{\lambda\},\Lambda^{\prime*}}\) fits into a recollement \[\mathcal{A}_{\Lambda^{\prime}\setminus\{\lambda\}}\]
3. If \(\Lambda^{\prime\prime}\subset\Lambda^{\prime}\) are lower subsets of \(\Lambda\), and \(\lambda\in\Lambda\) is a maximal element of both \(\Lambda^{\prime\prime}\) and \(\Lambda^{\prime}\), then the following diagram of functors commutes
We proceed with some important examples of recollements and stratifications.
**Example 2.5** (Constructible sheaves with respect to a stratification).: A _stratification_ of a quasiprojective complex variety \(X\) is a finite collection, \(\{X_{\lambda}\}_{\lambda\in\Lambda}\), of disjoint, smooth, connected, locally closed subvarieties, called _strata_, in which \(X=\bigcup_{\lambda\in\Lambda}X_{\lambda}\) and for each \(\lambda\in\Lambda\), \(\overline{X_{\lambda}}\) is a union of strata. In this case we equip \(\Lambda\) with the partial order
\[\mu\leq\lambda\text{ if }X_{\mu}\subset\overline{X_{\lambda}}.\]
We use \(\Lambda\) to refer to the stratification of \(X\).
For a variety \(X\), let \(\operatorname{Loc}^{ft}(X,\Bbbk)\) be the category of local systems on \(X\) of finite type with coefficients in a field \(\Bbbk\). Recall that, by taking monodromy, \(\operatorname{Loc}^{ft}(X,\Bbbk)\) is equivalent to the category, \(\Bbbk[\pi_{1}(X_{\lambda})]\text{-mod}_{fg}\), of finitely generated \(\Bbbk[\pi_{1}(X_{\lambda})]\)-modules (see e.g. [1, Theorem 1.7.9]).
Say that a sheaf \(\mathcal{F}\) on \(X\) is _constructible_ with respect to a stratification, \(\Lambda\), of \(X\) if \(\mathcal{F}|_{X_{\lambda}}\) is a local system of finite type for each \(\lambda\in\Lambda\). Write \(\mathcal{D}^{b}_{\Lambda}(X,\Bbbk)\) for the full triangulated subcategory of \(\mathcal{D}^{b}(X,\Bbbk)\) consisting of objects \(\mathcal{F}\) in which \(H^{k}(\mathcal{F})\) is constructible with respect to \(\Lambda\).
Say that a stratification, \(\Lambda\), of \(X\) is _good_ if for any \(\lambda\in\Lambda\) and any object \(\mathcal{L}\in\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)\), we have \(j_{\lambda*}\mathcal{L}\in\mathcal{D}^{b}_{\Lambda}(X,\Bbbk)\), where \(j_{\lambda}:X_{\lambda}\hookrightarrow X\) is the embedding, and \(j_{\lambda*}\) is the derived pushforward. It is difficult to tell in general whether a stratification is good (see [1, Remark 2.3.21] for a discussion of these difficulties). A stratification satisfying the _Whitney regularity conditions_[10] is good. In particular, if an algebraic group \(G\) acts on \(X\) with finitely many orbits (each connected), then the stratification of \(X\) by \(G\)-orbits is good (see e.g. [1, Exercise 6.5.2]).
Given a good stratification \(\Lambda\) on \(X\), the triangulated category \(\mathcal{D}^{b}_{\Lambda}(X,\Bbbk)\) has a stratification by \(\Lambda\) with strata categories
\[\mathcal{D}_{\lambda}:=\mathcal{D}^{b}(\operatorname{Loc}^{ft}(X_{\lambda}, \Bbbk))\simeq\mathcal{D}^{b}(\Bbbk[\pi_{1}(X_{\lambda})]\text{-mod}_{fg})\]
and Serre subcategories \(\mathcal{D}_{\Lambda^{\prime}}:=\mathcal{D}^{b}_{\Lambda^{\prime}}(\bigcup_{ \lambda\in\Lambda^{\prime}}X_{\lambda})\) for each lower \(\Lambda^{\prime}\subset\Lambda\).
For a perversity function \(p:\Lambda\to\mathbb{Z}\), the category \({}^{p}P_{\Lambda}(X,\Bbbk)\) of perverse sheaves on \(X\) with respect to the stratification \(\Lambda\) (and perversity function \(p\)) is the full subcategory of \(\mathcal{D}^{b}_{\Lambda}(X,\Bbbk)\) consisting of complexes \(\mathcal{F}\) in which for any strata \(h_{\lambda}:X_{\lambda}\hookrightarrow X\):
1. \(\mathcal{H}^{d}(h_{\lambda}^{*}\mathcal{F})=0\) if \(d>p(\lambda)\),
2. \(\mathcal{H}^{d}(h_{\lambda}^{!}\mathcal{F})=0\) if \(d<p(\lambda)\),
where \(\mathcal{H}^{d}(\mathcal{F})\) refers to the \(d\)-th cohomology sheaf of \(\mathcal{F}\). The category \(\mathcal{A}={}^{p}P_{\Lambda}(X,\Bbbk)\) is abelian and has a stratification by \(\Lambda\), with strata categories
\[\mathcal{A}_{\lambda}=\operatorname{Loc}^{ft}(X_{\lambda},\Bbbk)[p(\lambda) ]\simeq\Bbbk[\pi_{1}(X_{\lambda})]\text{-mod}_{fg}.\]
**Example 2.6** (\(G\)-equivariant perverse sheaves).: Another example of a stratification arises in the theory of equivariant perverse sheaves as defined in [1].
For a complex algebraic group \(G\) and quasiprojective complex \(G\)-variety \(X\), a \(G\)_-equivariant perverse sheaf_ on \(X\) is, roughly speaking, a perverse sheaf on \(X\) with a \(G\)-action compatible with the \(G\)-action on \(X\) (see e.g. [1, Definition 6.2.3] for a precise definition). The category, \(P_{G}(X,\Bbbk)\) of \(G\)-equivariant perverse sheaves is the heart of a \(t\)-structure on the \(G\)_-equivariant derived category_, \(\mathcal{D}_{G}(X,\Bbbk)\) defined by Bernstein-Lunts [1]. For a \(G\)-equivariant map of \(G\)-varieties \(h:X\to Y\), there are equivariant versions of the (proper) pushforward and (proper) pullback functors:
\(h_{*},h_{!},h^{!},h^{*}\). If \(i:Z\hookrightarrow X\) is the inclusion of a \(G\)-invariant closed subvariety with open complement \(j:U\hookrightarrow X\), then there is a recollement of triangulated categories
If \(X\) is a homogeneous \(G\)-variety, then every \(G\)-equivariant perverse sheaf is a finite type local system (shifted by \(\dim_{\mathbb{C}}X\)). Moreover, in this case,
\[P_{G}(X,\Bbbk)\simeq\Bbbk[G^{x}/(G^{x})^{\circ}]\text{-mod}_{fg}, \tag{2.8}\]
where \(G^{x}\subset G\) is the stabilizer of a point \(x\in X\), and \((G^{x})^{\circ}\) is the connected component of \(G^{x}\) containing the identity element (see e.g. [1, Proposition 6.2.13] for a proof of this statement).
Suppose \(G\) acts on \(X\) with finitely many orbits (each connected). Let \(\Lambda\) be a set indexing the set of \(G\)-orbits in \(X\), and write \(\mathcal{O}_{\lambda}\) for the orbit corresponding to \(\lambda\in\Lambda\). Consider \(\Lambda\) as a poset with the closure order: \(\lambda\leq\mu\) if \(\mathcal{O}_{\lambda}\subset\overline{\mathcal{O}_{\mu}}\). Then the category \(\mathcal{A}=P_{G}(X,\Bbbk)\) has a stratification with strata categories
\[\mathcal{A}_{\lambda}=P_{G}(\mathcal{O}_{\lambda},\Bbbk)\simeq\Bbbk[G^{x_{ \lambda}}/(G^{x_{\lambda}})^{\circ}]\text{-mod}_{fg},\]
where \(x_{\lambda}\in\mathcal{O}_{\lambda}\).
**Example 2.7** (Modules with idempotents).: For a ring \(A\), let \(\operatorname{Mod-}A\) be the category of all right \(A\)-modules, and \(\operatorname{mod-}A\) be the category of finitely presented right \(A\)-modules. Let \(e\) be an idempotent in \(A\), and define the inclusion functor \(i_{*}:\operatorname{Mod-}A/AeA\to\operatorname{Mod-}A\). Note that \(\operatorname{Mod-}A/AeA\) is equivalent to the Serre subcategory of \(\operatorname{Mod-}A\) consisting of modules annihilated by \(e\). There is a corresponding Serre quotient \(j^{*}:\operatorname{Mod-}A\to\operatorname{Mod-}eAe\) defined
\[j^{*}:=\operatorname{Hom}_{A}(eA,-)\simeq-\otimes_{A}Ae.\]
i.e. \(j^{*}M=Me\) for any object \(M\in\operatorname{Mod-}A\). These functors fit into a recollement of abelian categories
where for any right \(A\)-module \(M\):
1. \(i^{*}M\) is the largest quotient, \(N\), of \(M\) in which \(Ne=0\).
2. \(i^{!}M\) is the largest subobject, \(N\), of \(M\) in which \(Ne=0\).
Moreover \(j_{!}:=-\otimes_{eAe}eA\) and \(j_{*}:=\operatorname{Hom}_{eAe}(Ae,-)\).
If \(A\) is right artinian and has enough injectives then the inclusion \(i_{*}:\operatorname{mod-}A/AeA\to\operatorname{mod-}A\) fits into a recollement
in which \(j^{*}\) has left adjoint \(j_{!}=-\otimes_{eAe}eA\) (see e.g [16, Lemma 2.5]).
**Example 2.8** (Macpherson-Vilonen construction).: Let \(\mathcal{A}_{Z}\), \(\mathcal{A}_{U}\) be abelian categories, \(F:\mathcal{A}_{U}\to\mathcal{A}_{Z}\) be a right exact functor, \(G:\mathcal{A}_{U}\to\mathcal{A}_{Z}\) be a left exact functor, and let \(\varepsilon:F\to G\) be a natural transformation.
Macpherson and Vilonen [14] define a category, \(\mathcal{A}(\varepsilon)\), whose objects are tuples \((X_{U},X_{Z},\alpha,\beta)\), where \((X_{U},X_{Z})\in\operatorname{Obj}\mathcal{A}_{U}\times\operatorname{Obj} \mathcal{A}_{Z}\) and \(\alpha:F(X_{U})\to X_{Z}\), \(\beta:X_{Z}\to G(X_{U})\) are morphisms in \(\mathcal{A}_{Z}\) in which the following diagram commutes
A morphism \((X_{U},X_{Z},\alpha,\beta)\to(X_{U}^{\prime},X_{Z}^{\prime},\alpha^{\prime}, \beta^{\prime})\) is a pair
\[f=(f_{U}:X_{U}\to X_{U}^{\prime},f_{Z}:X_{Z}\to X_{Z}^{\prime})\in \operatorname{Mor}\mathcal{A}_{U}\times\operatorname{Mor}\mathcal{A}_{Z}\]
in which the following prism commutes:
Macpherson and Vilonen show [14, Proposition 1.1] that the category \(\mathcal{A}(\varepsilon)\) is abelian. Moreover they show that the category \(\mathcal{A}(\varepsilon)\) fits into a recollement
in which
\[i_{*}(X_{Z})=(0,X_{Z},0,0), j^{*}(X_{U},X_{Z},\alpha,\beta)=X_{U},\] \[i^{*}(X_{U},X_{Z},\alpha,\beta)=\operatorname{cok}\alpha, j_{!}(X_{U})=(X_{U},F(X_{U}),1_{F(X_{U})},\varepsilon_{X_{U}}),\] \[i^{!}(X_{U},X_{Z},\alpha,\beta)=\ker\beta, j_{*}(X_{U})=(X_{U},G(X_{U}), \varepsilon_{X_{U}},1_{G(X_{U})}).\]
Macpherson and Vilonen [14] use iterations of this formal construction to construct the category of perverse sheaves on a stratified variety. Franjou and Pirashvili [13, Theorem 8.7] give necessary and sufficient conditions for a recollement of abelian categories to be equivalent to a recollement built using the Macpherson-Vilonen construction.
Note that the functor \(i_{*}:\mathcal{A}_{U}\to\mathcal{A}(\varepsilon)\) has an exact retraction \(i^{!*}:\mathcal{A}(\varepsilon)\to\mathcal{A}_{Z}\) defined \(i^{!*}(X_{U},X_{Z},\alpha,\beta)=X_{Z}\). This is a special feature of recollements built using Macpherson-Vilonen's construction - the functor \(i_{*}\) does not usually have an exact retraction in a general recollement.
The functor \(j_{!*}:\mathcal{A}_{U}\to\mathcal{A}(\varepsilon)\) defined \(j_{!*}(X_{U})=(X_{U},\operatorname{im}\varepsilon_{X_{U}},\varepsilon_{X_{U}},1)\) is a special case of an intermediate-extension functor defined in Definition 3.1 below. Note that every simple object in \(\mathcal{A}(\varepsilon)\) is either of the form \(i_{*}L\) for a simple object \(L\) in \(\mathcal{A}_{Z}\), or of the form \(j_{!*}L\) for a simple object in \(\mathcal{A}_{U}\). This is a special case of Proposition 3.4 below.
## 3. The intermediate-extension functor
Consider again a recollement:
(3.1)
In this section we study the full subcategory, \(\mathcal{A}^{U}\hookrightarrow\mathcal{A}\), whose objects have no subobjects or quotients in \(\mathcal{A}^{Z}:=\operatorname{im}i_{*}\). The main result of this section (Proposition 3.3(ii)) is that the restricted functor \(j^{*}:\mathcal{A}^{U}\to\mathcal{A}_{U}\) is an equivalence of categories. The quasi-inverse \(j_{!*}:\mathcal{A}_{U}\to\mathcal{A}^{U}\) is defined as follows.
**Definition 3.1** (\(j_{!*}:\mathcal{A}_{U}\to\mathcal{A}^{U}\)).: For an object \(X\in\mathcal{A}_{U}\), let \(\overline{1_{X}}:j_{!}X\to j_{*}X\) be the morphism corresponding to the identity on \(X\) under the isomorphism
\[\operatorname{Hom}_{\mathcal{A}}(j_{!}X,j_{*}X)\simeq\operatorname{Hom}_{ \mathcal{A}_{U}}(X,j^{*}j_{*}X)\simeq\operatorname{Hom}_{\mathcal{A}_{U}}(X,X).\]
Define
\[j_{!*}X:=\operatorname{im}(\overline{1_{X}}:j_{!}X\to j_{*}X)\in\mathcal{A}.\]
It is easy to check that if \(X\in\mathcal{A}_{U}\) then \(j_{!*}X\in\mathcal{A}^{U}\). Indeed as \(i^{!}j_{*}X=0\), \(j_{*}X\) has no subobjects in \(\mathcal{A}^{Z}\). In particular, as \(j_{!*}X\) is a subobject of \(j_{*}X\) it cannot have any subobjects in \(\mathcal{A}^{Z}\). Likewise as \(j_{!*}X\) is a quotient of \(j_{!}X\) it cannot have any quotients in \(\mathcal{A}^{Z}\).
We call the functor \(j_{!*}:\mathcal{A}_{U}\to\mathcal{A}\) an _intermediate-extension functor_.
**Remark 3.2**.: Not every subquotient of an object in \(\mathcal{A}^{U}\) need be in \(\mathcal{A}^{U}\). In particular, an object in \(\mathcal{A}^{U}\) may still have simple composition factors in \(\mathcal{A}^{Z}\).
**Proposition 3.3**.: _Let \(\mathcal{A}\) be an abelian category with a recollement of abelian categories as in (3.1). Then_
* _If_ \(X\in\mathcal{A}\) _has no nonzero quotient objects in_ \(\mathcal{A}^{Z}\)_, and_ \(Y\in\mathcal{A}\) _has no nonzero subobjects in_ \(\mathcal{A}^{Z}\) _(i.e._ \(i^{*}X=0\) _and_ \(i^{!}Y=0\)_), then_ \[\operatorname{Hom}_{\mathcal{A}}(X,Y)\simeq\operatorname{Hom}_{\mathcal{A}_{U}} (j^{*}X,j^{*}Y).\]
* \(j^{*}:\mathcal{A}^{U}\to\mathcal{A}_{U}\) _is an equivalence of categories with quasi-inverse_ \(j_{!*}:\mathcal{A}_{U}\to\mathcal{A}^{U}\)_._
Proof.: If \(i^{*}X=0\) then (2.6) gives an exact sequence
\[0\to K\to j_{!}j^{*}X\to X\to 0\]
in which \(K\simeq i_{*}i^{!}K\). So applying \(\operatorname{Hom}(-,Y)\) we get the exact sequence
\[0\to\operatorname{Hom}(X,Y)\to\operatorname{Hom}(j_{!}j^{*}X,Y)\to \operatorname{Hom}(i_{*}i^{!}K,Y).\]
Applying adjunctions gives the exact sequence
\[0\to\operatorname{Hom}(X,Y)\to\operatorname{Hom}(j^{*}X,j^{*}Y)\to \operatorname{Hom}(i^{!}K,i^{!}Y).\]
Statement (i) follows as \(i^{!}Y=0\).
A corollary of statement (i) is that \(j^{*}:\mathcal{A}^{U}\to\mathcal{A}_{U}\) is fully-faithful. To show that \(j^{*}\) is essentially surjective it suffices to show that for any object \(X\in\mathcal{A}_{U}\), \(j^{*}j_{!*}X\simeq X\). Now, as \(j^{*}\) is exact:
\[j^{*}j_{!*}X=j^{*}\operatorname{im}(j_{!}X\to j_{*}X)\simeq\operatorname{im}(j ^{*}j_{!}X\to j^{*}j_{*}X)\simeq\operatorname{im}(\operatorname{Id}:X\to X)=X\]
and so (ii) follows.
If \(\mathcal{A}\) has a stratification by a finite poset \(\Lambda\), then for each \(\lambda\in\Lambda\), there is a functor \(j_{!*}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) defined by the composition
**Proposition 3.4**.: _Let \(\mathcal{A}\) be an abelian category with a stratification by a finite poset \(\Lambda\). Every simple object \(L\in\mathcal{A}\) is of the form \(j_{!*}^{\lambda}L_{\lambda}\), for a unique (up to isomorphism) simple object \(L_{\lambda}\in\mathcal{A}_{\lambda}\) and unique \(\lambda\in\Lambda\)._
Proof.: Suppose \(\mathcal{A}\) fits into a recollement as in (3.1). By Proposition 3.3, if \(L\in\mathcal{A}_{U}\) is a simple object, then \(j_{!*}L\) is a simple object in \(\mathcal{A}\). Moreover all the simple objects of \(\mathcal{A}\) are either of the form \(i_{*}L\) for a simple object \(L\in\mathcal{A}_{Z}\), or of the form \(j_{!*}L\) for a simple object \(L\in\mathcal{A}_{U}\). The statement follows via an induction argument on \(|\Lambda|\).
The following properties of the intermediate-extension functor will be useful.
**Proposition 3.5**.: _Let \(\mathcal{A}\) be an abelian category with a recollement of abelian categories as in (3.1). Then_
1. _The functor_ \(j_{!*}:\mathcal{A}_{U}\to\mathcal{A}\) _maps injective morphisms to injective morphisms and surjective morphisms to surjective morphisms._
2. _If_ \(X\in\mathcal{A}\) _has no nonzero quotient objects in_ \(\mathcal{A}^{Z}\) _then there is a canonical short exact sequence_ \[0\to i_{*}i^{!}X\to X\to j_{!*}j^{*}X\to 0\]
3. _If_ \(X\in\mathcal{A}\) _has no nonzero subobjects in_ \(\mathcal{A}^{Z}\) _then there is a canonical short exact sequence_ \[0\to j_{!*}j^{*}X\to X\to i_{*}i^{*}X\to 0\]
Proof.: Let \(f:X\to Y\) be a map in \(\mathcal{A}_{U}\) and define objects \(K_{1}\), \(K_{2}\) in \(\mathcal{A}\) by the exact sequence
\[0\to K_{1}\to j_{!*}X\to j_{!*}Y\to K_{2}\to 0\]
To prove statement (i) it suffices to show that if \(j^{*}K_{i}=0\) then \(K_{i}=0\). If \(j^{*}K_{i}=0\) then by (R4'), \(K_{1}\simeq i_{*}i^{*}K_{1}\) and \(K_{2}\simeq i_{*}i^{!}K_{2}\). Then each \(K_{i}=0\) since \(j_{!*}X\) and \(j_{!*}Y\) are in \(\mathcal{A}^{U}\).
To prove statement (ii), let \(X\in\mathcal{A}\) have no nonzero quotients in \(\mathcal{A}^{Z}\) and consider the short exact sequence
\[0\to i_{*}i^{!}X\to X\to K\to 0\]
Applying \(i^{!}\) to the sequence we see that \(i^{!}K=0\) and so \(K\in\mathcal{A}^{U}\). So \(K\simeq j_{!*}j^{*}K\) and (by applying \(j^{*}\) to this sequence) \(j^{*}X\simeq K\). Statement (ii) follows immediately. The proof of statement (iii) is similar.
Say that an abelian category is a _length category_ if every object has a finite filtration by simple objects.
**Proposition 3.6**.: _If \(\mathcal{A}\) be an abelian category with a stratification by a finite poset, then \(\mathcal{A}\) is a length category if and only if all the strata categories are length categories._
Proof.: Let \(\mathcal{A}\) be an abelian category fitting into a recollement of abelian categories as in (3.1). It suffices to show that \(\mathcal{A}\) is a length category if and only if both \(\mathcal{A}_{Z}\) and \(\mathcal{A}_{U}\) are length categories. The result follows from this statement by induction.
Let \(X\) be an object in \(\mathcal{A}\) and let \(K\) be defined by the short exact sequence
\[0\to i_{*}i^{!}X\to X\to K\to 0\]
Then \(i^{!}K=0\) and so applying Proposition 3.5(iii) we get the short exact sequence
\[0\to j_{*}j^{*}K\to K\to i_{*}i^{*}K\to 0\]
In particular if every object in \(\mathcal{A}_{Z}\) and every object in \(\mathcal{A}_{U}\) has a finite filtration by simple objects, then so does \(K\) and hence so does \(X\). The converse statement is obvious.
## 4. Recollements with enough projectives/injectives
In this section we study the relationship between projective covers of objects in the different categories of a recollement. More precisely, let \(\mathcal{A}\) be a category fitting into a recollement as in (3.1). Proposition 4.5 says that if \(\mathcal{A}\) has enough projectives/injectives then so does \(\mathcal{A}_{U}\). Proposition 4.6 says that if \(\mathcal{A}\) is a Krull-Schmidt category then if \(\mathcal{A}\) has enough projectives/injectives then so does \(\mathcal{A}_{Z}\). Proposition 4.7 says that if \(X\in\mathcal{A}_{U}\) has a projective cover \(P\) in \(\mathcal{A}_{U}\) then \(j_{!}P\) is a projective cover in \(\mathcal{A}\) of \(j_{!*}X\).
Unfortunately it is not easy to find a projective cover in \(\mathcal{A}\) of an object \(i_{*}X\in\mathcal{A}^{Z}\), even if a projective cover of \(X\) exists in \(\mathcal{A}_{Z}\). Theorem 4.9 gives sufficient conditions for such a projective cover to exist. A consequence of Theorem 4.9 (Corollary 4.11) is that a category \(\mathcal{A}\) with a stratification by a finite poset is equivalent to a category of finite dimensional modules of a finite dimensional algebra if and only if the same is true of the strata categories.
### Projective covers
Recall that a surjection \(\phi:X\to Y\) is _essential_ if for any morphism \(\alpha:X^{\prime}\to X\), if \(\phi\circ\alpha\) is surjective then \(\alpha\) is surjective. Equivalently \(\phi:X\to Y\) is essential if for any subobject \(U\subset X\), if \(U+\ker\phi=X\) then \(U=X\). If \(P\to X\) is an essential surjection and \(P\) is projective then we call \(P\) (or more accurately the morphism \(P\to X\)) a _projective cover_ of \(X\). The projective cover of \(X\) (if it exists) factors through every other essential cover of \(X\), and is unique up to isomorphism.
If \(L\in\mathcal{A}\) is a simple object and \(P\) is projective then \(\phi:P\to L\) is a projective cover if and only if the following equivalent conditions hold:
1. \(\ker\phi\) is the unique maximal subobject of \(P\).
2. The endomorphism ring of \(P\) is local.
See e.g. [15, Lemma 3.6] for a proof of these facts.
The dual concept of an essential surjection is called an _essential extension_. If \(X\to I\) is an essential extension and \(I\) is injective then this extension is called the _injective envelope_ of \(X\).
An abelian category has _enough projectives_ (resp. _enough injectives_) if every object has a projective cover (resp. injective envelope).
An abelian category \(\mathcal{A}\) is a _Krull-Schmidt category_ if every object in \(\mathcal{A}\) is a finite direct sum of objects with local endomorphism rings. For example, any abelian length category is a Krull-Schmidt category.
In a Krull-Schmidt category, the projective covers of simple objects are exactly the projective indecomposable objects. Moreover a Krull-Schmidt category \(\mathcal{A}\) has enough projectives if and only if every simple object has a projective cover. We will need the following well-known characterisation of projective covers of simple objects in Krull-Schmidt categories.
**Proposition 4.1**.: _Let \(\mathcal{A}\) be a Krull-Schmidt category. Let \(P\in\mathcal{A}\) be a projective object and \(L\in\mathcal{A}\) be a simple object. A map \(P\to L\) is a projective cover if and only if for any simple object \(L^{\prime}\)_
\[\dim_{\operatorname{End}_{\mathcal{A}}(L^{\prime})}\operatorname{Hom}_{ \mathcal{A}}(P,L^{\prime})=\begin{cases}1&\text{ if }L=L^{\prime}\text{,}\\ 0&\text{ otherwise.}\end{cases} \tag{4.1}\]
**Remark 4.2**.: Since \(\operatorname{End}_{\mathcal{A}}(B)\) is a division ring, any \(\operatorname{End}_{\mathcal{A}}(B)\)-module is free. For an \(\operatorname{End}_{\mathcal{A}}(B)\)-module \(M\), we write \(\dim_{\operatorname{End}_{\mathcal{A}}(B)}M\) for the rank of \(M\) as a free \(\operatorname{End}_{\mathcal{A}}(B)\)-module.
Proof of Proposition 4.1.: Let \(\phi:P\to L\) be a projective cover of a simple object. Since \(\ker\phi\) is the unique maximal subobject of \(\phi\), \(\operatorname{Hom}_{\mathcal{A}}(P,L^{\prime})=0\) whenever \(L\neq L^{\prime}\). To show equation (4.1), it remains to show that the \(\operatorname{End}_{\mathcal{A}}(L)\)-equivariant map
\[-\circ\phi:\operatorname{End}_{\mathcal{A}}(L)\to\operatorname{Hom}_{\mathcal{ A}}(P,L)\]
is an isomorphism. Since \(\phi\) is a surjection this map is injective. To show surjectivity, let \(f\in\operatorname{Hom}_{\mathcal{A}}(P,L)\) be nonzero. Then as \(\ker f\) is a maximal subobject of \(P\), \(\ker\phi\subset\ker f\), and so \(f\) factors through \(\phi\).
Conversely, if (4.1) holds, then if \(P=P_{1}\oplus P_{2}\), only one \(P_{i}\) can have a simple quotient and the other must be zero. In particular, \(P\) is indecomposable.
### Ext-finiteness
To state the main result of this section (Theorem 4.9) we need the concept of _\(\operatorname{Ext}\)-finiteness_. In this section we recall this definition and give two results about Ext-finiteness (Propositions 4.3 and 4.4) that will be needed in the discussion following Theorem 4.9.
For \(k\in\mathbb{N}\), say that an abelian category \(\mathcal{A}\) is _\(\operatorname{Ext}^{k}\)-finite_ if for any simple objects \(A,B\) in \(\mathcal{A}\),
\[\dim_{\operatorname{End}_{\mathcal{A}}(B)}\operatorname{Ext}^{k}_{\mathcal{A} }(A,B)<\infty.\]
Note that if \(\mathcal{A}\) is a \(\Bbbk\)-linear category, for some field \(\Bbbk\), then
\[\dim_{\operatorname{End}_{\mathcal{A}}(B)}\operatorname{Ext}^{k}_{\mathcal{A} }(A,B)=\frac{\dim_{\Bbbk}\operatorname{Ext}^{k}_{\mathcal{A}}(A,B)}{\dim_{ \Bbbk}\operatorname{End}_{\mathcal{A}}(B)}.\]
So \(\mathcal{A}\) is \(\operatorname{Ext}^{k}\)-finite whenever \(\dim_{\Bbbk}\operatorname{Ext}^{k}_{\mathcal{A}}(A,B)<\infty\) for every simple object \(A,B\). The converse is true if the endomorphism ring of every simple object has finite \(\Bbbk\)-dimension (e.g. if \(\Bbbk\) is algebraically closed).
The following two propositions give useful criteria for a category to be \(\operatorname{Ext}^{k}\)-finite.
**Proposition 4.3**.: _Any abelian length category with enough projectives is \(\operatorname{Ext}^{k}\)-finite for every \(k\in\mathbb{N}\)._
Proof.: Let \(\mathcal{A}\) be an abelian length category. Let \(X\) be an object in \(\mathcal{A}\), and let \(Y\) be a simple object in \(\mathcal{A}\). Consider a projective presentation of X:
\[0\to K\to P\to X\to 0\]
Since \(\operatorname{Ext}^{k}_{\mathcal{A}}(P,Y)=0\) there is a \(\operatorname{End}_{\mathcal{A}}(Y)\)-equivariant surjection of \(\operatorname{Ext}^{k-1}_{\mathcal{A}}(K,Y)\) onto \(\operatorname{Ext}^{k}_{\mathcal{A}}(X,Y)\) for each \(k>0\). Since \(\mathcal{A}\) is a length category,
\[\dim_{\operatorname{End}_{\mathcal{A}}(Y)}\operatorname{Hom}_{\mathcal{A}}(X,Y )<\infty.\]
The result follows by induction.
Say that a \(\Bbbk\)-linear abelian category is _finite over \(\Bbbk\)_ if \(\mathcal{A}\) is a length category with enough projectives, finitely many simple objects, and finite dimensional Homspaces. It is well-known that \(\mathcal{A}\) is finite over \(\Bbbk\) if and only if there is a finite-dimensional \(\Bbbk\)-algebra \(A\) in which \(\mathcal{A}\) is equivalent to the category, \(A\)-mod, of modules that are finite dimensional as \(\Bbbk\)-vector spaces. Indeed, if \(\{P_{\lambda}\}_{\lambda\in\Lambda}\) are the projective indecomposables in \(\mathcal{A}\) (up to isomorphism), then \(A=\operatorname{End}_{\mathcal{A}}(\bigoplus_{\lambda\in\Lambda}P_{\lambda})^{op}\) and \(\operatorname{Hom}_{\mathcal{A}}(\bigoplus_{\lambda\in\Lambda}P_{\lambda},-): \mathcal{A}\simeq A\text{-mod}\). Note that there is a contravariant equivalence \(\operatorname{Hom}_{\Bbbk}(-,\Bbbk):A\text{-mod}\to A^{op}\text{-mod}\). In particular, any finite abelian category has enough injectives.
**Proposition 4.4**.: _Let \(\Bbbk\) be a field and let \(\mathcal{A}\) be a \(\Bbbk\)-linear abelian category with a stratification by a finite poset in which every strata category is a finite abelian category. Then \(\mathcal{A}\) is \(\operatorname{Ext}^{1}\)-finite._
Proof.: By the assumptions on the strata categories, \(\mathcal{A}\) has finite dimensional Homspaces. Suppose \(\mathcal{A}\) has a recollement with objects and morphisms as in (3.1). Suppose \(\mathcal{A}_{U}\) and \(\mathcal{A}_{Z}\) have enough projectives, and \(\mathcal{A}_{U}\) has enough injectives. By Proposition 4.3, both \(\mathcal{A}_{Z}\) and \(\mathcal{A}_{U}\) are \(\operatorname{Ext}^{1}\)-finite. We show that \(\mathcal{A}\) is \(\operatorname{Ext}^{1}\)-finite. It suffices to show that \(\dim_{\Bbbk}\operatorname{Ext}^{1}_{\mathcal{A}}(X,Y)<\infty\) for all simple objects \(X,Y\).
Since \(\mathcal{A}^{Z}\) is a Serre subcategory of \(\mathcal{A}\), this is true whenever \(X\) and \(Y\) are both in \(\mathcal{A}^{Z}\). Let \(L\in\mathcal{A}_{U}\) be simple and let \(j_{!*}L\) have projective and injective presentations:
\[0\to K\to j_{!}P\to j_{!*}L\to 0\] \[0\to j_{!*}L\to j_{*}I\to K^{\prime}\to 0\]
The projective presentation implies that \(\operatorname{Hom}_{\mathcal{A}}(K,Y)\) surjects onto \(\operatorname{Ext}^{1}_{\mathcal{A}}(j_{!*}L,Y)\). The injective presentation implies that \(\operatorname{Hom}_{\mathcal{A}}(Y,K^{\prime})\) surjects onto \(\operatorname{Ext}^{1}_{\mathcal{A}}(Y,j_{!*}L)\). It follows that \(\mathcal{A}\) is \(\operatorname{Ext}^{1}\)-finite. The result follows by an induction argument.
### Main results
This section includes our original results about recollements and projective covers.
**Proposition 4.5**.: _Let \(\mathcal{A}\) be an abelian category with a recollement of abelian categories as in (3.1). Then_
1. _If_ \(X\to Y\) _is an essential surjection in_ \(\mathcal{A}\) _and_ \(i^{*}Y=0\) _then_ \(i^{*}X=0\) _and_ \(j^{*}X\to j^{*}Y\) _is an essential surjection._
2. _If_ \(P\in\mathcal{A}\) _is projective and_ \(i^{*}P=0\) _then_ \(j^{*}P\in\mathcal{A}_{U}\) _is projective. In particular if_ \(P\to X\) _is a projective cover in_ \(\mathcal{A}\) _and_ \(i^{*}X=0\) _then_ \(j^{*}P\to j^{*}X\) _is a projective cover in_ \(\mathcal{A}_{U}\)_._
_In particular, if \(\mathcal{A}\) has enough projectives then so does \(\mathcal{A}_{U}\)._
Proof.: Let \(\phi:X\to Y\) be an essential surjection in \(\mathcal{A}\) and suppose \(i^{*}Y=0\). To show that \(i^{*}X=0\) it suffices to show that the canonical map \(\epsilon_{X}:j_{!}j^{*}X\to X\) is surjective. This follows from the following commutative diagram since \(\phi\) is essential.
Let \(\alpha:X^{\prime}\to j^{*}X\) be a morphism in \(\mathcal{A}_{U}\), in which \(j^{*}(\phi)\circ\alpha:X^{\prime}\to j^{*}Y\) is surjective. Then \(\epsilon_{Y}\circ j_{!}j^{*}(\phi)\circ j_{!}\alpha:j_{!}X^{\prime}\to Y\) is surjective and so (since \(\phi\) is essential) \(\epsilon_{X}\circ j_{!}\alpha:j_{!}X^{\prime}\to X\) is surjective. Hence \(j^{*}(\epsilon_{X}\circ j_{!}\alpha)\simeq\alpha:X^{\prime}\to j^{*}X\) is surjective. This proves (i).
If \(P\in\mathcal{A}\) is projective and \(i^{*}P=0\) then the functor
\[\operatorname{Hom}_{\mathcal{A}_{U}}(j^{*}P,-)\simeq\operatorname{Hom}_{ \mathcal{A}}(j_{!}j^{*}P,j_{!}(-))\simeq\operatorname{Hom}_{\mathcal{A}}(P,j_ {!}(-)):\mathcal{A}_{U}\to\mathbb{Z}\text{-mod}\]
is exact. Here the last isomorphism follows from the sequence (2.6). It follows that \(j^{*}P\) is projective. Statement (ii) follows.
**Proposition 4.6**.: _Suppose \(\mathcal{A}\) is a Krull-Schmidt category with a recollement of abelian categories as in (3.1). If \(P\to L\) is a projective cover in \(\mathcal{A}\) of a simple object \(L\in\mathcal{A}^{Z}\), then \(i^{*}P\to i^{*}L\) is a projective cover in \(\mathcal{A}_{Z}\). In particular, if \(\mathcal{A}\) has enough projectives then so does \(\mathcal{A}_{Z}\)._
Proof.: Since \(i^{*}\) is the left adjoint of an exact functor it preserves projective objects. For any simple object \(L^{\prime}\in\mathcal{A}^{Z}\), \(\operatorname{Hom}_{\mathcal{A}_{Z}}(i^{*}P,i^{*}L^{\prime})=\operatorname{ Hom}_{\mathcal{A}}(P,i_{*}i^{*}L^{\prime})=\operatorname{Hom}_{\mathcal{A}}(P,L^{ \prime})\). The result follows.
**Proposition 4.7**.: _Let \(\mathcal{A}\) be an abelian category with a recollement of abelian categories as in (3.1). Let \(X\) and \(Y\) be objects in \(\mathcal{A}_{U}\). If \(X\to Y\) is an essential surjection in \(\mathcal{A}_{U}\) then the composition \(j_{!}X\to j_{!*}X\to j_{!*}Y\) is an essential surjection in \(\mathcal{A}\). In particular:_
1. _The canonical surjection_ \(j_{!}X\to j_{!*}X\) _is essential._
2. _If_ \(P\to X\) _is a projective cover of_ \(X\) _in_ \(\mathcal{A}_{U}\) _then_ \(j_{!}P\to j_{!}X\to j_{!*}X\) _is a projective cover of_ \(j_{!*}X\) _in_ \(\mathcal{A}\)_._
Proof.: Let \(\phi:X\to Y\) be an essential surjection in \(\mathcal{A}_{U}\). The map \(\phi^{\prime}:j_{!}X\to j_{!*}X\to j_{!*}Y\) is surjective by Proposition 3.5(i). Let \(\alpha:X^{\prime}\to X\) be a morphism in which \(\phi^{\prime}\circ\alpha\) is surjective. Now, \(j^{*}(\phi^{\prime})=\phi:X\to Y\) and since \(j^{*}\) is exact, \(j^{*}(\phi^{\prime}\circ\alpha)=\phi\circ j^{*}(\alpha):j^{*}X^{\prime}\to Y\) is surjective. Since \(\phi\) is essential it follows that \(j^{*}(\alpha):j^{*}X^{\prime}\to X\) is surjective in \(\mathcal{A}_{U}\) and so \(j_{!}j^{*}(\alpha):j_{!}j^{*}X^{\prime}\to j_{!}X\) is surjective in \(\mathcal{A}\). The surjectivity of \(\alpha\) follows from the commutative triangle
in which the downward arrow is the adjunction counit.
The following result holds by an almost identical argument.
**Proposition 4.8**.: _The intermediate-extension functor preserves essential surjections and essential extensions._
The following is the main result of this section.
**Theorem 4.9**.: _Let \(\mathcal{A}\) be an abelian length category with finitely many simple objects, and a recollement of abelian categories as in (3.1). If \(\mathcal{A}\) is \(\operatorname{Ext}^{1}\)-finite then \(\mathcal{A}\) has enough projectives if and only if both \(\mathcal{A}_{U}\) and \(\mathcal{A}_{Z}\) have enough projectives. Dually if \(\mathcal{A}^{op}\) is \(\operatorname{Ext}^{1}\)-finite then \(\mathcal{A}\) has enough injectives if and only if both \(\mathcal{A}_{U}\) and \(\mathcal{A}_{Z}\) have enough injectives._
Before giving the proof of this theorem we will explain one important ingredient: the _universal extension_. Let \(A,B\) be objects in an abelian category \(\mathcal{A}\) in which \(\operatorname{End}_{\mathcal{A}}(B)\) is a division ring and \(d:=\dim_{\operatorname{End}_{\mathcal{A}}(B)}\operatorname{Ext}^{1}_{ \mathcal{A}}(A,B)<\infty\). We form the _universal extension_\(\mathcal{E}\in\operatorname{Ext}^{1}_{\mathcal{A}}(A,B^{\oplus d})\) by the following process. First let \(E_{1},\ldots,E_{d}\in\operatorname{Ext}^{1}_{\mathcal{A}}(A,B)\) be an \(\operatorname{End}_{\mathcal{A}}(B)\)-basis. The diagonal map \(\Delta:A\to A^{\oplus d}\) induces a map \(\operatorname{Ext}^{1}_{\mathcal{A}}(A^{\oplus d},B^{\oplus d})\to \operatorname{Ext}^{1}_{\mathcal{A}}(A,B^{\oplus d})\). Let \(\mathcal{E}\) be the image of \(E_{1}\oplus\cdots\oplus E_{d}\) under this map. Note that the \(\operatorname{End}_{\mathcal{A}}(B)\)-equivariant map \(\operatorname{Hom}_{\mathcal{A}}(B^{\oplus d},B)\to\operatorname{Ext}^{1}_{ \mathcal{A}}(A,B)\) induced by the short exact sequence
\[0\to B^{\oplus d}\to\mathcal{E}\to A\to 0\]
is surjective (this is easy to check on the basis of \(\operatorname{Ext}^{1}_{\mathcal{A}}(A,B)\)).
When \(B_{1},\ldots,B_{n}\) are objects in \(\mathcal{A}\) in which each ring \(\operatorname{End}_{\mathcal{A}}(B_{i})\) is a division ring and \(d_{i}:=\dim_{\operatorname{End}_{\mathcal{A}}(B_{i})}\operatorname{Ext}^{1}_{ \mathcal{A}}(A,B_{i})<\infty\), we also talk about a _universal extension_\(\mathcal{E}\in\operatorname{Ext}^{1}_{\mathcal{A}}(A,\bigoplus_{i}B^{\oplus d _{i}}_{i})\) constructed in the following way. Let \(\mathcal{E}_{i}\in\operatorname{Ext}^{1}_{\mathcal{A}}(A,B^{\oplus d_{i}}_{i})\) be a universal extension (as defined in the previous paragraph) and define \(\mathcal{E}\) to be the image of \(\mathcal{E}_{1}\oplus\cdots\oplus\mathcal{E}_{n}\) under the map \(\operatorname{Ext}^{1}_{\mathcal{A}}(\bigoplus_{i}A,\bigoplus_{i}B^{\oplus d _{i}})\to\operatorname{Ext}^{1}_{\mathcal{A}}(A,\bigoplus_{i}B^{\oplus d_{i}})\) induced by the diagonal map \(\Delta:A\to A^{\oplus d}\). Then \(\mathcal{E}\) has the property that the short exact sequence
\[0\to\bigoplus_{i}B^{\oplus d_{i}}\to\mathcal{E}\to A\to 0\]
induces a surjection \(\operatorname{Hom}_{\mathcal{A}}(\bigoplus_{i}B^{\oplus d_{i}}_{i},B_{j})\to \operatorname{Ext}^{1}_{\mathcal{A}}(A,B_{j})\) for each \(j=1,\ldots,n\).
Dually, if \(\operatorname{End}_{\mathcal{A}}(A)^{op}\) is a division ring and \(\dim_{\operatorname{End}_{\mathcal{A}}(A)^{op}}\operatorname{Ext}^{1}(A,B)<\infty\) then one can form a universal extension \(\mathcal{E}^{\prime}\in\operatorname{Ext}^{1}_{\mathcal{A}}(A^{\otimes d},B)\) using the codiagonal map \(\delta:B^{\oplus d}\to B\) instead of the diagonal map.
Proof of Theorem 4.9.: Suppose that \(\mathcal{A}_{U}\) and \(\mathcal{A}_{Z}\) have enough projectives.
Suppose \(\mathcal{A}^{Z}\) has simple objects \(L_{1},\ldots,L_{m}\) with projective covers \(\bar{P}_{1},\ldots,\bar{P}_{m}\) in \(\mathcal{A}^{Z}\). Suppose \(\mathcal{A}^{U}\) has simple objects \(L_{m+1},\ldots,L_{m+n}\). By Proposition 4.7 every simple object in \(\mathcal{A}^{U}\) has a projective cover in \(\mathcal{A}\). It suffices to construct a projective cover in \(\mathcal{A}\) of each simple object in \(\mathcal{A}^{Z}\). This amounts to finding, for each \(1\leq t\leq m\), a projective object, \(P_{t}\), whose unique simple quotient is \(L_{t}\).
Fix \(1\leq t\leq m\).
_Step 1. Define \(P_{t}\)._ For simple object \(L_{m+k}\in\mathcal{A}^{U}\), let \(P_{m+k}\) denote its projective cover in \(\mathcal{A}\). Define \(Q\) to be a maximal length quotient of
\[P:=\bigoplus_{k=1}^{n}P_{m+k}^{\oplus\dim_{\operatorname{End}_{\mathcal{A}}(L_{m +k})}\operatorname{Ext}^{1}_{\mathcal{A}}(\bar{P}_{t},L_{m+k})}\]
in which there is an extension
\[0\to Q\to\mathcal{E}\to\bar{P}_{t}\to 0 \tag{4.2}\]
inducing an isomorphism \(\operatorname{Hom}_{\mathcal{A}}(Q,L)\simeq\operatorname{Ext}^{1}_{\mathcal{A}}( \bar{P}_{t},L)\) for each \(L\in\mathcal{A}^{U}\). That is \(0=\operatorname{Hom}_{\mathcal{A}}(\bar{P}_{t},L)\simeq\operatorname{Hom}_{ \mathcal{A}}(\mathcal{E},L)\) and \(\operatorname{Ext}^{1}_{\mathcal{A}}(\mathcal{E},L)\) injects into \(\operatorname{Ext}^{1}_{\mathcal{A}}(Q,L)\).
Let \(P_{t}\) be any choice of such \(\mathcal{E}\).
_Step 2. \(P_{t}\) is well-defined._ To show that the maximal quotient \(Q\) exists, we just need to find one quotient of \(P\) with the required property. Then since \(\mathcal{A}\) is a length category there exists a maximal length quotient with the required property. Since \(\mathcal{A}\) has finite \(\operatorname{Ext}^{1}\)-spaces, we can let
\[R=\bigoplus_{k=1}^{n}L_{m+k}^{\oplus\operatorname{dim}_{\operatorname{End}_{ \mathcal{A}}(L_{m+k})}\operatorname{Ext}^{1}_{\mathcal{A}}(\bar{P}_{t},L_{m+k})}\]
and form the universal extension
\[0\to R\to\mathcal{E}\to\bar{P}_{t}\to 0\]
Since this is a universal extension it induces a surjection \(\operatorname{Hom}_{\mathcal{A}}(R,L_{m+k})\to\operatorname{Ext}^{1}_{ \mathcal{A}}(\bar{P}_{t},L_{m+k})\) for each \(L_{m+k}\in\mathcal{A}^{U}\). This map is an isomorphism since
\[\operatorname{dim}_{\operatorname{End}_{\mathcal{A}}(L_{m+k})}\operatorname{ Hom}_{\mathcal{A}}(R,L_{m+k})=\operatorname{dim}_{\operatorname{End}_{ \mathcal{A}}(L_{m+k})}\operatorname{Ext}^{1}_{\mathcal{A}}(\bar{P}_{t},L_{m+k}).\]
_Step 3. \(P_{t}\) has unique simple quotient \(L_{t}\)._ By definition of \(P_{t}\) and by the \((i^{*},i_{*})\)-adjunction, for any simple module \(L\in\mathcal{A}\):
\[\operatorname{Hom}_{\mathcal{A}}(P_{t},L)\simeq\operatorname{Hom}_{\mathcal{ A}}(\overline{P}_{t},L)\simeq\operatorname{Hom}_{\mathcal{A}}(\overline{P}_{t},i_{*}i^{* }L)\]
and so the only simple quotient of \(P_{t}\) is \(L_{t}\) with multiplicity one.
_Step 4. \(P_{t}\) is projective._ We show that \(\operatorname{Ext}^{1}_{\mathcal{A}}(P_{t},L)=0\) for each simple \(L\in\mathcal{A}\). For any simple \(L\in\mathcal{A}\) there is an exact sequence
\[0\to\operatorname{Ext}^{1}_{\mathcal{A}}(P_{t},L)\to\operatorname{Ext}^{1}_{ \mathcal{A}}(Q,L)\to\operatorname{Ext}^{2}_{\mathcal{A}}(\bar{P}_{t},L) \tag{4.3}\]
Indeed if \(L\in\mathcal{A}^{U}\) then this holds because \(\operatorname{Hom}_{\mathcal{A}}(Q,L)\simeq\operatorname{Ext}^{1}_{\mathcal{A}} (\bar{P}_{t},L)\). If \(L\in\mathcal{A}^{Z}\) then (4.3) holds since \(\operatorname{Ext}^{1}_{\mathcal{A}}(\bar{P}_{t},L)=0\).
To show that \(P_{t}\) is projective it suffices to show that the third map in (4.3) is injective for any \(L\in\mathcal{A}\). Suppose for contradiction that there is a nontrivial extension
\[0\to L\to Q^{\prime}\to Q\to 0 \tag{4.4}\]
in the kernel of this map. Then there is an object \(\mathcal{E}\in\mathcal{A}\) fitting into the following diagram
(4.5)
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in which each row and column is exact. For each \(L^{\prime}\in\mathcal{A}^{U}\) the sequence (4.4) induces an exact sequence
\[0\to\operatorname{Hom}_{\mathcal{A}}(Q,L^{\prime})\to\operatorname{Hom}_{ \mathcal{A}}(Q^{\prime},L^{\prime})\to\operatorname{Hom}_{\mathcal{A}}(L,L^{ \prime})\]
Of course, \(\operatorname{Hom}_{\mathcal{A}}(L,L^{\prime})=0\) if \(L\neq L^{\prime}\). If \(L=L^{\prime}\) the third map must be zero. Indeed if \(f:L\to L\) factors through the inclusion \(\iota:L\to Q^{\prime}\) via a map \(g:Q^{\prime}\to L\), then \(f^{-1}\circ g:Q^{\prime}\to L\) is a retraction of \(\iota\). This contradicts the assumption that (4.4) does not split. Hence, for any \(L^{\prime}\in\mathcal{A}^{U}\), there is an isomorphism
\[\operatorname{Hom}_{\mathcal{A}}(Q^{\prime},L^{\prime})\simeq\operatorname{ Hom}_{\mathcal{A}}(Q,L^{\prime})\simeq\operatorname{Ext}_{\mathcal{A}}^{1}( \bar{P}_{t},L^{\prime}). \tag{4.6}\]
Since \(P\) is projective the quotient \(P\to Q\) fits into the diagram
Now \(\varphi\) cannot be surjective, as, by (4.6), this would contradict the maximality of \(Q\). Thus the image of \(\varphi\) is isomorphic to \(Q\) and so the sequence (4.4) splits. This is a contradiction. Hence \(P_{t}\) is projective.
**Corollary 4.10**.: _Let \(\mathcal{A}\) be an abelian category with a stratification in which every strata category is a length category, and has finitely many simple objects._
_If \(\mathcal{A}\) is \(\operatorname{Ext}^{1}\)-finite (respectively \(\mathcal{A}^{op}\) is \(\operatorname{Ext}^{1}\)-finite) then \(\mathcal{A}\) has enough projectives (respectively injectives) if and only if every strata category has enough projectives (respectively injectives)._
Proof.: By Proposition 3.6, every category \(\mathcal{A}_{\Lambda^{\prime}}\) (for lower \(\Lambda^{\prime}\subset\Lambda\)) satisfies the conditions of Theorem 4.9. So we can obtain a projective cover in \(\mathcal{A}\) of any simple object \(j_{!*}^{\lambda}L\) by repeatedly applying the construction in the proof of Theorem 4.9 to larger and larger Serre subcategories of \(\mathcal{A}\).
The following result follows immediately from Proposition 4.4 and Corollary 4.10.
**Corollary 4.11**.: _Let \(\mathcal{A}\) be a \(\Bbbk\)-linear abelian category with a stratification by a finite poset. Then \(\mathcal{A}\) is finite over \(\Bbbk\) if and only if the same is true of every strata category._
From this result we recover the following result of Cipriani-Woolf.
**Corollary 4.12** (Corollary 5.2 of [10]).: _Let \(P_{\Lambda}(X,\Bbbk)\) be the category of perverse sheaves (with coefficients in a field \(\Bbbk\)) on a variety \(X\) that is constructible with respect to a stratification, \(X=\bigcup_{\lambda\in\Lambda}X_{\lambda}\), with finitely many strata. Then \(P_{\Lambda}(X,\Bbbk)\) is finite over \(\Bbbk\) if and only if each category \(\Bbbk[\pi_{1}(X_{\lambda})]\text{-}\mathrm{mod}_{fg}\) is finite over \(\Bbbk\)._
For example, if \(X\) has a finite stratification, \(X=\bigcup_{\lambda\in\Lambda}X_{\lambda}\), in which each \(X_{\lambda}\) has finite fundamental group, then the category \(P_{\Lambda}(X,\Bbbk)\) is finite over \(\Bbbk\).
**Corollary 4.13**.: _Let \(G\) be an algebraic group and let \(X\) be a \(G\)-variety with finitely many orbits, each connected. Let \(\Bbbk\) be a field. The category, \(P_{G}(X,\Bbbk)\), of \(G\)-equivariant perverse sheaves is finite over \(\Bbbk\) if and only if for each \(G\)-orbit \(\mathcal{O}_{\lambda}\) and \(x\in\mathcal{O}_{\lambda}\), the category \(\Bbbk[G^{x}/(G^{x})^{\circ}]\text{-}\mathrm{mod}_{fg}\) is finite over \(\Bbbk\)._
## 5. Standard and costandard objects
In this section we focus on abelian length categories \(\mathcal{A}\) with finitely many simples, enough projectives and injectives, and admitting a stratification by a finite non-empty poset \(\Lambda\). Let \(B\) be a set indexing the simple objects of \(\mathcal{A}\) up to isomorphism. Write \(L(b)\) for the simple object corresponding to \(b\in B\), and write \(P(b)\) and \(I(b)\) for the projective cover and injective envelope of \(L(b)\).
For each \(\lambda\in\Lambda\), write \(\mathcal{A}_{\leq\lambda}:=\mathcal{A}_{\{\mu\in\Lambda\ |\ \mu\leq\lambda\}}\) and let \(j_{!}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) be the composition
Define \(j_{*}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) and \(j_{!*}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) likewise. Let \(j^{\lambda}:\mathcal{A}_{\leq\lambda}\to\mathcal{A}_{\lambda}\) denote the Serre quotient functor.
Define the _stratification function_\(\rho:B\to\Lambda\) that assigns to each \(b\in B\) the \(\lambda\in\Lambda\) in which \(L(b)=j_{!*}^{\lambda}L_{\lambda}(b)\). Let \(P_{\lambda}(b)\) and \(I_{\lambda}(b)\) be the projective cover and injective envelope of the simple object \(L_{\lambda}(b)\) in \(\mathcal{A}_{\lambda}\).
For \(b\in B\), define the _standard_ and _costandard_ objects
\[\Delta(b):=j_{!}^{\lambda}P_{\lambda}(b),\qquad\nabla(b):=j_{*}^{\lambda}I_{ \lambda}(b),\]
where \(\lambda=\rho(b)\).
The following original result follows from the proofs of Theorem 4.9 and Corollary 4.10.
**Porism 5.1**.: _Let \(\mathcal{A}\) be an abelian category with a stratification by a finite non-empty poset \(\Lambda\), in which every strata category is a length category with finitely many simple objects. Let \(\rho:B\to\Lambda\) be the stratification function for \(\mathcal{A}\)._
_If \(\mathcal{A}\) has enough projectives then for each \(\lambda\in\Lambda\) and \(b\in\rho^{-1}(\lambda)\), the projective indecomposable object \(P(b)\in\mathcal{A}\) fits into a short exact sequence_
\[0\to Q(b)\to P(b)\to\Delta(b)\to 0\]
_in which \(Q(b)\) has a filtration by quotients of \(\Delta(b^{\prime})\) satisfying \(\rho(b^{\prime})>\rho(b)\)._
_If \(\mathcal{A}\) has enough injectives then for each \(\lambda\in\Lambda\) and \(b\in\rho^{-1}(\lambda)\), the injective indecomposable object \(I(b)\in\mathcal{A}\) fits into a short exact sequence_
\[0\to\nabla(b)\to I(b)\to Q^{\prime}(b)\to 0\]
_in which \(Q^{\prime}(b)\) has a filtration by subobjects of \(\nabla(b^{\prime})\) satisfying \(\rho(b^{\prime})>\rho(b)\)._
Proof.: We just prove the first statement by induction on \(|\Lambda|\). The base case \(|\Lambda|=1\) is trivial.
Consider the projective cover, \(P(b)\), of simple object \(L(b)\) in \(\mathcal{A}\). If \(\rho(b)\) is maximal then \(P(b)\simeq\Delta(b)\) and the result holds. Suppose \(\rho(b)\) is not maximal, and let \(\mu\in\Lambda\) be a maximal element. Consider the recollement
Let \(P_{<\mu}(b)\) and \(\Delta_{<\mu}(b)\) be the projective indecomposable and standard object in \(\mathcal{A}_{<\mu}\) corresponding to the simple object \(i^{*}L(b)\in\mathcal{A}_{<\mu}\). By induction there is a short exact sequence
\[0\to Q_{<\mu}(b)\to P_{<\mu}(b)\to\Delta_{<\mu}(b)\to 0\]
in which \(Q_{<\mu}(b)\) has a filtration by quotients of standard objects \(\Delta_{<\mu}(b^{\prime})\) satisfying \(\rho(b^{\prime})>\rho(b)\). Since \(i_{*}\) is exact we get the following short exact sequence in \(\mathcal{A}\):
\[0\to i_{*}Q_{<\mu}(b)\to i_{*}P_{<\mu}(b)\to\Delta(b)\to 0 \tag{5.1}\]
and \(i_{*}Q_{<\mu}(b)\) has a filtration by quotients of standard objects \(\Delta(b^{\prime})\) satisfying \(\mu>\rho(b^{\prime})>\rho(b)\). By applying the construction in Step 1 of the proof of Theorem 4.9, \(P(b)\) fits into the short exact sequence in \(\mathcal{A}\):
\[0\to Q_{\mu}(b)\to P(b)\to i_{*}P_{<\mu}(b)\to 0 \tag{5.2}\]
and \(Q_{\mu}(b)\) is a quotient of a direct sum of standard objects of the form \(\Delta(b^{\prime})\) in which \(\rho(b^{\prime})=\mu\). Combining (5.1) and (5.2) gives the following diagram with exact rows and columns:
The result follows.
## 6. Brundan and Stroppel's \(\varepsilon\)-stratified categories
In this section we give necessary and sufficient conditions for a finite abelian category with a stratification by a finite poset to be \(\varepsilon\)-stratified in the sense of Brundan and Stroppel [1] (Theorem 6.4). This result specializes to a characterization of highest weight categories originally given by Krause [14, Theorem 3.4]. We recall this characterisation in Corollary 6.6.
Let \(\mathcal{A}\) be an abelian length category with finitely many simples, enough projectives and injectives, and with a stratification by a finite non-empty poset \(\Lambda\). Let \(\rho:B\to\Lambda\) be the stratification function corresponding to this stratification.
For \(b\in B\) and \(\lambda=\rho(b)\in\Lambda\), define _proper standard_ and _proper costandard_ objects
\[\overline{\Delta}(b):=j_{!}^{\lambda}L_{\lambda}(b),\qquad\overline{\nabla}(b) :=j_{*}^{\lambda}L_{\lambda}(b).\]
For a _sign function_\(\varepsilon:\Lambda\to\{+,-\}\), define the _\(\varepsilon\)-standard_ and _\(\varepsilon\)-costandard objects_
\[\Delta_{\varepsilon}(b):=\left\{\begin{array}{ll}\Delta(b)&\text{if } \varepsilon(\rho(b))=+\\ \overline{\Delta}(b)&\text{if }\varepsilon(\rho(b))=-\end{array}\right., \qquad\nabla_{\varepsilon}(b):=\left\{\begin{array}{ll}\overline{\nabla}(b )&\text{if }\varepsilon(\rho(b))=+\\ \nabla(b)&\text{if }\varepsilon(\rho(b))=-\end{array}\right..\]
Note that since \(j_{!}^{\lambda}\) and \(j_{*}^{\lambda}\) are fully-faithful, these objects have local endomorphism rings and are hence irreducible. Note also that if \(\rho(b)>\rho(b^{\prime})\) then for any
\(\varepsilon:\Lambda\to\{+,-\}\),
\[\operatorname{Hom}_{\mathcal{A}}(\Delta_{\varepsilon}(b),\Delta_{\varepsilon}(b^{ \prime}))=0=\operatorname{Hom}_{\mathcal{A}}(\nabla_{\varepsilon}(b^{\prime}), \nabla_{\varepsilon}(b)). \tag{6.1}\]
Indeed the only simple quotient of \(\Delta_{\varepsilon}(b)\) is \(L(b)\), and all simple subobjects, \(L(b^{\prime\prime})\), of \(\Delta_{\varepsilon}(b^{\prime})\) satisfy \(\rho(b^{\prime})\geq\rho(b^{\prime\prime})\). Likewise the only simple subobject of \(\nabla_{\varepsilon}(b)\) is \(L(b)\), and all simple quotients, \(L(b^{\prime\prime})\), of \(\nabla_{\varepsilon}(b^{\prime})\) satisfy \(\rho(b^{\prime})\geq\rho(b^{\prime\prime})\).
The following definition is due to Brundan and Stroppel [10].4
Footnote 4: In Brundan and Stoppel’s definition, \(\varepsilon\)-stratified categories need not be finite (they satisfy slightly weaker finiteness conditions).
**Definition 6.1** (\(\varepsilon\)-stratified category).: Let \(\mathcal{A}\) be a finite abelian category with a stratification by a poset \(\Lambda\) and stratification function \(\rho:B\to\Lambda\). Let \(\varepsilon:\Lambda\to\{+,-\}\) be a function. Say that \(\mathcal{A}\) is an _\(\varepsilon\)-stratified category_ if the following equivalent conditions are satisfied:
1. For every \(b\in B\), the projective indecomposable \(P(b)\) has a filtration by objects of the form \(\Delta_{\varepsilon}(b^{\prime})\), where \(\rho(b^{\prime})\geq\rho(b)\).
2. For every \(b\in B\), the injective indecomposable \(I(b)\) has a filtration by objects of the form \(\nabla_{\varepsilon}(b^{\prime})\), where \(\rho(b^{\prime})\geq\rho(b)\).
The equivalence of statements (\(\varepsilon\)-S1) and (\(\varepsilon\)-S2) is shown in [1, Theorem 2.2]. A proof of this fact can also be found in [10, Theorem 3.5].
**Remark 6.2**.: A finite abelian category \(A\)-mod is \(\varepsilon\)-stratified if and only if \(A\) is a _stratified algebra_ in the sense of [1].
The following definition is original.
**Definition 6.3** (\(\varepsilon\)-exact stratification).: For a function \(\varepsilon:\Lambda\to\{+,-\}\), say that a stratification of an abelian category \(\mathcal{A}\) by a finite non-empty poset \(\Lambda\) is _\(\varepsilon\)-exact_ if for all \(\lambda\in\Lambda\) the following hold:
1. If \(\varepsilon(\lambda)=+\) then the functor \(j_{*}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) is exact.
2. If \(\varepsilon(\lambda)=-\) then the functor \(j_{!}^{\lambda}:\mathcal{A}_{\lambda}\to\mathcal{A}\) is exact.
For \(k\in\mathbb{N}\), say that a recollement of abelian categories (as in (3.1)) is _\(k\)-homological_ if for all \(n\leq k\) and \(X,Y\in\mathcal{A}_{Z}\),
\[\operatorname{Ext}^{n}_{\mathcal{A}_{Z}}(X,Y)\simeq\operatorname{Ext}^{n}_{ \mathcal{A}}(i_{*}X,i_{*}Y).\]
Say that a recollement of abelian categories is _homological_ if it is \(k\)-homological for all \(k\in\mathbb{N}\). A study of homological recollements is given in [10].
Say that a stratification of an abelian category is _\(k\)-homological_ (respectively _homological_) if each of the recollements in the data of the stratification is \(k\)-homological (respectively homological).
The following theorem is the main result of this section.
**Theorem 6.4**.: _Let \(\mathcal{A}\) be a finite abelian category. Then, for any finite non-empty poset \(\Lambda\) and function \(\varepsilon:\Lambda\to\{+,-\}\), the following statements are equivalent:_
1. \(\mathcal{A}\) _has an_ \(\varepsilon\)_-exact homological stratification by_ \(\Lambda\)_._
2. \(\mathcal{A}\) _has an_ \(\varepsilon\)_-exact 2-homological stratification by_ \(\Lambda\)_._
3. \(\mathcal{A}\) _is_ \(\varepsilon\)_-stratified._
To prove this theorem we need the following lemma.
**Lemma 6.5**.: _Let \(\mathcal{A}\) be an abelian length category with finitely many simple objects, and fitting into a 2-homological recollement of abelian categories as in (3.1)._
_If \(\mathcal{A}\) has enough projectives then for any projective object \(P\in\mathcal{A}\), there is a short exact sequence_
\[0\to j_{!}j^{*}P\to P\to i_{*}i^{*}P\to 0\]
_If \(\mathcal{A}\) has enough injectives then for any injective object \(I\in\mathcal{A}\), there is a short exact sequence_
\[0\to i_{*}i^{!}I\to I\to j_{*}j^{*}I\to 0\]
Proof.: Let \(P\in\mathcal{A}\) be a projective indecomposable object in \(\mathcal{A}\). If \(i^{*}P=0\) then \(j_{!}j^{*}P\simeq P\) and the result holds.
Suppose that \(i^{*}P\neq 0\). Consider the exact sequence
\[0\to K\to j_{!}j^{*}P\to P\to i_{*}i^{*}P\to 0\]
where \(K\in\mathcal{A}^{Z}\). Since \(i^{*}P\) is projective in \(\mathcal{A}_{Z}\) and the recollement is 2-homological, \(\operatorname{Ext}_{\mathcal{A}}^{2}(i_{*}i^{*}P,K)=0\). In particular, there is an object \(\mathcal{E}\in\mathcal{A}\) and surjection \(q:\mathcal{E}\to P\) that fits into the following diagram in which all rows and columns are exact.
(6.2)
Since \(P\) is projective, there is a map \(\alpha:P\to\mathcal{E}\) making the following diagram commute.
Since \(P\) is indecomposable, the endomorphism \(q\circ\alpha:P\to P\) is either nilpotent or an automorphism. Since \(i_{*}i^{*}P\) is nonzero, this map is not nilpotent, and hence is an automorphism of \(P\). In particular, the first two rows of Diagram (6.2) split. Hence \(K\) is a quotient of \(j_{!}j^{!}P\). Since \(j_{!}j^{!}P\) has no nonzero quotient in \(\mathcal{A}^{Z}\), \(K=0\). The first result follows.
The second result holds by the dual argument.
Proof of Theorem 6.4.: The implication \((1)\implies(2)\) is obvious.
For the remainder of the proof, fix a maximal element \(\lambda\in\Lambda\) and recollement
(6.3)
\((2)\implies(3)\). Let \(P(b)\in\mathcal{A}\) be a projective indecomposable object. Suppose that every projective indecomposable, \(P(b^{\prime})\), in \(\mathcal{A}_{<\lambda}\) has a filtration by \(\varepsilon\)-standard objects, \(\Delta_{\varepsilon}(b^{\prime\prime})\), in which \(\rho(b^{\prime\prime})\geq\rho(b^{\prime})\). In particular \(i_{*}i^{*}P(b)\) has a filtration by \(\varepsilon\)-standard objects, \(\Delta_{\varepsilon}(b^{\prime})\), in which \(\rho(b^{\prime})\geq\rho(b)\).
By Lemma 6.5, if the recollement (6.3) is 2-homological then \(P(b)\) fits into a short exact sequence
\[0\to j_{!}j^{*}P(b)\to P(b)\to i_{*}i^{*}P(b)\to 0\]
If \(j_{*}\) is exact then \(j^{*}P(b)\) is projective (since \(j^{*}\) is the left adjoint of an exact functor) and so \(j_{!}j^{*}P(b)\) has a filtration by standard objects. If \(j_{!}\) is exact then \(j_{!}j^{*}P(b)\) has a filtration by proper standard objects.
In particular, in either case \(\varepsilon(\lambda)=\pm\), \(P(b)\) has a filtration by \(\varepsilon\)-standard objects, \(\Delta_{\varepsilon}(b^{\prime})\), in which \(\rho(b^{\prime})\geq\rho(b)\). The result follows by induction on \(|\Lambda|\).
\((3)\implies(1)\). Brundan-Stroppel show that if \(\mathcal{A}\) is \(\varepsilon\)-stratified then \(\mathcal{A}\) has an \(\varepsilon\)-exact stratification [1, Theorem 3.5] and
\[\operatorname{Ext}^{n}_{\mathcal{A}}(\Delta_{\varepsilon}(b),\nabla_{ \varepsilon}(b^{\prime}))=0 \tag{6.4}\]
for all \(b,b^{\prime}\in B\) and \(n\in\mathbb{N}\)[1, Theorem 3.14].
Let \(\mathcal{A}\) be \(\varepsilon\)-stratified abelian length category. Let \(P\) be a projective object in \(\mathcal{A}_{<\lambda}\), and \(I\) be an injective object in \(\mathcal{A}_{<\lambda}\). Then \(P\) and \(i_{*}P\) have a filtration by \(\varepsilon\)-standard objects, and \(I\) and \(i_{*}I\) have a filtration by \(\varepsilon\)-costandard objects. Since \(\mathcal{A}\) is a length category, then it follows from (6.4) that \(\operatorname{Ext}^{n}_{\mathcal{A}}(i_{*}P,i_{*}I)=0\). Since this is true for any projective object \(P\) and injective object \(I\) in \(\mathcal{A}_{<\lambda}\), the recollement (6.3) is homological (see e.g. [16, Theorem 3.9]). The result follows by induction on \(|\Lambda|\).
### Highest weight categories
Let \(\Bbbk\) be a field. Say that a \(\Bbbk\)-linear abelian category \(\mathcal{A}\) is a _highest weight category_ with respect to a finite poset \(\Lambda\) if \(\mathcal{A}\) is finite over \(\Bbbk\), and for every \(\lambda\in\Lambda\) there is a projective indecomposable, \(P_{\lambda}\), that fits into a short exact sequence in \(\mathcal{A}\):
\[0\to U_{\lambda}\to P_{\lambda}\to\Delta_{\lambda}\to 0\]
in which the following hold:
1. \(\operatorname{End}_{\mathcal{A}}(\Delta_{\lambda})\) is a division ring for all \(\lambda\in\Lambda\).
2. \(\operatorname{Hom}_{\mathcal{A}}(\Delta_{\lambda},\Delta_{\mu})=0\) whenever \(\lambda>\mu\).
3. \(U_{\lambda}\) has a filtration by objects \(\Delta_{\mu}\) in which \(\lambda<\mu\).
4. \(\bigoplus_{\lambda\in\Lambda}P_{\lambda}\) is a projective generator of \(\mathcal{A}\).
The following characterisation of highest weight categories is shown by Krause [18, Theorem 3.4]. We give a new proof using Theorem 6.4.
**Corollary 6.6**.: _Let \(\Bbbk\) be a field, and let \(\mathcal{A}\) be a \(\Bbbk\)-linear abelian category. The following statements are equivalent._
1. \(\mathcal{A}\) _is a highest weight category._
2. \(\mathcal{A}\) _has a homological stratification with respect to_ \(\Lambda\) _in which every strata category is equivalent to_ \(\operatorname{mod}\)_-_\(\Gamma_{\lambda}\) _for some finite dimensional division algebra_ \(\Gamma_{\lambda}\)_._
3. \(\mathcal{A}\) _has a 2-homological stratification with respect to_ \(\Lambda\) _in which every strata category is equivalent to_ \(\operatorname{mod}\)_-_\(\Gamma_{\lambda}\) _for some finite dimensional division algebra_ \(\Gamma_{\lambda}\)
Proof.: \((1)\implies(2)\). If \(\mathcal{A}\) is a highest weight category with respect to \(\Lambda\), then a homological stratification of \(\mathcal{A}\) by \(\Lambda\) is constructed as follows: For a lower subposet \(\Lambda^{\prime}\subset\Lambda\) define \(\mathcal{A}_{\Lambda^{\prime}}\) to be the Serre subcategory of \(\mathcal{A}\) generated by the standard objects \(\Delta_{\lambda}\) in which \(\lambda\in\Lambda^{\prime}\). Then for any maximal \(\mu\in\Lambda^{\prime}\) there is a homological recollement of abelian categories
in which \(j^{*}=\operatorname{Hom}_{\mathcal{A}_{\Lambda^{\prime}}}(\Delta_{\mu},-)\) (see the proof of [16, Theorem 3.4]).
\((2)\implies(3).\) This is obvious.
\((3)\implies(1).\) Suppose \(\mathcal{A}\) has a \(2\)-homological stratification with strata categories of the form \(\mathcal{A}_{\lambda}=\operatorname{mod-}\Gamma_{\lambda}\) for a finite dimensional division ring \(\Gamma_{\lambda}\). Then \(\mathcal{A}\) is finite (by Corollary 4.11). Let \(L_{\lambda}\) denote the unique simple object in \(\mathcal{A}_{\lambda}\). Define \(\Delta_{\lambda}:=j_{!}^{\lambda}L_{\lambda}\) and let \(P_{\lambda}\) be the projective cover of \(j_{!*}^{\lambda}L_{\lambda}\) in \(\mathcal{A}\). Statement (HW1) holds since \(j_{!}\) is fully-faithful, Statement (HW2) is exactly equation (6.1), Statement (HW3) follows from Theorem 6.4 (since each \(j_{!}^{\lambda}\) is exact), and Statement (HW4) is obvious.
| この論文は、アーティフィケーションと呼ばれる、 smallerabelian カテゴリーへ分解可能な abelian カテゴリーを研究しています。 例としては、Brundan-Stroppel (2018) の (等価性) perversesheaves のカテゴリーや epsilon-stratified カテゴリー (特に最高重量カテゴリー) が挙げられます。 このアーティフィケーションを持つ abelian カテゴリーが有限次元モジュールのカテゴリーと等価であるための必要十分条件を、Cipriani-Woolf (2022) の主結果を一般化して示しました。 また、このアーティフィケーションを持つカテゴリーが epsilon-stratified なための必要十分条件を、Krause (2017) の最高重量カテゴリーの特性化を一般化して示しました。
Please note: This translation is a bit awkward and verbose. |
2305.01718 | A tale of two faults: Statistical reconstruction of the 1820 Flores Sea
earthquake using tsunami observations alone | Using a Bayesian approach we compare anecdotal tsunami runup observations
from the 29 December 1820 Flores Sea earthquake with close to 200,000 tsunami
simulations to determine the most probable earthquake parameters causing the
tsunami. Using a dual hypothesis of the source earthquake either originating
from the Flores Thrust or the Walanae/Selayar Fault, we found that neither
source perfectly matches the observational data, particularly while satisfying
seismic constraints of the region. However, there is clear quantitative
evidence that a major earthquake on the Walanae/Selayar Fault more closely
aligns with historical records of the tsunami, and earthquake shaking. The
simulated data available from this study alludes to the potential for a
different source in the region or the occurrence of an earthquake near where
both faults potentially merge and simultaneously rupture similar to the 2016
Kaikoura, New Zealand event. | T. Paskett, J. P. Whitehead, R. A. Harris, C. Ashcroft, J. A. Krometis, I. Sorensen, R. Wonnacott | 2023-05-02T18:34:35 | http://arxiv.org/abs/2305.01718v1 | A tale of two faults: Statistical reconstruction of the 1820 Flores Sea earthquake using tsunami observations alone
###### Abstract
Using a Bayesian approach we compare anecdotal tsunami runup observations from the 29 December 1820 Flores Sea earthquake with close to 200,000 tsunami simulations to determine the most probable earthquake parameters causing the tsunami. Using a dual hypothesis of the source earthquake either originating from the Flores Thrust or the Walanea/Selayar Fault, we found that neither source perfectly matches the observational data, particularly while satisfying seismic constraints of the region. However, there is clear quantitative evidence that a major earthquake on the Walanea/Selayar Fault more closely aligns with historical records of the tsunami, and earthquake shaking. The simulated data available from this study alludes to the potential for a different source in the region or the occurrence of an earthquake near where both faults potentially merge and simultaneously rupture similar to the 2016 Kaikoura, New Zealand event.
## 1 Introduction
A thorough understanding of seismic history in tectonically active regions is necessary to determine the risk of future seismic hazards. The challenge of this need is that faults have seismic events at time scales that stretch back well beyond only a century of instrumental records. It is for this reason that there has been a focused effort to quantify past seismic events even though some observations may be unreliable Newcomb & McCann (1987); Sieh et al. (2008); Meltzner et al. (2010, 2012, 2015); Jankaew et al. (2008); Monecke et al. (2008); Bondevik (2008); Bryant et al. (2007); Grimes (2006); Reid (2016); Barkan & Ten Brink (2010); Tanioka & Sataka (1996); Nanayama et al. (2003); Liu & Harris (2014); Harris & Major (2016); Fisher & Harris (2016); Griffin et al. (2018); Martin et al. (2019); Ringer et al. (2021). A significant concern with the reconstruction of these historical events is the inherent uncertainty that is unavoidably tied to the nature of the observations themselves. Following the work of Ringer et al. (2021); Krometis et al. (2021) we apply a Bayesian framework to the task of quantitatively estimating the size and location of the Flores Sea Earthquake from 1820 that resulted in a devastating tsunami that was witnessed in four places throughout the Flores Sea region (Fig. 1).
As shown in Ringer et al. (2021); Krometis et al. (2021) the Bayesian approach provides a statistically justified method to generate several thousands of tsunami simulations, to determine the most probable source of the observed tsunami. Not only does this approach provide a phenomenological approach toward sampling the earthquake parameter space, but it also automatically yields estimates on the uncertainty in those estimates as demonstrated below. In the language of statistical inference, we are able to construct a posterior distribution on the set of earthquake parameters that best yields the observed tsunami characteristics. This posterior distribution provides far more information than simply specifying a single earthquake that best fits the observational data, but is actually a probability distribution on all potentially valid parameters, thus specifying correlations between the different parameters of the earthquake. In addition the resultant simulated data provides a quantitative probabilistic assessment for the danger posed by a repeated tsunamigenic event of the same magnitude.
The focus of this article is on the December 29, 1820 earthquake that rocked SW Sulawesi leading to a devastating tsunami. Historical records Wichmann (1918, 1922) document that the tsunami destroyed much of the settlement near Bulukumba on the SW arm of Sulawesi, and severely damaged the port city of Bima, Sumbawa over 300 kilometers away, as well as causing some tidal disturbance as far away as Sumenep on Madura Island off the NE coast of Java (Fig. 1). For observations of this event, we rely on translations of the Wichmann catalog Wichmann (1918, 1922); Harris and Major (2016), which details earthquakes and tsunamis of the Indonesian archipelago for parts of the 17th, 18th, and 19th centuries. This particular event is of significant interest seismically as there are two potential sources of the earthquake: the Flores back-arc thrust (a hypothesis that is investigated in Griffin et al. (2018) for shaking observations of this event), and the more recently quiescent Walanae/Selayar Fault that parallels Selayar Island. The impacts of such a major earthquake at either location on modern society would be devastating. However, it is critically important to determine which of these faults was the source of the 1820 event in order to gauge the potential for future seismic hazards, particularly since there is evidence of Quaternary deformation of the Selayar Island region, but no significant instrumental earthquakes. After constructing two posterior distributions, one for each potential fault source we quantitatively demonstrate that the Walanae/Selayar fault statistically is a far better fit to the observational data although it does not match the data perfectly.
The rest of this article proceeds as follows: The next section briefly reviews the Bayesian approach, and discusses the construction of prior distributions for the earthquake parameters i.e. models of the two disparate faults under consideration. Section 3 discusses the formation of a likelihood model that includes using the tsunami propagation model Geoclaw and the construction of the observational probability distributions. Section 4 presents the results of the 200,000+ tsunami simulations including the use of a binary classification scheme to quantitatively determine that the Walanae/Selayar Fault is 90% more likely to yield a match with modeled observations than the Flores thrust. Finally Section 5 concludes with a brief discussion and explanation of the hypothesis of a multi-fault rupture and/or presence of an underwater landslide near Bulukumba.
## 2 Bayesian Inverse Problems and Construction of the Prior Distribution
For the purposes of the current discussion, we briefly recall the basis for Bayes' Theorem and the use of Markov Chain Monte Carlo (MCMC) in identifying the earthquake parameters most likely associated with the 1820 event. Rather than reviewing all of the details, we will provide a succinct summary and focus on those aspects of the inverse problem particular to this event. A more detailed description of the approach taken here is provided in Krometis et al. (2021), and more generally in Gelman et al. (2014); Kaipio and Somersalo (2005). We do, however focus on the application of Bayes' Theorem to the problem at hand, determining a reasonable probability distribution on parameters meant to model an earthquake given statistical observations of the resultant tsunami wave height and arrival time at different locations.
### Earthquake parameterization
For earthquake induced tsunamis, we will consider earthquakes parameterized by the Okada model Okada (1985, 1992) which is dictated by a set of 9 model parameters in three distinct categories:
1. Magnitude (Mw): * length \(l\): the horizontal length of a rectangular rupture zone (typically measured in kilometers). * width \(w\): the width of the same rectangular rupture zone (typically measured in kilometers). * slip \(s\): the amount of movement the rectangular rupture zone sustained during the seismic event (typically measured in meters). Our model will assume a uniform slip distribution throughout the entire rectangular region.
The magnitude of the event can roughly be calculated as the logarithm of the product of these three variables. We will specifically use the rectangular Okada model so that all ruptures are assumed to be adequately described by a series of connecting rectangles.
2. Location: * latitude \(lat\): latitude coordinate of the earthquake centroid. * longitude \(lon\): longitude coordinate of the earthquake centroid. * depth \(d\): depth below the surface of the earth at which the centroid of the rupture occurs (typically measured in kilometers). We assume that the fault ruptures instantaneously so that the epicenter and centroid are identical. Further parameterization of a time-dependent, variable slip rupture is possible with the Okada model but we do not anticipate that our data is sufficiently robust to infer details for such a model.
3. Orientation/geometry: * strike \(\alpha\): orientation of the fault measured clockwise in degrees from north. * dip \(\beta\): angle of inclination of the fault from horizontal. * rake \(\gamma\): slip angle in degrees that the upper block of a fault (Hangingwall) moves relative to the strike angle, i.e. a rake of \(90^{\circ}\) corresponds to hanging wall slip up the fault parallel to the dip direction, which is a thrust fault.
### Bayesian inversion and MCMC
Referring to all of these model parameters as the vector \(\tilde{x}=(l,w,s,lat,lon,d,\alpha,\beta,\gamma)^{T}\), our goal is to determine a distribution on these nine parameters that best describes the 1820 earthquake by matching the historical record and our understanding of earthquake structure most closely. Hence we seek to identify, or at least approximate the conditional probability distribution \(\pi\left(\tilde{x}|\mathcal{O}\right)\), where \(\mathcal{O}\) are the historical observations that we have gleaned from the Wichmann catalog. In other words we are going to approximate the probability of a specific set of Okada earthquake parameters, given the observations from the historical record.
The natural way to compute \(\pi(\tilde{x}|\mathcal{O})\) is to apply Bayes' Theorem which states that this _posterior_ probability is proportional to the product of a _prior_\(\pi(\tilde{x})\) and _likelihood_\(L(\mathcal{O}|\tilde{x})\), i.e.
\[\pi(\tilde{x}|\mathcal{O})\propto\pi(\tilde{x})L(\mathcal{O}|\tilde{x}). \tag{1}\]
The prior \(\pi(\tilde{x})\) is a distribution that represents the _a priori_ expert knowledge of the potential distribution of earthquake parameters before examining the observational data, and \(L(\mathcal{O}|\tilde{x})\) represents the likelihood of the historical observations occurring given a specific set of earthquake parameters \(\tilde{x}\). Specifying the prior and likelihood will then fully describe the desired posterior distribution.
The discussion above details the computation of the relative posterior probability for a particular set of parameters. Approximating the full posterior distribution is a much more difficult task as (1) is only a relative proportionality i.e. the normalization of the full distribution is not available. To adequately approximate the full distribution we use Markov Chain Monte Carlo Gelman et al. (2014); Kaipio & Somersalo (2005) which generates a Markov chain whose stationary distribution converges to the desired posterior. For the computations performed here, we have utilized a random walk proposal kernel that takes randomized steps in parameter space between each proposed earthquake. For example, suppose that we are currently considering earthquake parameters \(\tilde{x}_{k}\), and have computed the prior probability \(\pi(\tilde{x}_{k})\), and after passing these parameters through our forward model (discussed below), also computed the likelihood \(L(\mathcal{O}|\tilde{x}_{k})\).
1. A new set of earthquake parameters \(\tilde{y}=\tilde{x}_{k}+\eta\) (referred to as the proposal) is proposed where \(\eta\) is a random variable with a prescribed covariance matrix (chosen to yield the optimal mixing and convergence of the Markov Chain).
2. The prior and likelihood of \(\tilde{y}\) are computed as well.
3. The proposal \(\tilde{y}\) is _accepted_ based on the relative probability: \[\alpha=\min\left(\frac{\pi(\tilde{y})L(\mathcal{O}|\tilde{y})}{\pi(\tilde{x}_{k })L(\mathcal{O}|\tilde{x}_{k})},1\right),\] (2) i.e. we accept the proposal if it has a relatively higher probability than the current sample \(\tilde{x}_{k}\), but may also accept the proposal (with lower probability) even if the posterior probability is less.
4. If the proposal is accepted then \(\tilde{x}_{k+1}=\tilde{y}\) and otherwise \(\tilde{x}_{k+1}=\tilde{x}_{k}\).
The sampling procedure introduced above is a bit too simplistic for the situation at hand. As noted in Ringer et al. (2021); Krometis et al. (2021), the standard Okada model parameters \(\tilde{x}\) are correlated with each other, and hence it is not practical to search over each of these parameters separately. Instead, we note that the geometry and depth of the fault explicitly depend on the latitude/longitude location of the epicenter, and the 3 magnitude parameters are highly correlated with respect to the magnitude itself. We address these issues by introducing _sample_ parameters \(x\) that we search over, from which the _model_ parameters \(\tilde{x}\) can be computed, i.e. \(\tilde{x}=f(x)\) for some map \(f\).
The sample parameters are derived from the two different observations noted above:
* The length, width, and slip are used to compute the magnitude and are highly correlated, i.e. the aspect ratio of an earthquake rupture zone follows a relatively deterministic log-linear relationship. As described in detail in Ringer et al. (2021) this allows us to sample instead from magnitude \(M\), and both of \(\Delta\log l\) and \(\Delta\log w\) which are deviations from the log-linear relationship between magnitude, length, and width which is identified from a log-linear fit to data from the past 70 years of earthquakes.
* The depth, strike, rake, and dip can be well approximated as functions of the latitude and longitude given previous fault plane solutions for more recent earthquakes along each of the two faults in question. However as we don't completely believe/trust the existing modern data nor the model selected to represent it (described below), we also introduce the offset sample parameters: depth_offset \(\Delta d\), strike_offset \(\Delta\alpha\), dip_offset \(\Delta\beta\), and rake_offset \(\Delta\gamma\) which represent adjustments to the modeled geometry of the fault.
Figure 1: A depiction of the latitude longitude placement of the Walanae/Selayar Fault (blue, striking N-S) and Flores Thrust (gtriking E-W). The red dots are the locations of tsunami observations with Sumenep to the far west, Belukumba and Nip-Nipa to the NE on the SW arm of Sulawesi, and Bima in Sumbawa (south of the Flores thrust).
### Modeling the faults
To develop the prior distribution for each fault we create a simplified model based on existing fault-plane solutions. This leads to two very different prior distributions as there is a substantial amount of data to constrain the Flores Thrust, but very little to constrain the geometry and location of the Walanae/Selayar Fault.
#### 2.3.1 Modeling and sampling from the Flores Thrust
The Flores Thrust forms in the backarc region of the eastern Sunda and Banda volcanic arcs due to distribution of strain away from the arc-continent collision occurring in the region Hamilton (1979); Silver et al. (1983); Harris (2011). The fault is inclined to the south and moves the volcanic arc northward over the Flores Sea ocean basin. This motion is driven by the high frictional resistance to subduction of the Australian continent beneath the volcanic arc. The amount of convergence between the Australian and Asian Plates that is partitioned to the Flores thrust increases eastward from 21-58 percent Nugroho et al. (2009). The two largest recorded earthquakes on the fault were in 1992 (Mw 7.8) and 2004 (Mw 7.5). Both of these earthquakes generated tsunamis, but neither impacted the areas inundated by the 1820 event. The USGS earthquake catalog lists over one hundred other recorded earthquakes along the Flores thrust, however a large number of these do not have full fault-plane solutions, or are missing some component of the needed fault geometry parameters. After filtering these data to restrict earthquakes exceeding \(5.0\) Mw, and with the following parameters defined:
1. latitude-longitude of the hypocenter
2. depth
3. dip
4. strike
we were left with 94 seismic events in the instrumental record. These fault plane solutions formed the basis for our prior distribution on Flores thrust fault geometry.
Due to the noisy and inherently irregular nature of this collected earthquake source data, we first created a multidimensional Gaussian process Williams & Rasmussen (2006) to represent/model the Flores thrust. This was done by considering the depth, dip, rake, and strike as independent functions of the hypocenter latitude and longitude of each instrumentally recorded event, and developing a statistical Gaussian process fit using a radial basis function (rbf) kernel with variance \(0.75\) and a normalized noise level in the data itself of \(1.0\) (see Algorithm 3.2 of Williams & Rasmussen (2006) for details). The benefit of using a Gaussian process rather than a standard regression technique is that under the assumed hyperparameters (variance of the kernel etc.) then the uncertainty is built into the regression. This is demonstrated in Figure 2 which depicts two depth surfaces that correspond to depths that are one standard deviation away from the mean predicted depth, i.e. roughly speaking we anticipate that approximately two thirds of the earthquakes on the Flores thrust will be contained between these two surfaces. Similar processes are constructed for the dip, rake and strike of the fault as well.
All parameters of the Flores thrust are modeled by these four Gaussian processes treated independently. The prior distribution is then selected to match this model. As discussed in Ringer et al. (2021) we develop a prior distribution on the latitude-longitude of the hypocenter by enforcing a distribution on the mean depth computed for our fault model. We use a Gaussian distribution on depth with mean \(30\) km and a standard deviation of \(5\) km with a truncation on the interval \([2.5,50]\) km. Hence each latitude-longitude coordinate is mapped through the model and the mean depth is then used to calculate a prior probability. The mean dip, rake, and strike are then computed from the Gaussian process model and we sample over the novel offset parameters: depth_offset, dip_offset, rake_offset, and strike_offset which allow for perturbations from the mean statistical model. To compute the final Okada earthquake parameters, we take the computed mean depth, dip, rake, and strike and then add the standard deviation of the Gaussian process at that point multiplied by the corresponding offset parameter.
In summary, in addition to the three parameters prescribed for the magnitude of the earthquake we introduce the following sample parameters: latitude, longitude, \(\Delta d,\ \Delta\alpha,\ \Delta\beta,\) and \(\Delta\gamma\). These are mapped through the Gaussian process fault model and then the offset parameters are used to produce the Okada earthquake parameters: latitude, longitude, depth, dip, rake, and strike.
#### 2.3.2 Walanae/Selayar Fault
Earthquakes are recorded for most of the Walanae/Selayar Fault (17 events \(>3.0\) Mw) including 3 quakes of Mw 5.0-5.9 since 1993 Jaya et al. (2020). However, the section of the fault south of Bulukumba (Belokumba), known as the Selayar Fault, which causes uplift of Selayer Island, is under-slipped with 5-10 mm/a of convergence to the ENE. This fault, which causes uplift of Quaternary coral terraces on Selayar Island, currently may be in a phase of interseismic elastic strain accumulation, but is capable of generating a tsunami,Sarsito et al. (2019); Simons et al. (2007) Cipta et al. (2017). Lack of instrumentally recorded earthquakes on the Selayar Fault hinders efforts to properly fit a Gaussian process to model the fault. Limited detail and constraint on the existing data lead us to make a simpler hypothesis for the fault parameters. We modeled the Walanae/Selayar fault as a plane following a default dip of \(25^{\circ}\) i.e. for a given latitude longitude the depth of the fault is calculated assuming that the fault interface dips \(25^{\circ}\). The fault strike is measured from different geographic points parallel to the fault and projected perpendicular to the fault line to points interior to the fault itself. We assume that the rake on the fault is centered at \(80^{\circ}\) throughout as there is no data to constrain the rake any further.
To account for the uncertainty in this over-simplified model of the Walanae/Selayar fault, we also introduce and search over \(\Delta d,\ \Delta\alpha,\ \Delta\beta\) and \(\Delta\gamma\), thus allowing for some strike-slip motion which is evident on the Walanae section of the fault. In contrast to the Flores thrust, the final Okada parameters are then obtained from simply adding the offset parameters to those computed from the planar model (the offsets in the Flores thrust are first multiplied by the corresponding standard deviation from the Gaussian process fit). This leads to the final set of Okada parameters required by the forward model.
Figure 2: The two surfaces defining one standard deviation away from the mean fit for the depth of the Flores thrust. The Gaussian process for the depth defines the most probable depth as the region between these two surfaces. Note the significant difference in scales between the two axes: this represents a change of only one degree in latitude but 10 degrees in longitude. The difference in scales explains the apparent ‘ridged’ behavior of the two surfaces along the longitudinal direction.
Construction of a Likelihood function
### Observational probabilities
As described above, we make use of extremely anecdotal observational accounts that present a high level of uncertainty. For instance the historical account records that in southern Sulawesi (see Wichmann (1918, 1922)...there was after a weak shock, vibrations becoming gradually more powerful, such that the flat of the command in Fort Bulekomba fluctuated to and fro. The six-pounders set up in bastion number 2 hopped from their mounting. After the 4-5 minute long quake, shots were believed to be heard in the west, coming from the sea. Barely had the sent envoy returned with the news that ships were nowhere to be seen, than did the sea, under a both whistling and thunder-like rumble, come in, formed as a 60-80 foot high wall, and flooded everything.
This particular account is unique because it yields both an arrival time (after the main earthquake) of the initial wave and an approximate wave height. This also clearly illustrates the anecdotal and uncertain nature of the observational data that we are using. There is very little in the way of definitive measurements that can be used to pin down the exact nature of either the earthquake or the subsequent tsunami. The hypothesis is that a combination of several such observations will be enough to adequately constrain some of the earthquake parameters to at least partially glean information on the causal earthquake.
We have chosen to focus on observations of the tsunami alone, as shaking intensity is notoriously a highly uncertain prediction Abrahamson et al. (2016) particularly without extensive knowledge of VS30 at each observation site. Precise measurements and careful study of the entire Flores Sea region may yield a set of Ground Motion Prediction Equations (GMPE) that fits the ground motion, but to date no such data is available (see Griffin et al. (2018) where such a study is carried out with a generic GMPE). As we have a physics-based and rigorously validated Berger et al. (2011) forward model for tsunami propagation, we are more confident in inferring earthquake parameters from observations of the tsunami. Although we do not make direct use of the shaking observations, the historical record of shaking intensity can be used to validate our results as discussed in Section 5.
As already described, the textual observation cited above illustrates the two types of observations that we make use of for the 1820 tsunami at different geographic locations:
* Wave arrival time: The time it takes for the initial wave to reach a specific location.
* Maximum wave height: The maximal wave height at a specific location.
For the 1820 tsunami we identified 4 distinct geographic locations around the Flores Sea where the tsunami was observed (Fig. 1). There are a few things worth pointing out about these observation locations before we consider the actual observations themselves.
1. Bulukumba and Nipa-Nipa are both on the southern tip of SW Sulawesi and 20 km apart. The historical record reports that the earthquake lasted 4.5 minutes that was followed by tsunami 18-24 m high at the Fort Bulukumba that inundated 300-400 m inland destroying villages around Nipa Nipa and carrying ships off the coast into rice fields.
2. Sumenep is over 700 kilometers WSW of Bulukumba over a relatively shallow sea (much of the Flores Sea is less than \(300m\) deep) so a wave that reaches both locations would dissipate a significant amount, and take a long time to propagate that far.
3. Bima, on Sumbawa Island (the southern most observation location), is deep inside a narrow inlet that opens into a bay. It is well known that inlets and bays _can_ amplify tsunamis, but the angle of incidence in such a case is critical to capture the effects accurately and capturing such an effect may require simulations at a higher resolution than the available bathymetry allows.
With all of these considerations in mind, we define the observational probability distributions for each observation on a case-by-case basis, some of which are illustrated specifically here.
* To begin, the account quoted above that refers to the wave height being 18-24 m near Bulukumba is likely an over-exaggeration than under so the observational probability distribution on wave height at Bulukumba is a normal distribution centered at \(18m\) with a standard deviation of \(5m\).
* Similarly the wave arrival time at Bulukumba is prescribed as a normal distribution centered at 15 minutes with a standard deviation of 10 minutes (truncated at a 0 minute or instantaneous arrival). This is based on the proximity of Fort Bulukumba to the coast.
* The observation at Bima is given by Bima on Sumbawa. Violent quake of a good 2 minutes in duration, which was followed by a violent rumbling and then a flood wave that flung anchored ships far inland and over roof tops. As there is no time given we make use of the observation of wave height only. Although flinging "anchored ships far inland" is very graphic, it's not very quantitative. The fact the ships were anchored seems to indicate they were larger than say, just canoes or other small boats. This observation, and the fact that they were flung "far" inland and over roof tops, indicates a sizeable wave. We don't think that waves smaller than 1 meter are plausible. So, for Bima's wave height we chose a truncated Gaussian likelihood with mean 10 meters, standard deviation 4 meters, and a lower bound of 1 meter.
* The account from Nipa-Nipa has no estimate of wave height but only inundation, which leads to an observational distribution with an assigned mean of 3 meters. The tsunami striking Sumenep was observed without any detail so we select a truncated distribution centered around 1.5 meters (basically guaranteeing a wave of some sort is noticed at Sumenep).
* The final observation is the wave arrival time at Sumenep. In this case the historical record indicates that the wave arrived at Sumenep 5 hours after the earthquake was felt in Bulukumba and Bima. The issue with this particular observation is that Bima and Bulukumba are currently (and according to Dutch records was at the time) in a different time zone than Sumenep. In particular, the record indicates that the earthquake was felt close to 10:00 hours, but the wave arrived in Sumenep at 15:00 hours. The issue is that Sumenep is on the very eastern edge of its time zone, and at different times in the 1800s, was either in the same time zone as Bulukumba and Bima, or 30 minutes or 60 minutes off. In addition to the concerns over the time zones which were not standardized in Indonesia until 1912 Nguyen et al. (2015), the definitive times of 10:00 and 15:00 hours are rather ambiguous (if the record had instead cited 10:12 and 15:27 for instance, then we would take more credence to the precise time interval). All of this is to say that although this observation does indicate that the wave took a very long time to reach Sumenep, the exact timing of the wave's arrival is very clearly uncertain. From preliminary estimates of the wave speed across the Flores Sea (recall that in open water tsunamis travel very near the linear phase speed \(\sqrt{gH}\) where \(g\) is the gravitational constant, and \(H\) is the water depth), we were unable to legitimately justify a wave originating from any location on either proposed fault and taking even close to 5 hours to reach Sumenep. Hence to construct the observational probability distribution for the arrival time at Sumenep, we went with the hypothesis that Sumenep was in a different time zone than the other observation locations which would put the observed time interval at 4 hours rather than 5. With this in mind, we selected a normal distribution with a mean of 240 minutes (4 hours) and a standard deviation of 45 minutes.
The final observational probability distributions are illustrated in Figure 9 as the continuous red curves.
### The forward model
As discussed in more detail in Ringer et al. (2021) we make use of Geoclaw as the forward model which takes the required earthquake parameters as inputs, and applies the Okada model Okada (1985, 1992) to generate
an idealized seafloor deformation which is then used as an initial condition for the fully nonlinear shallow water equations. Geoclaw has the capability of rendering both rectangular and triangular faults, but we only take advantage of the former. Unlike the Banda Arc studied in Ringer et al. (2021) both the Flores Thrust and Walanae/Selayar Fault are fairly geographically linear and hence are easily modeled by a small number of rectangular faults. In particular we use three rectangular faults to model the full rupture zone of each fault.
The Okada rectangular rupture regions are identified via the following process which is a simplification of that employed for the 1852 event in Ringer et al. (2021).
1. The latitude-longitude centroid location is identified via the random walk Monte Carlo step, and the total width and length of the rupture are computed from the sampled magnitude and \(\Delta\log l\) and \(\Delta\log w\) as described above.
2. The length is split into 3 and the rupture is specified as three different rectangular regions, each with the same width. The centroid of each of these rectangles is identified along a line of equal depth according to the model specified for each fault (the Gaussian process for the Flores Thrust etc.) and the orientation is parallel to the modeled fault.
3. The Okada model is employed for each of the three sub-rectangles for a simultaneous, instantaneous rupture.
Following the formation of the seafloor deformation from the 3-rectangular rupture via the Okada model, Geoclaw uses a finite volume formulation Berger et al. (2011) with a dynamically adaptive spatial mesh to simulate the propagation of the resultant tsunami via the nonlinear shallow water equations. We leave most parameters in Geoclaw as their default values including bottom drag and friction coefficients, and carefully tune the adaptive mesh as described below.
The forward propagation of a tsunami wave critically depends on accurately resolving the bathymetry (underwater topography), which is a difficult and pressing issue for all tsunami simulations and studies. For bathymetry we primarily relied on the 1-arcminute etopo datasets available from the open access NOAA database ([https://www.ngdc.noaa.gov/mgg/global/global.html](https://www.ngdc.noaa.gov/mgg/global/global.html)), and for the coastline near each observational point we utilize higher resolution Digital Elevation Models (DEM) from the Consortium for Spatial Information (CGIAR-CSI, [http://srtm.csi.cgiar.org/srtmdata/](http://srtm.csi.cgiar.org/srtmdata/)). These higher resolution topographical files yield a 3-arcsecond resolution on land, but give no additional information on the sub-surface bathymetry. In addition we also took advantage of detailed sounding maps available from the Badan National Penangulangan Bencana (BNPB or Indonesian National Agency of Disaster Countermeasure, see [http://inarisk.bnpb.go.id](http://inarisk.bnpb.go.id)). To convert these data into digitally accessible information, contours were taken from images exported from the website and then traced and interpolated in arcGIS to produce approximate depths in the same regions as the DEM files. This approach provides a set of bathymetric files that are accurate to around 10-15 arcseconds near each observation location with a maximum possible resolution of 3 arcseconds.
We make use of six different levels of refinement, starting with a resolution of 6 arcminutes in the open ocean going down to 3 arcseconds (the maximum resolution allowed by our bathymetric data) around those parts of the wave that will impact the observation locations directly (see Ringer et al. (2021) for a more thorough description of the same adaptive mesh). The mesh refinement is activated whenever the solution of the linearized backward adjoint equation Davis & LeVeque (2016) exceeds a specified threshold at the same time that the forward solution does as well. The linearized adjoint solution is computed on a global mesh of 15 arcseconds, initialized with an endpoint condition corresponding to pointwise Gaussian sea surface perturbations at each observation location so that the adjoint solution solved backward in time will identify when and where the forward tsunami will be that directly affects each of the observation locations. This dictates where the mesh is refined. The benefit of using the adjoint driven adaptive mesh is that because every one of our Monte Carlo samples uses the same observation locations, then we need only run the linearized adjoint solver one time (hence the global 15 arcsecond resolution, while expensive, is a one time cost), and save the corresponding output to be used with the forward runs.
In addition to the dynamically adaptive mesh, we include several statically refined regions at the highest (3 arcsecond) resolution. Each of these regions is specified as a series of rectangular (Geoclaw requires specification of regions in rectangular latitude-longitude coordinates) sub-regions that encapsulate each observation location. This is meant to ensure that the incoming wave is accurately captured as it approaches each observation location. For instance, Bima in Sumbawa is located deep inside a bay that must be accurately captured in order to simulate the tsunami reaching Bima, and so we defined several statically defined regions that encapsulate the bay and surrounding coastline as much as possible without unnecessarily refining the grid on land at the same time.
We ran each tsunami simulation for at least 4 hours in physical time (we initially ran the tsunamis for 5 hours, but none of the waves required more than 4 hours to reach Sumenep, so we allowed the samples to run for 4 hours only to save compute time). Running on 24 cores on a single node each of these simulations took approximately 10-12 minutes of wall-clock time, i.e. 240-288 minutes of compute time. Wave heights and arrival times were extracted from the Geoclaw output using the previously developed tsunamibayes package Whitehead (2023) and wrapped into the MCMC method to create the optimal sampling strategy.
## 4 Results
### Statistical summary
For each fault we initialized ten different chains with five unique latitude-longitude locations geographically spread across the entire fault and with two different magnitudes: \(8.0\) and \(8.5\) for a total of ten initial earthquakes. After running each chain for two thousand samples a piece, we resampled all ten chains according to their final posterior probability and restarted each chain accordingly. In this process most of the chains were eliminated, as most had still not achieved a finite log likelihood (most of the chains were unable to generate a noticeable tsunami wave that reached Sumenep). After resampling, each chain was run for a minimum of 9,000 samples via random walk MCMC. In total, we simulated 104,970 tsunamis originating from the Walanae/Selayar fault and 127,690 originating from the Flores thrust. This cost an estimated 110 years of total compute time spread over 24 cores at a time and twenty chains, for nearly 2.5 months of real time computational cost.
Figure 3: The rectangular regions where the resolution is fixed at the highest grid level near the port of Bima. As Geoclaw requires each region to be specified as a rectangular region, we specified several sub-regions (shown as the red rectangles) that depict the regions of interest. Similar highly resolved regions are defined for all of the other observation locations as well.
the following for each sample parameter:
* latitude (degrees): 0.086 (Flores), 0.05 (Walanae/Selayar)
* longitude (degrees): 0.11 (Flores), 0.04 (Walanae/Selayar)
* magnitude (Mw): 0.075 (Flores), 0.045 (Walanae/Selayar)
* \(\Delta\log l\): 0.0132 (Flores), 0.012 (Walanae/Selayar)
* \(\Delta\log w\): 0.0132 (Flores), 0.012 (Walanae/Selayar)
* \(\Delta d\) (km): 0.525 (Flores), 0.55 (Walanae/Selayar)
* \(\Delta\beta\) (degrees): 2.7 (Flores), 2.55 (Walanae/Selayar)
* \(\Delta\gamma\) (degrees): 3.7 (Flores), 3.55 (Walanae/Selayar)
* \(\Delta\alpha\) (degrees): 3.15 (Flores), 3.05 (Walanae/Selayar)
This covariance matrix was adjusted slightly (covariance values for dip offset \(\Delta\beta\) and rake offset \(\Delta\gamma\) as well as the longitude for Flores only were increased partway through the sampling) with the goal of getting close to a \(0.25\) acceptance ratio for both sets of chains. The averaged acceptance ratio for each different set of chains is depicted in Figure 4. Note that the acceptance ratio for the Walanae/Selayar chains is slightly below the desired value, but the acceptance ratio for the Flores chains is still quite high, indicating that the sampling may be more aggressive on the Flores thrust then the covariance matrix described above. Despite this high acceptance rate, all ten chains were mixing very nicely in all of the relevant variables.
To verify the inter-chain mixing and ensure that the approximated posterior distribution is adequately converged, we computed the Gelman-Rubin diagnostic Gelman et al. (1992, 2014) for all of the parameters from the posterior distribution as shown in Figure 5. Note that the Flores posterior mixes at a slightly faster rate (the Gelman-Rubin diagnostic drops below \(1.1\) at a lower number of total samples), but in either case the diagnostic clearly indicates sufficient mixing between chains to satisfy the necessary invariance properties to anticipate that the posterior distributions are converging.
### Summary of posterior distribution
The primary description of the desired posterior distribution can be visualized via Figures 6 and 7. In particular, Figure 6 displays a histogram of the sampled centroids for both faults with the left colorbar representing
Figure 4: The acceptance ratio averaged over a 2000 sample interval for each fault’s set of chains.
Figure 5: The Gelman-Rubin diagnostic displaying the relative inter-chain vs. intra-chain variance for each constructed posterior distribution. The dashed horizontal line is at the value of \(1.1\). As described in Gelman et al. (1992, 2014) the diagnostic should drop below this line to indicate appropriate mixing of the sampled posterior. Note that this occurs for both posteriors before 5,000 samples are collected (for every observable computed here) so we do not show the rest of the chain’s data for samples beyond 5,000.
Figure 6: The posterior distribution of the earthquake centroid locations for both faults. Note that those earthquakes originating on the Walanae/Selayar fault will be oriented primarily north-south (in line with the fault line itself) whereas those earthquakes along the Flores thrust will be oriented primarily east-west. The prior distribution for the centroid of both of these faults is centered at 20km deep on the interior of the fault, i.e. to the west of the Walanae/Selayar fault line (blue curve) and south of the Flores fault line.
the density of samples on the Flores thrust and the right colorbar representing the density of samples on Walanae/Selayar. There are several items to note from this Figure alone:
* The centroid location along the Walanae/Selayar fault is in a very concentrated location near \(120^{\circ}\) longitude and \(-6.5^{\circ}\) latitude. In contrast the sampling along the Flores thrust is far less focused, with preferred centroid locations spanning a wide range of longitudinal values, and a relatively wide range of latitudes near \(119.5^{\circ}-120^{\circ}\) longitude.
* The prior distribution on centroid location for the Flores thrust did not force the earthquake centroid to be on the 'correct' side of the fault line (south of the blue curve in Fig. 6). This allowed for a surprising number of earthquake samples that were on the physically infeasible side of the fault (north of the blue curve), a region that appeared to actually be preferred to some extent by the sampling strategy employed here (there is a high concentration of centroids north of the blue curve in Fig. 6). This may indicate either that the Gaussian process prior is not sufficiently restrictive or (as discussed further below) the observational data prefers earthquakes centered north of the actual Flores fault.
* The centroids for the posterior on the Walanae/Selayar fault are on the 'correct' side of the fault, but they are further south than the prior prefers, indicating that observations are better matched with an earthquake centroid further south than the modeled Walanae/Selayar fault extends.
* In addition, the most preferred centroid locations for the Flores thrust (at least those that lie on the 'correct' side of the fault itself) line up with the curvature of the Walanae/Selayar fault. That is, the centroid locations from the two faults nearly line up in a north-south line as if the Walanae/Selayar fault extended all the way to the Flores thrust.
Figure 7 depicts histograms of the sampled posterior distribution for all of the other sample parameters (omitting latitude and longitude which are depicted in Fig. 6). We first note that the prior and posterior distributions for all four offset parameters are nearly identical for the Walanae/Selayar fault (even though the prior distribution is _NOT_ shown here). The prior distribution for the Flores thrust is not independent in each of these parameters, as the sampled values are multiplied by the variance of the Gaussian process at each centroid location, and hence plotting a one-dimensional pdf of the prior would require integrating the full prior against the centroid position which is computationally prohibitive for visualization purposes alone. For this reason we are unable to draw definitive conclusions about the influence of the observed data on the geometry of the Flores fault, however it is clear that the geometry of the Walanae/Selayar fault is not constrained by the data, i.e. the posterior simply recreates the prior distribution for these parameters. Although we are unable to form the same comparison for the Flores fault we anticipate a similar result. The rub of the matter is, our limited tsunami observations are not sufficient to constrain the geometry of the fault.
On the other hand, there is a clear signal in both \(\Delta\log l\) and \(\Delta\log w\) that indicates that the observational data is a better match for smaller values of both of these parameters. Smaller values of these parameters for a fixed magnitude corresponds to a larger slip length than expected, i.e. this indicates that the earthquakes that best match the data have very large slip as seen in Fig. 8. We see that the most probable slip that matched the data for both faults was over \(10m\) with a definite preference for larger slip. The slip on the Walanae/Selayar posterior is slightly smaller, with a maximum probability estimate close to \(8m\) rather than \(10m\), and a slightly less positive bias toward larger slip, i.e. the Walanae/Selayar posterior is slightly more seismically sound. This tendency toward an unexpectedly large slip was noticed in Ringer et al. (2021) for the 1852 Banda Sea earthquake in Eastern Indonesia where the Bayesian technique employed here was first introduced. Future studies will consider the potential discretization effects and selection of hyper-parameters in the forward model that could lead to a preference for smaller rectangular area, large slip ruptures.
Fig. 8 also displays the other earthquake parameters derived from the posterior distribution. Note in particular that a hypothesized Flores earthquake is less constrained in size as the length and width have a significantly wider histogram that extends to much larger values than earthquakes hypothesized for the Walanae/Selayar fault. This result is likely because, as discussed in more detail below, the Flores posterior tends to favor extremely high magnitude earthquakes. It is also interesting to note that in contrast to
Figure 7: The posterior samples for all of the relevant sample parameters from both faults compared against the prior distributions. We do not reproduce the prior distribution for the four offset parameters because the Flores prior for the offset parameters is centroid location dependent, i.e. the Gaussian process which models the Flores thrust yields an estimate of uncertainty at every point along the fault which is used to weight the prior distribution on the offset parameters for the Flores model. This means that we can _NOT_ represent the prior distribution for the Flores fault along the offset parameters as a one-dimensional distribution without integrating against the position (a very costly exercise).
the magnitude derived parameters, the depth of the Flores posterior is more constrained than the depth for Walanae/Selayar. This is likely a result of a more data-driven prior distribution on the Flores fault whereas the Walanae/Selayar prior is a nearly linear fit and hence the depth values of the modeled fault are highly suspect.
This brings us to the final comparison between the two posterior distributions. As previously described, the prior distribution on magnitude was the exponential Richter distribution that exponentially decays with growing magnitude as indicated by the red curve in the upper left plot of Fig. 7. Due to the size of the Flores and Walanae/Selayar faults, we also truncated the magnitude at \(9.0\)Mw to ensure physically reasonable earthquakes were observed. As shown the observational accounts best matched with earthquakes of extremely large magnitude, particularly along the Flores thrust where the most probable magnitude is near \(8.8\)Mw, with a clear preference toward the cutoff magnitude of \(9.0\)Mw. In contrast, although the earthquakes sampled from the Walanae/Selayar posterior were also quite large for the size of the Walanae/Selayar fault with the most likely value near \(8.5\)Mw, the Walanae/Selayar posterior did not have as much of a positive bias toward extremely high magnitude events. In fact, the Walanae/Selayar posterior preferred earthquakes around \(8.5\)Mw, which although large for the fault in question, is far more likely than an \(8.8\)Mw event.
The high magnitude preference for both posterior distributions is in line with the observation made previously that the slip was quite large for both of these earthquakes. Neither the Walanae/Selayar fault nor the Flores thrust are large enough to sustain an \(8.5\)Mw earthquake with the standard relationship maintained between length, width and slip. However the observational data indicates that large magnitude events are necessary for the tsunami observations to match. The apparent trade off here is satisfied with large (but not extreme) magnitude earthquakes that are shorter and narrower, but with very high average slip length (recall that our model of slip requires an instantaneous, uniform slip distribution across the entire rupture). An alternative hypothesis is that the earthquake was triggered on both the Walanae/Selayar and Flores faults, perhaps allowing for a smaller magnitude event on both nearly simultaneously.
Figure 8: Histograms of the posterior distribution on the actual earthquake parameters (rather than sample parameters) for each fault. The strike is omitted as it is fundamentally different for each fault (Walanae/Selayar runs north-south while Flores runs primarily east-west), and a comparison between the strike for the two posterior distributions is not informative.
### Comparison of posterior predictive to observation probabilities
Figure 9 depicts the histograms of the posterior predictive (output of each simulation for the observational data points) relative to the original observational probabilities. Each of the six observations have particular characteristics that are of interest in this setting:
* The extreme wave height from the observation in Bulukumba is clearly not achieved for either posterior distribution, leading us to believe that either the historical observation which claimed a wave height of 60-80 feet (18-24 meters) was over-exaggerated, or some other nonlinear, local effects were at play. In particular, it is possible that a submarine landslide caused by the earthquake could generate a wave of this magnitude at least locally. This hypothesis is reasonable when we consider that the wave heights at Nipa-Nipa generated from either fault are near the observational probabilities, noting that Nipa-Nipa is geographically very close to Bulukumba so that it is highly unlikely that Bulukumba would have a wave near 20 m whereas Nipa-Nipa only sustained one of 4-5 m.
* The arrival time in Bulukumba has a few peculiarities. The large number of arrival times at time \(0\) for the Walanae/Selayar posterior arise because a significant number of the Walanae/Selayar earthquakes have a rupture zone overlapping the observation locations at Bulukumba so that the sea surface is instantaneously shifted i.e. the wave seemingly arrives instantaneously although this arrival time isn't the actual wave arriving, but just the initial disturbance from the earthquake. Beside these events, it is apparent that the posterior predictive from the Walanae/Selayar fault matches the observational distribution quite well for the Bulukumba arrival time whereas the Flores posterior indicates a much longer arrival time to Bulukumba than anticipated.
* The observational distribution for wave height at Sumenep appears to better match the Flores posterior predictive except that for this particular observation we must recall that the only statement was that the wave was observed at Sumenep, i.e. the observational distribution at this location is not very precise.
* The arrival time at Sumenep clearly doesn't agree well with either posterior predictive, but it is also certain that the Walanae/Selayar posterior is a much better fit than the Flores as waves originating from the Walanae/Selayar fault take over 3 hours to arrive at Sumenep while most tsunamis originating from Flores arrive just under 3 hours. This particular comparison should not be weighted too heavily though, as neither fault generates a tsunami whose initial wave arrives in 4 hours which is the observational value only after assuming that Eastern Java and Sulawesi are in two different time zones. A partial explanation for this is that the initial wave is not the one recorded in the historical record, but that a secondary wave is the one observed in Sumenep. We did not collect the arrival times for the secondary waves for all \(200,000+\) simulated earthquakes, but from repeat simulation of a few events we did note that some of the later waves from both the Walanae/Selayar and Flores faults were larger than the initial wave with
Figure 9: Posterior samples from both hypothesized faults plotted against the observational probabilities assigned to each observation.
corresponding arrival times exceeding 210 minutes (for Flores) and 240 minutes (for Walanae/Selayar). Low resolution bathymetry may also play a role in faster predicted versus actual arrival times.
* The posterior predictive wave height at Bima is also not a great match with the observational data for either fault, although earthquakes generated from the Flores thrust match the observation better. In essence the generated earthquakes are under-estimating the wave height in Bima. There are several potential reasons for this, one of which may simply be that the bathymetric resolution isn't sufficient to capture the amplification of the wave entering the bay. Even with the type of amplification that may occur, the Walanae/Selayar posterior clearly underestimates the wave height at Bima as it is hard to imagine a \(2m\) wave having sufficient buoyancy and force to 'fling' ships far inland.
#### Which fault fits the observations better?
The discussion above gives credence from some observations for each of the potential posteriors, i.e. some parts of the posterior predictive are supportive of the Walanae/Selayar hypothesis and some support the Flores Thrust. To make a quantifiable comparison between the two hypotheses, we propose a _novel_ approach using the dual posterior predictive combined with the observational probabilities to obtain a relative probability of which fault best matches the observational data.
1. Using all of the data from the two posterior predictives, we trained a binary classifier whose input was the wave heights and arrival times for all 6 observed data points. The data was divided into two distinct classes defined by the fault the earthquake originated from.
2. Once we trained a sufficiently accurate classifier, we drew samples from the observational probabilities (as shown in red in Fig 9) and use the previously trained classifer to determine which fault these observations were most likely generated from.
To train the binary classifier, we first took all samples from both faults, removed those that did _not_ have finite log posterior, and randomly selected \(70\%\) of the remaining samples for the training set and the remaining \(30\%\) for the testing set to give:
* 162,862 samples for the training set: 89,554 originating from the Flores fault and 73,308 from Walanae/Selayar.
* 69,798 samples for the testing set: 38,136 from the Flores fault and 31,662 from Walanae/Selayar.
Both the training and testing sets were normalized using sklearn's preprocessing package to reduce the amount of bias in the fit. Then rather than rely on a single classifier, we trained two different classifiers of different architectures to ensure the results were consistent:
* We trained a binary logistic classifier via XGBoost (with all default settings). This classified all but 5 of the samples from the test set correctly for an accuracy of 99.993%.
* We also trained a random forest classifier again with all default settings from sklearn, which only misclassified 3 samples from the test set for an accuracy of 99.996%.
To identify which fault best matched the source we sampled 1 million data points from the observational probabilities depicted in Fig. 9. After normalization, each set of these data points was then fed through the classifier and a fault source was prescribed according to the classifier. The random forest classifier selected Walanae/Selayar as the source 94.4% of the time and the logistic regression classifier selected Walanae/Selayar 98.4% of the time.
These probabilities should be considered in the proper context however. We can summarize the above results in the following probabilistic statement:
Given the hypothesis that either the Walanae/Selayar or Flores faults were the source of the 1820 tsunami, and provided that the observational probabilities depicted in Fig. 9 are realistic, the random forest classifier is 94% confident that the Walanae/Selayar fault was the source.
This does not say that the authors are 94% confident that the Walanae/Selayar fault is the source, but that if the only hypothesis considered is that one of the two faults separately generated the tsunami, then we are. In other words, if we are certain that one of these two faults generated the observations, we are quite certain it had to originate from the Walanae/Selayar fault and _NOT_ the Flores thrust. The rub of the matter is that this yields a conditional probability, i.e. we have a roughly 94% probability that Walanae/Selayar was the source of the tsunami, given that the source was either Walanae/Selayar or Flores. Even so, as noted above in our explicit discussion on different parts of the posterior predictive, this is not a completely satisfactory hypothesis.
Careful investigation of either classifier described above implies that the wave arrival time in Bulukumba seems to be the determining factor. While all the other observables may be mildly more or less probable to identify with either fault the Bulukumba arrival time histograms for the Walanae/Selayar and Flores faults are clearly separated with the Walanae/Selayar data matching the observational distribution much better.
#### So who is at fault?
In summary:
* Neither posterior appears to match the wave height at Bulukumba, which we anticipate is a result of an over-inflated observation or an additional local source.
* Neither fault appears to match the arrival time at Sumenep although Walanae/Selayar certainly is closer than Flores.
* Neither fault matches the wave height at Bima well, but Flores is much closer than Walanae/Selayar.
* Finally the arrival time at Bulukumba is clearly better fit with an earthquake originating from the Walanae/Selayar fault.
Hence, from purely viewing the posterior predictive, the selection of which fault was the most likely source of the tsunami, is uncertain at best. In overall probability (likelihood) it is apparent that the Walanae/Selayar fault is a better fit to the assigned observational distributions, but claiming the tsunami was generated on Walanae/Selayar would severely discount the detailed observation at Bima. On the other hand, a tsunami generated on Flores clearly misses the wave arrival time in Bulukumba and has a significantly poorer fit to the arrival time in Sumenep.
As a final note, we recognize that the Walanae/Selayar fault is a more likely candidate to match the shaking intensity recorded at Bulukumba, where as noted above, the canons were said to jump from their mounting. This implies a peak ground acceleration exceeding 1 G at a minimum, which would be highly unlikely for an earthquake originating over 200km away (the distance from Bulukukumba to the Flores thrust). Hence, although we have not made use of the shaking observations in our computations, our final result that indicates the earthquake was more likely generated along the Walanae/Selayar fault, is consistent with those observations.
## 5 Discussion and Conclusion
The Bayesian approach toward identifying characteristics of earthquakes using anecdotal historical accounts of tsunamis first introduced in Ringer et al. (2021) has been applied to the 1820 south Sulawesi earthquake and consequent tsunami. Using the Bayesian framework, we have simulated close to 200,000 different events, searching through parameter space for earthquakes that probabilistically best match the interpreted historical record. Hypothesizing that the earthquake originated purely from either the Flores or Walanae/Selayar faults does not yield a posterior distribution that appears to match the data perfectly, although we have strong statistical reasoning to assert that the Walanae/Selayar fault was far more likely to be the source than the Flores thrust.
To further investigate this event and hopefully ascertain the source of the recorded tsunami, we will next consider two potential hypotheses:
* A landslide near where the Walanae/Selayar Fault goes offshore (likely very near Bulukumba itself) that can produce the significant wave heights recorded near the Fort.
* The dual rupture of both the Walanae/Selayar and Flores faults. Although a time-dependent rupture is clearly more physically relevant, we will restrict our attention to an instantaneous rupture of both faults simultaneously, as it is unlikely that our limited observations of tsunami impacts will provide enough detail to constrain a more sophisticated rupture model.
Further extensions of this Bayesian approach to historical tsunamis will be carried out both to improve the sampling procedure, and to investigate other events with the goal of providing a thorough description of the past seismicity in the Indonesian region.
## Acknowledgments
Beyond the authors listed on this manuscript, there are several students who have come through BYU that have contributed to the formation of these results. These include but are not limited to: H. Ringer, S. Giddens, M. Morrise, C. Noorda, J. Callahan, K. Lightheart, C. Kesler, and C. Herrera. In particular, we wish to thank N. Thompson for his assistance with improving the presentation of some of the figures reported here. We also would like to thank N. Glatt-Holtz for inspiring this work several years ago, and useful conversations with R. LeVeque, V. Martinez, C. Mondaini, A. Holbrook, as well as a host of other colleagues who have provided useful feedback and guidance over the years.
We also acknowledge generous financial support from the BYU Mathematics and Geology Departments as well as the College of Physical and Mathematical Sciences. JAK would like to acknowledge the support of the National Science Foundation under grant DMS-2108791.
## Data Availability
All of the relevant code that generated the data provided in this article appears in Whitehead (2023). The data itself can be viewed via the graphical interface provided at [http://tsunami.byu.edu/whitehead-lab](http://tsunami.byu.edu/whitehead-lab).
| Bayesianアプローチを用いて、1820年12月29日フィロース海地震のAnecdotal tsunami runup observationsと、200,000以上の津波シミュレーションを比較し、津波を引き起こす最も可能性の高い地震パラメータを決定します。Flores ThrustまたはWalanae/Selayar Faultのどちらかという、双方の仮説に基づき、これらの仮説は観測データに完璧に一致しません。特に、この地域の震度制約を満たします。しかし、Walanae/Selayar Faultの地震は、津波の記録と地震の揺れに密接に関連しており、この研究からのシミュレーションデータは、この地域における別のソースの可能性や、両Faultが潜在的に合流し同時に破壊する可能性を示唆しています。 2016年のカイカウラ、ニュージーランドの地震のように、両Faultが同時に破損する可能性があります。 |
2305.19157 | Sensor Fault Detection and Compensation with Performance Prescription
for Robotic Manipulators | This paper focuses on sensor fault detection and compensation for robotic
manipulators. The proposed method features a new adaptive observer and a new
terminal sliding mode control law established on a second-order integral
sliding surface. The method enables sensor fault detection without the need to
know the bounds on fault value and/or its derivative. It also enables fast and
fixed-time fault-tolerant control whose performance can be prescribed
beforehand by defining funnel bounds on the tracking error. The ultimate
boundedness of the estimation errors for the proposed observer and the
fixed-time stability of the control system are shown using Lyapunov stability
analysis. The effectiveness of the proposed method is verified using numerical
simulations on two different robotic manipulators, and the results are compared
with existing methods. Our results demonstrate performance gains obtained by
the proposed method compared to the existing results. | S. Mohammadreza Ebrahimi, Farid Norouzi, Hossein Dastres, Reza Faieghi, Mehdi Naderi, Milad Malekzadeh | 2023-05-30T15:58:56 | http://arxiv.org/abs/2305.19157v2 | # Highlights
###### Abstract
We propose a new method for solving the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new optimal solution for the problem of finding a new solution for the problem of finding a new optimal solution for the problem of finding a new solution for the problem of finding a new solution for the problem of finding a new optimal solution for the problem
# Sensor Fault Detection and Compensation with Performance Prescription for Robotic Manipulators
S. Mohammadreza Ebrahimi
Farid Norouzi
Hossein Dastres
Reza Faieghi
Mehdi Naderi
Milad Malekzadeh
Department of Electrical Engineering, Babol Noshirvani University of Technology, Babol, Iran Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran Department of Aerospace Engineering, Toronto Metropolitan University, Toronto, Canada School of Production Engineering and Management, Technical University of Crete, Chania, Greece
###### Abstract
This paper focuses on sensor fault detection and compensation for robotic manipulators. The proposed method features a new adaptive observer and a new terminal sliding mode control law established on a second-order integral sliding surface. The method enables sensor fault detection without the need to impose known bounds on fault value and/or its derivative. It also enables fast and fixed-time fault-tolerant control whose performance can be prescribed beforehand by defining funnel bounds on the tracking error. The ultimate boundedness of the estimation errors for the proposed observer and the fixed-time stability of the control system are shown using Lyapunov stability analysis. The effectiveness of the proposed method is verified using numerical simulations on two different robotic manipulators, and the results are compared with existing methods. Our results demonstrate performance gains obtained by the proposed method compared to the existing results.
keywords: Fault detection, fault-tolerant control, fixed-time stability, performance prescription, robot manipulator +
Footnote †: journal: Journal of Robotics and Automation
## 1 Introduction
A fault in a robotic system can be defined as an unexpected or unplanned deviation from the normal behavior of the system, which can result in incorrect or unintended actions being taken. One of the most common types of faults is the sensor fault which occurs when a sensor fails to accurately measure a variable or provides incorrect data. As surveyed in [1], the common approach to detect sensor faults is to use state observers. Examples in the context of robotic manipulators include bank of linear observers [2], \(H_{\infty}\)-based observer [3], sliding mode observer and its higher-order variations [4; 5], and adaptive observer [6; 7]. One notable mention is [8] in which the sensor fault detection problem is transformed into a virtual actuator fault detection framework to use the rich body of literature on actuator fault detection and compensation [1].
Once a fault is detected, fault-tolerant control (FTC) methods are required to mitigate the effects of the fault. As pointed out in [1], one popular FTC method for nonlinear systems is sliding mode control (SMC) [9; 10; 11; 12; 13; 14; 15; 16; 17]. SMC offers simplicity and robustness; however, it suffers from chattering in control input. This has been addressed by variations of SMCs like boundary-layer SMC, fuzzy SMC, higher-order SMC, etc. [18; 19; 20; 21]. Most variations of SMC are designed based on asymptotic stability without a guarantee for finite convergence time. One variation of SMC that can address this challenge is Terminal SMC (TSMC)
[22; 23; 24; 25; 26; 27; 28; 29]. Although early TSMC methods guaranteed finite-time convergence, their convergence time heavily relied on the initial conditions. The farther the initial conditions are from the desired states, the longer the convergence time is [25; 30; 31]. With the emergence of performance prescription control techniques, new TSMC designs achieved fixed-time convergence irrespective of the initial conditions [32].
Most fixed-time control methods suffer from singularity issues which can be partially alleviated using discontinuous control laws [33; 34; 35]. Several recent studies have developed continuous and non-singular fixed-time control laws [36; 37; 38]. However, such methods rely on using the sign function in the control law which leads to chattering. Replacing the sign function with saturation or hyperbolic tangent can resolve the chattering issue, but leads to a slower convergence rate. A faster chattering-free controller is developed in [39]; however, as it will be shown in our numerical simulations, this controller has a steady-state error in the presence of sensor faults. We will build upon [39] to develop a controller with similar fast convergence characteristics but superior fault tolerance.
Overall, the objective of the present paper is to develop a fast and fixed-time sensor fault detection and compensation technique for robotic manipulators. Inspired by [8], we will represent sensor faults as virtual actuator faults. Then, we will design an adaptive observer to detect faults and estimate the actual system states. The prominent feature of this observer is that it can guarantee the ultimate boundedness of state estimation error without the need to impose known bounds on the fault and/or its rate of change. Furthermore, we will develop a new chattering-free fixed-time TSMC control law.
The control law lends well to the performance prescription control concept and enables imposing a boundary for the transient response of the system. We will integrate this control law with the aforementioned fault estimation observer and show system stability using Lyapunov stability analysis.
The major contribution of this paper is that it takes a holistic approach to sensor fault estimation and compensation of robot manipulators, enabling fault estimation and compensation with fast, fixed-time, and performance prescription capabilities. In achieving this, several new features are introduced both in our observer and control design approach, particularly in the design of adaptation laws for the observer, and also in the construction of the sliding surface for our TSMC control law. These design decisions can be extended to other control problems.
**Notation.** Throughout the paper, vectors, and matrices, if not explicitly stated, are assumed to have appropriate dimensions. For a given matrix \(A\), \(\lambda_{max}(A)\) and \(\lambda_{min}(A)\) indicate the largest and smallest eigenvalues of \(A\). \(O_{n\times m}\) denote \(n\times m\) zero matrix and \(I_{n}\) denotes the identity matrix. \(\|\cdot\|\) denotes the 2-norm of a vector, and \(L_{\infty}\) represents the set of vectors with bounded \(\infty\)-norm.
## 2 Problem Overview
Let us consider the dynamics of an \(n\)-link robotic manipulator described as follows
\[M\left(q\right)\ddot{q}+D\left(q,\dot{q}\right)\dot{q}+G\left(q\right)=\tau, \tag{1}\]
where \(q\in\mathbb{R}^{n}\) is the joints angular position, \(\tau\in\mathbb{R}^{n}\) is the vector of torques applied to each joint, \(M\left(q\right)\in\mathbb{R}^{n\times n}\) is the inertia matrix,
is the Centrifugal and Coriolis terms matrix, and \(G\left(q\right)\in\mathbb{R}^{n}\) is the gravity vector. Equation (1) can be written in the following nonlinear form
\[\dot{x}=Ax+H\left(x,u\right),\;y=Cx, \tag{2}\]
where \(x=(x_{1},x_{2})^{T}\) with \(x_{1}=q\), \(x_{2}=\dot{q}\), \(u=\tau\), \(y=x_{1}\),
\[\begin{split}& A=\left(\begin{array}{cc}O_{n\times n}&I_{n}\\ O_{n\times n}&O_{n\times n}\end{array}\right),\\ & H(x,u)=\left(\begin{array}{c}O_{n\times 1}\\ M^{-1}\left(u-D(x)x_{2}-G(x)\right)\end{array}\right),\;\text{and}\\ & C=\left(I_{n},\;O_{n\times n}\right).\end{split} \tag{3}\]
In [7], it is shown that \(H(x,u)\) is a Lipschitz function in \(x\) such that
\[\left\|H(x,u)-H(\hat{x},u)\right\|\leq\kappa\left\|x-\hat{x}\right\|, \tag{4}\]
where \(\kappa>0\). When a sensor fault occurs, the measurement is represented by
\[y_{f}\left(t\right)=y\left(t\right)+Ef\left(t\right), \tag{5}\]
where \(y_{f}\) is the faulty measurement, \(f\in\mathbb{R}^{m}\) is the fault value, and \(E\in\mathbb{R}^{n\times m}\) is a constant matrix representing the propagation of fault in the output variable. We assume that \(f\) is time-varying and satisfies the following inequalities
\[\left\|f\right\|^{2}\leq F\quad\text{and}\quad\left\|\dot{f}\right\|^{2}\leq F _{d}, \tag{6}\]
where \(F\) and \(F_{d}\) are two unknown positive constants.
In this paper, we focus on both fault estimation and compensation. For fault estimation, our objective is to design an observer that can estimate
fault and also recover \(y\) from faulty measurements \(y_{f}\). Since we assume \(F\) and \(F_{d}\) are unknown, we will use an adaptive observer to achieve the above objective. For fault compensation, our objective is to design a fast fixed-time TSMC law that allows performance prescription on the trajectory tracking error. This control law must use the estimated states from the observer and must guarantee the system stability. Figure 1 illustrates an overview of the system architecture. In the next two sections, we will explain the details of our observer and controller design.
## 3 Observer Design
To design the observer, we first represent the sensor faults as virtual actuator faults. As mentioned in Section 1, this representation is inspired by [8], and facilitates handling sensor faults.
The sensor fault can be represented as follows,
\[\dot{x}_{v}=-A_{v}x_{v}+A_{v}y_{f}, \tag{7}\]
Figure 1: Block diagram of the proposed fault estimation and compensation method
where \(x_{v}\in\mathbb{R}^{n}\) is the virtual actuator state vector, and \(-A_{v}\in\mathbb{R}^{n\times n}\) is a constant Hurwitz matrix.
Having defined sensor fault as a virtual actuator fault, an augmented system can be formed using (3) and (7) such that \(x_{a}=(x,x_{v})^{T}\) is the state vector of the augmented system with the following dynamics
\[\dot{x}_{a}=A_{a}x_{a}+H_{a}\left(x_{a},u\right)+E_{a}f,\;y_{a}=C_{a}x_{a}=x_{v}, \tag{8}\]
where
\[\begin{array}{c}A_{a}=\left(\begin{array}{cc}A&O_{2n\times n}\\ A_{v}C&-A_{v}\end{array}\right),\\ H_{a}(x_{a},u)=\left(\begin{array}{c}H(x,u)\\ O_{n\times 1}\end{array}\right),\\ E_{a}=\left(\begin{array}{c}O_{2n\times m}\\ A_{v}E\end{array}\right),\;\mbox{and}\\ C_{a}=\left(\begin{array}{cc}O_{n\times 2n}&I_{n}\end{array}\right).\end{array} \tag{9}\]
Note that \(A_{v}\) must be chosen such that the pair \((A_{a},C_{a})\) is observable.
To design the observer, we use \(\hat{x}_{a}\) to denote the estimation of \(x_{a}\). Then, the estimated output will be \(\hat{y}_{a}=C_{a}\hat{x}_{a}=\hat{x}_{v}\). We also use the notations \(\tilde{x}_{a}=x_{a}-\hat{x}_{a}\) and \(\tilde{y}_{a}=y_{a}-\hat{y}_{a}\) to indicate the estimation errors. We propose the following observer
\[\dot{\hat{x}}_{a}=A_{a}\hat{x}_{a}+H_{a}\left(\hat{x}_{a},u\right)+E_{a}\hat{f }\left(t\right)+L\tilde{y}+\Lambda\;\upsilon, \tag{10}\]
where \(L\in R^{3n\times n}\) and \(\Lambda\in R^{3n\times n}\) are design parameters, \(\hat{f}\) is the estimation of fault, and \(\upsilon\in R^{n}\) is a term that will be introduced shortly.
Let us first explain how \(\hat{f}\) is computed. We represent the sensor fault as
\[f=\gamma d, \tag{11}\]
and aim to estimate \(d\). Here, \(d\in\mathbb{R}^{m}\), and \(\gamma\) is a constant positive definite matrix in \(\mathbb{R}^{m\times m}\).
**Remark 1:**_Note that \(\gamma\) serves as a scaling factor that adds an extra degree of freedom to tune the convergence rate of the fault estimator. While \(\gamma\) can be simply set to the identity matrix in many cases, our experiences with observer design reveal that this additional degree of freedom is generally useful in system tuning._
We use the following expressions to estimate \(d\).
\[\hat{d}=\hat{\pi}+\Gamma E^{T}\hat{y}_{a}, \tag{12}\]
where \(\hat{\pi}\) is an adaptive term with the following dynamics
\[\dot{\hat{\pi}}=-\Gamma E^{T}C_{a}\left(A_{a}\hat{x}_{a}+H_{a}(\hat{x}_{a},u)+ E_{a}\hat{f}\right), \tag{13}\]
where \(\Gamma\in\mathbb{R}^{m\times m}\) is a positive-definite design parameter. As it will be revealed in our Lyapunov analysis, \(\hat{\pi}\) can be regarded as the estimate of the value represented by
\[\pi=d-\Gamma E^{T}y_{a}. \tag{14}\]
The adaptation law (13) helps the observer to deal with the unknown upper bound of \(f\) i.e. \(F\).
As we deal with the challenging case in which the upper bound of \(\dot{f}\), i.e. \(F_{d}\), is unknown too, we introduce the following expression for \(\upsilon\)
\[\upsilon=\Upsilon\tilde{y}+\hat{\beta}\tilde{y}\|\tilde{y}\|^{-2}\text{tanh}^ {2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right), \tag{15}\]
where \(\Upsilon\in\mathbb{R}^{n\times n}\) and is positive-definite, and \(\rho_{1}>0\). Similar to \(\hat{\pi}\), \(\hat{\beta}\) is an adaptive parameter but with the following adaptation law
\[\dot{\hat{\beta}}=2\text{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right)- \rho_{2}\hat{\beta}, \tag{16}\]
where \(\rho_{2}>0\). Again, as it will be revealed in our Lyapunov analysis, \(\hat{\beta}\) is an estimate of a parameter defined as
\[\beta=0.5\lambda_{\min}^{-1}\left(\gamma\right)\lambda_{\max}\left(\left(\Gamma^ {-1}\right)^{T}\Gamma^{-1}\right)F_{d}. \tag{17}\]
Therefore, by using (16), the observer can deal with unknown \(F_{d}\).
**Remark 2:**_As it will become clear shortly, the reason to use \(\tanh^{2}\left(\cdot\right)\) is the desirable properties of this function in the vicinity of the origin. Our stability analysis below will lead to the evaluation of terms in the form of \(1-2\tanh^{2}\left(\cdot\right)\) where we can apply Lemma 3 (given in Appendix A) to establish stability criteria for design parameters while ensuring the smoothness of parameters evolution with time._
To determine the stability criteria for the design parameters, let us first write the estimation error dynamics using (8), (10), and \(\tilde{x}_{a}=x_{a}-\hat{x}_{a}\).
\[\dot{\tilde{x}}_{a}=(A_{a}-LC_{a})\tilde{x}_{a}+E_{a}\tilde{f}+H_{a}(x_{a},u) -H_{a}(\hat{x}_{a},u)-\Lambda\upsilon, \tag{18}\]
where \(\tilde{f}=f-\hat{f}\) is the fault estimation error. Using (11), (12), and (14), we have
\[\tilde{f}=\gamma\left(\tilde{\pi}+\Gamma E^{T}C_{a}\tilde{x_{a}}\right), \tag{19}\]
where \(\tilde{\pi}=\pi-\hat{\pi}\). Let us define the following Lyapunov function
\[V_{1}=\tilde{x}_{a}^{T}P\tilde{x}_{a}+0.5\tilde{\pi}^{T}\Gamma^{-1}\tilde{\pi }+0.5\tilde{\beta}^{2}, \tag{20}\]
where \(P>0\), and \(\tilde{\beta}=\beta-\hat{\beta}\). Taking the time derivative of \(V_{1}\) and substituting (18) yields
\[\begin{split}\dot{V}_{1}&=\tilde{x_{a}}^{T}P\dot{\tilde{x }}_{a}+\dot{\tilde{x}}_{a}^{T}P\tilde{x}_{a}+\tilde{\pi}^{T}\Gamma^{-1}\dot{ \tilde{\pi}}+\tilde{\beta}\dot{\tilde{\beta}}\\ &=\tilde{x}_{a}^{T}\left(P\left(A_{a}-LC_{a}\right)+\left(A_{a}- LC_{a}\right)^{T}P\right)\tilde{x}_{a}+\tilde{x}_{a}^{T}PE_{a}\tilde{f}+\tilde{f}^{T}E_{a}^ {T}P\tilde{x}_{a}\\ &+\tilde{x}_{a}^{T}P\left(H_{a}(x_{a},u)-H(\hat{x}_{a},u)\right) +\left(H_{a}(x_{a},u)-H_{a}(\hat{x}_{a},u)\right)^{T}P\tilde{x}_{a}\\ &-\tilde{x}_{a}^{T}P\Lambda v-\upsilon^{T}\Lambda^{T}P\tilde{x}_{ a}+\tilde{\pi}^{T}\Gamma^{-1}\dot{\tilde{\pi}}+\tilde{\beta}\dot{\tilde{\beta}}, \end{split} \tag{21}\]
In the following, we intend to find upper bounds on some of the terms on the right-hand side of (21). Our developments rely on the application of Young's inequality given in Appendix A.
For the second and third terms in the right-hand side of (21), we apply (A.1) to write
\[\tilde{x}_{a}^{T}PE_{a}\tilde{f}+\tilde{f}^{T}E_{a}^{T}P\tilde{x}_{a}\ \leq\frac{2}{3}\tilde{x}_{a}^{T}PP^{T}\tilde{x}_{a}\ +3\ \tilde{x}_{a}^{T}\varepsilon_{1} \varepsilon_{1}^{T}\tilde{x}_{a}\ +3\ \tilde{\pi}^{T}\varepsilon_{2} \varepsilon_{2}^{T}\tilde{\pi}, \tag{22}\]
where \(\varepsilon_{1}=C_{a}^{T}E\Gamma^{T}\gamma^{T}{E_{a}}^{T}\) and \(\varepsilon_{2}=\ \gamma^{T}E_{a}^{T}\). For the fourth term, we apply (A.1) and use the Lipschitz assumption (4) to write
\[\begin{split}\left(H_{a}\left(x_{a},u\right)-H_{a}\left(\hat{x}_{ a},u\right)\right)^{T}P\tilde{x}_{a}\ +\tilde{x}_{a}^{T}P\left(H_{a}\left(x_{a},u\right)-H_{a}\left(\hat{x}_{a},u \right)\right)\\ \leq\frac{1}{3}\tilde{x}_{a}^{T}PP^{T}\tilde{x}_{a}+3\ \kappa^{2} \tilde{x}_{a}^{T}\tilde{x}_{a}.\end{split} \tag{23}\]
For \(\dot{\tilde{\pi}}\), we use (8), (13), (14), and \(\tilde{\pi}=\pi-\hat{\pi}\) to write
\[\dot{\tilde{\pi}}=\dot{d}-\Gamma E^{T}C_{a}\left(A_{a}\tilde{x}_{a}+E_{a} \tilde{f}\right)-\Gamma E^{T}C_{a}\left(H_{a}\left(x_{a},u\right)-H_{a}\left( \hat{x}_{a},u\right)\right). \tag{24}\]
Therefore, \(\tilde{\pi}^{T}\Gamma^{-1}\dot{\tilde{\pi}}\) in (21) becomes
\[\begin{split}\tilde{\pi}^{T}\Gamma^{-1}\dot{\tilde{\pi}}& =\tilde{\pi}^{T}\Gamma^{-1}\dot{d}-\tilde{\pi}^{T}E^{T}C_{a}A_{a} \tilde{x}_{a}-\tilde{\pi}^{T}E^{T}C_{a}E_{a}\tilde{f}\\ -\tilde{\pi}^{T}E^{T}C_{a}\left(H_{a}\left(x_{a},u\right)-H_{a} \left(\hat{x}_{a},u\right)\right).\end{split} \tag{25}\]
Again, by applying Young's inequality (A.1) we can derive upper bounds for each term on the right-hand side of (25). To this end, the first term leads to
\[\tilde{\pi}^{T}\Gamma^{-1}\dot{d}\leq\frac{1}{2}\tilde{\pi}^{T}\tilde{\pi}+\frac{ 1}{2}\dot{d}^{T}\big{(}\Gamma^{-1}\big{)}^{T}\Gamma^{-1}\dot{d}\leq\frac{1}{2} \tilde{\pi}^{T}\tilde{\pi}+\beta, \tag{26}\]
where \(\beta\) was defined in (17). Similarly, we have
\[-\tilde{\pi}^{T}E^{T}C_{a}A_{a}\tilde{x}_{a}\leq\frac{1}{2c_{1}}\tilde{\pi}^{T }\varepsilon_{3}\varepsilon_{3}^{T}\tilde{\pi}+\frac{c_{1}}{2}\tilde{x}_{a}^ {T}\tilde{x}_{a}, \tag{27}\]
where \(\varepsilon_{3}=E^{T}C_{a}A_{a}\), and \(c_{1}\) is an arbitrary positive constant. We can also write
\[-\tilde{\pi}^{T}E^{T}C_{a}E_{a}\gamma\;\Gamma E^{T}C_{a}\tilde{x}_{a}\leq\frac {1}{2c_{2}}\tilde{\pi}^{T}\varepsilon_{4}\varepsilon_{4}^{T}\tilde{\pi}+\frac {c_{2}}{2}\tilde{x}_{a}^{T}\tilde{x}_{a}, \tag{28}\]
where \(\varepsilon_{4}=E^{T}C_{a}E_{a}\gamma\Gamma E^{T}C_{a}\) and \(c_{2}\) is an arbitrary positive constant. Next, we have
\[-\tilde{\pi}^{T}E^{T}C_{a}\left(H_{a}\left(x_{a},u\right)-H_{a}\left(\hat{x}_ {a},u\right)\right)\leq\kappa^{2}\;\tilde{x}_{a}^{T}\tilde{x}_{a}+\frac{1}{4} \tilde{\pi}^{T}\varepsilon_{5}\varepsilon_{5}^{T}\tilde{\pi}, \tag{29}\]
where \(\varepsilon_{5}=E^{T}C_{a}\). Substituting the expressions from (22) to (29) in (21) leads to
\[\dot{V}_{1}\leq-\tilde{x}_{a}^{T}Q_{1}\tilde{x}_{a}-\tilde{\pi}^{T}Q_{2} \tilde{\pi}-\tilde{x}_{a}^{T}P\Lambda v-\upsilon^{T}\Lambda^{T}P\tilde{x}_{a} +\beta+\tilde{\beta}\dot{\tilde{\beta}}, \tag{30}\]
where
\[Q_{1}=-\left(\left(A_{a}-LC_{a}\right)^{T}P+P\left(A_{a}-LC_{a}\right)+P^{2}+ 4\kappa^{2}I_{3n}+3\varepsilon_{1}\varepsilon_{1}^{T}+\frac{c_{1}}{2}+\frac{ c_{2}}{2}\right), \tag{31}\]
and
\[Q_{2}=-\left(-E^{T}C_{a}E_{a}+3\varepsilon_{2}\;\varepsilon_{2}^{T}+\frac{1} {2}+\frac{1}{2c_{1}}\varepsilon_{3}\;\varepsilon_{3}^{T}+\frac{1}{2c_{2}} \varepsilon_{4}\;\varepsilon_{4}^{T}+\frac{1}{4}\varepsilon_{5}\;\varepsilon _{5}^{T}\right). \tag{32}\]
Now, let us choose
\[\Lambda=P^{-1}C_{a}^{T}. \tag{33}\]
It follows from \(\tilde{y}=C_{a}\tilde{x}_{a}\) that
\[-\tilde{x}_{a}^{T}P\Lambda\upsilon-\upsilon^{T}\Lambda^{T}P\tilde{x}_{a}=-2 \tilde{x}_{a}^{T}P\Lambda\upsilon=-2\tilde{y}^{T}\upsilon. \tag{34}\]
Substituting (34) in (30), adding and subtracting \(2\beta\tanh^{2}\left(\rho_{1}^{-1}\left\|\tilde{y}\right\|^{2}\right)\), and using \(\tilde{\beta}=\beta-\hat{\beta}\rightarrow\dot{\tilde{\beta}}=-\dot{\tilde{ \beta}}\) yields
\[\begin{split}\dot{V}_{1}&\leq-\tilde{x}_{a}^{T}Q_{1} \tilde{x}_{a}-\tilde{\pi}^{T}Q_{2}\tilde{\pi}-2\tilde{y}^{T}\Upsilon\tilde{y}+ \tilde{\beta}\left(2\text{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right) -\dot{\tilde{\beta}}\right)\\ &\quad+\beta\left(1-2\text{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{ y}\|^{2}\right)\right).\end{split} \tag{35}\]
For the fourth term in the right-hand side of (35), we can use (16) and Young's inequality (A.1) to write
\[\begin{split}\tilde{\beta}\left(2\text{tanh}^{2}\left(\rho_{1}^{ -1}\|\tilde{y}\|^{2}\right)-\dot{\tilde{\beta}}\right)=\rho_{2}\tilde{\beta} \hat{\beta}=\rho_{2}\tilde{\beta}\left(\beta-\tilde{\beta}\right)\\ \leq-0.5\left(\rho_{2}\tilde{\beta}^{2}-\rho_{2}\beta^{2}\right). \end{split} \tag{36}\]
Therefore,
\[\dot{V}_{1}\leq-\tilde{x}_{a}^{T}Q_{1}\tilde{x}_{a}-\tilde{\pi}^{T}Q_{2} \tilde{\pi}+0.5\rho\beta^{2}-0.5\rho\tilde{\beta}^{2}+\beta\left(1-2\text{tanh }^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right)\right), \tag{37}\]
which can be re-written in the form of
\[\dot{V}_{1}\leq-\alpha V_{1}+\delta \tag{38}\]
where \(\alpha=\left\{\frac{\lambda_{\min}(Q_{1})}{\lambda_{\max}(P)},\frac{2\lambda_ {\min}(Q_{2})}{\lambda_{\max}(\Gamma^{-1})},\rho\right\}\), and
\[\delta=\left\{\begin{array}{ll}0.5\rho_{2}\beta^{2},&\text{if}\left\|\tilde {y}\right\|\geq 0.8814\rho_{1},\\ 0.5\rho_{2}\beta^{2}+\beta,&\text{if}\left\|\tilde{y}\right\|<0.8814\rho_{1}. \end{array}\right. \tag{39}\]
If \(Q_{1}>0\), \(Q_{2}>0\), and \(\rho_{2}>0\), then from Lemma 4 (given in Appendix A) we have
\[V_{1}(t)\leq V_{1}(0)e^{-\alpha t}+\int_{0}^{t}e^{-\alpha(t-\tau)}\delta^{2}( \tau)d\tau. \tag{40}\]
Then, from (20) it can be inferred that \(\tilde{x}_{a}\) is ultimately bounded. Therefore, the estimation error will remain in an adjustable neighborhood of the origin, and this guarantees that with the appropriate choice of design parameters, \(\hat{x}_{a}\) can become close to \(x_{a}\).
**Remark 3:**_The key feature of the proposed observer (10) is the adaptation laws (13) and (16) that are designed in a way that can estimate \(f\) without the need to know \(F\) and \(F_{d}\). The two adaptation laws also provide a higher degree of freedom in tuning the system which can lead to improved estimation performance._
## 4 Controller Design
This section presents the controller design. As mentioned in Section 1, our objective is to design a fault-tolerant controller that can integrate well with the fault detection observer, and establish a performance-prescribed, fast, and fixed-time convergence.
Let us first decompose \(L\) and \(\Lambda\) as \(L=\left(L_{1},L_{2},L_{3}\right)^{T}\) and \(\Lambda=\left(\Lambda_{1},\Lambda_{2},\Lambda_{3}\right)^{T}\). Based on the observer (10), we have
\[\left\{\begin{array}{l}\dot{\hat{x}}_{1}=\hat{x}_{2}+L_{1}\tilde{y}+\Lambda_ {1}\ \upsilon\\ \dot{\hat{x}}_{2}=M^{-1}\left(\hat{x}_{1}\right)\left(u-D\left(\hat{x}_{1}, \hat{x}_{2}\right)\hat{x}_{2}-G\left(\hat{x}_{1}\right)\right)+L_{2}\tilde{y} +\Lambda_{2}\ \upsilon\end{array}\right. \tag{41}\]
Let \(y_{d}\) be the desired trajectory, and \(e=\hat{x}_{1}-y_{d}\) be the tracking error.
Then, the error dynamics takes the following form
\[\left\{\begin{array}{l}\dot{e}=\hat{x}_{2}+L_{1}\tilde{y}+\Lambda_{1}\;v-\dot{y} _{d}\\ \ddot{e}=M^{-1}\left(\hat{x}_{1}\right)\left(u-D\left(\hat{x}_{1},\hat{x}_{2} \right)\hat{x}_{2}-G\left(\hat{x}_{1}\right)\right)+L_{2}\tilde{y}+L_{1}\dot{ \tilde{y}}+\Lambda_{2}v+\Lambda_{1}\dot{v}-\ddot{y}_{d}\end{array}\right. \tag{42}\]
Using (7), (10), and (15), we have
\[\dot{y}=\dot{x}_{v}=-A_{v}x_{v}+A_{v}y_{f}, \tag{43}\]
\[\dot{\hat{y}}=\dot{\hat{x}}_{v}=\left(A_{v}C\;-A_{v}\right)\hat{x}+A_{v}E\hat{ f}+L_{3}\tilde{y}+\Lambda_{3}v, \tag{44}\]
and
\[\dot{v}=\Upsilon\dot{\tilde{y}}+\dot{\hat{\beta}}\tilde{y}\|\tilde{y}\|^{-2} \mbox{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right)+\hat{\beta}\dot{ \tilde{y}}\|\tilde{y}\|^{-2}\mbox{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{ 2}\right)+\hat{\beta}\tilde{y}\Psi, \tag{45}\]
where
\[\Psi=\|\tilde{y}\|^{-4}\left(\Theta_{1}-\Theta_{2}\right), \tag{46}\]
\[\Theta_{1}=2\rho_{1}^{-1}\|\tilde{y}\|^{2}\tanh\left(\rho_{1}^{-1}\|\tilde{y} \|^{2}\right)\left(1-\mbox{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right) \right)\left(\dot{\tilde{y}}^{T}\tilde{y}+\tilde{y}^{T}\dot{\tilde{y}}\right), \tag{47}\]
and
\[\Theta_{2}=\mbox{tanh}^{2}\left(\rho_{1}^{-1}\|\tilde{y}\|^{2}\right)\left( \dot{\tilde{y}}^{T}\tilde{y}+\tilde{y}^{T}\dot{\tilde{y}}\right). \tag{48}\]
### Performance Prescription
To enable performance prescription, let us define a prescribed performance function \(\mu(t)\) as follows
\[\mu(t)=\left(\mu_{0}-\mu_{\infty}\right)e^{-lt}+\mu_{\infty}, \tag{49}\]
where \(\mu_{0}>e(0)\), and \(\mu_{\infty}>0\), and \(l>0\). Note that \(\mu(t)\) is a positive and descending function. If we find a way to ensure the following criteria
\[-\mu(t)\leq e(t)\leq\mu(t), \tag{50}\]
then we can shape the tracking error trajectory using the definition of \(\mu(t)\). As such, the inequality (50) is our performance prescription control objective.
To facilitate dealing with the inequality (50), we define a new variable as follows
\[z\left(t\right)=T^{-1}\left(\omega\right) \tag{51}\]
where \(\omega=\frac{e\left(t\right)}{\mu\left(t\right)}\), and \(T\left(z\left(t\right)\right)\) is a strictly monotonic ascending function with the following properties:
\[T\left(0\right)=0,\lim_{z\rightarrow+\infty}T\left(z\right)=+1,\lim_{z \rightarrow-\infty}T\left(z\right)=-1,\text{ and }\forall z\in L_{\infty} \rightarrow\left\|T(z)\right\|<1. \tag{52}\]
Let \(\underline{z}\) and \(\bar{z}\) be the lower and upper bounds on \(z\). Since \(T\) is a strictly uniform ascending function and \(\mu\left(t\right)\) is always positive, we can write
\[\mu\left(t\right)T\left(\underline{z}\right)\leq\mu\left(t\right)T\left(z \left(t\right)\right)\leq\mu\left(t\right)T\left(\overline{z}\right). \tag{53}\]
According to (51), \(e\left(t\right)=\mu\left(t\right)T\left(z\left(t\right)\right)\). Therefore, (50) can be fulfilled by ensuring \(z\) is bounded. This implies that our performance prescription objective is to find an appropriate mapping \(T\) and a control law that guarantees the boundness of \(z\).
A suitable choice for the function \(T\) is \(\tanh\left(\cdot\right)\). Using its inverse, we have
\[z=T^{-1}(\omega)=0.5\ln\left(\frac{1+\omega}{1-\omega}\right). \tag{54}\]
As such, we will use (54) in designing our TSMC control law.
### Terminal Sliding Mode Dynamics
Our TSMC developments here are inspired by [39]. The TSMC approach presented in [39] has the advantage of fast fixed-time convergence; however,
as it will be shown in our numerical simulations, it suffers from steady-state error in the presence of sensor fault. To address this, we propose a new integral second-order sliding surface, as opposed to the first-order sliding surface developed in [39].
**Remark 4:**_As it will be shown in our theoretical developments and numerical simulations, using the new integral second-order sliding surface reduces the steady-state error while maintaining similar fast convergence characteristics of its first-order counterpart in [39]._
**Remark 5:**_Note that we will base our developments on \(z\) instead of \(e\) to enable performance prescription. This is another key difference of our work compared to [39]._
Let us define the following sliding surface
\[\sigma=\dot{\eta}+\bar{g}\left(\eta\right), \tag{55}\]
where
\[\eta=z+\int_{0}^{t}\bar{g}\left(z\left(\tau\right)\right)d\tau. \tag{56}\]
In this definition, \(g\left(\cdot\right)\) is a scalar function defined as follows
\[g\left(\chi\right)=\frac{1}{\varphi\left(\chi\right)}\left(\underline{\lambda }\,\operatorname{sgn}\left(\chi\right)\left|\chi\right|^{p^{*}}+\bar{\lambda} \,\operatorname{sgn}\left(\chi\right)\left|\chi\right|^{\frac{p}{q}}\right), \tag{57}\]
where \(\chi\) is an arbitrary scalar serving as the argument for \(g\), and
\[p^{*}=\frac{1}{2}+\frac{\underline{p}}{2\underline{q}}+\left(\frac{\underline {p}}{2\underline{q}}-\frac{1}{2}\right)\operatorname{sgn}\left(\left|\chi \right|-1\right), \tag{58}\]
\[\varphi\left(\chi\right)=a+\left(1-a\right)e^{-b\left|\chi\right|^{c}}, \tag{59}\]
\(\underline{\lambda}\), \(\bar{\lambda}\), \(a<1\), and \(b\) are positive constants, \(c\) is an even positive integer, \(\underline{p}\), \(\underline{q}\), \(\bar{p}\), and \(\bar{q}\) are positive odd integers such t \(\underline{p}>\underline{q}\) and \(\bar{p}<\bar{q}\). To construct the
sliding surface, we define two vectors as \(\bar{g}\left(z\right)=\left(g\left(z_{1}\right),g\left(z_{2}\right),\cdots,g \left(z_{n}\right)\right)^{T}\) and \(\bar{g}\left(\eta\right)=\left(g\left(\eta_{1}\right),g\left(\eta_{2}\right), \cdots,g\left(\eta_{n}\right)\right)^{T}\).
As it will be revealed in our Lyapunov analysis shortly, the use of (55) will lead to the following terminal sliding dynamics
\[\left\{\begin{array}{l}\chi_{2}=\chi_{1}+\int_{0}^{t}\frac{1}{\varphi\left( \chi_{1}\left(\tau\right)\right)}(\ \underline{\lambda}\,\mathrm{sgn}\left(\chi_{1}\left(\tau\right)\right)|\chi_{ 1}\left(\tau\right)|^{p^{*}}+\bar{\lambda}\ \mathrm{sgn}(\chi_{1}(\tau))|\chi_{1}(\tau)|^{\frac{ \bar{p}}{q}})d\tau,\\ \dot{\chi}_{2}=-\frac{1}{\varphi\left(\chi_{2}\right)}(\ \underline{\lambda}\, \mathrm{sgn}\left(\chi_{2}\right)|\chi_{2}|^{p^{*}}+\bar{\lambda}\ \mathrm{sgn}(\chi_{2})|\chi_{2}|^{\frac{ \bar{p}}{q}}),\end{array}\right. \tag{60}\]
where \(\chi_{1}=z_{i}\) and \(\chi_{2}=\eta_{i}\) for \(i=1,\cdots,n\). We will establish our convergence results based on findings in [39] which is summarized as Lemma 6 in Appendix A of this paper.
First, let us take the derivative of the first equation in (60). We have
\[\dot{\chi}_{2}=\dot{\chi}_{1}+\frac{1}{\varphi\left(\chi_{1}\right)}\left( \underline{\lambda}\ \mathrm{sgn}\left(\chi_{1}\right)|\chi_{1}|^{p^{*}}+\bar{ \lambda}\mathrm{sgn}\left(\chi_{1}\right)|\chi_{1}|^{\frac{\bar{p}}{q}}\right). \tag{61}\]
According to Lemma 6 and the second equation of (60), \(\chi_{2}\) will converge to the origin in fixed time. When \(\chi_{2}=0\) and \(\dot{\chi}_{2}=0\), (61) becomes
\[\dot{\chi_{1}}=-\frac{1}{\varphi\left(\chi_{1}\right)}\left(\underline{\lambda }\ \mathrm{sgn}\left(\chi_{1}\right)|\chi_{1}|^{p^{*}}+\bar{\lambda}\ \mathrm{sgn}\left(\chi_{1}\right)|\chi_{1}|^{\frac{ \bar{p}}{q}}\right). \tag{62}\]
Again, based on Lemma 6, we infer that \(\chi_{1}\) will converge to the origin in fixed time.
Note that in the definition of the function \(g\) and its design parameters, we follow the criteria given by Lemma 6. Since \(\dot{\chi}_{1}\) and \(\dot{\chi}_{2}\) have the same dynamics as in (A.11), the convergence time for them will be in the form of (A.12), which is identical to [39].
### Control Law and Stability Analysis
This section presents our fault-tolerant control law based on the sliding surface (55).
Similar to most sliding mode control designs, let us consider the Lyapunov function
\[V_{2}=\frac{1}{2}\sigma^{T}\sigma, \tag{63}\]
Taking the time-derivative of (63) yields \(\dot{V}_{2}=\sigma^{T}\dot{\sigma}\). In the following, we calculate \(\dot{\sigma}\) and substitute in \(\dot{V}_{2}\).
Evaluating (55) leads to the following expression
\[\dot{\sigma}=\ddot{z}+\dot{\bar{g}}\left(z\right)+\dot{\bar{g}}\left(\eta \right). \tag{64}\]
For the first term in the right-hand side of (64), we take the derivative of (54). It follows that
\[\dot{z}=\frac{\partial T^{-1}}{\partial\omega}\dot{\omega}=r\left(\dot{e}-se \right), \tag{65}\]
where
\[r=\mbox{diag}\left(\left(\mu\;\left(1-{\omega_{1}}^{2}\right)\right)^{-1}, \ldots,\left(\mu\;\left(1-{\omega_{n}}^{2}\right)\right)^{-1}\right), \tag{66}\]
and \(s=\dot{\mu}\mu^{-1}I_{n}\). Next,
\[\ddot{z}=\dot{r}\left(\dot{e}-se\right)+r\left(\ddot{e}-\dot{s}e-s\dot{e} \right), \tag{67}\]
where
\[\dot{s}=\mbox{diag}\left(\frac{\ddot{\mu}\;\mu-\dot{\mu}^{2}}{\mu^{2}},\ldots,\frac{\ddot{\mu}\;\mu-\dot{\mu}^{2}}{\mu^{2}}\right), \tag{68}\]
and
\[\dot{r}=\mbox{diag}\left(\frac{-\dot{\mu}\left(1-{\omega_{1}}^{2}\right)+2\; \mu\omega_{1}\dot{\omega}_{1}}{\left(\mu\;\left(1-{\omega_{1}}^{2}\right) \right)^{2}},\ldots,\frac{-\dot{\mu}\;\left(1-{\omega_{n}}^{2}\right)+2\mu\; \omega_{n}\dot{\omega}_{n}}{\left(\mu\;\left(1-{\omega_{n}}^{2}\right)\right) ^{2}}\right). \tag{69}\]
By defining \(R=\dot{r}\left(\dot{e}-se\right)-r\left(\dot{s}e+s\dot{e}\right)\), \(\ddot{z}\) can be written as
\[\ddot{z}=R+r\ddot{e}. \tag{70}\]
For the second and third terms in the right-hand side of (64), we take the derivative of (57). This leads to the following expression
\[\begin{array}{c}\dot{g}\left(\chi\right)=-\frac{\dot{\varphi}\left(\chi\right)} {\varphi^{2}\left(\chi\right)}\left(\ \lambda\,{\rm sgn}\left(\chi\right)\left|\chi\right|^{p^{*}}+\bar{\lambda}\,{ \rm sgn}\left(\chi\right)\left|\chi\right|^{\frac{p}{q}}\right)\\ \qquad\qquad+\frac{1}{\varphi\left(\chi\right)}\left(\lambda p^{*}\left|\chi \right|^{p^{*}-1}+\bar{\lambda}\frac{\bar{p}}{\bar{q}}\left|\chi\right|^{ \frac{p}{q}-1}\right)\dot{\chi},\end{array} \tag{71}\]
where
\[\dot{\varphi}\left(\chi\right)=-\left(1-a\right)b{\rm csgn}\left(\chi\right) \left|\chi\right|^{c-1}\dot{\chi}e^{-b\left|\chi\right|^{c}}. \tag{72}\]
Substituting (70) and (71) into (64) yields
\[\dot{\sigma}=\ddot{z}+\dot{g}\left(z\right)+\dot{g}\left(\eta\right)=R+r\ddot{ e}+\dot{g}\left(z\right)+\dot{g}\left(\eta\right). \tag{73}\]
Then, from (42) it follows that
\[\begin{array}{c}\dot{\sigma}=rM^{-1}\left(\hat{x}_{1}\right)\left(u-D\left( \hat{x}_{1},\hat{x}_{2}\right)\hat{x}_{2}-G\left(\hat{x}_{1}\right)\right)\\ \qquad+M\left(\hat{x}_{1}\right)\left(L_{2}\tilde{y}+L_{1}\dot{\tilde{y}}+ \Lambda_{2}\,\,v+\Lambda_{1}\,\,\dot{v}-\ddot{y}_{d}\right)\\ \qquad+R+\dot{g}\left(z\right)+\dot{g}\left(\eta\right).\end{array} \tag{74}\]
Let us define the control law as follows
\[\begin{array}{c}u=D\left(\hat{x}_{1},\hat{x}_{2}\right)\ \hat{x}_{2}+G\left(\hat{x}_{1}\right)-M\left(\hat{x}_{1}\right)\ \left(L_{2}\tilde{y}+L_{1}\dot{\tilde{y}}+\Lambda_{2}\,\,v+\Lambda_{1}\,\, \dot{v}-\ddot{y}_{d}\right)\\ \qquad-M\left(\hat{x}_{1}\right)r^{-1}\left(R+\dot{g}\left(z\right)+\dot{g} \left(\eta\right)+k_{1\sigma}\sigma+c_{1\sigma}\sigma^{p_{\sigma}/q_{\sigma}} +\bar{c}_{1\sigma}\sigma^{\bar{p}_{\sigma}/\bar{q}_{\sigma}}\right),\end{array} \tag{75}\]
where \(k_{1\sigma}\), \(c_{1\sigma}\) and \(\bar{c}_{1\sigma}\) are positive constants, and \(p_{\sigma}\), \(q_{\sigma}\), \(\bar{p}_{\sigma}\) and \(\bar{q}_{\sigma}\) are positive odd numbers such that \(p_{\sigma}>q_{\sigma}\) and \(\bar{p}_{\sigma}<\bar{q}_{\sigma}\). Furthermore, \(\sigma^{p_{\sigma}/q_{\sigma}}=\left(\sigma_{1}{}^{p_{\sigma}/q_{\sigma}}, \ldots,\sigma_{n}{}^{p_{\sigma}/q_{\sigma}}\right)^{T}\) and \(\sigma^{\bar{p}_{\sigma}/\bar{q}_{\sigma}}=\left(\sigma_{1}{}^{\bar{p}_{\sigma }/\bar{q}_{\sigma}},\ldots,\sigma_{n}{}^{\bar{p}_{\sigma}/\bar{q}_{\sigma}} \right)^{T}\).
Substituting (75) in (74) and using \(\dot{V}_{2}=\sigma^{T}\dot{\sigma}\) yields
\[\dot{V}_{2}=-k_{1\sigma}\sigma^{T}\sigma-c_{1\sigma}\sigma^{T}\sigma^{p_{ \sigma}/q_{\sigma}}-\bar{c}_{1\sigma}\sigma^{T}\sigma^{\bar{p}_{\sigma}/\bar{ q}_{\sigma}}, \tag{76}\]
where
\[\sigma^{T}\sigma^{p_{\sigma}/q_{\sigma}}=\sigma_{1}{}^{1+p_{\sigma}/q_{\sigma} }+\ldots+\sigma_{n}{}^{1+p_{\sigma}/q_{\sigma}}, \tag{77}\]
\[\sigma^{T}\sigma^{\bar{p}_{\sigma}/\bar{q}_{\sigma}}=\sigma_{1}^{1+\bar{p}_{\sigma }/\bar{q}_{\sigma}}+\ldots+\sigma_{n}^{1+\bar{p}_{\sigma}/\bar{q}_{\sigma}}. \tag{78}\]
Based on the definition of \(V_{2}\) in (63), (A.2) and (A.3), we have
\[\alpha_{\sigma}V_{2}^{\beta_{v}}\leq\sigma^{T}\sigma^{p_{\sigma}/q_{\sigma}} \tag{79}\]
where \(\beta_{v}=(\ p_{\sigma}+q_{\sigma})/2q_{\sigma}\), \(\alpha_{\sigma}=2^{(\ p_{\sigma}+q_{\sigma})/2q_{\sigma}}\ n^{(\ p_{\sigma}-q_{ \sigma})/2q_{\sigma}}\), and
\[\bar{\alpha}_{\sigma}V_{2}^{\bar{\beta}_{v}}\leq\sigma^{T}\sigma^{\bar{p}_{ \sigma}/\bar{q}_{\sigma}}, \tag{80}\]
with \(\bar{\beta}_{v}=\frac{(\ \bar{p}_{\sigma}+\bar{q}_{\sigma})}{2\bar{q}_{\sigma}}\) and \(\bar{\alpha}_{\sigma}=2^{\frac{(\ \bar{p}_{\sigma}+\bar{q}_{\sigma})}{2\bar{q}_{\sigma}}}\). Thus, \(\dot{V}_{2}\) satisfies
\[\dot{V}_{2}\leq-2k_{1\sigma}\ V_{2}-c_{1\sigma}\alpha_{\sigma}\ V_{2}^{\beta_{ v}}-\bar{c}_{1\sigma}\bar{\alpha}_{\sigma}\ V_{2}^{\bar{\beta}_{v}}. \tag{81}\]
Let \(\alpha_{\nu}=c_{1\sigma}\alpha_{\sigma}\) and \(\bar{\alpha}_{\nu}=\bar{c}_{1\sigma}\bar{\alpha}_{\sigma}\). Then, it follows that
\[\dot{V}_{2}\leq-\alpha_{\nu}V_{2}^{\beta_{v}}-\bar{\alpha}_{\nu}V_{2}^{\bar{ \beta}_{v}}, \tag{82}\]
which guarantees fixed-time stability of \(\sigma\) according to Lemma 5 (given in Appendix A). This implies that \(\eta\) and \(z\) are fixed-time stable.
Furthermore, based on (54), we infer that \(e\to 0\) when \(z\to 0\). Since \(\tilde{x}_{1}=x_{1}-\hat{x}_{1}\), we can write \(x_{1}=\tilde{x}_{1}+\hat{x}_{1}=\tilde{x}_{1}+e+y_{d}\). Since \(e\to 0\) and \(\tilde{x}_{1}\) is ultimately bounded, it can be inferred that \(x_{1}\) converges to a neighborhood of \(y_{d}\).
We can also show that \(x_{2}\) converges to a neighborhood of \(\dot{y}_{d}\). To this end, from (55) and (56), we note that \(\sigma=\dot{z}+\bar{g}\left(z\right)+\bar{g}\left(\eta\right)\). Since \(\bar{g}(0)=0\), at the time that \(\sigma\to 0\), we have \(\dot{z}\to 0\). It follows from (65) that \(\dot{e}\to e\). When \(\sigma\to 0\), we have \(e\to 0\). Therefore, \(\dot{e}\to 0\).
With that in mind, we can now examine the first equation in (42) as follows. Since the observer guarantees the ultimate boundedness of estimation
error, \(\tilde{y}\) is arbitrarily small. This implies that \(\upsilon\) is arbitrarily small as it is given by (15). We also established that \(\dot{e}\to 0\) in the above paragraph. Putting all these together and evaluating the first equation in (42) implies that \(\hat{x}_{2}\) will converge to a neighborhood of \(\dot{y}_{d}\). Finally, we note that \(x_{2}=\tilde{x}_{2}\) + \(\hat{x}_{2}\), and since \(\tilde{x}_{2}\) is arbitrarily small, we can infer that \(x_{2}\) will converge to a neighborhood of \(\dot{y}_{d}\).
## 5 Numerical Simulations
This section presents numerical simulations to evaluate the effectiveness of the proposed method.
### Example 1: Two-Link Rigid Robotic Manipulator
In this example, we consider a two-link rigid robotic manipulator (Fig. 2), and compare our method with [39] and [7]. The parameters of the robot model are selected according to [39] to minimize confounding errors in comparisons.
Figure 2: Two-link rigid robotics manipulator system
Before the comparison, we would like to highlight the benefits of performance prescription. Figures 3 and 4 show the tracking error of the robot controlled by our method in nominal conditions. Two different performance prescription functions are considered, one with \(\mu_{0}=5\), \(\mu_{\infty}\) = 2, and \(l=0.1\) (Fig. 3), and the other with \(\mu_{0}=1\), \(\mu_{\infty}\) = 0.01 and \(l=10\) (Fig. 4). The rest of the design parameters are set as \(\underline{\lambda}=1\), \(\bar{\lambda}=2\), \(a=0.7\), \(b=1\), \(c=2\), \(\underline{p}=25\), \(\underline{q}=23\), \(\bar{p}=23\), \(\bar{q}=25\), \(k_{1\sigma}=10\), \(c_{1\sigma}=10\), \(\bar{c}_{1\sigma}=10\), and the robot joints are tasked to follow sinusoidal trajectories.
Figure 3: Tracking error within the bounds of the performance prescription function with \(\mu_{0}=5\), \(\mu_{\infty}\) = 2, and \(l=0.1\).
It is evident that in both cases the tracking error is bounded within the funnels set by the prescription functions. Clearly, setting tighter bounds in Fig. 4 has not violated the boundaries and the controller has managed to keep \(\|e_{i}\|\leq 0.01\) with a fast decay rate. This shows the degree of flexibility that our method offers in prescribing the tracking error performance, a feature that is missing from the benchmark methods that will be discussed shortly and also from many other methods in fault-tolerant control literature.
Let us now compare our method with methods proposed in [39] and [7]. We set the design parameters for our controller as mentioned above. For the method by Gao et al., we used the same parameters that were proposed in [39]. For the method by Ma et al., there was a need to fine-tune the parameters. After several simulation trials, we chose the design parameters according to the original paper [7] with the exceptions of \(G_{1}=8\) and \(\theta=12\). Furthermore, we set the initial conditions of all adaptation laws to zero. We tasked the robot joints to follow sinusoidal trajectories and considered a step
Figure 4: Tracking error within the bounds of the performance prescription function \(\mu_{0}=1\), \(\mu_{\infty}\) = 0.01, and \(l=10\).
fault on the first angular position sensor with \(E=\left(1\;0\right)^{T}\). This fault occurs at \(t=25s\) and remains in the rest of the simulation.
Figures 5 and 6 demonstrate the trajectory-tracking performance and control inputs of all three methods. When the sensor fault occurs at \(t=25s\), Gao et al.'s method fails to compensate for the fault's effect and undergoes a large steady-state error. This observation is aligned with our discussion in Section 4.2. Our second-order sliding mode surface which was designed to address this steady-state error can effectively compensate for the effect of sensor fault and maintain accurate tracking. Note that Ma et al.'s method is also capable of compensating for the fault effect, but with a larger overshoot and slower convergence rate. The superiority of our method partially stems from our adaptive observer design which is able to estimate the fault effects significantly faster than the Ma et al. method as shown in Fig. 7.
Figure 5: Time trajectory of robot joint positions for the proposed method, Gao et al.[39] and Ma et al.[7]
Overall, the above simulation results show that our method (i) provides the ability for prescribing the performance of the trajectory-tracking error, (ii) addresses the steady-state error issue present in the Gao et al. method while maintaining similar fast convergence, and (iii) enables faster detection of fault and subsequently more effective fault compensation compared to the Ma et al. method.
Figure 6: Control input for the proposed method, Gao et al.[39] and Ma et al.[7]
Figure 7: Fault estimation for the proposed method and Ma et al.[7]
### Example 2: Three-Degrees-of-Freedom Robotic Manipulator
To further examine the effectiveness of the proposed method, we consider another example in which the trajectory control of a three-degrees-of-freedom manipulator is considered. This manipulator is a newly designed system to be used as a solar tracker base, as shown in Fig. 8. The model of the manipulator is given in Appendix B.
We study the system performance under different forms of sensor faults: (i) sinusoidal in the form of \(0.5\left(\sin t+\sin 3.5t\right)\), (ii) an offset of \(0.5\ rad\) with additive noise, and (iii) continuously increasing fault represented by a ramp signal. We apply all these faults to the position sensor of all three joints at \(t=25\ s\). We set the design parameters as \(\underline{\lambda}=0.01\), \(\bar{\lambda}=0.01\), \(a=0.7\), \(b=1\), \(c=2\), \(\underline{p}=11\), \(\underline{q}=9\), \(\bar{p}=9\), \(\bar{q}=11\), \(k_{1\sigma}=1\), \(c_{1\sigma}=1\), \(\bar{c}_{1\sigma}=1\).
Figure 8: Solar tracker system
Figure 9 presents the tracking error of all three joints under different faults. It is evident that the occurrence of faults has degraded the tracking error momentarily; however, our proposed fault estimation and control scheme has managed to recover the system states and restore zero steady-state error. Of note, even in the transient phase, the estimation error remains within the prescribed funnels.
Figure 10 illustrates the state estimation error in each simulation scenario, showing the proposed observer's effectiveness to recover system states shortly after the occurrence of faults. This performance is attributed to the ability of the observer to converge to the actual fault values shown in Fig. 11. The fault estimation part of the observer relies on two adaptive parameters \(\hat{\pi}\) and \(\hat{\beta}\) whose time trajectories are shown in Fig. 12. Note that since the results for different joints in each scenario are similar, we only present the results
Figure 9: Tracking error in different fault scenarios
for the first joint.
## 6 Conclusion
This paper presented new results for sensor fault detection and compensation in robotic manipulators. Our Lyapunov-based stability analysis and simulation experiments verified the proposed method both theoretically and numerically. Our results conclude that the representation of sensor faults as virtual actuator faults combined with adaptive observer design given in (10) can be a viable technique for sensor fault detection. In addition, the new TSMC law given in (75) proved to be effective in compensating for the sensor fault effects.
The strengths of the proposed method include the ability to detect faults without the need to impose known bounds on the fault value or its derivative, and also fault compensation with a fast and fixed-time transient response, and the ability to prescribe system performance. The above is achieved with only joint position measurements, despite many existing methods that require a measure of joint velocities in addition to position measurements.
Future research directions on the proposed method include the extension of the results to under-actuated systems, and the incorporation of delays, input saturation, and/or dynamic uncertainty.
## Appendix A Mathematical Background
### Useful Inequalities
**Lemma 1 - Young's Inequality**[40]: _For any given \(a,b\in\mathbb{R}^{n}\) we have_
\[2a^{T}SQb\leq a^{T}SPS^{T}a+b^{T}Q^{T}P^{-1}Qb,\] (A.1)
_where \(P>0\), \(S\), and \(Q\) have appropriate dimensions._
**Lemma 2**[33]: _For \(v=\left(v_{1},v_{2},\cdots,v_{N}\right)^{T}\in\mathbb{R}^{n}\), and the constants \(0<a_{1}<1\) and \(a_{2}>1\), we have_
\[\sum_{i=1}^{n}v_{i}^{a_{1}}\geq\left(\sum_{i=1}^{n}v_{i}\right)^{a_{1}},\] (A.2)
_and_
\[\sum_{i=1}^{n}v_{i}^{a_{2}}\geq n^{1-a_{2}}\left(\sum_{i=1}^{n}v_{i}\right)^{a _{2}}.\] (A.3)
**Lemma 3**[41]: _Given two positive scalars \(a\) and \(b\), we have_
\[|a|\geq 0.8814b\to 1-2\tanh^{2}\left(\frac{a}{b}\right)\leq 0,\] (A.4)
_and_
\[|a|<0.8814b\to 0<1-2\tanh^{2}\left(\frac{a}{b}\right)<1.\] (A.5)
### Ultimate Boundedness
**Lemma 4:**[42] _Let \(V\) and \(\rho\) be real-valued positive definite functions, and let \(\alpha\) and \(\beta\) be positive constants. If they satisfy the differential inequality_
\[\dot{V}\leq-\alpha V+\beta\rho^{2},\ \ v(0)\geq 0,\] (A.6)
_then we have_
\[V(t)\leq V(0)e^{-\alpha t}+\beta\int_{0}^{t}e^{-\alpha(t-\tau)}\rho(\tau)^{2}d\tau.\] (A.7)
_A.3. Fixed-Time Stability_
**Lemma 5**[43]: _Consider the following nonlinear system_
\[\dot{\chi}\left(t\right)=f\left(\chi\left(t\right)\right),\ \ \ \chi\left(0\right)=\chi_{0},\] (A.8)
_where \(\chi\in\mathbb{R}^{n}\), and \(f\left(\chi\left(t\right)\right):\mathbb{R}^{n}\ \rightarrow\ \mathbb{R}^{n}\) is a continuous function. The system (A.8) is said to be fixed-time stable if there exists a continuous positive definite function \(V\left(\chi\right)\) such that_
\[\dot{V}\left(\chi\right)\leq-a\ V\left(\chi\right)^{\alpha}-b\ V\left(\chi \right)^{\vartheta}+\zeta,\] (A.9)
_where \(a>0\), \(b>0\), \(0<\alpha<1\), \(\vartheta>1\), and \(0<\zeta<\infty\). The convergence region is_
\[\Delta=\left\{\chi|\ V\left(\chi\right)\leq\min\left\{\left(\frac{\zeta}{\left( 1-\theta\right)a}\right)^{\frac{1}{\alpha}},\left(\frac{\zeta}{\left(1-\theta \right)b}\right)^{\frac{1}{\vartheta}}\right\}\right\},\] (A.10)
_where \(0<\theta<1\), and the settling time is \(T\left(\chi_{0}\right)\) such that \(T\left(\chi_{0}\right)<T_{max}\), and \(0<T_{max}\leq\frac{1}{a\left(1-\alpha\right)}+\frac{1}{b\left(1-\vartheta \right)}\)._
**Lemma 6**[39]:_Consider the following scalar system_
\[\dot{\chi}=-\frac{1}{\varphi\left(\chi\right)}\left(\underline{\lambda} \mathrm{sgn}^{p^{*}}\left(\chi\right)+\bar{\lambda}\mathrm{sgn}^{\frac{p}{q}} \left(\chi\right)\right),\] (A.11)
_where \(\varphi\left(\chi\right)=a_{1}+\left(1-a_{1}\right)e^{-b_{1}\left|\chi\right| ^{c_{1}}}\), \(p^{*}=0.5\left(\underline{p}\big{/}\underline{q}+\left(\underline{p}\big{/} \underline{q}-1\right)\mathrm{sgn}\left(\left|\chi\right|-1\right)\right)\), \(\underline{\lambda}>0\), \(\bar{\lambda}>0\), \(0<a_{1}<1\), \(b_{1}>0\), \(c_{1}\) is a positive even integer, and \(\underline{p}>0\), \(\underline{q}>0\), \(\bar{p}>0\), \(\bar{q}>0\), \(\underline{p}>\underline{q}\), and \(\bar{p}<\bar{q}\) are odd integers. The system (A.11) is fixed-time stable with the following convergence time_
\[T_{s1}\left(\chi_{0}\right)<\frac{\underline{q}}{\underline{\lambda}\left( \underline{p}-\underline{q}\right)}+\frac{\bar{q}}{\bar{q}-\bar{p}}\frac{1}{ \underline{\lambda}}\ln\left(1+\frac{\underline{\lambda}}{\bar{\lambda}} \right)\!.\] (A.12)
## Appendix B Three-Degrees-of-Freedom Robotic Manipulator Model
The elements of the \(M(q)\) matrix include
\[m_{11}=s_{2}^{2}\left(m_{2}\ell_{2}^{2}+m_{3}(c_{3}\ell_{3}+L_{2}) ^{2}+I_{y_{2}}+I_{y_{3}}\right)+m_{3}s_{3}^{2}\ell_{3}^{2}+I_{z_{1}}\] \[+c_{2}^{2}\left(I_{z_{2}}+s_{3}^{2}I_{x_{3}}+c_{3}^{2}I_{z_{3}} \right),\]
\[m_{12}=m_{21}=s_{3}c_{2}\left(c_{3}\left(I_{x_{3}}-I_{z_{3}}\right)-m_{3}\ell_ {3}\left(c_{3}\ell_{3}+L_{2}\right)\right),\]
\[m_{13}=m_{31}=s_{2}\left(m_{3}\ell_{3}\left(\ell_{3}+c_{3}L_{2}\right)-I_{y_{3 }}\right),\]
\[m_{22}=m_{2}\ell_{2}^{2}+m_{3}\left(c_{3}\ell_{3}+L_{2}\right)^{2}+I_{x_{2}}+c _{3}^{2}I_{x_{3}}+s_{3}^{2}I_{z_{3}}\]
\[m_{23}=m_{32}=0,\]
and
\[m_{33}=m_{3}l_{3}^{2}+I_{y_{3}}.\]
The elements of the \(D(q,\dot{q})\) include
\[d_{11}=\left(s_{3}c_{3}\left(m_{3}l_{3}^{2}+c_{2}^{2}(I_{x_{3}}- I_{z_{3}})-s_{2}^{2}\left(m_{3}s_{3}\ell_{3}\left(c_{3}\ell_{3}+L_{2}\right) \right)\right)\dot{q}_{3}\] \[+s_{2}c_{2}\left(m_{2}\ell_{2}^{2}+m_{3}\left(c_{3}\ell_{3}+L_{2} \right)^{2}+I_{y_{2}}+I_{y_{3}}-I_{z_{2}}-s_{3}^{2}I_{x_{3}}-c_{3}^{2}I_{z_{3} }\right)\dot{q}_{2},\]
\[d_{12}=s_{2}c_{2}\left(m_{2}\ell_{2}^{2}+m_{3}\left(c_{3}\ell_{3}+L_{2} \right)^{2}+I_{y_{2}}+I_{y_{3}}-I_{z_{2}}-s_{3}^{2}I_{x_{3}}-c_{3}^{2}I_{z_{3} }\right)\dot{q}_{1}\]
\[-s_{2}s_{3}\left(c_{3}\left(I_{x_{3}}-I_{z_{3}}\right)-m_{3}\ell_{3}\left(c_{ 3}\ell_{3}+L_{2}\right)\right)\dot{q}_{2}\]
\[+c_{2}\left(m_{3}s_{3}^{2}\ell_{3}^{2}+c_{3}^{2}\left(I_{x_{3}}-I_{z_{3}} \right)+\frac{1}{2}\left(I_{z_{3}}-I_{y_{3}}-I_{x_{3}}\right)\right)\dot{q}_{3},\]
\[d_{13}=\left(s_{3}c_{3}(m_{3}l_{3}^{2}+c_{2}^{2}(I_{x_{3}}-I_{z_{3}}))-s_{2}^{ 2}\left(m_{3}s_{3}\ell_{3}\left(c_{3}\ell_{3}+L_{2}\right)\right)\right)\dot{q }_{1}\]
\[+c_{2}\left(m_{3}s_{3}^{2}\ell_{3}^{2}+c_{3}^{2}\left(I_{x_{3}}-I_{z_{3}} \right)+\frac{1}{2}\left(I_{z_{3}}-I_{x_{3}}-I_{y_{3}}\right)\right)\dot{q}_{2}\]
\[-m_{3}s_{2}s_{3}\ell_{3}L_{2}\dot{q}_{3},\]
\[d_{21}=-s_{2}c_{2}\left(m_{2}\ell_{2}^{2}+m_{3}\left(c_{3}\ell_{3}+L_{2} \right)^{2}+I_{y_{2}}+I_{y_{3}}-I_{z_{2}}-s_{3}^{2}I_{x_{3}}-c_{3}^{2}I_{z_{3} }\right)\dot{q}_{1}\]
\[+c_{2}\left(c_{3}^{2}\left(I_{x_{3}}-I_{z_{3}}\right)-m_{3}c_{3}\ell_{3}\left( c_{3}\ell_{3}+L_{2}\right)+\frac{1}{2}\left(I_{y_{3}}+I_{z_{3}}-I_{x_{3}}\right) \right)\dot{q}_{3},\]
\[c_{22}=s_{3}\left(c_{3}\left(I_{z_{3}}-I_{x_{3}}\right)-m_{3}l_{3}\left(c_{3}l_{3}+ L_{2}\right)\right)\dot{q}_{3}\]
\[d_{23}=c_{2}\left(c_{3}^{2}\left(I_{x_{3}}-I_{z_{3}}\right)-m_{3}c_{3}\ell_{3} \left(c_{3}\ell_{3}+L_{2}\right)+\frac{1}{2}\left(I_{y_{3}}+I_{z_{3}}-I_{x_{3} }\right)\right)\dot{q}_{1}\]
\[+s_{3}\left(c_{3}\left(I_{z_{3}}-I_{x_{3}}\right)-m_{3}l_{3}\left(c_{3}l_{3}+L _{2}\right)\right)\dot{q}_{2},\]
\[d_{31}=\left(s_{2}^{2}\left(m_{3}s_{3}\ell_{3}\left(c_{3}\ell_{3}+L_{2}\right) \right)-s_{3}c_{3}\left(m_{3}l_{3}^{2}+c_{2}^{2}\right.\left(I_{x_{3}}-I_{z_{ 3}}\right)\right)\dot{q}_{1}\]
\[+c_{2}\,\,\left(m_{3}c_{3}\ell_{3}\left(c_{3}\ell_{3}+L_{2}\right)-c_{3}^{2} \left(I_{x_{3}}-I_{z_{3}}\right)-\frac{1}{2}\left(I_{y_{3}}+I_{z_{3}}-I_{x_{3} }\right)\right)\dot{q}_{2},\]
\[d_{32}=c_{2}\left(m_{3}c_{3}\ell_{3}\left(c_{3}\ell_{3}+L_{2}\right)-c_{3}^{2} \left(I_{x_{3}}-I_{z_{3}}\right)-\frac{1}{2}\left(I_{y_{3}}+I_{z_{3}}-I_{x_{3} }\right)\right)\dot{q}_{1}\]
\[+s_{3}\left(m_{3}l_{3}\left(c_{3}l_{3}+L_{2}\right)-c_{3}\left(I_{z_{3}}-I_{x_ {3}}\right)\right)\dot{q}_{2},\]
and \(d_{33}=0\), where \(s_{i}\) and \(c_{i}\) stand for \(\sin(q_{i})\) and \(\cos(q_{i})\), respectively.
The vector of the effect of gravitational force is expressed as
\[G\left(q\right)=-g\left(\begin{array}{c}0\\ s_{2}\left(m_{2}l_{2}+m_{3}\left(c_{3}\ell_{3}+L_{2}\right)\right)\\ m_{3}l_{3}s_{3}c_{2}\end{array}\right)\]
The parameter values of the robot are given in Tab. B.1.
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline Parameter & Value & Parameter & Value & Parameter & Value \\ \hline \(m_{1}\) & \(27.387kg\) & \(l_{1}\) & \(0.07m\) & \(L_{1}\) & \(0.410m\) \\ \(m_{2}\) & \(15.843kg\) & \(l_{2}\) & \(0.085m\) & \(L_{2}\) & \(0.254m\) \\ \(m_{3}\) & \(40.53kg\) & \(l_{3}\) & \(0.326m\) & \(L_{3}\) & \(0.5m\) \\ \(I_{x_{1}}\) & \(0.285kg.m^{-2}\) & \(I_{y_{1}}\) & \(0.458kg.m^{-2}\) & \(I_{z_{1}}\) & \(0.427kg.m^{-2}\) \\ \(I_{x_{2}}\) & \(0.254kg.m^{-2}\) & \(I_{y_{2}}\) & \(0.254kg.m^{-2}\) & \(I_{z_{2}}\) & \(0.229kg.m^{-2}\) \\ \(I_{x_{3}}\) & \(2.161kg.m^{-2}\) & \(I_{y_{3}}\) & \(0.9491kg.m^{-2}\) & \(I_{z_{3}}\) & \(3.341kg.m^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table B.1: Parameter values of the three-degrees-of-freedom robot manipulator | この論文では、ロボットマニピュレータのセンサ故障検出と補償に焦点を当てています。提案される方法は、新たな適応的オベッサと、2次積分滑らかな表面に基づいた新しい末端のスライドモード制御法を特徴としています。この方法では、故障値の値やその微分値に関する境界を必要とせず、センサ故障検出が可能となります。また、この方法により、故障に耐えつつ、速くかつ一定の時間で制御を行うことが可能となり、この制御の性能は、追跡誤差に対する funnel bound を定義することで予め決められます。提案されたオベッサの推定誤差の極限性と、制御システムの固定時間安定性が、Lyapunov 安定性分析を用いて示されています。提案された方法の効果は、2つの異なるロボットマニピュレータに関する数値シミュレーションを用いて検証され、既存の方法と比較して |
2308.07395 | Text Injection for Capitalization and Turn-Taking Prediction in Speech
Models | Text injection for automatic speech recognition (ASR), wherein unpaired
text-only data is used to supplement paired audio-text data, has shown
promising improvements for word error rate. This study examines the use of text
injection for auxiliary tasks, which are the non-ASR tasks often performed by
an E2E model. In this work, we use joint end-to-end and internal language model
training (JEIT) as our text injection algorithm to train an ASR model which
performs two auxiliary tasks. The first is capitalization, which is a
de-normalization task. The second is turn-taking prediction, which attempts to
identify whether a user has completed their conversation turn in a digital
assistant interaction. We show results demonstrating that our text injection
method boosts capitalization performance for long-tail data, and improves
turn-taking detection recall. | Shaan Bijwadia, Shuo-yiin Chang, Weiran Wang, Zhong Meng, Hao Zhang, Tara N. Sainath | 2023-08-14T18:28:04 | http://arxiv.org/abs/2308.07395v1 | # Text Injection for Capitalization and Turn-Taking Prediction in Speech Models
###### Abstract
Text injection for automatic speech recognition (ASR), wherein unpaired text-only data is used to supplement paired audio-text data, has shown promising improvements for word error rate. This study examines the use of text injection for auxiliary tasks, which are the non-ASR tasks often performed by an E2E model. In this work, we use joint end-to-end and internal language model training (JEIT) as our text injection algorithm to train an ASR model which performs two auxiliary tasks. The first is capitalization, which is a de-normalization task. The second is turn-taking prediction, which attempts to identify whether a user has completed their conversation turn in a digital assistant interaction. We show results demonstrating that our text injection method boosts capitalization performance for long-tail data, and improves turn-taking detection recall.
Shaan Bijwadia\({}^{1}\), Shuo-yjin Chang\({}^{1}\), Weiran Wang\({}^{1}\), Zhong Meng\({}^{1}\), Hao Zhang\({}^{1}\), Tara N. Sainath\({}^{1}\)\({}^{1}\)Google, USA
{shaanb, shuoyjin, weiranwang, haozhang, zhongmeng, tsainath}@google.com
**Index Terms**: speech recognition, text injection, auxiliary tasks
## 1 Introduction
Automatic speech recognition (ASR) has long been an integral part of important technologies, including voice dictation, digital assistants, and video captioning [1]. While ASR systems are typically evaluated based on word error rate (WER), this is not the only metric of concern in production applications; several "auxiliary tasks" must be integrated with the ASR task in a full system. These tasks may include: capitalization and punctuation, which improves readability; voice activity detection (VAD) and end-of-query (EOQ) detection, which are important for implementing low-latency systems; and natural conversation understanding, which involves predicting the cadence and turn-taking aspects of an ongoing conversation. In this study, we focus on improving the quality of such auxiliary tasks in an end-to-end (E2E) ASR setting via text injection.
We build on two recent capabilities for speech models. First is the E2E integration of auxiliary tasks with the ASR task into a single model. In the past, auxiliary tasks were usually performed by separate models downstream of ASR [2, 3, 4, 5]. Recent work has successfully explored integrating auxiliary tasks, such as endpointing [6, 7, 8], capitalization [9], natural conversation understanding [10], and speaker diarization [11] into the same model as ASR prediction. E2E integration of ASR and auxiliary tasks has a key drawback, however. When folded into an E2E ASR model, pure text-to-text tasks (such as capitalization) can no longer be trained on plentiful text-only data (i.e., text data with no associated audio); instead, their training examples will be limited to the transcripts available in paired audio-text labeled data. This puts E2E methods at a disadvantage, since text-only data is generally more plentiful and easier to obtain than labeled audio data, and can be used to more easily expose the model to rare words and other long-tail phenomena which may be difficult to collect in labeled audio form [12].
The second capability enabling the current study is the use of "text injection" as a means of improving ASR quality [13]. An ASR model's internal language model (ILM) is the notional part of the network that predicts the next token given the previous token history, independent of audio input. Though it is usually infeasible to exactly separate the influence of audio input from previous token predictions, several methods have been developed to estimate ILM scores [14, 15]. Text-only data can then be used to refine the ILM capabilities of the ASR network [16, 17].
In this work, we propose a method to utilize text injection techniques for improving auxiliary task performance in an E2E ASR model. Doing so allows auxiliary tasks to access the multi-task learning benefits of co-training with ASR while still including rich text-only data in their training corpora. We focus our study on two tasks: capitalization and conversational turn-taking prediction. The former is a strongly text-based task, since capitalization is merely a form of de-normalization from spoken to written domain, and capitalized words are not pronounced differently. The latter task clearly involves combining linguistic and acoustic understanding -- the prosody of the input speech as well as the semantics of the current recognition are both informative for predicting whether a pause is only momentary or if the user has finished speaking. We integrate these tasks into a production-ready model, streaming E2E RNN-T ASR model [18, 19]. We show results demonstrating that text injection can meaningfully improve auxiliary task performance, particularly in long-tail settings.
## 2 Related Work
While auxiliary tasks are usually performed by separate models from ASR [20, 21], E2E approaches to auxiliary task modeling have been recently popular for production-grade systems. Joint training of ASR with endpointing [7], capitalization [9, 22], intended query detection [23, 24], sentence segmentation [25], and more, have been explored. Our work builds most closely on Wang et al.[9], who co-train ASR, capitalization, and turn-taking prediction by building multiple parallel label sequences. To our knowledge, this is the first attempt to refine auxiliary tasks in an E2E ASR model using text-only data.
There has long been interest in utilizing unpaired text data for the ASR task. Several approaches to LM fusion, the use of an external LM to improve ASR recognition quality, have been proposed [26]. These methods have the drawback of increasing total parameter count (due to the size of the LM model), and computation cost during inference. Text injection [13] solves these issues by using LM-style unpaired text data to train the
ASR model itself. Some methods focus on fine-tuning an existing ASR model trained on audio-text data; ILM adaptation of the ASR decoder has been shown to work well [27, 28, 29]. The text injection method we employ here is joint end-to-end and ILM training (JEIT), which was introduced by Meng et al [30]. We choose JEIT as our method due to its lightweight nature; its primary focus on refining the ASR decoder makes comparison to standard methods straightforward, since the behavior of the audio encoder is preserved. Other methods inject text data directly into the encoder, with fixed and learned duration models to align text and audio sequences [16, 17]. All of the above works focus on improving ASR quality, both for standard and long-tail data; to the best of our knowledge, adapting these techniques for auxiliary tasks is a novel contribution to the literature.
## 3 Auxiliary Tasks
### Capitalization
Capitalization is the process of restoring the correct case (uppercase or lowercase) of noisy text. Notably, capitalization is specific to the written domain, and has no marker in spoken speech. This task is important for maintaining readability in ASR output, especially for long-form captioning cases.
### Conversational turn-taking
Turn-taking is an active area of research for E2E speech modeling [10, 31]. While humans typically adjust their speech when interacting with voice assistants [31], natural human speech patterns during conversation are often filled with natural disfluencies. For digital assistant products, it is desirable that voice assistants have the ability to predict when the speaker is expecting a response, versus when they merely pause with the intention to resume speaking. We model this phenomenon similar to Chang et al. [10], who classify pauses in speech as being within a complete thought, or after having a finished complete thought. That is, when a user stops speaking, the model should predict whether they will continue speaking after a brief pause or whether a system response is expected. Because the active region of interest is pauses in the audio, we refer to this task in this paper as "pause prediction."
## 4 Model
### Multi-output HAT decoder
HAT is a decoder structure for RNN-T in which the (blank) probability is computed separately from next token prediction, facilitating more accurate ILM estimation [14]. Wang et al. [9] propose a variant of HAT decoder which introduces multiple joint networks, one for each task (in our case, these are ASR, capitalization, and pause prediction). All of the parallel joint networks are conditioned on features from both the prediction network and audio encoders.
The model is trained using an RNN-T objective [18], where at each timestep the model may choose to emit a wordpiece token, or to insert a special token (blank) which indicates non-emission. Formally, let \(X\) be the input utterance and \(Y\) be the label sequence. The ASR output space \(\mathcal{Y}_{\text{ASR}}\) consists of \(\{y^{0}=(\text{blank}),y^{1},y^{2},...\}\). Let \(T=|X|\) be the number of input audio frames and \(U=|Y|\) be the length of the transcript. The acoustic encoder produces \(f(X)=[f_{0},...,f_{T-1}]\), \(f_{t}\in\mathcal{R}^{D_{a}}\), and the prediction network produces \(g(X)=[g_{0},...,g_{U-1}]\), \(g_{u}\in\mathcal{R}^{D_{p}}\). As in the original HAT implementation, the joint network fuses \(f_{t}\) and \(g_{u}\) with a "project and sum" operation to produce a hidden representation \(h_{t,u}\), which is then passed through a non-linear activation and a final linear layer to produce \(s_{t,u}\):
\[h_{t,u}=P\cdot f_{t}+Q\cdot g_{u}+b_{h} \in\mathcal{R}^{D_{h}} \tag{1}\] \[s_{t,u}=A\cdot\text{tanh}(h_{t,u})+b_{s} \in\mathcal{R}^{V}. \tag{2}\]
where \(P\), \(Q\), and \(A\) are learned weight matrices with dimensions determined by \(D_{a}\), \(D_{p}\), \(D_{h}\), and \(V\) is the size of the vocabulary. As this is a HAT model, the 0-th logit of \(s_{t,u}\) is used individually to compute the probability of emission \(b_{t,u}\):
\[b_{t,u}:=P_{t,u}(\langle\texttt{blank}\rangle|f_{0:t},g_{0:u})=\sigma(s_{t,u}[ 0]) \tag{3}\]
where \(\sigma(x)=1/(1+\exp(-x))\) is the sigmoid activation. Probabilities over the ASR tokens are computed by feeding all remaining logits to a softmax function. The probability of each ASR token \(y_{v}\) in the vocabulary is:
\[\hat{y}_{v;t,u} =P_{t,u}(\hat{y}_{v}|f_{0:t},g_{0:u})\] \[=\text{softmax}(s_{t,u}[1:])[v-1] \tag{4}\]
Thus the predicted probability distribution over all output tokens is the emission probability, followed by the probabilities of each token given emission:
\[\hat{y}_{t,u}=[b_{t,u}, (1-b_{t,u})\cdot\hat{y}_{0:t,u},\] \[..., (1-b_{t,u})\cdot\hat{y}_{V-1:t,u}] \tag{5}\]
Thus far we have referred to the mechanism above in terms of ASR prediction. Capitalization and pause predictions are made in the exact same way, where each task independently computes Eqs. (1) and (2) based on the shared representations \(f_{t}\) and \(g_{u}\) (note that each auxiliary task is exposed to the label history of the ASR output, not its own prediction history).
Since capitalization tokens must be strictly aligned with ASR tokens, the capitalization posterior borrows the blank logit
Figure 1: Model diagram for JEIT training. The blue arrows denote the data flow for paired audio-text data. The red arrows denote the path that unpaired text data takes through the network. Baseline experiments are trained using only the blue paths, while the proposed system is trained using both.
from the ASR prediction. Thus, a capitalization token will only be emitted when an ASR token is emitted as well. Capitalization has output space \(\mathcal{Y}_{\text{Cap}}=\{\langle\text{cap}\rangle,\langle\text{non-cap}\rangle\}\) and its posterior is:
\[\hat{y}_{t,u}^{\text{Cap}}=[b_{t,u}^{\text{ASR}},\quad(1-b_{t,u}^{ \text{ASR}})\cdot P_{t,u}(\langle\text{cap}\rangle),\] \[(1-b_{t,u}^{\text{ASR}})\cdot P_{t,u}(\langle\text{non-cap} \rangle)] \tag{6}\]
At inference time, we estimate \(P(\langle\text{cap}\rangle)\) every time an ASR token is emitted and predict a capitalization if it is above a threshold (in this work, we use 0.5).
Pause tokens do not need to be strictly aligned with the ASR transcript prediction, since they are likely to be predicted during non-speech periods in the audio during inference, so the turn-taking sequence has its own blank posterior. The pause prediction output space is \(\mathcal{Y}_{\text{Pause}}=\{\langle\text{blank}\rangle,\langle\text{non- pause}\rangle,\langle\text{eos}\rangle\}\) and its posterior is computed in the same way as Eq. (5).
## 5 Training
### Jeit
Joint end-to-end model and ILM training (JEIT) was proposed by Meng et al. [30] as a way to train an RNN-T ASR model on paired audio-text data while simultaneously training the HAT decoder ILM on text-only data. For paired dataset \(\mathcal{D}_{\text{paired}}\), training is conducted in the usual way; the model is given the audio sequence as input and predicts \(P_{\text{EE}}(Y|X)\). This is converted to a loss \(\mathcal{L}_{\text{EE}}^{\text{ASR}}\) via the RNN-T objective [18].
The text-only dataset \(\mathcal{D}_{\text{paired}}\) contains transcripts with capitalization and pause annotations (see SS5.2). Similar to HAT ILM adaptation (ILMA) [27], we feed the transcript as the previous token history to the prediction network, and mock the encoder output with vectors full of zeros: \(\forall_{t\in 0:T}:f_{t}=\mathbf{0}\). Since the audio sequence does not exist, we simply ignore the blank posterior, and the predicted next token probabilities are given directly by the softmax output in Eq. (4). With previous token history as input and next token probabilities as output, this allows us to estimate \(P_{\text{ILM}}(y_{t}:y_{0:t-1})\). ILM loss is defined as the negative log probability of each label token given the label sequence prefix:
\[\mathcal{L}_{\text{ILM}}^{\text{ASR}}=-\sum_{u=1}^{U}\log P(y_{u}^{\text{ASR} }|\hat{y}_{0:u-1}^{\text{ASR}}) \tag{7}\]
The losses \(\mathcal{L}_{\text{EEE}}\) and \(\mathcal{L}_{\text{ILM}}\) are averaged over their respective datasets \(\mathcal{D}_{\text{paired}}\) and \(\mathcal{D}_{\text{paired}}\), then combined in a weighted average to obtain the total JETI loss:
\[\mathcal{L}_{\text{JEIT}}^{\text{ASR}}(\mathcal{D}_{\text{paired }},\mathcal{D}_{\text{paired}})=\] \[\mathcal{L}_{\text{EEE}}^{\text{ASR}}(\mathcal{D}_{\text{paired }})+\beta\mathcal{L}_{\text{ILM}}^{\text{ASR}}(\mathcal{D}_{\text{ unpaired}}) \tag{8}\]
where \(\beta\) is a hyperparameter controlling the weight given to ILM training (in this work, we use \(\beta=0.2\) to match Meng et al.'s original study).
Adapting JEIT to include auxiliary tasks is straightforward. As described in SS4.1, each auxiliary task makes a sequence prediction \(Y_{\text{Aux}}\) based on the predicted ASR sequence \(Y_{\text{ASR}}\). Thus, each auxiliary task predicts \(P_{\text{EE}}(Y_{\text{Aux}}|\hat{Y}_{\text{ASR}};X)\) to produce \(\mathcal{L}_{\text{EEE}}^{\text{Aux}}\). Similarly, the ILM loss is
\[\mathcal{L}_{\text{ILM}}^{\text{Aux}}=-\sum_{u=1}^{U}\log P(y_{u}^{\text{Aux} }|\hat{y}_{0:u-1}^{\text{ASR}}) \tag{9}\]
The full JEIT loss for each task is defined in the same way as Eq. (8). Total loss is a linear combination of all tasks: (datasets omitted for clarity):
\[\mathcal{L}_{\text{IEIT}}^{\text{Total}}= \mathcal{L}_{\text{EEE}}^{\text{ASR}}+\beta\mathcal{L}_{\text{ILM} }^{\text{ASR}}+\alpha_{\text{Cap}}(\mathcal{L}_{\text{EEE}}^{\text{Cap}}+ \beta\mathcal{L}_{\text{ILM}}^{\text{Cap}})\] \[+\alpha_{\text{Pune}}(\mathcal{L}_{\text{EEE}}^{\text{Due}}+ \beta\mathcal{L}_{\text{ILM}}^{\text{Pune}}) \tag{10}\]
where each \(\alpha\) is a loss weight for the corresponding task. Matching Wang's original study, we use \(\alpha_{\text{Cap}}=0.1\) and \(\alpha_{\text{Pune}}=0.3\). Figure 1 shows the data flow for paired and unpaired data through the ASR model.
### Transcript annotation
While a small amount of our paired training corpus is hand-labeled and capitalized, most of our paired data and all of our unpaired text data have lowercase transcripts. For the lowercase transcripts, we use a text-based increasing RNN teacher model similar to [32] to produce capitalization predictions.
Producing pause prediction labels requires different approaches for paired and unpaired data. For paired audio-text data, we use the approach taken by Chang et al. [10], which uses heuristics based on a forced alignment [33] to insert pause tokens into the transcript. There are two such tokens: (pause) denotes a brief stop by the speaker in the middle of a full thought, and \(\langle\text{eos}\rangle\) (end of sentence) is inserted at the end of the full thought, i.e. a full conversational turn.
For unpaired text-only data, the above strategy is impossible, since we do not have access to the associated audio. Instead, we rely on the fact that our text-only data comes from short-query sources (see SS6.2). We simply append the \(\langle\text{eos}\rangle\) token to the end of the transcript.
### Multi-task label structure
A common approach to transcript labeling for auxiliary tasks would be to embed special tokens corresponding to each task in the transcript itself [7]. However, this is not ideal for inference, since the extra tokens must be expanded in-line with the ASR tokens; if predictions on competing beams differ only in their special tokens, lattice diversity is reduced because the ASR prediction would be identical. To solve for this, we follow Wang et al. [9], factorizing the auxiliary task tokens into parallel sequences of equal length, one for each task. The ASR task is trained on the lowercase transcript sequence, segmented into wordpieces. The capitalization sequence is defined as follows: each token is either \(\langle\text{cap}\rangle\) (capitalized) or \(\langle\text{non-cap}\rangle\)
Figure 2: Data preparation for auxiliary tasks. Wordpieces that begin with _denote word boundaries. In this example, we assume that the speaker takes a verbal pause as follows: ”Driving time to... San Francisco,” to illustrate the \(\langle\text{pause}\rangle\) logic.
(not capitalized), based on the corresponding wordpiece in the ASR transcript. Similarly, the turn-prediction sequence is populated with \(\langle\texttt{pause}\rangle\) and \(\langle\texttt{eos}\rangle\) tokens corresponding to the wordpieces immediately preceding the corresponding predicted pauses in the transcript. All other token slots are filled with \(\langle\texttt{non-pause}\rangle\). The successive steps of label generation are shown in Figure 2.
## 6 Experimental Details
### Model architecture
We use a 128-dimensional log-mel feature frontend computed on 32ms windows with a 10ms stride. We stack four consecutive frames together and sub-sampled by a factor of 3, resulting in 512-dim features at a 30ms framerate. This vector is then concatenated with a 16-dim one-hot domain ID vector [34]. As our ASR backbone we use a 2-pass cascaded encoder model [35]. The first encoder consists of 7 conformer layers [36] with causal convolution and left-context attention. The second encoder consists of 10 conformer layers with a 900ms lookahead. Each conformer layer uses 512-dim 8-head self-attention and a kernel size of 15, and the final layer emits \(D_{a}=384\)-dim encodings. The prediction network of each decoder is a \(V^{2}\) embedding lookup table, which computes \(D_{p}=640\)-dim features based on embeddings of the previous two wordpiece tokens. Each joint network has hidden dimension \(D_{h}=384\), and predictions are made over a vocabulary of \(V=4096\) wordpieces. For evaluation, we report only 2nd pass WER.
In total, our model has \(\sim\)160M params. It is implemented in Tensorflow using the Lingvo toolkit, and is trained on proprietary specialized hardware for 500k steps using batch size 4096 for paired and unpaired data.
### Data
#### 6.2.1 Paired training data
Our training set of audio-text pairs consists of a dataset of 650 million English multi-domain examples, drawn from search, dictation, online video, and telephony domains. A small subset of these utterances are anonymized and hand-transcribed, and the rest are pseudo-labeled by a 600M parameter bidirectional teacher model. To increase model robustness, we apply simulated noise to utterances, as well as SpecAug [37].
#### 6.2.2 Unpaired training data
Our text-only data selection pipeline is designed in the style of Sentence-Select by Huang et al [12]. Text query data (\(\sim\) 100B utterances) is collected from web search, maps search, app store search, and online video search domains. This data is filtered for rare words and contrastive filtering based on perplexity is applied. Because the data is selected to include rare words, we expect improvements at the tails of the evaluation distribution.
#### 6.2.3 Evaluation Data
WER is reported on \(\sim\) 17k utterances representative of real-world voice dictation traffic. Ground truth transcript and auxiliary task annotations are obtained via human labeling. We also report uppercase error rate (UER) on this set, which is calculated by removing all lowercase letters from the ground truth label and the predicted transcript and computing standard WER with upper case letters as words. Since our text-only data focuses on long-tail traffic, we also report UER on a set of \(\sim\)300 utterances with transcripts containing rare words.
For pause prediction, we use a testset of \(\sim\)2500 utterances containing hesitations and multiple consecutive commands. Pauses in the audio are hand-annotated as continuation pauses or final pauses. The metrics reported are average precision and recall of the \(\langle\texttt{eos}\rangle\) token.
## 7 Results
We evaluate the proposed method (E1) against a baseline (B1) which uses an identical model but is trained on paired data only (Table 1). On the large voice search test set on which it is evaluated, WER does not change, while UER regresses slightly on the voice dictation dataset (1.6% relative). For long tail data, UER improves by a relative 2.0%. Table 2 shows example transcripts demonstrating our proposed method's better capability at recognizing capitalized named entities. | テキストインジェクションによる自動音声認識（ASR）における、ペアリングされた音声-テキストデータに加えて、単一テキストのみを用いたデータの活用が、単語誤差率の改善に実用的な成果を示している。本研究では、テキストインジェクションを用いた補助タスクの適用について考察する。補助タスクは、E2Eモデルがしばしば実行する非ASRタスクである。本研究では、テキストインジェクションアルゴリズムとして、エンドツーエンドと内部言語モデルのトレーニング（JEIT）を用いて、ASRモデルを訓練する。訓練対象の第一段階は、デノormal化タスクである capitalizati on である。第二段階は、ユーザーがデジタルアシスタントインタラクションにおける会話のターンを完了したかどうかを予測するターンを取る予測である。結果を提示し、テキストインジェクション手法は、長尾データにおける capitalizati on の性能を向上させ、 |
2301.04397 | TBV Radar SLAM -- trust but verify loop candidates | Robust SLAM in large-scale environments requires fault resilience and
awareness at multiple stages, from sensing and odometry estimation to loop
closure. In this work, we present TBV (Trust But Verify) Radar SLAM, a method
for radar SLAM that introspectively verifies loop closure candidates. TBV Radar
SLAM achieves a high correct-loop-retrieval rate by combining multiple
place-recognition techniques: tightly coupled place similarity and odometry
uncertainty search, creating loop descriptors from origin-shifted scans, and
delaying loop selection until after verification. Robustness to false
constraints is achieved by carefully verifying and selecting the most likely
ones from multiple loop constraints. Importantly, the verification and
selection are carried out after registration when additional sources of loop
evidence can easily be computed. We integrate our loop retrieval and
verification method with a fault-resilient odometry pipeline within a pose
graph framework. By evaluating on public benchmarks we found that TBV Radar
SLAM achieves 65% lower error than the previous state of the art. We also show
that it's generalizing across environments without needing to change any
parameters. | Daniel Adolfsson, Mattias Karlsson, VladimÃr Kubelka, Martin Magnusson, Henrik Andreasson | 2023-01-11T10:50:24 | http://arxiv.org/abs/2301.04397v3 | # TBV Radar SLAM - trust but verify loop candidates
###### Abstract
Robust SLAM in large-scale environments requires fault resilience and awareness at multiple stages, from sensing and odometry estimation to loop closure. In this work, we present TBV (Trust But Verify) Radar SLAM, a method for radar SLAM that introspectively verifies loop closure candidates. TBV Radar SLAM achieves a high correct-loop-retrieval rate by combining multiple place-recognition techniques: tightly coupled place similarity and odometry uncertainty search, creating loop descriptors from origin-shifted scans, and delaying loop selection until after verification. Robustness to false constraints is achieved by carefully verifying and selecting the most likely ones from multiple loop constraints. Importantly, the verification and selection are carried out after registration when additional sources of loop evidence can easily be computed. We integrate our loop retrieval and verification method with a fault-resilient odometry pipeline within a pose graph framework. By evaluating on public benchmarks we found that TBV Radar SLAM achieves 65% lower error than the previous state of the art. We also show that it's generalizing across environments without needing to change any parameters.
## I Introduction
Robust localization is key to enabling safe and reliable autonomous systems. Achieving robustness requires careful design at multiple stages of a localization pipeline, from environment-tolerant sensing and pose estimation, to place recognition and pose refinement. At each stage, a localization and mapping pipeline should be designed for fault awareness to detect failures and fault resilience to mitigate failures as they inevitably occur. Today, active exteroceptive sensors such as lidar and radar are suitable when robust uninterrupted localization is required. Of these two, radar perception is significantly less affected when operating within dusty environments or under harsh weather conditions. It is currently debated how the sensing properties affect localization and mapping performance [1, 2]. Compared to lidar, limited work focuses on robust and accurate radar SLAM, and none of the existing methods include introspective fault detection.
In this letter, we propose TBV (Trust But Verify) Radar SLAM - a 2D pose-graph localization and mapping pipeline which integrates fault resilience and fault-aware robustness at multiple stages. A radar-only odometry front-end adds pose nodes to the graph. In parallel, a robust loop-detection module adds loop closure constraints such that the SLAM back-end can optimize the graph to correct drift. TBV Radar SLAM uses radar odometry (further only _odometry_), place descriptors, and scan alignment measures to retrieve, verify, and select between loop constraints. We achieve a high correct-loop-retrieval rate by combining: a tightly coupled place similarity and odometry uncertainty search, creating place descriptors computed from origin-shifted scans, and by delaying loop selection until after verification. Robustness, with a high loop detection rate, is achieved by unifying the process of place recognition and verification. Rather than rejecting candidates early during place recognition, we register and verify multiple loop constraints in parallel. Our verification combines place similarity, odometry consistency, and an alignment quality assessment automatically learned from odometry and scans. We integrate our loop retrieval and verification module with a robust method for radar odometry, into a full-fledged SLAM pipeline visualized in Fig. 1. The contributions of this letter are:
* A combination of techniques for a high rate of correct loop retrievals, including: coupled place similarity and odometry uncertainty search, creating place descriptors from origin-shifted scans, and selection between multiple competing loop constraints.
* A unified loop retrieval and verification step that jointly considers place similarity, odometry uncertainty, and alignment quality after registration. Multiple loop candidates are retrieved, registered, and verified.
* We integrate these techniques with a robust odometry estimator into a SLAM framework that pushes state of the art in radar SLAM accuracy while generalizing between environments without parameter tuning.
## II Related work
Early methods for radar SLAM employ filtering [3] and landmark graph SLAM [4]. We instead use pose-graph SLAM based on odometry and loop constraints obtained by
Fig. 1: Overview and demonstration of TBV Radar SLAM
-registering scans. Holder et al. [5] proposed a pose graph SLAM based on automotive radar, using GLARE [6] for place recognition. Their method uses gyroscope and wheel odometry to provide an initial alignment guess to Iterative Closest Point (ICP) for scan matching. Our method uses a pose graph back-end but relies solely on a spinning radar.
Recent progress in the development of spinning 2D radar has quickly inspired numerous works on radar odometry estimation [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], place recognition [20, 21, 22, 23, 24, 25], topological localization [21, 25, 2, 21] localization in overhead imagery [26, 27] and SLAM [10, 28, 1]. Most similar to ours is the preliminary work by Wang et a. [28], and the seminal work on radar SLAM by Hong et al. [1]. Hong et al. [1] estimates odometry and registers loop constraints using KLT tracker keypoints. For place recognition, they use adaptive thresholding to filter data, and M2DP [29] to compute and match descriptors. The trajectory is corrected using pose graph optimization. Our work brings a larger focus on the introspective verification of loop constraints. Martini et al. [21] proposed a teach-and-repeat localization framework using a hierarchical approach. First, a place candidate is retrieved via nearest neighbor search within a learned metric space using the method by Saftescu et al. [20]. Second, sensor pose is estimated (without the need for an initial guess) by finding and registering the (globally) optimal set of landmark matches as described by Cen et al. [7]. Unlike [21], we use a fast local registration method, motivated by the access of an initial alignment guess from the place recognition module. However, we carefully verify registration success before accepting new loop constraints.
Verification of pose estimates is essential for safety. E.g., such tests could have prevented a reported accident that occurred during an automated driving test [30]. Holder et al. [5] verify the detected loop candidates using radar data by the condition that ICP residuals cannot exceed a set threshold. Rather than verifying loops as a final step (separated from loop detection), we unify loop retrieval and verification. Some works [21, 31] use the landmark matching compatibility matrix [7] to assess quality. Martini et al. [21] reject place candidates based on the quality score of the matrix. Aldera et al. [31] learn detection of pose estimate failures from features extracted from the eigenvectors of the compatibility matrix. Training labels are provided by an external ground truth system. We build on the method CorA [32] to detect false loop candidates or constraints. In the lidar domain, Behley et al. [33] accept only reappearing loop candidates that are consistent with odometry over multiple consecutive scans, we instead focus on verification using only individual scans.
Some methods attempt to measure how observations would constrain registration in the current scene; a low level of constraints in one direction suggests registration could be unstable. The measure has been used to predict the risk of registration failure [34, 35], abstain from closing loops when risk is high [1], or prioritize the inclusion of loop closures accordingly [36]. We instead use odometry uncertainty as a prior, which we combine with additional loop evidence computed after registration. Finally, a range of methods aims at verifying pose estimates by detecting point cloud misalignment [30, 37, 32]. We fuse misalignment detection [32] together with place similarity and odometry uncertainty to achieve robust unified loop detection and verification.
## III TBV Radar SLAM
An overview of TBV Radar SLAM is presented in Fig. 2. In this section, we detail the components including CFEAR Radar Odometry (Sec. III-A), place recognition (Sec. III-B), verification of loop candidates (Sec. III-D), and pose graph optimization (Sec. III-E). The self-supervised training of alignment verification (Sec. III-C) runs once during a learning phase and is not required for online SLAM.
### _CFEAR Radar Odometry_
We use the radar odometry pipeline _CFEAR Radar Odometry_[17] (specifically the _CFEAR-3_ configuration). This method takes raw polar radar sweeps as input and estimates sensor pose and velocity. As an intermediate step, the method computes a set of radar peaks \(P^{t}\) and a set of oriented surface points \(\mathcal{M}^{t}\) at time \(t\). These representations will be referred to in the later stages of our pipeline. CFEAR filters the radar data by keeping the \(k\)-strongest range bins per azimuth. The filtered point set is compensated for motion distortion and transformed into Cartesian space, - this is also the point set from which we extract radar peaks (\(P^{t}\)). A grid method is then used to compute a set of oriented surface points \(\mathcal{M}^{t}\) from the filtered set. Odometry is estimated by finding the pose \(\mathbf{x}^{t}\) in SE(2) which minimizes the sum of surface point distances between the current scan \(\mathcal{M}^{t}\) and the \(|\mathcal{K}|\) latest keyframes \(\mathcal{M}^{\mathcal{K}}\) jointly as
\[f(\mathcal{M}^{\mathcal{K}},\mathcal{M}^{t},\mathbf{x}^{t})=\!\!\!\sum_{k\in \mathcal{K}}\sum_{\forall\{i,j\}\in\mathcal{C}_{corr}}\!\!\!\!w_{i,j}\rho\big{(} g(m_{j}^{k},m_{i}^{t},\mathbf{x}^{t})\big{)}, \tag{1}\]
where \(w_{i,j}\) are surface point similarity weights, \(\rho\) is the Huber loss function, and \(g\) is the pairwise distance between surface points in the correspondence set \(\mathcal{C}_{corr}\). A keyframe \(k\in\mathcal{K}\) is created when the distance to the previous exceeds \(1.5\) m. This technique reduces drift and removes excessive scans acquired when the robot remains stationary. Upon
Fig. 2: Overview of TBV Radar SLAM. The main contribution is the Loop retrieval and verification module.
creation of a new keyframe, odometry constraints are created from the relative alignment between the current pose and the latest keyframe. Odometry constraints are added to \(\mathcal{C}_{odom}\) and used to correct trajectory (via pose graph optimization) as detailed in Sec. III-E.
### _Place recognition_
We attempt to unify the stages of loop closure, from detection to constraint verification. Accordingly, we retrieve, register, verify multiple candidates, of which one can is selected. For that reason, we do not discard potential loop candidates based on place similarity. Instead, we trust multiple candidates to be meaningful until verified. At this point, we select the verifiably best loop candidate, if such exist.
#### Iii-B1 Scan Context
We build upon the place recognition method _Scan Context_ by Giseop Kim et al. [38, 39]. Their method detects loops and relative orientation by matching the query (\(q\)) scan with candidates (\(\{c\}\)) stored in a database using scan descriptors. Scenes are encoded by their polar representations into 2D descriptors \(\mathbf{I}_{ring\times sec}\). The 2D descriptor is in turn used to create a rotation-invariant 1D descriptor (ring key) via a ring encoding function. Loops are detected via a two-step search: First, the query 1D ring key is matched with the top 10 candidates via a fast nearest neighbor (NN) search. Second, for each of the top candidates, a sparse optimization finds the relative rotation that minimizes a distance metric \(d_{sc}(\mathbf{I}^{q},\Gamma^{c})\): the sum of cosine distances between descriptors columns. The candidate \(c\) with the lowest score (\(d_{sc}\)) which does not exceed the fixed threshold \(\tau\) is selected as loop candidate.
\[c=\operatorname*{argmin}_{c\in\mathcal{C}}d_{sc}(\mathbf{I}^{q},\mathbf{I}^{ c}),s.t.\,d_{sc}<\tau. \tag{2}\]
In our implementation, query descriptors are created, matched, and stored in the database for each keyframe rather than for each scan.
#### Iii-B2 Descriptor generation
As described in [28, 40], a raw polar representation such as those produced by a spinning 2D radar, can be used directly as a Scan Context descriptor. However, we believe that doing so poses multiple challenges, including sensor noise, motion distortion, scene dynamics and translation sensitivity. Thus, we create our descriptor from multiple filtered and motion-compensated scans. Conveniently, such processed scans are already provided by the CFEAR. We aggregate the peak detections from keyframe \(q\) and its two neighbours in the odometry frame.
Having the radar scan represented as a sparse point cloud in Cartesian space allows us to address translation sensitivity in place recognition by applying the data augmentation step (Augmented PC) from [39] before computing place descriptors. We perform data augmentation by shifting the sensor origin, i.e. by transforming \(\mathcal{P}^{\tilde{q}}\) with \(\pm 2\) and \(\pm 4\) m lateral translation offsets. The original, and the 4 augmented point clouds, are each used to compute and match descriptors, after which the best matching pair of query/candidate is selected. Note that by using the aggregated sparse point cloud, rather than the dense raw radar scan, we can efficiently compute all augmentations and corresponding descriptors. As such, the main computational load from the augmentation technique is due to matching of additional descriptors and not the computation of these. The descriptor itself is created by populating the Scan Context \(\mathbf{I}\) with radar intensity readings. Specifically, for each grid cell \(\mathbf{I}(i,j)\) we sum the intensity of all coinciding point intensities (of radar peaks) divided by \(1000\). Empty cells are set to \(\mathbf{I}(i,j)=-1\), which we found increased descriptiveness compared to \(\mathbf{I}(i,j)=0\).
#### Iii-B3 Coupled odometry/appearance matching
When retrieving loop candidates, odometry information can be used to filter unlikely candidates. This could be done by rejecting unlikely loop constraints. For example, if the likelihood of the loop constraint \(\mathbf{x}_{loop}^{q,c}\) (given the estimated odometry trajectory \(\mathbf{v}_{odom}^{cq}\) between \(c\) and \(q\) ) is close to zero:
\[p(\mathbf{x}_{loop}^{q,c}\mid\mathbf{v}_{odom}^{cq})\approx 0. \tag{3}\]
While this strategy may provide higher tolerance to spatial aliasing by rejecting false positives, it does not provide means to detect the correct candidate under such circumstances. For that reason, we propose a coupled place similarity / odometry uncertainty search, which combines Eq. 2 and Eq. 3. Candidates are thus selected jointly by the similarity of appearance \(d_{sc}(\mathbf{I}^{q},\mathbf{I}^{c})\) and the similarity of odometry \(d_{odom}^{q,c}\):
\[\begin{split} c=\operatorname*{argmin}_{c\in\mathcal{C}}d_{sc}( \mathbf{I}^{q},\mathbf{\Gamma}^{c})+d_{odom}^{q,c},\\ d_{odom}^{q,c}=1-p(\mathbf{x}_{loop}^{q,c}\mid\mathbf{v}_{ odom}^{cq})\,.\end{split} \tag{4}\]
We estimate \(p(\mathbf{x}_{loop}^{q,c}\mid\mathbf{v}_{odom}^{cq})=\exp{(-\frac{t_{err}^{2} }{2\sigma^{2}})}\).
\[t_{err}=\frac{\max(||transl(\mathbf{x}^{q})-transl(\mathbf{x}^{c})||-\epsilon, 0)}{dist(\mathbf{v}_{odom}^{cq})}. \tag{5}\]
Here, \(transl\) is the translation component, \(\epsilon\) is the expected maximum spacing between loop candidates (fixed to \(\epsilon=5\) i.e. slightly higher than the lateral augmentation distance), and \(dist(\mathbf{v}_{odom}^{cq})\) is the traversed distance between the query and loop candidate estimated by the odometry. Note that \(t_{err}\) quantifies the _relative final position error_, thus \(\sigma\) can be chosen according to expected odometry quality to penalize unlikely loops. We refrained, however, from making strong assumptions on odometry quality, and fixed \(\sigma=0.05\); i.e., a pessimistic assumption of \(5\%\) relative translation error.
Note that the two-step search of Scan Context requires that odometry uncertainty is integrated already in the 1D NN search. We do this by extending all 1D descriptors (of size \(ring=40\)) with odometry similarity scores (\(d_{odom}\)) as an extra element. (\(d_{odom}\)) is scaled with a factor \((ring/4)\) to balance odometry uncertainty and appearance similarity.
### _Automatic learning of alignment verification_
To improve loop closure verification, we build upon the system _CorAl_[32] which learns to detect alignment errors between two registered scans. CorAl allows us to determine if a loop constraint is correct by formulating loop constraint verification as a misalignment detection problem. Specifically, a loop (between scan nr \(q\) and \(c\)) is verified as _correct_ only if the scans are correctly aligned via the relative alignment \(\mathbf{x}_{loop}^{q,c}\). During the learning phase, CorAl automatically
generates labeled training data. The input to CorAl is pairs of odometry estimates, radar peak detections (\(\mathcal{P}\)), and computed sets of oriented surface points (\(\mathcal{M}\)). These entities are used to extract alignment quality residuals from which alignment quality can be assessed. After the learning phase, CorAl can verify loops by detecting alignment errors (caused e.g. by heavy motion distortion or incorrect convergence). CorAl also aids in distinguishing between places that differ by small geometric details. We generate training data by repeating the following process for each pair of consecutive keyframes. Positive training labels (\(y_{aligned}\!=\!true\)) and training data \(X_{quality}\) are computed using the scan alignment provided by the odometry. For each pair, the alignment quality measures in Eq. 6 are extracted. Negative training labels (\(y_{aligned}\!=\!false\)) and training data are extracted similarly. However, before extracting the alignment quality, an error is induced in the alignment in both translation and rotation. This allows us to learn the detection of different types of errors. Specifically, we distribute 12 translation errors symmetrically in either the positive or negative x or y-axis. We use 4 small (\(\pm 0.5\) m), 4 medium (\(\pm 1\) m) and 4 large (\(\pm 2\) m) errors. To these errors, we additionally induce a clockwise rotation with matching rotation errors: small (\(0.5^{\circ}\)), medium (\(2^{\circ}\)) or large (\(15^{\circ}\)). Note that the class ratio 1:12, between positive to negative training is alleviated during learning by assigning weights according to the inverse of class frequency.
#### Iii-C1 Alignment measure
We extract the following alignment measures between each pair of scans:
\[\mathbf{X}_{quality}=[H_{j}\;H_{s}\;H_{o}\;C_{f}\;C_{o}\;C_{a}\;1]^{T}. \tag{6}\]
The joint entropy (\(H_{j}\)) and separate entropy (\(H_{s}\)) are average per-point differential entropies, extracted from point cloud pairs of radar peak detections (\(\mathcal{P}^{q},\mathcal{P}^{c}\)). These metrics are described in-depth in [32]. We complement these measures with a measure of overlap \(H_{o}\): (\(H_{overlap}\)), defined as the portion of peaks in \(\mathcal{P}^{q}\) or \(\mathcal{P}^{c}\) with a neighboring point within the radius \(r\) in the other point cloud.
In this work, we combine these CorAl measures with additional ones, extracted from (\(\mathcal{M}^{q},\mathcal{M}^{c}\)), i.e. from pairs of scans represented as oriented surface points. The measures are obtained from the registration cost(Eq. 1), but with a single keyframe and with the point-to-line cost function (\(g_{P2L}\)[17]). Note that these measures are already computed during the final iteration of the registration, and this step brings little computational overhead. Specifically, from Eq. 1 we reuse \(C_{f}\): \(f(\mathcal{M}^{q},\mathcal{M}^{c},\mathbf{x}^{q,c})\), the number of correspondences (overlap) \(C_{o}\): \(|\mathcal{C}_{corr}|\), and average surface point set size \(C_{a}\): \(1/2(|\mathcal{M}^{q}|+|\mathcal{M}^{c}|)\). The intuition of combining these quality measures is that the CorAl measures (which use small-region data association) are specialized in detecting small errors whereas the _CFEAR measures_ are more suitable for larger alignment errors. We refer to [32] for details.
#### Iii-C2 Assessing alignment
Once training data has been computed, we train a logistic regression classifier
\[p_{align} =1/(1+e^{-d_{align}}), \tag{7a}\] \[d_{align} =\beta\mathbf{X}_{quality}, \tag{7b}\]
where \(\beta_{1\times 7}\) are the learned model parameters. We train on discrete alignment classification here as we consider all visible errors to be undesired. However, \(d_{align}\) is passed to our loop verification module rather than \(p_{align}\). We found \(d_{align}\) to be more suitable, as the sigmoid output \(p_{align}\) is insensitive to alignment change close to \(0\) or \(1\).
### _Verification of loop candidates_
We allow for multiple competing loop candidates \(c_{k}\) per query \(q\) as illustrated in Fig. 1. Each of the \(N_{cand}=3\) best matching pairs \(\{(q,c_{k})\}\) provided by the place recognition module is used to compute and verify potential loop constraints. A constraint is computed by finding the relative alignment \(\mathbf{x}_{loop}^{q,c_{k}}\) that minimizes Eq. 1 i.e. the distance between correspondences, similarly to the odometry module. As an initial guess, we use the relative orientation provided by the place recognition module. If the loop candidate was retrieved from a match with an augmented query scan, the corresponding augmented lateral offset is used together with the rotation as an initial guess. Note that the local registration method is motivated by the access to an initial guess, required for convergence. After registration, we extract and assess the alignment quality \(d_{align}=\beta\mathbf{X}_{quality}^{q,c_{k}}\) following the procedure in Sec. (III-C1&III-C2). Each constraint is finally verified by combining the Scan Context distance (\(d_{sc}\)) with odometry uncertainty (\(d_{odom}\)) and alignment quality (\(d_{align}\)) with a logistic regression classifier
\[\begin{split}& y_{loop}^{q,c_{k}}=\frac{1}{1+e^{-\boldsymbol{\Theta }\mathbf{X}_{loop}^{q,c_{k}}}},\;s.t.\;y_{loop}^{q,c_{k}}>y_{th},\\ &\mathbf{X}_{loop}^{q,c_{k}}=[d_{odom}\;d_{sc}\;d_{align}\;1]^{T}. \end{split} \tag{8}\]
The model parameters \(\boldsymbol{\Theta}\) can be learned via ground truth loop labels, or simply tuned as the 4 parameters have intuitive meaning. \(y_{th}\) is the sensitivity threshold - we rarely observe false positives when fixed to 0.9.
We investigated two strategies for selecting loop closures after a successful verification: (i) We select the first candidate retrieved from the place recognition module (\(N_{cand}=1\)) - the lowest Scan Context distance score \(d_{sc}\). (ii) We use \(N_{cand}=3\) candidates and select the best candidate according to our verifier - the highest probability \(y_{loop}^{q,c_{k}}\). The intuition for strategy (i) is that the first retrieved place candidate is often the best guess, without considering registration. However, there are cases where one of the latter candidates is preferred. For example, registration of query and the first candidate may fail, or subtle scene differences that distinguish places can be hard to detect until a more thorough local analysis of alignment has been carried out. Thus, selecting the verifiably better loop constraint candidate is desired. We compare these two strategies in Sec. IV-B ( T.6 and T.8) Once a loop constraint has been verified, the loop is added to \(\mathcal{C}_{loop}\).
### _Pose Graph Optimization_
We correct the odometry by solving a sparse least squares optimization problem. We do so by minimizing Eq. 9, using
the odometry and loop constraints \(\mathcal{C}_{odom}\), \(\mathcal{C}_{loop}\):
\[J(\mathbf{Y})=\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
sequences, and Fig. 3(b) for the nine MulRan sequences. Loop detections are visualized in Fig. 3 for the Oxford sequence 16-13-09 with \(y_{th}=0.9\). The raw Scan Context ( T.1) achieves higher recall compared to the sparse local map (T.2). The difference is larger in MulRan, where scans are acquired in the same moving direction, and motion distortion is less of a challenge. Additionally, we noted that the difference is highest within the feature-poor Riverside sequences. This suggests that maintaining information-rich radar data is largely advantageous compared to using a sparse local map, especially when features are scarce. Note however that our local mapping technique is primarily motivated by the need for a sparse Cartesian point cloud for efficient augmentation.
Oxford is more challenging compared to MulRan as a majority of the revisits occur in opposite road lanes and directions. However, the augmentation technique (T.3) allows the detection of additional candidates with higher lateral displacement, and as expected, increases the highest top recall, yet at a cost of lower precision. This loss of precision can however be alleviated via alignment loop verification (T.4). The improvement is larger in Oxford and we believe the more structured scenes are favorable for alignment analysis. The decoupled odometry approach (T.5), which extends verification by including odometry uncertainty, gives a higher tolerance to false positives. At this point, the decision boundary can be chosen such that almost all candidates are correctly classified. Unified verification is preferred over separate verification (T.7). Selecting the candidate with the highest probability (T.8), rather than the first place recognition candidate (T.6) yields a clear improvement in Oxford. We believe this improvement is because our alignment quality aids in distinguishing between places and detecting registration failures, especially in structured scenes.
### _SLAM performance - comparative evaluation_
We compare TBV Radar SLAM to previous methods for radar, lidar, and visual SLAM within the Oxford and Mulran dataset. We primarily compare methods over full trajectories i.e. Absolute Trajectory Error (ATE) \(ATE_{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{i=n}||trans(\mathbf{x}_{i}^{est})-trans( \mathbf{x}_{i}^{gt}))||^{2}}\). Additionally, we provide the KITTI odometry metric [48], which computes the relative error between 100-800 m, e.g. error over a shorter distance. ATE metrics for method
Fig. 4: Loop closure performance over all sequences.
Fig. 5: Oxford trajectories using the proposed method TBV Radar SLAM, compared with CFEAR-3 [17] (odometry only) and Ground truth. Initial and final pose are marked with \(\times\) and \(\square\). Trajectories can be directly compared to [10, 17].
\begin{table}
\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline \multicolumn{1}{c|}{**Dependability**} & \multicolumn{1}{c|}{**Method**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} & \multicolumn{1}{c|}{**Evaluation**} \\ \cline{2-15} \multicolumn{1}{c|}{**SLAMness**} & \multicolumn{1}{c|}{OROR-SLAM [24]} & [45] & 7.96 & 2.90 & 7.84 & 24.63 & 12.17 & 7.30 & 3.54 & 9.72 & 1007 \\ \hline \multirow{3}{*}{ATE} & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{\begin{tabular}{} \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{} \end{tabular} } \\ \cline{1-1} \cline{6-15} \cline{7-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15} \cline{15-15-15}
MAROAM [28] was kindly computed and provided by Wang et al. for this letter. We tuned our parameters on the Oxford sequence 10-12-32 and evaluated the performance of SLAM on all other Oxford and MulRan sequences.
The estimated trajectories are depicted in Fig. 5 and Fig. 6(a-i). We found that TBV effortlessly closes loops and corrects odometry in all sequences. ATE is substantially corrected over the full trajectory, with slightly reduced drift (Tab. I & Tab. II). TBV outperforms previous methods for radar SLAM in terms of ATE and drift over all sequences.
Hence, we conclude that our method improves the state of the art in radar SLAM. Surprisingly, we did not observe any improvement from using dynamic covariance (dyn-cov) compared to fixed. The Hessian-approximated covariance occasionally under- or over-estimates the odometry uncertainty [49] and thus deteriorates the optimization process.
### _Generalization to off-road environments_
Finally, we tested TBV on the sequences Kvarntorp and VolvoCE from the _Diverse ORU Radar Dataset [16]_, see footnote for a demo1. Kvarntorp is an underground mine with partly feature poor sections, while VolvoCE is a mixed environment with forest and open fields. Trajectories are visualized in Fig. 6.(j-k). We found that TBV was able to produce smooth and globally consistent maps, through substantially different environments, including challenging road conditions - without any parameter changes.
Footnote 1: ORU dataset download: [https://tinyurl.com/radarDataset](https://tinyurl.com/radarDataset). Demo video: [https://tinyurl.com/TBV-KvarntorpVolvo](https://tinyurl.com/TBV-KvarntorpVolvo).
## V Conclusions
We proposed TBV Radar SLAM - a real-time method for robust and accurate large-scale SLAM using a spinning 2D radar. We showed that loop candidate retrieval can be largely improved by origin-shifting, coupled place similarity/odometry uncertainty search, and selecting the most likely loop constraint as proposed by our verification model. A high level of loop robustness was achieved by carefully verifying loop constraints based on multiple sources of information, such as place similarity, consistency with odometry uncertainty, and alignment quality assessed after registration. We evaluated TBV on two public datasets and demonstrated a substantial improvement to the state of the art in radar SLAM, making radar an attractive option to lidar for robust and accurate localization. Quantitative and qualitative experiments demonstrated a high level of generalization across environments. Some findings in our ablation study suggest that heavy filtering is undesired as it discards details that are important for place recognition. Thus, in the future, we will explore building detailed and dense representations of scenes, fully utilizing the geometric information richness, uniquely provided by spinning FMCW radar.
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{2}{|c|}{Type-Bradar} & \multicolumn{3}{c|}{Evaluation} & \multicolumn{3}{c|}{\(\lambda_{0.551}\), \(\lambda_{0.552}\), \(\lambda_{0.553}\), \(\lambda_{0.554}\), \(\lambda_{0.555}\), \(\lambda_{0.556}\)} & \multicolumn{3}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{3}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.559}\)} & \multicolumn{3}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} \\ \hline \multicolumn{2}{|c|}{} & \multicolumn{1}{c|}{\(\lambda_{0.551}\), \(\lambda_{0.552}\)} & \multicolumn{1}{c|}{\(\lambda_{0.553}\), \(\lambda_{0.554}\)} & \multicolumn{1}{c|}{\(\lambda_{0.556}\), \(\lambda_{0.556}\)} & \multicolumn{1}{c|}{\(\lambda_{0.556}\), \(\lambda_{0.557}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.559}\), \(\lambda_{0.557}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.557}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.558}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.558}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.558}\), \(\lambda_{0.558}\)} & \multicolumn{1}{c|}{\(\lambda_{0.558}\), \(\lambda_{0.558}\), \(\lambda_{0. | 大規模環境における頑健なSLAMには、感知とオドメトリー推定からループクローズに至るまでの複数の段階で耐障害性と認識能力が必要です。この論文では、TBV (信頼するが検証する) rada SLAMを提案しました。これは、ループクローズ候補を自己検証する Rada SLAMの方法です。TBV Radar SLAMは、複数の場所認識技術を組み合わせることでループクローズの正確な回収率を達成します。それは、緊密に結合された場所の類似性とオドメトリー不確実性探索、起源のシフトされたスキャンからループ説明を作成し、ループ選択を検証後に遅らせることで達成しています。誤った制約に対する robustness は、複数のループ制約の中から最も可能性が高いものを検証と選択することによって達成されます。重要なのは、検証と選択は、ループの証拠の追加情報が簡単に計算できる登録後に実行されることです。私たちは |
2305.16264 | Scaling Data-Constrained Language Models | The current trend of scaling language models involves increasing both
parameter count and training dataset size. Extrapolating this trend suggests
that training dataset size may soon be limited by the amount of text data
available on the internet. Motivated by this limit, we investigate scaling
language models in data-constrained regimes. Specifically, we run a large set
of experiments varying the extent of data repetition and compute budget,
ranging up to 900 billion training tokens and 9 billion parameter models. We
find that with constrained data for a fixed compute budget, training with up to
4 epochs of repeated data yields negligible changes to loss compared to having
unique data. However, with more repetition, the value of adding compute
eventually decays to zero. We propose and empirically validate a scaling law
for compute optimality that accounts for the decreasing value of repeated
tokens and excess parameters. Finally, we experiment with approaches mitigating
data scarcity, including augmenting the training dataset with code data or
removing commonly used filters. Models and datasets from our 400 training runs
are freely available at https://github.com/huggingface/datablations. | Niklas Muennighoff, Alexander M. Rush, Boaz Barak, Teven Le Scao, Aleksandra Piktus, Nouamane Tazi, Sampo Pyysalo, Thomas Wolf, Colin Raffel | 2023-05-25T17:18:55 | http://arxiv.org/abs/2305.16264v4 | # Scaling Data-Constrained Language Models
###### Abstract
The current trend of scaling language models involves increasing both parameter count and training dataset size. Extrapolating this trend suggests that training dataset size may soon be limited by the amount of text data available on the internet. Motivated by this limit, we investigate scaling language models in data-constrained regimes. Specifically, we run a large set of experiments varying the extent of data repetition and compute budget, ranging up to 900 billion training tokens and 9 billion parameter models. We find that with constrained data for a fixed compute budget, training with up to 4 epochs of repeated data yields negligible changes to loss compared to having unique data. However, with more repetition, the value of adding compute eventually decays to zero. We propose and empirically validate a scaling law for compute optimality that accounts for the decreasing value of repeated tokens and excess parameters. Finally, we experiment with approaches mitigating data scarcity, including augmenting the training dataset with code data or removing commonly used filters. Models and datasets from our 400 training runs are freely available at [https://github.com/huggingface/databases](https://github.com/huggingface/databases).
Figure 1: _Return and Allocation when repeating data. (Left):_ Loss of LLMs (4.2B parameters) scaled on repeated data decays predictably (\(\lx@sectionsign\)6). _(Right):_ To maximize performance when repeating, our data-constrained scaling laws and empirical data suggest training smaller models for more epochs in contrast to what assuming Chinchilla scaling laws [42] hold for repeated data would predict (\(\lx@sectionsign\)5).
Introduction
Recent work on compute-optimal language models [42] shows that many previously trained large language models (LLMs, which we define as having more than one billion parameters) could have attained better performance for a given compute budget by training a smaller model on more data. Notably, the 70-billion parameter Chinchilla model [42] outperforms the 280-billion parameter Gopher model [86] while using a similar compute budget by being trained on four times more data. Extrapolating these laws for compute allocation (hereafter "Chinchilla scaling laws") to a 530 billion parameter model, such as the under-trained MT-NLG model [96], would require training on a massive 11 trillion tokens, corresponding to more than 30 terabytes of text data. For most languages, available data is several orders of magnitude smaller, meaning that LLMs in those languages are already data-constrained. Villalobos et al. [109] estimate that even high-quality English language data will be exhausted by the year 2024 given the Chinchilla scaling laws and the trend of training ever-larger models. This motivates the question [109; 78]: what should we do when we run out of data?
In this work we investigate scaling large language models in a data-constrained regime, and whether training an LLM with multiple epochs of repeated data impacts scaling. Using multiple epochs is, of course, standard in machine learning generally; however, most prior large language models have been trained for a single epoch [51; 15] and some work explicitly advocates against reusing data [40]. An exception is the recent Galactica models [105] that were trained for 4.25 epochs and exhibit continually decreasing validation loss and improving downstream performance throughout training. However, the experiments of Galactica do not compare this setup to an alternative non-data-constrained model trained for one epoch on unique data. Without this comparison, it is difficult to quantify the trade-off between additional compute versus additional data collection.
Our main focus is to quantify the impact of multiple epochs in LLM training such that practitioners can decide how to allocate compute when scaling models. Toward this end, we assembled a battery of empirical training runs of varying data and compute constraints. Specifically, we train more than 400 models ranging from 10 million to 9 billion parameters for up to 1500 epochs and record final test loss. We use these results to fit a new _data-constrained scaling law_ that generalizes the Chinchilla scaling law [42] to the repeated data regime and yields a better prediction of loss in this setting. Figure 1 summarizes our main results targeting the value of repeated data (_Return_) and optimal allocation of resources in that regime (_Allocation_). We find that, while models trained for a single epoch consistently have the best validation loss per compute, differences tend to be _insignificant_ among models trained for up to 4 epochs and do not lead to differences in downstream task performance. Additional epochs continue to be beneficial, but returns eventually diminish to zero. We find that, in the data-constrained regime, allocating new compute to both more parameters and epochs is necessary, and that epochs should be scaled slightly faster. These findings suggest a simple way to continue scaling total training compute budgets further ahead in the future than the previously anticipated limits.
Finally, given the challenges imposed by data constraints, we consider methods complementary to repeating for improving downstream accuracy without adding new natural language data. Experiments consider incorporating code tokens and relaxing data filtering. For code, English LLMs, such as PaLM [19] or Gopher [86], are trained on a small amount of code data alongside natural language data, though no benchmarking was reported to justify that decision. We investigate training LLMs on a mix of language data and Python data at 10 different mixing rates and find that mixing in code is able to provide a 2\(\times\) increase in effective tokens even when evaluating only natural language tasks. For filtering, we revisit perplexity and deduplication filtering strategies on both noisy and clean datasets and find that data filtering is primarily effective for noisy datasets.
## 2 Background
Predicting the scaling behavior of large models is critical when deciding on training resources. Specifically, two questions are of interest: (_Allocation_) What is the optimal balance of resources? (_Return_) What is the expected value of additional resources? For scaling LLMs, the resource is compute (measured in FLOPs), and it can be allocated to training a larger model or training for more
steps.1 The metric used to quantify progress is the model's loss on held-out data, i.e. the ability to predict the underlying data as measured in the model's cross-entropy [2; 42]. We aim to minimize the loss (\(L\)) subject to a compute resource constraint (\(C\)) via optimal allocation to \(N\) and \(D\) as:
Footnote 1: In this work we use [46]’s approximation for the compute cost: FLOPs\((N,D)\approx 6ND\), where N denotes the number of model parameters and D denotes the number of tokens processed.
\[\operatorname*{argmin}_{N,D}L(N,D)\text{ s.t. FLOPs}(N,D)=C \tag{1}\]
Currently, there are established best practices for scaling LLMs. _Return_ follows a power-law: loss scales as a power-law with the amount of compute used for training [39; 46; 6; 35; 7; 41]. _Allocation_ is balanced: resources are divided roughly equally between scaling of parameters and data [42]. These scaling laws were established empirically by training LLMs and carefully extrapolating behavior.
Chinchilla [42] uses three methods for making scaling predictions:
* (_Fixed Parameters_) Train with a fixed model size but on varying amounts of data.
* (_Fixed FLOPs_) Train with fixed computation while parameters and training tokens vary.
* (_Parametric Fit_) Derive and fit a formula for the loss.
For the parametric fit, the loss (\(L\)) is a function of parameters (\(N\)) and training tokens (\(D\)):
\[L(N,D)=\frac{A}{N^{\alpha}}+\frac{B}{D^{\beta}}+E \tag{2}\]
Where \(\{A,\alpha,B,\beta,E\}\) are learned variables fit using the training runs from the first two approaches [42]. Using these learned variables, they propose calculating the optimal allocation of compute (\(C\)) to \(N\) and \(D\) as follows:
\[\begin{split} N_{opt}(C)=G(C/6)^{a}\quad D_{opt}(C)=G^{-1}(C/6) ^{b}\\ \text{where}\quad G=\left(\frac{\alpha A}{\beta B}\right)^{\frac{ 1}{\alpha+\beta}}\quad a=\frac{\beta}{\alpha+\beta}\quad b=\frac{\alpha}{ \alpha+\beta}\end{split} \tag{3}\]
These methods lead to the conclusion that \(\alpha\approx\beta\) and hence \(N\) and \(D\) should be scaled proportionally for compute-optimal training. As loss can be an imperfect proxy for performance on natural language tasks [120; 94; 102], they also validate their conclusions on various downstream tasks.
## 3 Method: Data-Constrained Scaling Laws
We are interested in scaling behavior in the data-constrained regime. Specifically, given a limited amount of unique data, what is the best _Allocation_ of and _Return_ for computational resources. Prior work [46; 42] assumes that the necessary data to support scaling is unlimited. Our aim is therefore to introduce a modified version of Equation 2 that accounts for data constraints and fit the terms in the modified scaling law to data from a large body of experiments.
The primary method we consider is _repeating_ data, i.e. allocating FLOPs to multiple epochs on the same data. Given a budget of unique data \(D_{C}\), we split the Chinchilla total data term \(D\) into two parts: the number of unique tokens used, \(U_{D}\), and the number of repetitions, \(R_{D}\) (i.e. epochs - 1). Given total training tokens \(D\) and data budget \(D_{C}\) these terms are simply computed as \(U_{D}=\min\{D_{C},D\}\) and \(R_{D}=(D/U_{D})-1\). When training for a single epoch like done in prior scaling studies, \(R_{D}=0\). We are thus interested in minimizing Equation 1 with the additional constraint of a data budget \(D_{C}\):
\[\operatorname*{argmin}_{N,D}L(N,D)\text{ s.t. FLOPs}(N,D)=C,U_{D}\leq D_{C} \tag{4}\]
Symmetrically, for mathematical convenience, we split the parameter term \(N\) into two parts: the base number of parameters needed to optimally fit the unique tokens \(U_{N}\), and the number of times
to "repeat" this initial allocation, \(R_{N}\). We compute \(U_{N}\) by first rearranging Equation 3 to find the optimal compute budget for the unique tokens used (\(U_{D}\)). We input this value into the \(N_{opt}\) formula of Equation 3 to get \(\bar{U}_{N}=\min\{N_{opt},N\}\). \(U_{N}\) thus corresponds to the compute-optimal number of parameters for \(U_{D}\) or less if \(N<N_{opt}\). Once we have \(U_{N}\), we compute the repeat value as \(R_{N}=(N/U_{N})-1\).
To empirically explore the scaling behavior in a data-limited setting we train LLMs under these constraints. We consider three different experimental protocols in this work:
* (_Fixed Unique Data_) In SS5 we fix the data constraint \(D_{C}\) and train models varying epochs and parameters. These experiments target _Allocation_, specifically tradeoff of \(D\) or \(N\).
* (_Fixed FLOPs_) In SS6 we fix the computation available and vary \(D_{C}\) (and thus also \(U_{D}\) and \(U_{N}\)). These experiments target _Return_, i.e. how well does repeating scale compared to having more unique data.
* (_Parametric Fit_) We fit a formula introduced in SS3.1 on all our training runs and evaluate its predictive capability throughout SS5 and SS6.
Before discussing experimental results we describe the parametric assumptions.
### Parametric Fit
To extrapolate scaling curves, it is necessary to incorporate repetition into the Chinchilla formula (Equation 2). We generalize Equation 2 by replacing \(D\) and \(N\) with terms corresponding to the _effective data_ (\(D^{\prime}\)) and _effective model parameters_ (\(N^{\prime}\)).
\[L(N,D)=\frac{A}{N^{\prime\alpha}}+\frac{B}{D^{\prime\beta}}+E\]
Intuitively, \(D^{\prime}\) should be smaller or equal to \(D\) where \(D\) is the total number of processed tokens since repeated tokens provide less useful information to the model than new ones. We use an _exponential decay_ formulation, where the value of a data token processed loses roughly \((1-1/R_{D}^{*})\) fraction of its value per repetition, where \(R_{D}^{*}\) is a learned constant. After some derivations and approximations (see Appendix A), this boils down to
\[D^{\prime}=U_{D}+U_{D}R_{D}^{*}(1-e^{\frac{-R_{D}}{R_{D}^{*}}})\;. \tag{5}\]
Note that for \(R_{D}=0\) (no repetitions), \(D^{\prime}=U_{D}=D\). For \(R_{D}\ll R_{D}^{*},e^{-R_{D}/R_{D}^{*}}\approx 1-\frac{R_{D}}{R_{D}^{*}}\) and so
\[D^{\prime}\approx U_{D}+U_{D}R_{D}^{*}(1-1+R_{D}/R_{D}^{*})=U_{D}(1+R_{D})=D\]
and hence in this case, repeated data is worth almost the same as fresh data. (This is also consistent with the predictions of the "deep bootstrap" framework [73].) As \(R_{D}\) grows, the value of repeated tokens tends to zero, and the effective data \(D^{\prime}\) becomes much smaller than \(D\). The formula implies that no matter how many times we repeat the data, we will not get a better loss than could be obtained with a single epoch on \(U_{D}+U_{D}R_{D}^{*}\) fresh tokens.
Just as processing repeated tokens yields a diminishing return, both intuitively and empirically, models with sizes that vastly outstrip the available data also offer diminishing returns per parameter. Hence we use a symmetric formula for the number of effective parameters, where again \(R_{N}^{*}\) is learned,
\[N^{\prime}=U_{N}+U_{N}R_{N}^{*}(1-e^{\frac{-R_{N}}{R_{N}^{*}}})\;. \tag{6}\]
The learned constants \(R_{D}^{*}\), \(R_{N}^{*}\) roughly correspond to the "half-life" of repeated data and excess parameters. For example, at \(R_{D}=R_{D}^{*}\), the number of effective tokens \(D^{\prime}\) is \(U_{D}+U_{D}R_{D}(1-e^{-1})\) which means that the \(U_{D}R_{D}\) repeated tokens are worth on average \(1-1/e\) fraction of fresh ones.
Using a methodology similar to [42], \(R_{N}^{*}\) and \(R_{D}^{*}\) can be fit on empirical measurements, which yields data-driven estimates. See Appendix A for more details on the derivations and the fitting procedure.
## 4 Experimental Setup
For all experiments, we train transformer language models with the GPT-2 architecture and tokenizer [85]. Models have up to 8.7 billion parameters and are trained for up to 900 billion total tokens. Following [42] we use cosine learning rate schedules that decay 10\(\times\) over the course of training for each model (different schedules led to different estimates in [46]). Unlike [46], we do not use early stopping to also explore the extent of overfitting when repeating. Other hyperparameters are based on prior work [86; 42] and detailed in Appendix S. Models are trained on subsets of C4 [87]. The data constraints are carefully defined to ensure maximal overlap as shown in Figure 2. Unlike [40], we always repeat the entire available data rather than subsets of it. Data is shuffled after each epoch. As repeating data can result in extreme overfitting (see Appendix H), we report loss on a held-out test set unless otherwise specified (see Appendix K). This contrasts training loss used in [42], but should not alter our findings as the held-out data stems from the same underlying dataset.
## 5 Results: Resource Allocation for Data-Constrained Scaling
Our first experimental setting considers scaling in a setting where all models have the same data constraint. For these experiments, the unique training data budget \(D_{C}\) is fixed at either 100M, 400M or 1.5B tokens. For each data budget, we train a set of language models with increasing amounts of compute that is allocated to either more parameters or more epochs on the unique training data.
Figure 3 (left) shows the main results for scaling with 100M unique tokens2 (see Appendix C for 400M and 1.5B tokens). For 100M tokens, the corresponding one-epoch compute-optimal model
Figure 3: **IsoLoss contours for 100 million unique tokens. (_Left_): 93 models trained with varying parameters and epochs on a fixed dataset. Contours show an interpolation of results with the same final test loss. (_Right_): Comparison with the loss predictions from our proposed scaling laws for the same budget of 100 million unique tokens and the predicted efficient frontier. The diminishing returns from training on repeated data can be seen in the increase in distance of the contour curves.**
Figure 2: **Dataset setup. Training runs with different epochs reuse subsets of the same data to ensure different training data is not a confounding factor.**
according to scaling laws from [42] has \(U_{N}\) of approximately 7M parameters (see Appendix B for the scaling coefficients we use). Results show that more than a 50% reduction in loss can be attained by training for several epochs (\(R_{D}>0\)) and increasing model size beyond what would be compute optimal for 100M tokens (\(R_{N}>0\)). We find the best loss to be at around 20-60\(\times\) more parameters and epochs, which corresponds to spending around 7000\(\times\) more FLOPs. These results suggest that one-epoch models significantly under-utilize their training data and more signal can be extracted by repeating data and adding parameters at the cost of sub-optimal compute utilization.
Figure 3 (right) shows the predicted contours created by fitting our data-constrained scaling laws on 182 training runs. In the single-epoch case (\(R_{D}=0\)) with near compute-optimal parameters (\(R_{N}=0\)) our scaling equation (SS3.1) reduces to the Chinchilla equation. In this case, both formulas predict the optimal allocation of compute to parameters and data to be the same, resulting in overlapping efficient frontiers. As data is repeated for more than a single epoch, our fit predicts that excess parameters decay faster in value than repeated data (\(R_{N}^{*}<R_{D}^{*}\)). As a result, the data-constrained efficient frontier suggests allocating most additional compute to more epochs rather than more parameters. This contrasts the Chinchilla scaling laws [42], which suggest equally scaling both. However, note that they do not repeat the entire training data and their parametric fit explicitly relies on the assumption that models are trained for a single epoch only. Thus, there is no guarantee that their scaling predictions hold for repeated data.
For all three data budgets, our results suggest that _Allocation_ is optimized by scaling epochs faster than additional parameters. We confirm this at scale by training the data-constrained compute-optimal model for \(9.3\times 10^{21}\) FLOPs and 25 billion unique tokens as suggested by our efficient frontier. Despite having 27% less parameters, this model achieves better loss than the model suggested by the Chinchilla scaling laws (Figure 1, right). Similarly, the 120 billion parameter Galactica model trained on repeated data should have been significantly smaller according to data-constrained scaling laws (Appendix G). An additional benefit of using a smaller model is cheaper inference, though adding parameters can make it easier to parallelize training across GPUs.
Figure 4: **Validation Loss for Different Data Constraints (IsoFLOP). Each curve represents the same number of FLOPs spent on an equal size model. Colors represent different numbers of epochs due to repeating because of data constraints. Parameters and training tokens are set to match the single-epoch compute-optimal configurations for the given FLOPs. Models trained on data that is repeated for multiple epochs have consistently worse loss and diverge if too many epochs are used.**
Adding parameters and epochs causes the loss to decrease and eventually increase again, suggesting that too much compute can hurt performance. Results from [46] also show that loss can increase when too many parameters are used, even with early stopping. However, we expect that appropriate regularization (such as simply removing all excess parameters as an extreme case) could prevent this behavior. Thus, our formula presented in SS3 and its predicted isoLoss contours in Figure 3 do not model the possibility that excess epochs or parameters could hurt performance.
## 6 Results: Resource Return for Data-Constrained Scaling
Next, consider the question of _Return_ on scaling. To quantify this value, we run experiments with three FLOP budgets across eight respective data budgets to compare return on FLOPs.
Figure 4 shows the configurations and validation curves for models trained on the same number of total tokens. Conforming to intuition and prior work on deduplication [55], repeated data is worth less, thus models trained on less unique data (and, correspondingly, more epochs) have consistently higher loss. However, the loss difference for a few epochs is negligible. For example, the \(N=8.7\) billion parameter model trained for four epochs (\(D_{C}=44\) billion unique tokens) finishes training with only 0.5% higher validation loss than the single-epoch model (\(D_{C}=178\) billion unique tokens).
In Figure 5 (left), we compare the final test loss of each model to predictions from our parametric fit. The data-constrained scaling laws can accurately measure the decay in the value of repeated data as seen by the proximity of empirical results (dots) and parametric fit (lines). We note however that it significantly underestimates the final test loss of failing models where loss increases midway through training, such as models trained for 44 epochs (not depicted).
In Figure 5 (right), we extrapolate the three budgets by further scaling compute while keeping the data constraints (\(D_{C}\)) at 55B, 84B, and 178B tokens, respectively. The parameter \(R_{D}^{*}\) introduced in SS3 represents roughly the "half-life" of epochs: specifically the point where repeated tokens have lost \(\frac{1}{e}\) of their value. Through our fitting in Appendix A, we found \(R_{D}^{*}\approx 15\), corresponding to 15 repetitions (or 16 epochs). Graphically, this can be seen by the stark diminishing returns in the proximity of the 16-epoch marker and the flattening out soon after.
Overall, the _Return_ when repeating data is relatively good. Meaningful gains from repeating data can be made up to around 16 epochs (\(R_{D}^{*}\)) beyond which returns diminish extremely fast.
Figure 5: **Empirical and Extrapolated loss with constrained data.** _(Left):_ Loss as a function of repeated tokens for three different training budgets each with fixed number of parameters. Loss curves predicted by our data-constrained scaling laws are shifted to exactly match the loss at 100% unique data. Return on FLOPs decays with repeated data in a regular pattern. _(Right):_ Extrapolating from the proposed data-constrained scaling law shows that at small numbers epochs are benign, but at large number of epochs loss stops improving.
## 7 Results: Complementary Strategies for Obtaining Additional Data
While repeating data is effective, it has diminishing returns. We therefore consider strategies for scaling \(D\) targeting improved downstream performance as opposed to directly minimizing loss.
Figure 6 (left) illustrates the strategies: **(a) Code augmentation:** We use Python code from The Stack [49] to make up for missing natural language data. The combined dataset consisting of code and natural language samples is shuffled randomly. **(b) Adapting filtering:** We investigate the performance impact of deduplication and perplexity filtering, two common filtering steps that can severely limit available data. Removing such filtering steps can free up additional training data.
For these experiments, we set a maximum data budget (\(D_{C}\)) of 84 billion tokens. For repetition and code filling, only a subset of \(D_{C}\) is available and the rest needs to be compensated for via repeating or adding code. For both filtering methods, we start out with approximately twice the budget (178 billion tokens), as it is easier to gather noisy data and filter it than it is to gather clean data for training. For perplexity filtering, we select the top 25% samples with the lowest perplexity according to a language model trained on Wikipedia. This results in 44 billion tokens that are repeated for close to two epochs to reach the full data budget. For deduplication filtering, all samples with a 100-char overlap are removed resulting in 21 billion tokens that are repeated for four epochs during training. See Appendix N for more details on the filtering procedures.
When comparing across data strategies, loss ceases to be a good evaluation metric as the models are trained on different data distributions. We thus evaluate models on 19 natural language tasks with zero to five in-context few-shot exemplars [15] producing 114 scores per model. As our evaluation tasks cover different metrics and random baselines, we re-scale all scores to be in the same range to better reflect performance ranges before averaging. Details on the evaluation datasets are in Appendix K.
In Figure 6 (right) we compare the downstream performance of all strategies. For repeating data, differences in downstream performance are insignificant for up to around 4 epochs (25% budget) and then start dropping, which aligns with our results on test loss in SS6. Filling up to 50% of data with code (42 billion tokens) also shows no deterioration. Beyond that, performance decreases quickly on natural language tasks. However, adding more code data may benefit non-natural language tasks, which are not considered in the benchmarking. Two of the tasks benchmarked, WebNLG [17; 34], a generation task, and bAbI [119; 57], a reasoning task, see jumps in performance as soon as code is added, possibly due to code enabling models to learn long-range state-tracking capabilities beneficial for these tasks.
Of the filtering approaches, we find perplexity-filtering to be effective, while deduplication does not help. Prior work found deduplication was able to improve perplexity [55]; however, it did not evaluate on downstream tasks. Deduplication may have value not captured in our benchmark, such as reducing memorization [45; 40; 16; 10]. We also investigate filtering on a different noisier dataset in Appendix O, where we find it to be more effective. Overall, in a data-constrained regime, we recommend reserving filtering for noisy datasets and using both code augmentation and repeating to
Figure 6: **Strategies for data-constrained settings and their downstream performance. (_Left_): Schematic showing alternative data use strategies of code filling and filtering. _(Right):_\(N=4.2\) billion parameter models trained for a total of \(D=84\) billion tokens with varying budgets \(D_{C}\). For repeating and filling with code, five models with different seeds are trained for each dot and the standard deviation is visualized as the shaded area.**
increase data tokens. For example, first doubling the available data by adding code and then repeating the new dataset for four epochs results in 8\(\times\) more training tokens that are expected to be just as good as having had 8\(\times\) more unique data from the start.
## 8 Related Work
**Large language models** Scaling up transformer language models [108] across parameter count and training data has been shown to result in continuous performance gains [19]. Starting with the 1.4 billion parameter GPT-2 model [85], a variety of scaled-up language models have been trained, commonly referred to as large language models (LLMs). They can be grouped into dense models [15, 47, 58, 86, 20, 13, 129, 106, 100, 105, 127, 92, 56] and sparse models [30, 128, 28, 132] depending on whether each forward pass makes use of all parameters. These models are generally pre-trained to predict the next token in a sequence, which makes them applicable to various language tasks directly after pre-training [15, 115, 50, 69, 99] by reformulating said NLP tasks as context continuation tasks (see [65] for an earlier proposal on this topic). We focus on the most common scenario, where a dense transformer model is trained to do next-token prediction on a large corpus and evaluated directly after pre-training using held-out loss or zero- to few-shot prompting.
**Scaling laws** Prior work has estimated an optimal allocation of compute for the training of LLMs. Kaplan et al. [46] suggested a 10\(\times\) increase in compute should be allocated to a 5.5\(\times\) increase in model size and a 1.8\(\times\) increase in training tokens. This first scaling law has led to the creation of very large models trained on relatively little data, such as the 530 billion parameter MT-NLG model trained on 270 billion tokens [96]. More recent work [42], however, showed that model size and training data should rather be scaled in equal proportions. These findings called for a renewed focus on the scaling of pre-training data rather than scaling model size via complex parallelization strategies [95, 88, 9, 75]. Up-sampling is often employed when pre-training data is partly limited, such as data from a high-quality domain like Wikipedia or text in a rare language for training multilingual LLMs [60, 79]. Hernandez et al. [40] study up-sampling of data subsets and find that repeating only 0.1% of training data 100 times significantly degrades performance. In contrast, our work focuses on repeating the entire pre-training corpus for multiple epochs rather than up-sampling parts of it.
**Alternative data strategies** Large pre-training datasets are commonly filtered to remove undesired samples or reduce noise [98]. Perplexity-based filtering, whereby a trained model is used to filter out samples with high perplexity, has been found beneficial to reduce noise in web-crawled datasets [118]. Mixing of data is employed for the pre-training data of multilingual LLMs, where text data from different languages is combined [23, 123, 97, 71]. However, both for code and natural language models, mixing different (programming) languages has been reported to under-perform monolingual models [77, 110]. Some work has investigated mixing code and natural language data for prediction tasks, such as summarizing code snippets [44] or predicting function names [4]. Several pre-training datasets for LLMs include low amounts of code data [31, 86, 92]. However, these past works generally do not provide any ablation on the drawbacks of including code or the benefits for natural language task performance. We perform a detailed benchmarking of mixing Python and natural language in LLM pre-training at 10 different mixing rates.
## 9 Conclusion
This work studies data-constrained scaling, focusing on the optimal use of computational resources when unique data is limited. We propose an extension to the Chinchilla scaling laws that takes into account the decay in value of repeated data, and we fit this function using a large set of controlled experiments. We find that despite recommendations of earlier work, training large language models for multiple epochs by repeating data is beneficial and that scaling laws continue to hold in the multi-epoch regime, albeit with diminishing returns. We also consider complementary approaches to continue scaling models, and find that code gives the ability to scale an additional 2\(\times\). We believe that our findings will enable further scaling of language models to unlock new capabilities with current data. However, our work also indicates that there are limits on the scaling horizon. In addition to collecting additional data, researchers should explore using current data in a more effective manner.
## Acknowledgments and Disclosure of Funding
This work was co-funded by the European Union under grant agreement No 101070350. The authors wish to acknowledge CSC - IT Center for Science, Finland, for generous computational resources on the LUMI supercomputer.3 We are thankful for the immense support from teams at LUMI and AMD, especially Samuel Antao. Huugging Face provided storage and additional compute instances. This work was supported by a Simons Investigator Fellowship, NSF grant DMS-2134157, DARPA grant W911NF2010021, and DOE grant DE-SC0022199. We are grateful to Harm de Vries, Woojeong Kim, Mengzhou Xia and the EleutherAI community for exceptional feedback. We thank Loubna Ben Allal for help with the Python data and Big Code members for insightful discussions on scaling laws. We thank Thomas Wang, Helen Ngo and TurkuNLP members for support on early experiments.
Footnote 3: [https://www.lumi-supercomputer.eu/](https://www.lumi-supercomputer.eu/)
| 言語モデルの規模を拡大する現在のトレンドは、パラメータ数とトレーニングデータのサイズを増加させることです。このトレンドを extrapolation することは、インターネット上のテキストデータの量に制限される可能性があるという結論に導きます。この制限に動機付けられ、データ制限された環境での言語モデルの拡張を調査します。特に、データの重複度合いを変化させ、計算予算を最大9000億の訓練トークンと90億のパラメータのモデルまで実行します。データの重複度合いが増加すると、固定された計算予算での訓練では、ユニークなデータと比較して、損失のわずかな変化を伴うように、4エポックの繰り返しデータで訓練を行います。しかし、より多くの重複があると、追加の計算の価値はゼロになります。私たちは、繰り返しトークンと過剰のパラメータの価値を考慮した、計算の最適性を考慮するスケール定理を提案し |
2306.14123 | Privacy and Fairness in Federated Learning: on the Perspective of
Trade-off | Federated learning (FL) has been a hot topic in recent years. Ever since it
was introduced, researchers have endeavored to devise FL systems that protect
privacy or ensure fair results, with most research focusing on one or the
other. As two crucial ethical notions, the interactions between privacy and
fairness are comparatively less studied. However, since privacy and fairness
compete, considering each in isolation will inevitably come at the cost of the
other. To provide a broad view of these two critical topics, we presented a
detailed literature review of privacy and fairness issues, highlighting unique
challenges posed by FL and solutions in federated settings. We further
systematically surveyed different interactions between privacy and fairness,
trying to reveal how privacy and fairness could affect each other and point out
new research directions in fair and private FL. | Huiqiang Chen, Tianqing Zhu, Tao Zhang, Wanlei Zhou, Philip S. Yu | 2023-06-25T04:38:19 | http://arxiv.org/abs/2306.14123v1 | # Privacy and Fairness in Federated Learning: on the Perspective of Trade-off
###### Abstract.
Federated learning (FL) has been a hot topic in recent years. Ever since it was introduced, researchers have endeavored to devise FL systems that protect privacy or ensure fair results, with most research focusing on one or the other. As two crucial ethical notions, the interactions between privacy and fairness are comparatively less studied. However, since privacy and fairness compete, considering each in isolation will inevitably come at the cost of the other. To provide a broad view of these two critical topics, we presented a detailed literature review of privacy and fairness issues, highlighting unique challenges posed by FL and solutions in federated settings. We further systematically surveyed different interactions between privacy and fairness, trying to reveal how privacy and fairness could affect each other and point out new research directions in fair and private FL.
Federated learning, data privacy, model fairness +
Footnote †: c) 2023 Association for Computing Machinery.
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+
Footnote †: c) 2023 Association for Computing Machinery.
XXXX-XXXX/2023/6-ART $15.00
[https://doi.org/10.1145/nnnnnnnnnnnn](https://doi.org/10.1145/nnnnnnnnnnnn)
+
Footnote †: c) 2023 Association for Computing Machinery.
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## 1. Introduction
Machine learning has changed our lives and will undoubtedly bring us more excitement. However, its success is closely tied to the availability of large-scale training data, and as new learning models keep emerging, the demand for more data persists relentlessly. One worrisome issue with collecting massive amounts of data is the risks that present to privacy. FL (Huiqiang et al., 2019) has emerged as an attractive learning paradigm to meet privacy requirements.
Unlike traditional centralized machine learning, FL trains models in a distributed and parallel way such that different clients collectively train a model with their training data. This technique offers two enormous benefits. First, it saves companies the costly process of collecting large-scale data because the clients provide their local data. Second, it preserves the client's privacy by keeping data locally. With such benefits, it is no surprise that the industry has already leaped to put FL into practice, such as Gboard (Ghoard et al., 2019).
### Privacy and Fairness in FL
Great achievements have been made with FL. However, the paradigm of FL still suffers from ethical issues surrounding data use. One of those ethical issues is privacy. Although raw data never leave the device, the uploaded gradients/parameters still carry local information. Therefore, a trained model could hold the client's data distribution. Consequently, an adversary can infer information about what data are included in the training set (Zhu et al., 2020; Zhang et al., 2021) or, even worse, reconstruct the training data (Zhu et al., 2020). As such, the research community is endeavoring to identify all potential privacy risks by launching different privacy attacks (Zhu et al., 2020). Accordingly, defenses for all these attacks are also being proposed to secure private data (Zhu et al., 2020). This wargaming between attack and defense is leading us to more private FL environments.
Another ethical issue in FL is fairness, which refers to reducing the model's bias towards disadvantaged groups, such as ethnic minorities, women, or the aged. Fairness in FL is defined at two different levels. The first pertains to _algorithmic fairness_(Kolmogorov, 1969; Kolmogorov, 1969), where model output should not skew towards disadvantaged groups defined by some sensitive attributes. The second pertains to _client fairness_(Kolmogorov, 1969; Kolmogorov, 1969). In vanilla FL (Zhu et al., 2020), models trained on the larger dataset are given higher importance during aggregation. Hence, the global model will be optimized to capture the data distributions of clients with a larger dataset. Therefore the model performance will vary significantly among clients, which imposes unfairness at the client level.
Privacy and fairness are two crucial ethical notions, and violating either of them is unacceptable. However, to date, the research community has primarily considered these two issues separately, yet they are inextricably entwined. For example, it is well known that privacy comes at the cost of accuracy. What is surprising is that the cost is not consistent across all groups as expected, where disadvantaged groups often suffer more of an accuracy decrease than the other groups due to data scarcity (Zhu et al., 2020; Zhang et al., 2021). In other words, ensuring privacy can exacerbate the inequities between groups. Fairness, in turn, may negatively affect privacy. For example, to ensure a classification model is fair, the server usually needs to know the underlying distribution of the training dataset to eliminate bias existing in either the training data (Zhu et al., 2020; Zhang et al., 2021; Zhang et al., 2021) or the model (Zhu et al., 2020; Zhang et al., 2021; Zhang et al., 2021). This means the client will share more data with the server, increasing the privacy risk.
Therefore, in addition to reviewing privacy and fairness issues in FL, another motivation of this survey is to explore the possible interactions between privacy and fairness and to discuss the relationships between fairness and privacy. It is worth noting that the issue of client fairness adds an extra layer of complexity to the federated setting. To the best of our knowledge, this survey is the first attempt to examine the relationships between privacy and fairness in the federated setting.
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{Reference} & \multicolumn{2}{c}{Privacy-preserving} & \multicolumn{2}{c}{Fairness-aware} & Interactions \\ \cline{2-5} & Privacy & Defense & Algorithmic & Client & between privacy \\ & attack & & fairness & fairness & and fairness \\ \hline (Zhu et al., 2020) & ✓ & ✓ & & & \\ (Zhu et al., 2020) & ✓ & ✓ & & & \\ (Zhu et al., 2020) & ✓ & ✓ & & & \\ (Zhu et al., 2020) & ✓ & ✓ & ✓ & ✓ & \\ Our work & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table}
Table 1. Comparison to Related Surveys on Privacy or Fairness in FL
### Main Contribution
This survey provides a broad view of privacy and fairness issues in FL. We first illustrated that FL is not as private as it claimed to be. Adversarial clients and server have several new attack vectors at their disposal, and we outlined several techniques for preserving privacy against these attacks. Turning to fairness, we explained the two lines of fairness notions adopted in FL and the corresponding debiasing strategies. Lastly, we discussed interactions between privacy and fairness. Our contributions are as follows:
* This is the first survey that provides a comprehensive overview of privacy, fairness, and the interactions between the two.
* We present a detailed survey of privacy attacks and defenses in FL, discuss how these privacy attacks could damage privacy in FL and highlight the assumptions and principle methods of these attack strategies.
* Following a rigorous enumeration of the sources of bias in the FL pipeline, we discuss the fairness notions adopted in FL and summarize fairness-aware FL approaches.
* We point out several future research directions toward training private and fair FL models.
## 2. Background Knowledge
### Definition of FL
The goal of FL is to train a global model in a distributed way. The objective function is formulated as:
\[\min_{w}f\left(w\right)=\sum_{k=1}^{m}p_{k}F_{k}\left(w\right) \tag{1}\]
where \(m\) is the number of clients, \(p_{k}>0\) is the aggregating weight of client \(k\), satisfying \(\sum_{k}p_{k}=1\). \(F_{k}(w)\) is the empirical risk on client \(k\)'s data. In a trivial setting, \(p_{k}\) is the ratio of local samples to total samples of all clients. The process of FL consists of two stages: the training and inference stages. Three actors are involved: 1) clients, each of which has a local dataset and will use it to contribute to the global model's training by uploading local gradients/parameters, noting that each client's dataset may vary from the others; 2) a server that coordinates the learning process; and 3) users, who will use the final well-trained model.
In each iteration of FL, selected clients download the global model and perform learning algorithms locally. They communicate their updates to the server for aggregation and model updating. This interaction between clients and the server repeats until the model converges. At the inference stage, a well-trained model is deployed to users, where users can infer the model via black-box access. This step is no different from a traditional data center approach.
### Privacy Disclosure in FL
Recent research verified the privacy risk of FL. Adversaries can glean the participants' training data. For example, Zhu et al. (Zhu et al., 2021) fully reconstructed training data from the victim client through the uploaded gradients as shown in Fig.1(a). In the course of the FL pipeline, several attack interfaces exist for an adversary, who could be the server or a client in FL. The attack could occur during the training or inference stage, within or outside the FL. The attack targets include membership, property, class representative, and the raw data (Zhu et al., 2020).
#### 2.2.1. Membership Inference Attacks
Membership Inference Attacks (MIA) aim to identify whether a given sample was used to train the target model. These types of attacks can pose privacy risks to individuals. For example, confirming a patient's clinical record was used to train a model associated with a particular disease would reveal that patient's health condition. MIA was initially investigated
by Shokri et al. (2016). The attack models are essentially binary classifiers. Given an instance \(X\) and a target model \(F_{t}\), the goal of the MIA model is to identify whether or not \(X\) is contained within the training dataset \(D\) of the target model \(F_{t}\).
#### 2.2.2. Property Inference Attacks
Property inference attacks aim to recover some property of the training set, which may be irrelevant to the main tasks. Such as the property of "wearing glasses" against a gender classifier or the composition of the training dataset. This kind of attack also leads to privacy issues. With proper prior knowledge, the adversary can infer the presence of a specific sample in the training set.
#### 2.2.3. Model Inversion Attacks
Model inversion attacks aim to recover class-specific features or construct class representatives by accessing the target model and other possible auxiliary information. The recovered data is a representing sample (usually a synthetic sample) that only reflects some aspects of the training data and is not a member of the training set.
#### 2.2.4. Reconstruction Attacks
Reconstruction attacks (Shokri et al., 2016) aim to reconstruct a probabilistic version of samples in the training set. Success in a reconstruction attack is measured by comparing the reconstruction with the original data. If the two are similar, the attack has been successful. Fig.1(b) (Zhou et al., 2017) shows an example. Unlike the model inversion attacks, the recovered data here is almost the same as the training dataset at the pixel level and belongs to the training dataset.
### Privacy-preserving Techniques
In the realm of FL, plenty of studies have shown how to break the basic privacy assurances, such as determining a client's membership in the training set (Shi et al., 2017), ascertaining the class representations of the client's training data (Shi et al., 2017; Shi et al., 2018; Shokri et al., 2018), and, the worst case of all, procuring the raw training data (Shi et al., 2017). Several privacy-preserving techniques can help stop these privacy leakages, including cryptographic techniques and the perturbation approach.
#### 2.3.1. Cryptographic Approach
Secure computation is a cryptographic technique in which functions are evaluated based on a set of distributed inputs without revealing additional information, e.g., the parties' inputs or intermediate results. Secure multi-party computation, homomorphic encryption, and secret sharing are the most common choices for a secure computing platform.
Multi-party computation (Zhou et al., 2017) was first introduced to secure the private inputs of multiple participants while they jointly compute an agreed-upon model or function. Formally, \(n\) participants
Figure 1. Reconstruction attack in FL.
\(p_{1},p_{2},\ldots\), and \(p_{n}\) can collaboratively compute \(y=f\left(x_{1},\ldots,x_{n}\right)\), where \(x_{i}\) is a secret input that belongs to participants \(p_{i}\), This form of secure computing offers both correctness and privacy. After all, no participant learns anything about the others' data other than the final result.
Homomorphic encryption (Kumar et al., 2017) allows certain mathematical operations, such as addition and multiplication, to be performed directly on ciphertexts. These can then be used as the basis for more complex arbitrary functions.
#### 2.3.2. Perturbation Approach
With privacy concerns in mind, one needs to be cautious of how much information about a participating client is revealed during training. The perturbation approach arises as a natural way of preventing information leaks. By injecting the proper amount of artificial noise into the original data, the statistical information calculated from the perturbed data will be statistically indistinguishable from the original data.
There are three types of widely used perturbation techniques: differential privacy (DP), additive perturbation, and multiplicative perturbation. DP proposed by Dwork (Dwork, 1998), is the gold standard. The intuition behind DP is to mask the contribution of any individual user by a sufficient level of uncertainty. A randomized mechanism \(\mathcal{M}\) is said to be \((\epsilon,\delta)\)-differentially private if, for any pair of neighboring datasets \(\mathcal{D}\) and \(\mathcal{D}^{\prime}\), and for every set of \(S\subseteq Range\) (\(\mathcal{M}\)), if \(\mathcal{M}\) satisfies:
\[\Pr\left[\mathcal{M}\left(\mathcal{D}\right)\in S\right]\leq\exp\left(\epsilon \right)\cdot\Pr\left[\mathcal{M}\left(\mathcal{D}^{\prime}\right)\in S\right]+\delta \tag{2}\]
The parameter \(\epsilon\) is defined as privacy budget, which measures how alike the two adjacent datasets \(\mathcal{D}\) and \(\mathcal{D}^{\prime}\) are to each other. A smaller \(\epsilon\) indicates a stronger privacy guarantee. If \(\delta=0\), then the randomized mechanism \(\mathcal{M}\) is degraded into \(\epsilon\)- DP.
### Fairness in FL
Algorithms are widely used to assist in making recommendations, assessing loan applications, etc. Several studies have identified the unfairness in different algorithmic scenarios (Kumar et al., 2017). One example is the hate speech detector designed to rate the toxicity score of the given phrases to help companies like Twitter recognize harmful speech. The detector relies on a tool called _Perspective_, which is trained on labeled data. The detector behaves differently towards phrases written in African American English in a racially biased way, as Fig. 2 shows (Kumar et al., 2017).
Figure 2. Racial disparities in classifier predictions on tweets written in African-American English and in Standard American English (reproduced from (Kumar et al., 2017))
#### 2.4.1. Bias in FL
Fairness can be eroded by bias. According to Olteanu et al. [143], bias can slip into data flows from generation to collection to processing [143]. The distributed learning paradigm of FL brings new and unique challenges to our efforts to build fair models. One challenge is that the independent and identical distribution (i.i.d.) assumption no longer holds [89]. In FL, bias can also be introduced by either the client or the server.
* prejudice, underestimation, negative legacies, etc. [93]. This bias is then integrated into the global model through client and server interactions. Second, bias can strike when clients are dropped out of the federated setting due to device shutdowns or communication limitations. In these cases, the global model will find it hard to fit the clients' data properly. The massively distributed data also incurs bias. In this case, the client does not have enough data to capture an underlying distribution, and moreover, the underlying distributions of different clients are probably not the same. [18, 111, 228]
* **Server-introduced bias**. The server can also add bias. As the coordinator of the learning scheme, the server will sample clients in each round to train a global model with their local data [139]. However, the sampling process is prone to producing bias if it is not done with careful consideration. First, in terms of efficiency, only a fraction of clients are selected in each round [129]. Yet the data distribution of only a few selected clients forms an inadequate representation of the actual population distribution. Second, the sampling may be skewed toward certain clients. For instance, to speed up convergence, the server prefers clients that meet specific criteria [29, 142].
#### 2.4.2. Fairness Notions in FL
To date, researchers have proposed several definitions of fairness, see, e.g., [182, 131]. These definitions vary from scenario to scenario, and it is unlikely that there will ever be one particular definition of fairness that fits all circumstances. Table 2 lists common algorithmic fairness notions adopted in centralized machine learning. At a high level, two families of definitions exist the _individual_ notion and _statistical_ notion [31].
* **Individual notion**. The individual notions ensure fairness between specific pairs of individuals: "_Give similar predictions to similar individuals_." [43]. Formally, for a set of samples \(V\), a distance metric is defined as \(d:V\times V\to R\) to measure the similarity. A function \(\mathcal{M}:V\rightarrow\Delta A\) maps the samples \(V\) to the probability distributions over outcomes, and another distance \(D\) metric measures the distance between the distributions of outputs. Fairness is achieved if and only if \(D\left(\mathcal{M}\left(x\right),\mathcal{M}\left(y\right)\right)\leq d\left( x,y\right)\). This family of definitions provides a meaningful guarantee. However, they are at the cost of making significant assumptions, some of which are non-trivial problems in fairness.
* **Statistical notion**. The statistical notions provide fairness assurance at a statistical level. For the protected demographic groups \(G\) (such as racial minorities), some statistical measures are required to be equal across all of these groups. These statistical measures include positive classification rates [94, 20, 43], false positive and false negative rates [73, 103], and positive predictive value [211, 30]. Detailed enumeration can be found in [182, 12]. This family of definitions requires no assumption over data and can be easily verified. However, statistical notions are insufficient as a fairness constraint, which does not give meaningful guarantees to individuals or structured subgroups of the protected demographic groups. Jiang et al. [85] generalized the demographic parity [106] to continuous sensitive attribute.
Apart from algorithmic fairness, which is measured on sensitive attributes, fairness in FL can also be made from a client's view since clients are naturally grouped by attributes like geographic location, gender and income [40]. At a client level, fairness can be evaluated by different metrics.
**Definition 1** (_Good-intent fairness_(Kumar et al., 2017)).: The training procedure does not overfit a model to any device at the expense of other clients in FL.
This metric improves the worst-case performance. Li et al. (Li et al., 2019) took a further step. They tried to ensure a fair FL model for all clients by producing a more uniform model performance across all clients. Fairness is defined as the uniformity of the accuracy distribution across clients in FL.
**Definition 2** (_Accuracy parity_(Li et al., 2019)).: Consider two trained models, \(f(w)\) and \(f(\tilde{w})\). The model that provides the most uniform performance across all clients will also provide the fairest solution to the FL objective in Eq. (1).
### Interactions between Privacy and Fairness
Both fairness and privacy are important ethical notions in machine learning and have been extensively studied. However, the majority of current studies in the research community consider fairness and privacy separately. However, the interactions between privacy and fairness are bilateral.
* **Privacy degrades fairness**. Several works have observed inconsistent reductions in accuracy caused by private mechanisms on classification (Kumar et al., 2017) and generative tasks (Zhu et al., 2019). It turns out that privacy mechanisms affect the underrepresented group more than other groups.
* **Fairness increases privacy risk**. Fairness comes at the cost of privacy. To ensure fairness, a model is trained to perform equally on data from different groups, even though the underrepresented group didn't have enough data in the training set, which incurs overfit and increases the privacy risk (Zhu et al., 2019).
## 3. Privacy in FL
With the advent of FL, many claims that user data are now secure. However, even sharing a small fraction of gradients (Zhu et al., 2019; Li et al., 2019) with a server would raise privacy concerns. In FL, there are several unexplored types of privacy attacks: _membership inference attacks, property inference attacks, model inversion attacks_, and _reconstruction attacks_. This section will outline these attacks before moving on to mitigation techniques and discussions.
\begin{table}
\begin{tabular}{p{113.8pt} p{113.8pt} p{113.8pt}} \hline
**Fairness notion** & **Definition** & **Explanation** \\ \hline Individual Fairness (Kumar et al., 2017) & \(D(M(x),M(y))\leq d(x,y)\) & Similar samples receive similar treatment \\ \hline \multirow{2}{*}{Eqal Opportunity (Kumar et al., 2017)} & \multirow{2}{*}{\(\Pr[\hat{Y}=1|A=0,Y=1]=\Pr[\hat{Y}=1|A=1,Y=1]\)} & Equal true positive rates for protected/unprotected groups \\ & & \\ \hline Equal Accuracy (Kumar et al., 2017) & \multirow{2}{*}{\(\Pr[\hat{Y}=Y|A=0]=\Pr[\hat{Y}=Y|A=1]\)} & Equal prediction accuracy for protected/unprotected groups \\ \hline Equal Odds (Kumar et al., 2017) & \multirow{2}{*}{\(\Pr[\hat{Y}=1|A=1,Y=y]=\Pr[\hat{Y}=1|A=0,Y=y]\)} & Equal positive rates for protected/unprotected groups \\ & & \\ \hline Treatment Equality (Kumar et al., 2017) & \multirow{2}{*}{Equal false negatives and false positives for protected/unprotected groups} \\ & & \\ \hline Demographic Parity (Kumar et al., 2017) & \multirow{2}{*}{\(\Pr[\hat{Y}|A=0]=\Pr[\hat{Y}|A=1]\)} & Outcome is independent of the protected attribute \\ \hline \end{tabular}
\end{table}
Table 2. Definitions of Algorithmic Fairness Notions
### Membership Inference Attacks
The adversary's goal in FL is to determine whether a given sample belongs to a single client's private training data or of any participants (Levy et al., 2017). MIAs take occur in different ways in FL.
#### 3.1.1. White-box and Black-box MIAs
Based on the access granted to the adversary, MIAs can be divided into black-box and white-box (Levy et al., 2017). In the black-box setting, the adversary can only obtain a prediction vector computed by the target model while the internal parameters remain secret. MIAs in this setting exploit the statistical differences between a model's predictions on its training set versus unseen data (Levy et al., 2017). Truex et al. (2017) described a systematic approach to constructing a black-box MIA model and the general formulation of each component in the attack model.
However, since the global model is shared with all participants for local training, it is often assumed that an adversary has white-box access in FL. The white-box access renders much more information to the adversary, such as the internal parameters of each layer. This enables the adversary to calculate the outputs of each layer. Nasr et al. (Levy et al., 2017) designed a deep learning attack model that separately processes the gradients extracted from different layers of the target model and combines this information to compute the membership probability of a target data point.
#### 3.1.2. Training and Inference MIAs
MIAs can be launched during the training stage or once the model is complete in the inference stage in FL.
In the training stage, the adversary could be the server or any client participating in the training. Both characters have white-box access to the global model and can easily save the snapshots of the global model at each iteration during training. In this way, the adversary obtains multiple versions of the target model over time and acquires the updated information to infer private data (Levy et al., 2017). In addition to passively collecting the updated information, the adversary may further modify the information to allure the victim clients to reveal more information.
In the inference phase, the FL model is well-trained and fixed. The adversary can only perform an inference attack passively. In this case, MIA in FL resembles that in a centralized setting. The attack's success largely depends on the information that is revealed to the adversary. Melis et al. (Melis et al., 2017) investigated privacy leaks concerning membership during the inference phase and showed that positions of words in a batch could be revealed from a deep learning model.
#### 3.1.3. Active and Passive MIAs
The adversary can conduct MIAs against the FL model actively or passively. For instance, the server can either adaptively modify the aggregate parameters or honestly calculate the global model and passively conduct MIAs (Meliis et al., 2017).
Melis et al. (Melis et al., 2017) designed MIAs against models operating on non-numerical data (e.g., natural-language text). An embedding layer is equipped for the target model, transforming the inputs into a lower-dimensional vector representation. The adversary passively saves a snapshot of the joint model parameters \(\mathbf{w}_{t}\). The difference between the consecutive snapshots \(\Delta\mathbf{w}_{t}=\mathbf{w}_{t}-\mathbf{w}_{t-1}=\Sigma_{k}\Delta\mathbf{ w}_{t}^{k}\) reveals the aggregated updates from all participants and hence reveals the membership.
Nasr et al. (Nasr et al., 2017) performed active MIAs on FL models by reversing the stochastic gradient descent algorithm and extracting membership information. If the target data point belongs to the training dataset, the attacker's modifications will be nullified since the target model will descend the model's gradient for training samples. However, if the target data sample is not used during the training, the target model will not respond to the attacker's modification. Thus, membership can be deduced. In (Turaev et al., 2017), a malicious client actively mislabels the training sample to fool the victim into releasing private information.
#### 3.1.4. Insider and Outsider MIAs
FL involves two types of actors who can access model information: internal actors (participating clients and the server) and external actors (model consumers and
eavesdroppers). Therefore, FL systems must withstand potential adversaries within and outside the protocol.
The inner adversary could be a client or a server. Clients are picked at random to participate in a training round. When training with hundreds or millions of clients, malicious clients are highly likely involved, who will attempt to deduce the sensitive information of others (Krishnan et al., 2017). The real-time nature of FL added to the inner attacker's strength. For example, Zhang et al. (Zhang et al., 2019) trained a GAN as a malicious client during training to infer the data of other clients in FL. A malicious central server poses a greater threat than a malicious client. Because it can manipulate the global model supplied to victims and obtain more information. Nasr et al. (Nasr et al., 2019) launched MIAs from both the client and server sides and witnessed a higher inference accuracy as a curious central server than as a malicious client.
In addition to internal threats, FL also faces potential attacks from adversaries outside the system. Once the FL training is finished and the model is deployed to users, these users may conduct both black- and white-box attacks depending on their access.
#### 3.1.5. Discussion
The attacks mentioned above demonstrate the vulnerability of FL to privacy attacks, and these privacy risks stem from two assumptions made within the FL protocols: 1) _The server is trustworthy_. FL gives the server access to each participant's updates in the form of gradients or model parameters containing clients' private information. The server can even purposefully send a modified model to steal information. 2) _Clients are honest_. A malicious client can collect several copies of the global model from the rounds it participates in. In this way, inference phase attacks on data privacy are also plausible during the learning phase. Additionally, adversarial clients may influence and shift the bounds of the model during development rather than just abusing the boundaries of a model's service while it is in production.
### Property Inference Attacks
The adversary in property inference attacks attempts to infer the specific property of the subset of the training dataset. The target property may be irrelevant to the classification task (e.g., "wearing glasses" in a gender classification task) and do not characterize the whole class. The attack is made at the population level as opposed to a single sample in MIA. In terms of when the attack is launched, property inference attacks can be classified as _static_ or _dynamic_ attacks. The static attack is applied after the training phase has concluded and the target training set is fixed. The dynamic attack typically occurs during the training phase in FL. In this instance, the training set is changing dynamically.
#### 3.2.1. Static Attacks
The research of property inference attacks dates back to (Ateniese et al., 2017). Ateniese et al. performed a property inference attack against Hidden Markov Models and Support Vector Machine based on a meta-classifier. A set of shadow classifiers were trained on a dataset similar to the target model except for the target property. The meta-classifier is trained with shadow classifiers as the input to find the classifiers trained on the dataset with the target property. Ganju et al. (Ganju et al., 2018) extended this attack to a fully connected neural network case. They shared a similar idea, using the gradient of shadow classifiers to train a meta-classifier. Different from (Ateniese et al., 2017), their research focuses on improving the attack efficiency by taking permutation invariance into account.
#### 3.2.2. Dynamic Attacks
In every communication round of FL, clients are selected at random. This means the training data is dynamically changing, which weakens the property inference attack because the target property appears unpredictable, thereby diminishing the distinguishability of model updates (Wang et al., 2018). Wang et al. (Wang et al., 2018) explored property inference attacks within the FL framework. Inspired by the relationship between the changing of neuron weights in the output
layer and the sample label, the authors proposed three attacks as an eavesdropper to infer the labels' quantity composition proportion. Recently, Wang et al. [188] presented a poisoning-assisted property inference attack in FL from the client's viewpoint, aiming at inferring if and when a sensitive property emerges. The authors built their attacks around the realization that regular model updates reflect the shift in data distribution and, in particular. A binary classifier is trained to make predictions based on these periodic model updates. A property-specific poisoning attack is proposed to distort the decision boundary of the shared model on target attribute data. Thus, model updates have a better discerning ability to infer target property.
#### 3.2.3. Discussion
The MIAs and reconstruction attacks represent two ends of a spectrum of privacy invasion. Property inference attacks locate in the middle and seek to determine if the attackers' target property is present in the training samples. This type of attack is more complex than MIA since the target property doesn't always match the attributes that characterize the classes of the FL model. Nonetheless, a property inference attack poses a greater threat than MIA. Using the real-time nature of FL, the adversary can even infer when the target property appears.
### Model Inversion Attacks
Fredrikson et al. [55] initiated model inversion attacks on tabular data. A subsequent work [54] extended it to the image data. The attack is formulated as an optimization problem to synthesize the input for a given label: \(y:\max_{x}\log T_{y}\left(x\right)\), where \(T_{y}\left(x\right)\) is the probability of the model \(T\) outputs label \(y\) for input \(x\). The access could be black-box [54] or white-box [27, 225].
#### 3.3.1. Black-box Attacks
In the black-box setting, the attacker can only make prediction queries to the model. Fredrikson et al. [54] built attack algorithms following the maximum a posterior principle. Their attack recovered a recognizable image of a person given only API access to a facial recognition system and a specific name of a target person. Yang et al. [203] engineered an inversion model a perform the inversion attacks. The adversary composed an auxiliary set assumed generic enough to retain meaningful information to regularize the ill-posed inversion problem [154]. However, the target model is usually assumed to be simple networks, and the generalization to complex models is not trivial. The inversion problem of a neural network is non-convex, and the optimization suffers minimal local problems, which leads to poor attack performance.
#### 3.3.2. White-box Attacks
n the white-box setting, the attacker has complete knowledge of the model. Zhang et al. [225] sketched a generative model to learn an informative prior from the public dataset. This prior is then used to regulate the inversion problem. Benefiting from this, the authors revealed private training data of DNNs with high fidelity. Chen et al. [27] boosted [225]'s methods. They leveraged the target model to label a public dataset, and a GAN model was trained to distinguish not only real and synthesized samples but also labels. They also modeled the private data distribution to reconstruct representative data points better. The success of model inversion attack benefits from an informative prior.
#### 3.3.3. Discussion
MIAs can be performed with either black-box access or white-box access. When given black-box access, the attack's success heavily relies on the auxiliary dataset, which is assumed to share the same generic features as the private target dataset. Furthermore, the target models in this category are usually simple due to limited access. In the white-box case, the target models extend to DNNs. Most attacks implement GAN to synthesize samples to mimic the private samples regarding the soft labels. This kind of attack is less common compared with reconstruction attacks in FL.
### Reconstruction Attacks
Unlike MIAs, reconstruction attacks attempt to retrieve training data and pose a much more severe threat to privacy. As demonstrated by Aono et al. (Aono et al., 2018), the gradient of the weights is proportional to that of the bias in the first layer of the model, and their ratio approximates the training input. Geiping et al. (Geiping et al., 2018) demonstrated that it is possible to faithfully reconstruct images at high resolution given knowledge of the parameter gradients. Such a privacy break is possible even for deep neural networks. Huang et al. (Huang et al., 2019) evaluated existing reconstruction attacks and defenses. Gupta et al. (Gupta et al., 2019) extended this attack to text data and successfully reconstructed single sentences with high fidelity for large batch sizes. To date, we know of two kinds of reconstruction attacks, namely _optimization-based attacks_ (Opt-based) and _closed-form attacks_.
#### 3.4.1. Optimization-based Attack
Raw data can be reconstructed from gradients by solving an optimization problem. Given a machine learning model \(f(w)\) and the gradient \(g=\frac{1}{b}\sum_{j=1}^{b}\nabla_{w}L_{w}\left(x_{j}^{*},y_{j}^{*}\right)\) computed on a private batch \((x^{*},y^{*})\in\mathbb{R}^{b\times d}\times\mathbb{R}^{b}\) with bath size \(b\). The adversary tries to reconstruct \(x\in\mathbb{R}^{b\times d}\) as an approximation of the true data \(x^{*}\) by solving the following optimization problem:
\[\arg\min_{x}\mathcal{L}_{grad}\left(x;w,g\right)+\alpha\mathcal{R}_{aux}\left( x\right) \tag{3}\]
The first part of Eq. 3 pushes the recovered gradients towards the true gradients \(g\), hence deducing a better approximation. The regularization term \(\mathcal{R}_{aux}\left(x\right)\) is used to incorporate the prior knowledge to further improve reconstruction.
Zhu and Han (Zhu and Han, 2018) proposed _DLG_ attack with \(l_{2}\) distance as the reconstruction loss \(\mathcal{L}_{grad}\). _DLG_ starts with randomly generated dummy samples and then iteratively optimizes the dummy samples and labels until they converge. Finally, _DLG_ achieves pixel-wise recovery accuracy for image classification and token-wise recovery accuracy for a masked language model. Zhao et al. (Zhao et al., 2019) improved Zhu and Han's work (Zhu and Han, 2018) on convergence speed and reconstruction fidelity by leveraging the relationship between the ground-truth labels and the signs of the gradients.
The optimization problem Eq. 3 is often under-determined (Zhu and Han, 2018), as the information in the gradients \(g\) is usually insufficient to recover the training data \(x^{*}\). Even when the gradient size is substantially bigger than the input data dimension. As demonstrated by Zhu and Blaschko (Zhu and Han, 2018), when the learning model is huge, there may be a pair of separate data sharing the same gradient.
In response to this, one may introduce prior knowledge as a regularization term to narrow the search space, making it more consistent with the underlying distribution of training data. Yin et al. (Yin et al., 2019) utilized the local batch norm as the regularization term since adjacent pixels in natural photographs are likely to have comparable values. They achieved precise recovery of the high-resolution images on complex datasets, deep networks, and large batch sizes. Hatamizadeh et al. (Hatamizadeh et al., 2019) extended (Yin et al., 2019)'s approach to vision transformers and discovered that, because of the attention mechanism, vision transformers are substantially more sensitive than previously researched CNNs. In the image domain, total variance is another choice (Zhu and Han, 2018; Zhu and Han, 2018).
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline & \begin{tabular}{c} **Theoretical** \\ **guarantee** \\ \end{tabular} & \begin{tabular}{c} **Convergence** \\ **convergence** \\ \end{tabular} & \begin{tabular}{c} **Running** \\ **time** \\ \end{tabular} & \begin{tabular}{c} **Recovered** \\ **image** \\ \end{tabular} &
\begin{tabular}{c} **Applicability** \\ **in** \\ \end{tabular} \\ \hline Opt-based & No & Local optimal & Slow & With artifacts & No limitation & No \\ \hline Closed-form & Yes & / & Fast & Original & Limited & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 3. Comparison between two reconstruction attack categories
Selecting a proper loss function can also contribute to attack efficiency. By combining mean square error and Wasserstein distance (Beng et al., 2017), Ren et al. (2018) achieved a better reconstruction result than (Zhou et al., 2018; Ren et al., 2019) in terms of batch size and reconstruction fidelity. Geiping et al. (2019) adopted cosine distances to better capture the observation that the angle between two data points quantifies the change in prediction. With this method, they rebuilt a single high-resolution image and a series of low-resolution photos with a maximum batch size of 100. Jeon et al. (2019) systematically investigated ways to best utilize gradients, including using them to extract prior information. Wei et al. (2019) conducted a thorough analysis of how different hyper-parameter setups and settings for attack algorithms influence the effectiveness of the attack and its cost. In an effort to investigate the worst-case attack and evaluate the effectiveness of defense methods for reconstruction attacks, Balunovic et al. (2019) formulated the gradient leakage problem in a Bayesian framework and analyzed the condition for a Bayes optimal adversary.
#### 3.4.2. Closed-form Attack
In an attempt to provide a theoretical understanding of how and when gradients lead to the remarkable recovery of original data, several studies investigated the possibility of recovering the input of a learnable affine function from gradients (Beng et al., 2017; Ren et al., 2019; Ren et al., 2019). Aono et al. (2019) initiated the closed-form attack based on gradients. In certain circumstances, an honest-but-curious server could directly calculate individual data from the gradients uploaded by clients (Beng et al., 2017; Ren et al., 2019).
Consider a fully connected layer \(Wx+b=z\) with \(l=l(f(x),y)\) as the loss function, where \(x\) and \(z\) are the input and output vectors, respectively. Private data \(x\) can be derived from \(l\)'s gradients w.r.t \(W\) and \(b\), i.e.,: \(x^{T}=\frac{\partial l}{\partial W}\oslash\frac{\partial l}{\partial b}\). Here \(\oslash\) denotes entry-wise division. The assumption of a fully connected layer holds for the last prediction layers in many popular architectures. As a result, the prediction modules' input, which is the output of the preceding layers, can be rebuilt. These outputs typically include some information about the training data, making them vulnerable to attackers. In this light, the ability to recover ground truth label information from the gradients of the final fully-connected layer, as stated in Zhao et al. (2018), is very intriguing.
Despite its inspiring nature, Aono et al.'s work (2019) has some limitations. First, it does not apply to convolutional neural networks due to a mismatch in dimensions. Second, it cannot deal with batch inputs. For the batch input \(\left\{x_{j}\right\}_{j=1}^{b}\), all derivatives are summed over the batch dimension \(b\) and the recovered \(\bar{x}\) is merely proportional to the average of batch inputs \(\sum_{j=1}^{b}x_{j}\).
To fix the dimension mismatch issue, Zhu and Blaschko (2019) converted the convolutional layer into a fully connected layer using circulant matrix representation (Zhu and Blaschko, 2019) of the convolutional kernel (Zhu and Blaschko, 2019). The gradients of each layer were interpreted as the _gradient constraints_. Finally, they recursively reconstructed the layer-wise input. However, their implementation can only recover low-resolution images in settings where the batch size equals 1. For the batch input, the algorithm returns a linear combination of the training data.
To address these difficulties with the batch size, Fowl et al. (2019) suggested making minor but malicious changes to the global model to reconstruct the client's data from a batch of gradient updates. The key idea is to separate the batch data by some quantity \(h\), such as image brightness. To this end, an imprint module is added to the global model, which acts as a filter that separates a batch of samples based on quantity \(h\). Qian and Hansen (2019) found bias term in the output layer is the key to the success of reconstruction. A fully-connected neural network requires just one node in one hidden layer for single-input reconstruction. In contrast, mini-batch reconstruction requires that the hidden units exceed the input size. Pan et al. (2019) conducted an analytic investigation of the security boundary of the reconstruction attacks. Given a batch input, the secure/insecure boundary of the reconstruction attack was characterized by the number of Exclusively Activated Neurons (ExANs), where the more ExANs, the more likely the attack's success.
#### 3.4.3. Discussion
The closed-form attacks outperform optimization-based attacks in several aspects. First, the closed-form attacks provide a theoretical guarantee of convergence. In contrast optimization-based attacks suffer from the local optimum problem since a non-convex optimization may not always converge to a correct solution (Shi et al., 2018). Further, optimization-based attacks are sensitive to initialization (Shi et al., 2018), whereas closed-form attacks do not. Second, the deterministic algorithms run by closed-form attacks are faster than optimization-based attacks. Third, closed-form attacks recover the data more accurately, while optimization-based methods, like GradInversion (Zhu et al., 2019), recover data with artifacts. Jin et al. (Jin et al., 2020) made a good comparison between different reconstruction attacks in FL. Table 4 summarizes their finding and includes some additional results from this study.
### Privacy-preserving Techniques
Privacy-preserving machine learning approaches can be roughly classified as cryptographic approaches and perturbation approaches. Cryptographic approaches enable computation over encrypted data and provide rigorous privacy guarantees in the training process. However, they come at a high computational cost compared to the non-encryption alternatives (Jin et al., 2020). This computation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Objective function**} & \multicolumn{1}{c}{**Maximal**} & \multicolumn{1}{c}{**Opt-based/**} & \multicolumn{1}{c}{**Theoretical**} & \multicolumn{1}{c}{**Additional**} \\ \cline{2-7} & \(\mathcal{L}_{grad}\) & \(\mathcal{R}_{aux}\) & **batch size** & **Closed-form** & **guarantee** & **information** \\ \hline iDLG (Shi et al., 2018) & \(l_{2}\) distance & / & 8 & Opt-based & No & No \\ \hline DLG (Shi et al., 2018) & \(l_{2}\) distance & / & 8 & Opt-based & Yes & No \\ \hline Inverting gradients(Shi et al., 2018) & \begin{tabular}{c} Cosine \\ similarity \\ \end{tabular} & Total variance & 100 & Opt-based & Yes &
\begin{tabular}{c} Local updates; \\ BN statistics \\ \end{tabular} \\ \hline
[192] & \(l_{2}\) distance & \begin{tabular}{c} Label-based \\ regularizer \\ \end{tabular} & 8 & Opt-based & Yes & No \\ \hline SAPAG (Jin et al., 2020) & \begin{tabular}{c} Gaussian \\ kernel \\ based function \\ \end{tabular} & / & 8 & Opt-based & No & No \\ \hline R-GAP (Shi et al., 2018) & \begin{tabular}{c} Recursive \\ gradients \\ \end{tabular} & / & 5 & Closed-form & No & No \\ \hline \begin{tabular}{c} Theory- \\ oriented (Shi et al., 2018) \\ \end{tabular} & \(l_{2}\) distance & \begin{tabular}{c} \(l_{1}\) distance of \\ feature map \\ \end{tabular} & 32 & Closed-form & Yes & \begin{tabular}{c} Exclusive \\ activated \\ neurons \\ \end{tabular} \\ \hline \begin{tabular}{c} GradInversion \\ (Zhu et al., 2019) \\ \end{tabular} & \(l_{2}\) distance & \begin{tabular}{c} Group \\ consistency \\ \end{tabular} & 48 & Opt-based & No & BN statistics \\ \hline CAFE (Jin et al., 2020) & \(l_{2}\) distance & Total variance & 100 & Opt-based & Yes & Batch indices \\ \hline GIAS\&GIM (Shi et al., 2018) & \begin{tabular}{c} Negative \\ cosine \\ \end{tabular} & \begin{tabular}{c} \(l_{2}\) distance in \\ latent space \\ \end{tabular} & 4 & Opt-based & No & No \\ \hline Imprint & \begin{tabular}{c} One-shot \\ module (Shi et al., 2018) \\ \end{tabular} & / & 16384 & Closed-form & Yes & CDF \\ \hline GradViT (Jin et al., 2020) & \(l_{2}\) distance & \begin{tabular}{c} Image prior; \\ Auxiliary Regularization \\ \end{tabular} & 64 & Opt-based & No &
\begin{tabular}{c} Auxiliary \\ networks \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 4. Comparison of reconstruction attacks in FL
overhead limits their application in some learning scenarios, particularly in deep neural networks with huge amounts of parameters. As a result, most state-of-the-art privacy-preserving methods are perturbation-based. The perturbation can be accomplished by adding artifact noise into the dataset, such as DP mechanism (Kumar et al., 2017; Zhang et al., 2018). Or by representing the raw dataset with a surrogate dataset (Zhang et al., 2018; Zhang et al., 2018) or abstracting the dataset via sketch techniques (Zhang et al., 2018; Zhang et al., 2018).
#### 3.5.1. Cryptographic Approaches
Secure multi-party computation is a sub-field of cryptography that executes calculations on data dispersed among multiple parties in such a way that the computation results are only revealed to the participants (Zhang et al., 2018). It can take the form of homomorphic encryption (HE) or secret sharing.
As one of the defacto privacy-preserving solutions, homomorphic encryption (HE) provides perfect privacy protection in the face of a malicious server. It allows clients to encrypt their updates in such a way that the server may directly aggregate ciphertexts without divulging anything about the plain text underneath. The downside is that encryption followed by decryption will inevitably impose both a computation and a communications overhead. Phong et al. (Phong et al., 2018) used _additively homomorphic encryption_ to ensure no information was leaked to a malicious server. The encrypted aggregation was formulated as follows:
\[\mathbf{E}(\mathbf{W}_{\text{global}}):=\mathbf{E}(\mathbf{W}_{\text{global}}) +\mathbf{E}(-\alpha\cdot\mathbf{G}_{\text{local}}) \tag{4}\]
where \(\mathbf{E}\) is a homomorphic encryption operator that supports addition over ciphertexts, \(\mathbf{G}_{\text{local}}\) is the aggregated gradient. The decryption key is public to the clients and private to the server, and thus, the client's information is secured. Due to the additively homomorphic property of \(\mathbf{E}\), each client is still able to receive the correct updated model \(W_{global}\) via decryption.
\[\mathbf{E}(\mathbf{W}_{\text{global}})+\mathbf{E}(-\alpha\cdot\mathbf{G}_{ \text{local}})=\mathbf{E}(\mathbf{W}_{\text{global}}-\alpha\cdot\mathbf{G}_{ \text{local}}) \tag{5}\]
However, the amount of data transferred between clients and the server is inflated by two orders of magnitude over the vanilla setting (Zhang et al., 2018). To reduce the communication load, Zhang et al. (Zhang et al., 2018) chose a distributed selective stochastic gradient descent (DSSGD) method in the local training phase to achieve distributed encryption and reduce the computation costs. Zhang et al. (Zhang et al., 2018) presented a BatchCrypt as a simple batch encryption technique. Clients first quantize their local gradients and then encode a batch of quantized updates into a long integer. As a result, the communication overhead is reduced by up to 101 times. Jiang et al. (Jiang et al., 2018) further reduced communication overhead by sending only a sparse subset of local states to the server.
Another drawback is that all participants share the same private key for decryption since homomorphic operations require all values to be encrypted with the same public kewhich degrades privacy protection in the face of malicious clients. To counter this problem, Park and Lim (Park and Lim, 2018) sketched a privacy-preserving FL scheme based on a distributed homomorphic cryptosystem that allows clients to have their own unique private key for the homomorphic encryption scheme.
Secret sharing (Zhang et al., 2018) is another kind of cryptographic technique. It splits a secret data \(\mathcal{D}\) into \(n\) pieces such that the secret \(\mathcal{D}\) can be easily reconstructed with at least \(k\) pieces. However, any set containing less than \(k\) piece reveals no information about \(\mathcal{D}\). It enables the server to aggregate at least a certain number of clients' updates without disclosing any individual client's contribution. Bonawitz et al. (Bonawitz et al., 2018; Zhang et al., 2018) proposed a secure aggregation method for FL based on \(t\)-out-of-\(n\) secret sharing. The key idea is to mask the raw data in a symmetric way. Thus, when aggregated by the server, the introduced noise will be nullified. Liu et al. (Liu et al., 2018) incorporated secure sharing into their federated transfer learning framework to protect privacy. Based on an investigation of how secure aggregation parameters influence communication efficiency, Bonawitz et al. (Bonawitz et al., 2018) used quantization to build a communication-efficient secure aggregation scheme. So et al. (So et al., 2018) designed Turbo-Aggregate that leverages additive secret sharing and Lagrange coding to reduce the secure
aggregation overhead. Shao et al. [161] shared a similar ideal, utilizing Lagrange coding to secretly share private datasets among clients.
Even though FL based on a homomorphic encryption scheme can prevent privacy leaks during training, it remains vulnerable to attacks in the inference stage. The trained model embodies the distribution of training data to a certain extent and the privacy risk still exists. Model inversion attack gives such an example. Given the white-box access to a trained model, Zhang et al. [225] successfully discovered the sensitive features \(x\) associated with a specific label \(y\).
#### 3.5.2. Perturbation Methods
Due to its theoretical guarantee of privacy protection and its low computational and communication complexity, the DP technique [42] has emerged as the most popular choice for privacy protection among a variety of options. In differential privacy, a proper amount of noise is added to the raw data [67], the model [60, 118], the output [23], or the gradients [172] to protect privacy.
Geyer et al. [60] applied a differentially private mechanism to the FL scenario, where they approximated the averaging operations with a randomized mechanism that provided client-level privacy. Truex et al. [178] presented an alternative approach that draws on both DP and secure multi-party computation. The clients and the server communicate through a secure channel. Upon receiving a request from the server, the clients upload their answers following the principles of DP. Xu et al. [195] took a different approach. They approximated the objective function of a regression problem via polynomial representation and then added Laplace noise to the polynomial coefficients to protect privacy. Khalili et al. [98] exploited an exponential mechanism [130] to privately select applicants based on the qualification scores predicted by a pre-trained model.
One concern with perturbation techniques is the trade-off between privacy, accuracy and convergence. A significant noise perfectly protects privacy at the cost of accuracy and convergence. Conversely, a weak noise is futile to privacy attacks [75]. Wei et al. [190] conducted a theoretical analysis of the convergence behavior of FL with DP and identified the trade-off between convergence performance and privacy protection levels.
Many studies have been published on ways to deal with this trade-off [48, 165, 223]. Shokri and Shmatikov [165] suggested randomly selecting and sharing a small fraction of gradient elements (those with large magnitudes) to reduce privacy loss. Fan et al. [48] leveraged element-wise adaptive gradient perturbations to defeat reconstruction attacks and maintain high model accuracy. In a similar manner, Wei and Liu [191] used dynamic privacy parameters. Introducing noise with a greater variance at the beginning of training and progressively decreasing the amount of noise and variance as training progresses.
Huang et al. [79] proposed _InstaHide_, a combination of cryptographic and perturbation approaches to provide rigorous privacy protection at the cost of minor effects on accuracy. _InstaHide_ encrypts the raw image by mixing it with multiple random images from a large public dataset. After that, it randomly flips the signs of the pixels before using it to train the model.
Yang created et al. [201] NISS to avoid the trade-off between accuracy and privacy by permitting clients to collaborate on reducing the total amount of injected noise. In particular, each client's noise is neutralized and distributed to other clients. Theoretically, if all clients are trustworthy, the locally introduced noise can be perfectly offset by the server's aggregation, completely avoiding the privacy accuracy trade-off. A similar idea can be found in Yang et al. [202].
#### 3.5.3. Trusted Execution Environment
Some researchers use Trusted Execution Contexts (TEEs) like Intel SGX and ARM TrustZone to secure ML training in untrusted environments [134, 65, 175]. With hardware and software safeguards, TEEs secure critical code from other programs. Compared with purely cryptography methods, TEEs provide much better performance since it only requires extra operations to create the trusted environment and communicate between trusted and untrusted
components. Gu et al. [65] partitioned DNN models and solely encased the first layers in an SGX-powered TEE to protect input information. Hynes et al. [80] investigated speeding up the training using Graphics Processing Units (GPU). Tramer et al. [175] shared the same concept and offered effective privacy-preserving neural network inference utilizing trusted hardware that delegated matrix multiplication to an untrusted GPU. However, this work does not translate well to FL due to the possible adversary server and limited computation power of the client device. To remedy this, Mo et al. [134] advocated using the TEE of client devices in tandem with model partitioning to defend against MIA. The model is divided into two halves, and the final layers are calculated within TEE. Kato et al. [97] proposed to combine DP with TEE in FL in the presence of an untrusted server. The models are aggregated within the TEE of the server's device. [28]
#### 3.5.4. Discussion
Table 5 summarized and compared the existing defense techniques. Cryptographic approaches preserve privacy to a great extent while suffering from computational complexity and are less feasible. The perturbation approaches trade-off privacy for model performance. Several inspiring works demonstrate that it may be possible to avoid that trade-off through either client collaborations to neutralize locally added noise on the server side or by using a surrogate dataset to protect the raw data without adding noise. The cryptographic approaches only ensure that no information will leak during training. They do not protect privacy during the inference stage. In contrast, the perturbation approaches (e.g., DP) protect privacy in both the training and inference stages. One may combine cryptographic and perturbation approaches to obtain better privacy protection throughout the machine learning pipeline.
### Discussion of privacy attacks and defenses in FL
This section reviews existing privacy attacks and defense approaches in FL. Table 6 summarized the existing privacy attacks in FL. From the attacker's perspective, FL differentiates from the centralized counterparts in sever aspects: 1) _The active attacker in FL_. Due to the collaboration between clients
\begin{table}
\begin{tabular}{|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|} \hline
**Attack** & \begin{tabular}{c} **Defense** \\ **method** \\ \end{tabular} & \multicolumn{1}{p{56.9pt}|}{**Rationale**} & \multicolumn{1}{p{56.9pt}|}{**Advantage**} & \multicolumn{1}{p{56.9pt}|}{**Disadvantage**} \\ \hline \multirow{8}{*}{RA} & HE [5, 86, 163] & Gradients are encrypted & Accurate & \begin{tabular}{c} 1. Vulnerable if there are multiple colluding entities; \\ 2. Ineffective at inference \\ \end{tabular} \\ \cline{2-5} & Secret sharing [15, 137, 161] & Hiding information about clients’ individual update, except for their sum & 1. Accurate; 2. Robust to users dropping out & Ineffective at inference \\ \cline{2-5} & \begin{tabular}{c} Variational \\ bottleneck \\ [159] \\ \end{tabular} & \begin{tabular}{c} Using surrogate gradient \\ to protect privacy. \\ \end{tabular} & \begin{tabular}{c} Keep training process \\ and performance intact \\ \end{tabular} & \begin{tabular}{c} Limit to optimization-based \\ attack \\ \end{tabular} \\ \cline{2-5} & Gradient compression [115, 173, 231] & \begin{tabular}{c} Compressing gradients to \\ prevent reconstruct private \\ data by matching gradients \\ \end{tabular} & \begin{tabular}{c} 1. Easy to implement; \\ 2. Reduce communication \\ \end{tabular} & \begin{tabular}{c} Requires considerable noise, \\ degrades model performance, \\ and increases convergence \\ time \\ \end{tabular} \\ \hline \multirow{4}{*}{\begin{tabular}{c} RA, \\ MA, \\ and PIA \\ \end{tabular} } & DE [26, 125, 134] & \begin{tabular}{c} Hiding private information \\ by injecting noise to the raw data, model, or output \\ \end{tabular} & \begin{tabular}{c} 1. Easy to implement; \\ 2. Long-term protection \\ \end{tabular} & \begin{tabular}{c} Requires considerable noise, \\ degrades model performance, \\ and increases convergence \\ time \\ \end{tabular} \\ \cline{1-1} \cline{2-5} & TEEs [97, 134] &
\begin{tabular}{c} Isolating part of networks \\ from the untrusted environments \\ \end{tabular} & Reduce computation & Limited memory space \\ \hline \end{tabular}
* RA: Reconstruction attack; MIA: Membership inference attack; PIA: Property inference attack.
\end{table}
Table 5. Privacy-preserving methods in FL
and the server, an adversary could actively attack for victim's private data. For example, the attacker may maliciously reverse the gradients [140] or mislabel the training sample [75] to neutralize the benign clients' efforts and fool them into revealing more information about their private data. Hence making them more venerable compared to centralized machine learning. 2) _The real-time nature of FL strengthens the attacker's ability._ During the training process, the adversary could adaptive change their strategy to infer the victim's private data. As a result, the adversary can
\begin{table}
\begin{tabular}{|p{5.6pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|} \hline \hline
* BB: black-box; WB: white-box
\end{table}
Table 6. Comparison of privacy attacks in FL
even infer a specific clients' data (Kamiran and Calders, 2017) when the target features appear in FL (Kamiran and Calders, 2017; Kamiran and Calders, 2017), which is way more severe than centralized setting. 3) _Gradients are shared between clients and server_. Unlike the centralized counterpart, where the adversary could at most access the white-box access to the target model. In FL, gradients are repeatedly shared between clients and the server. Which enables gradient-based privacy attacks. As shown by (Kamiran and Calders, 2017; Kamiran and Calders, 2017), the malicious server can reconstruct clients' training data at the pixel level by minimizing the distance to the target gradients.
From the defender's perspective, protecting privacy in FL is also different from that in the centralized scenario. 1) _Malicious could be the server or any client_. FL allows clients to keep private data local. A central server is designed to orchestrate the training process. Which complicated privacy protection. The adversary could be the server (Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017) or the client (Kamiran and Calders, 2017). The malicious adversary is able to infer the target client's privacy passively or actively. For example, sending the modified global model to the target client to probe private data (Kamiran and Calders, 2017). This brings challenges to defending against potential privacy attacks. DP is a prevailing choice, but it degrades performance. The cryptographic approaches, like HE, and MPC, retain both privacy and performance at the cost of computation overhead, which is a more severe issue in FL since most clients' devices are limited in computation power. 2) _Training and inference stage privacy attacks_. Different from centralized machine learning, where the major privacy leakage happens at the inference stage, i.e., malicious users probe private training data by inferring the target model. In FL, the attacks could happen during or after the training. This requires the defenders to be aware of both possibilities. The cryptographic approaches proved provable privacy protection during the training stage. However, fail at the inference stage since training distribution is embedded in the trained model's parameters. The perturbation approaches, e.g., DP, provides long-term protection and covers both the training and inference stage. One can hide sensitive information from adversaries by adding appropriate noise to the training data.
## 4. Fairness in FL
Fairness, as discussed in the centralized setting, is mainly defined at either the group level (Kamiran and Calders, 2017; Kamiran and Calders, 2017) or the individual level (Kamiran and Calders, 2017). In the FL scenario, fairness has a broader definition. Beyond the long-established _algorithmic fairness_(Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017) arises as a new challenge in FL.
### Algorithmic Fairness
Algorithmic fairness is commonly used to describe the discrepancies in algorithm decisions made across distinct groups as defined by a sensitive attribute. FL often involves a deep neural network with redundant parameters and is pruned to overfit the privileged groups. Various debiasing methods have been devised for different applications, including machine learning (Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017), representation learning (Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017), and natural language processing (Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017). These methods vary in detail but share similar principles. Following the data flow, debiasing methods can be grouped into _pre-processing_, _in-processing_ and _post-processing_ categories, which address the discriminate issues at three distinct stages of the data's handling (Kamiran and Calders, 2017).
#### 4.1.1. Pre-processing
Pre-processing tries to remove the underlying discrimination from the data typically by 1) altering the values of the sensitive attributes/class labels; 2) mapping the training data to a new space where the sensitive attributes and class labels are no longer relevant (Kamiran and Calders, 2017; Kamiran and Calders, 2017; Kamiran and Calders, 2017); or 3) reweighting the samples in the training dataset to compensate for skewed treatment (Kamiran and Calders, 2017).
Intuitively, by training a classifier on discrimination-free data, it is likely that the resulting predictions will be discrimination-free. Inspired by this idea, Kamiran and Calders (Kamiran and Calders, 2017) proposed three types of pre-processing solutions to learn a fair classification, _messaging, reweighing and
sampling_. Feldman et al. (2019) investigated the problem of identifying and removing disparate impacts in the data. Xu et al. (2019) proposed FairGAN, which generates fair data from the original training data and uses the generated data to train the model. Abay et al. (2019) proposed two reweighting methods for the FL setting (Kirshman et al., 2017), _local reweighing_ and _global reweighing with DP_.
Notably, these pre-processing techniques require access to the training data, which violates the privacy principles of FL. As a result, these types of techniques can only be deployed locally on each client. However, in the presence of data heterogeneous among clients, local debiasing cannot provide fair performance for an entire population (Zhu et al., 2019).
#### 4.1.2. In-processing
In-processing modifies traditional learning algorithms to address discrimination (Berk et al., 2011; Krizshman et al., 2017; Krizshman et al., 2018; Krizshman et al., 2019; Krizshman et al., 2019). Such as adding a regularization term to the loss function. Berk et al. (2011), for example, incorporated a family of fairness regularizers into the objective function for regression problems. These regularizers span the range from notions of group fairness to individual fairness. They also create a trade-off between accuracy and fairness.
Another in-processing option is imposing constraints. Zhang et al. (2014) used a GAN to constrain the bias in a model trained on biased data. During training, the scheme simultaneously tries to maximize the accuracy of the predictor while minimizing the ability of the adversary to predict the protected variable. In FL, G'alvez et al. (2019) studied the notion of group fairness as an optimization problem with fairness constraints. Papadaki et al. (2019) formulated a min-max optimization problem to investigate group fairness in scenarios where population data were distributed across clients. Ezzeldin et al. (2019) replaced the aggregation protocol FedAvg with FairFed, which adaptively updates the aggregating weights in each round to improve group fairness. Clients whose local measurements match the global fairness measure are given preferential treatment. Khedr et al. (2019) add a regularizer term to minimize the average loss in fairness across all training data.
#### 4.1.3. Post-processing
Post-processing addresses discrimination issues after the model is trained and doesn't need to change the training process. The general methodology of post-processing algorithms is to take a subset of samples and change their predicted labels to meet a group fairness requirement (Berk et al., 2011; Krizshman et al., 2018; Krizshman et al., 2019; Krizshman et al., 2019; Krizshman et al., 2019).
Hardt et al. (2019) proposed a post-processing technique to construct a non-discriminating predictor \(\tilde{Y}\) from a learned discriminatory binary predictor \(\hat{Y}\). Only access to the prediction \(\hat{Y}\), the protected attribute \(A\) and target label \(Y\) in the data are required, while details of the mapping of features \(X\) to prediction \(\hat{Y}\) are not needed. Canetti et al. (2018) and Pleiss et al. (2019) shared the key characteristics as Hardt et al.'s (2019) work. Lohia et al. (2019) designed a post-processing method to increase both individual and group fairness. Salvador et al. (2019) introduced a conditional calibration method for fair face verification. Their method clusters images into different sets and assigns distinct thresholds to different sets.
#### 4.1.4. Discussion
Three different kinds of debiasing methods are at hand in centralized machine learning. However, solutions in the centralized setting cannot be applied directly in the FL scenario due to limitations with the training data. More specifically, in federated settings, the clients usually have limited amounts of data. Hence, a single client can't accurately represent the true distribution over all clients. Consequently, debiasing data before training is not an option. Another limitation is that direct access to local data is prohibited on the server side. Nevertheless, canny researchers have found inspiration from and workarounds to these issues. Galvez et al. (2019), for example, bypassed this access restriction by using statistics to guide the model's training instead of the raw data.
### Client Fairness
Client fairness in FL is another different fairness notion than algorithmic notions. Ideally, the models produced from FL should capture clients' data distributions and generalize well when deployed on the client side. However, data distribution usually varies among clients. As a result, the global model has inconsistent performance on different clients' dataset. At the client level, a FL protocol is considered to be fair if the performance fluctuates within a limited range, i.e., the variance in the model's performance across clients falls under a predefined threshold. To this end, two lines of research exist to mitigate fairness issues in FL. These are the _single model approach_ and the _personalized models approach._
#### 4.2.1. Single Model Approach
The single model approach trains a single global model for all clients as a standard FL scheme. Here, the focus is on solving any statistical heterogeneity during the training phase rather than smoothing the distribution difference.
* **Data augmentation** is a straightforward solution to statistical heterogeneity. It increases data diversity on the client side. Several researchers have studied ways to enhance the statistical homogeneity of local data in FL (Zhao et al., 2018; Wang et al., 2019; Zhao et al., 2020). Zhao et al. (2020) suggested a data share scheme, which creates a globally-shared dataset that is balanced by class. The experiment shows a 30% improvement on accuracy with only 5% globally shared data. Jeong et al. (2020) proposed _FAug_. Clients first collectively train a GAN model, which is then distributed to clients to augment their local data towards yielding an i.i.d dataset.
* **Client Selection** is another strategy that focuses on sampling data from a homogeneous distribution. Wang et al. (2019) proposed a control framework to actively select the best subset of clients in each training round. In Yang et al.'s (2020) method, the local data distribution is estimated first by comparing local updated gradients and gradients inferred from a balanced proxy dataset. The client selection algorithm based on a combinatorial multi-armed bandit was designed to minimize the effect of class imbalances.
* **Agnostic approach** trains a robust model against a possible unknown testing distribution. Mohri et al. (2019) modeled testing distributions as an unknown mixture of all \(m\) clients' data. The global model is optimized for all possible target distributions. This makes the global model more robust to an unknown testing distribution. Du et al. (2020) introduced a fairness constraint into Mohri et al.'s method (Mohri et al., 2019) and proposed _AgnosticFair_, a fairness-aware FL framework. Their method can provide both _Good-intent fairness_ and _demographic parity_
* **Reweighting** tries to train a fair model by assigning suitable aggregating weights \(p_{k}\) in Eq. 1 to clients. Inspired by \(\alpha\)-fairness notions (Li et al., 2019; Li et al., 2019), Li et al. (2019) sketched _q-Fair FL_ (_q_-FFL) to foster fairer accuracy distribution across all clients by up-weighing clients with lower performance during aggregation. Huang et al. (2020) shared a similar idea where, for each round of aggregation, clients with lower accuracy or less training participant times are assigned higher aggregation weights.
#### 4.2.2. Personalized Models Approach
Instead of smoothing the statistical heterogeneity, in _personalized FL_, multiple distinct models are trained for clients with different data distributions. A global model is first trained collaboratively and then personalized to clients using private data. In this way, clients can benefit from other clients' data and solve the issue of statistical heterogeneity. Mansour et al. (2019) designed and analyzed three approaches to learning personalized models to learn personalized models. Kulkarni et al. (2019) conducted a brief overview of personalized FL. Chen et al. (2020) provided a comprehensive benchmark of various personalized FL methods. Tan et al. (2020) systematically reviewed this topic and classified personalized FL techniques in terms of data-based and model-based approaches. Here, we summarize their conclusions.
* **Multi-task learning** treats building models for each client as different tasks. Smith et al. [168] pioneered this approach and explored personalized FL via a multi-task learning framework. [3, 116] followed this principle. Dinh et al. [38] proposed FedU, which incorporates a Laplacian regularization term into the optimization problem to leverage relationships between clients.
* **Model interpolation** trains local and global models simultaneously, where the global model is used for its generalization ability, and the local model is used to improve local performance. Hanzely and Richtarik [70] formulated an optimization problem that learns a mixture of the global and local models. The local model is trained solely on each client's private data. Softly-enforced similarity from multi-task learning is borrowed to discourage the local model from departing too much from the mean model. Deng et al. [36] and Mansour et al. [126] adopt a similar formulation to determine the optimal interpolation of the local and global models. In Zhang et al.'s [222] work, clients are given access to multiple models uploaded by other clients to evaluate how much they will benefit from these models. An optimal combination is then used as a personal update. Lin et al. [117] investigated the trade-offs between local and global models.
* **Parameter decoupling** learns local parameters as an independent task performed locally. The local model is designed to assist in personalizing the global model to local distributions. Liang et al. [116] devised the local-global federated averaging algorithm, which jointly learns compact local representations for each client and a global model across all devices. Chen and Chao [25] decomposed a FL model as a generic predictor, which is trained globally, along with a personalized predictor that is trained locally. The personalized predictor is formulated as a lightweight, adaptive module on top of the generic predictor.
* **Transfer learning** is a practical training paradigm that leverages knowledge from a source domain to help train a model in a target domain. The performance of transfer learning depends on the similarity between the two domains. Federated transfer learning was first introduced by Liu et al. [119]. Since clients in the same federation usually share the same domain, a FL scheme would make a suitable partner for transfer learning. Li and Wang [110] subsequently proposed FedMD, which combines transfer learning and knowledge distillation. Each client performs transfer learning by training a model to converge on a public dataset and subsequently fine-tune it on local data.
* **Clustering** arranges clients into different groups and trains a specific model for each group. Ghosh et al. [61] iteratively determines the membership of each client to a cluster and optimizes each of the cluster models via gradient descent in a distributed setting. Sattler et al. [158] clusters clients according to the cosine similarity between the clients' gradient updates. This allows clients with a similar distribution to profit from one another while minimizing detrimental interference from others. In Briggs et al.'s [18] method, a clustering step is periodically inserted into the training process to cluster clients based on their local updates. The clusters are then trained individually and in parallel on specialized models. Mansour et al. [126] proposed hypothesis-based clustering, partitioning clients into \(q\) clusters and finding the best hypothesis for each cluster.
* **Regularization** prevents overfitting when training models and has been used in several studies to remedy the weight divergence problem in FL settings. Li et al. [113] introduced a proximal term that considers the differences between global and local models to limit the effect of local updates. Yao et al. [205] considered parameter importance in the regularised local loss function by using elastic weight consolidation [102]. In addition, a regularization term is introduced to penalize the deviation of the local model from the global model.
* **Meta-learning** aims to leverage prior experience with other tasks to facilitate the learning process. The resulting models are highly-adaptable to new heterogeneous tasks [51, 141].
Fallah et al. (Fallah et al., 2017) studied a personalized variant of FedAvg based on model-agnostic meta-learning formulation. The proposed Per-FedAvg algorithm looks for an initial model that performs well after one step of the local gradient update on each client's data. Others have interpreted FedAvg as a meta-learning algorithm, breaking it into two stages of training and fine-tuning to optimize personalized performance and model convergence (Fallah et al., 2017; Fallah et al., 2018).
#### 4.2.3. Discussion
In addition to algorithmic fairness, client fairness is another concern in the FL community. Table 7 enumerated various works on these two topics. Regarding client fairness, the single model approach focuses on smoothing data heterogeneity, where it is easy to implement and can be added to the general FL paradigm since it only needs modest modification. On the downside, the single model approach is less effective than personalized approaches in terms of capturing local data distribution and may be insufficient when the data distributions vary significantly between clients. Additionally, the single-model approach does not allow clients to customize their models.
### Discussion of fairness in FL
There are two definitions of fairness in FL, _client fairness_ and _algorithmic fairness_. _Algorithmic fairness_ has been extensively studied in centralized machine learning. These algorithms presuppose centralized access to data, however, one virtue of FL is data never leaves the device. This means neither the server nor any client gains centralized access to the training data. Therefore, generalizing the fair learning algorithms to FL is not trivial. On the one hand, data is stored locally in FL. The server cannot directly access the local data of clients. Hence, server-side debiasing is not a viable solution. On the other hand, debiasing on the client side is ineffective due to the inadequate data, which can hardly represent the global data distribution (Krishnan et al., 2018). There is no guarantee that model debiased with local data will generalize to the global distribution. The non-i.i.d data distributions further complicated this problem (Krishnan et al., 2018).
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Reference** & **Single** & **Personalized** & **Algorithmic** & **Client** & **Method** \\ & **Model** & **Model** & **Fairness** & **Fairness** & **Fairness** \\ \hline (Fallah et al., 2017; Fallah et al., 2018; Krishnan et al., 2018) & ✓ & & & ✓ & Data Augmentation \\ (Fallah et al., 2018; Fallah et al., 2018; Krishnan et al., 2018) & ✓ & & & ✓ & Client Selection \\ (Fallah et al., 2018) & ✓ & & ✓ & & Agnostic approach \\ (Fallah et al., 2018) & ✓ & & ✓ & ✓ & Agnostic approach \\ (Fallah et al., 2018; Krishnan et al., 2018) & ✓ & & & ✓ & Agnostic approach \\ (Fallah et al., 2018) & ✓ & & ✓ & & Reweight \\ (Fallah et al., 2018; Fallah et al., 2018) & ✓ & & & ✓ & Regularization \\ (Fallah et al., 2018; Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Cluster \\ (Fallah et al., 2018; Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Model interpolation \\ (Fallah et al., 2018; Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Multi-task learning \\ (Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Parameter decoupling \\ (Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Transfer learning \\ (Fallah et al., 2018; Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Regularization \\ (Fallah et al., 2018; Fallah et al., 2018; Fallah et al., 2018) & ✓ & & ✓ & Meta-learning \\ \hline \hline \end{tabular}
\end{table}
Table 7. Summary of Fairness-aware FL
_Client fairness_ is tailored to FL and stems from the non-i.i.d data. Each client sampled the training data from a distinct distribution. In this case, the vanilla FL protocol, _FedAvg_, fails to train a model to fits clients' data distribution. Various methods have been proposed to alleviate this. From the data aspect, (Kal
the non-private alternative at the cost of communication overhead. The trade-offs between privacy and efficiency are the main concern in this category.
However, as argued by [81], the cryptographic approach does not guarantee privacy at the inference stage. It only ensures the training data remain private during training and cannot prevent the adversary from inferring training samples from the neural network parameters [170; 226]. On the contrary, DP guarantees a fair model will not leak anything beyond what could be carried out from "population level" correlations. As such, the majority of works focus on learning fair, and DP model [9; 34; 81; 196]. Thus, this subsection focus on DP as the privacy-preserving techniques.
#### 5.1.1. Empirical Findings
The impact of privacy on fairness was initially observed in empirical studies. Bagdasaryan et al. [9] first observed that the reduction in accuracy caused by deep DP
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & **Privacy** & **Fairness** & \multicolumn{2}{c}{**Techniques to achieve**} & **Trade-off** \\ & **notion** & **notion** & **Privacy** & **Fairness** & **type** \\ \hline
[34] & \(\epsilon\)-DP & \(\alpha\)-Discrimination & Exponential mechanism & Minimize discrimination scores & I \\ \hline
[196] & \(\epsilon\)-DP & Decision boundary fairness & Functional mechanism & Fairness constraints & I \\ \hline
[138] & \(\epsilon\)-DP & \(\alpha\)-Equal opportunity & Local DP & Post-processing & I \\ \hline
[107] & \(\epsilon\)-DP & Equal odds \& Class conditional mechanism & Fairness constraints & I \\ \hline
[37] & \(\epsilon\)-DP \& Decision boundary fairness & Functional mechanism mechanism & Fairness constraints & I \\ \hline
[81] & \((\epsilon,\delta)\)-DP & \(\alpha\)-Equal opportunity & Exponential mechanism & Fairness constraints & / \\ & & & \& Laplace noise & \\ \hline
[122] & \((\epsilon,\delta)\)-DP & Equal odds \& Demographic parity & DP-SGDA & ERMI regularizer & II \\ \hline
[45] & \((\epsilon,\delta)\)-DP & Excessive risk gap & DPSGD-Global-Adapt Gradient correction & II \\ \hline
[176] & \((\alpha,\epsilon_{p})\)- & Equal odds, Accuracy parity & DP-SGD & Fairness constraints & II \\ & Rényi DP & \& Demographic parity & & \\ \hline
[101] & / & Equal accuracy & MPC & Fairness constraints & II \\ \hline
[66] & / & Equal opportunity & Proxy attribute & Post-processing & II \\ \hline
[186] & / & Demographic parity & Noisy attribute & Fairness constraints & II \\ \hline
[8] & / & Equal odds & Noisy attribute & Post-processing & II \\ \hline \hline \end{tabular}
* I: Trade fairness for privacy. Relaxing fairness notions to achieve purely DP
* II: Trade privacy for fairness. Adopting relaxed DP notion to accommodate exact fairness
\end{table}
Table 8. Private and Fair Learning
models negatively impacts underrepresented subgroups disproportionately. DP-SGD strengthens the model's "bias" toward the most prominent features of the distribution that is being learned. Kuppam et al. (2015) reached a similar conclusion when they examined the effects of DP on fairness in three real-world tasks involving sensitive public data. When the noise added by a private algorithm is negligible in relation to the underlying statistics, the costs of adopting a private technique may be minor. When stronger privacy is implemented or when a task entails a small population, significant disparities may emerge. Farrand et al. (2019) demonstrated that even minor differences and weak privacy protections could result in disparate outcomes. Ganev et al. (2018) shifted the emphasis to generative models and tabular synthetic data. Three DP generative models PrivBayes (2018), DP-WGAN (Beng et al., 2019), and PATE-GAN (Petersson et al., 2019), were involved. They witnessed a disparate effect on the accuracy of classifiers trained on synthetic data generated by all generative models. The losses are greater and/or more dispersed for underrepresented groups. Uniyal et al. (2018) compared DP-SGD and Private Aggregation of Teacher Ensembles (PATE) (Farrand et al., 2019), an alternative DP mechanism for discrerely training a deep neural network, in terms of fairness. They discovered that PATE has a disparate effect, but it is considerably less severe than DP-SGD.
#### 5.1.2. Theoretical Explanations
Several works have attempted to determine the mechanism underlying the well-known relationship between privacy and unfairness. Bagdasaryan et al. (2019) associated the impact with the gradient clipping operation in DP-SGD. During training, the model generates larger gradients for samples from underrepresented subgroups; consequently, clipping slows their learning rate. Therefore, the model learns less from the underrepresented subgroups, and its performance on those subgroups is negatively impacted more. Tran et al. (2019) conducted an in-depth study into this phenomenon with output perturbation (Tran et al., 2019) and DP-SGD as the private mechanism. By measuring fairness with _excessive risk gap_, Tran et al. proved that output perturbation mechanisms incur unfairness when the local curvatures of the loss functions of different groups differ substantially. For DP-SGD, Tran et al. found that the clipping bound, the norm of inputs, and the group's distance to the decision boundary collectively contributed to the unfairness raised by DP-SGD. Esipova et al. (2019) examined the same issue from the gradient perspective. They proved that the gradient misalignment caused by DP-SGD is the main reason for unfairness. If the clipping operation disproportionately and sufficiently increases the direction error for group \(a\) relative to group \(b\), then group \(a\) incurs larger excessive risk due to gradient misalignment.
#### 5.1.3. Mitigation Strategies
Diverse methods have been proposed to mitigate the effect of private mechanisms on fairness. Xu et al. (2019) proposed DP-SGD-F, a variant of DP-SGD that reduces the divergent impact on different populations. By adaptively designating clipping bounds for each group, DP-SGD-F achieves a level of privacy proportional to each group's utility-privacy trade-off. For the group whose clipping bias is greater (due to large gradients), a larger clipping bound is adopted to mitigate for their greater privacy cost. Tran et al. (2019) formulated a regularized optimization problem that minimizes empirical loss while satisfying two additional constraints. The first constraint equalizes the averaged non-private and private gradients, while the second constraint penalizes the difference between the local curvatures of distinct groups' loss functions. Esipova et al. (2019) modified DP-SGD and developed DP-SGD-Global-Adapt to preserve gradient direction. It is assumed that a hyperparameter \(Z\) is the upper bound for most gradients. Gradients less than \(Z\) are uniformly scaled, whereas gradients greater than \(Z\) are trimmed to \(Z\).
### Fairness Increases Privacy Risk
Fairness in turn presents challenges for privacy mechanisms. Chang and Shokri (Chang and Shokri, 2018) observed an elevated privacy risk for underprivileged subgroups in a fair model. In order to ensure fairness, the model must perform equally well for all subgroups. However, limited data availability for
underprivileged subgroups can lead to overfitting of the training data for unprivileged subgroups in a fair model, thereby increasing the privacy risk. Previous works on fairness-aware machine learning often assume that the sensitive features are reliable and accessible. This assumption unavoidably introduces privacy risks. However, achieving precise notions of fairness, such as demographic parity, becomes unattainable without access to sensitive attributes, specifically the membership information of sensitive groups. To address this issue, several techniques have been proposed to safeguard the privacy of sensitive attributes during the training of fair models.
#### 5.2.1. Training Fair Models with Noisy Representation
Researchers in this field train approximately fair models using noisy sensitive attributes to protect privacy. Gupta et al. (Gupta et al., 2019) substituted protected groups with proxy groups. To achieve fairness, the proxy groups need to align with the true positive group and even overlap with the ground-truth groups. Thus, the fairness guarantee comes at the cost of privacy. Several studies have explored fairness with imperfect group information. (Lamy et al., 2019; Padala et al., 2020; Padala et al., 2020; Padala et al., 2020). Lamy et al. (Lamy et al., 2020) introduced a mutual contaminated model to simulate a noisy distribution with corrupted attributes. Under this framework, they demonstrated that the fairness constraint on the clean distribution is equivalent to a scaled fairness constraint on the noisy distribution. To protect the privacy of sensitive attributes, they added class conditional noise to release the noisy dataset. Awasthi et al. (Awasthi et al., 2020) addressed the challenging problem of training a fair model with perturbed sensitive attribute values, where each attribute is independently flipped to its complementary value with probability \(\gamma\). They identified conditions on the perturbation under which the classifier, denoted as \(\hat{Y}\), obtained by Hardt et al.'s method (Hardt et al., 2019), is fairer than the vanilla classifier, denoted as \(\tilde{Y}\), trained on accurate attributes. They further provided a formal guarantee of effectiveness under the necessary conditions. Wang et al. (Wang et al., 2020) trained a fair binary classifier based on a noisy label \(\hat{G}\in\{1,...,\hat{m}\}\), i.e., \(\hat{G}\) could be _"country of residence"_ as a noisy representation of the true group labels \(G=\)_"language spoken at home"_.
#### 5.2.2. Training Fair Models with DP
Works in this area protect privacy by adding noise to the private characteristic (Yang et al., 2020). The trade-off between privacy and fairness depends on the amount of noise added, with no noise and excessive noise representing the two extremes. In the case of no noise, the model's performance remains unaffected but could lead to information breaches. Conversely, high levels of noise are effective in preserving privacy but can compromise the model's utility. Tran et al. (Tran et al., 2020) proposed a constrained optimization problem to address both private and fair learning tasks. Their framework ensures \((\alpha,\epsilon_{p})\)-Renyi DP (Jagielski et al., 2020) for the sensitive attributes by solving the constrained problem with DP-SGD. Jagielski et al. (2020) extended Agarwal et al.'s approach (Agarwal et al., 2020) by incorporating privacy considerations. They formulated a two-player zero-sum game, played between a "learner" and an "auditor," to derive a fair classifier. Laplacian noise (Zavak et al., 2019) and the exponential mechanism (Jagielski et al., 2020) were utilized separately for the "learner" and the "auditor". As a result, the learned model satisfies \((\epsilon,\delta)\)-DP and achieves equalized odds.
### Fair and Private FL
In centralized machine learning, one entails centralized access to training data (either the true data or noisy data). However, this is invalid in FL, where neither the server nor clients have access to others' data. Therefore, one cannot simply apply centralized fair learning algorithms in FL tasks. This raises a question: _How can we promote algorithmic fairness in FL without accessing clients' data in FL_? Several studies made progress in response to this challenge.
**Using a surrogate model to preserve privacy**. Padala et al. (Padala et al., 2020) tried to satisfy both \((\epsilon,\delta)\)-local DP and demographic fairness through a fair and private FL framework. To circumvent the access restriction, they decomposed the learning into two phases. First, each client learns a fair and
accurate model on a local dataset, where the fairness constraint acts as a regularization term in the loss function. Then, every client trains a surrogate model to match the fair predictions from the first model with a DP guarantee. Finally, only the surrogate model is communicated to the server.
**Privacy through secure aggregation**. Zhang et al. (Zhang et al., 2016) investigated classification problems in FL through multiple goal optimization problems with privacy constraints. The objective is to minimize the accuracy loss and the discrimination risk. To this end, a team Markov game was designed to select participating clients at each communication round. In each round, clients decide whether or not to participate based on the global model's state, which is characterized by bias level and accuracy. Further, a secure aggregation protocol is designed to estimate the global model's status based on polynomial interpolation (Zhou et al., 2017) for privacy concerns. Under this protocol, the server is able to calculate the discrimination status without accessing the local data.
**Achieve fairness based on statistics**. Galvez et al. (Galvez et al., 2016) formulated a constrained optimization problem that is solved by the differential multiplier. Local statistics are provided to the server for debiasing the global model. To further protect privacy, client updates are clipped and perturbed by Gaussian noise before being sent to the server. Finally, their solution is able to provide the approximate group fairness notion over multiple attributes and \((\epsilon,\delta)\)-DP.
**Fairness through agnostic learning**. Shifts in distribution is one source of bias in FL. The global model is trained on the data of all clients (source distribution), but each client's local data distribution (target distribution) may differ. When deployed to the client, unfavorable outcomes occur. Du et al. (Du et al., 2018) proposed treating the client data distribution in an agnostic way. An adversary generates any possible unknown local data distribution to maximize the loss, while the learner aims to optimize the accuracy and fairness.
**Calculate fairness violations locally**. Chu et al. (Chu et al., 2018) formulated a constraint optimization problem to learn a fair and private model in FL. Each client locally calculates fairness violations to avoid impinging on the data privacy of any client. Chu et al. (Chu et al., 2018) further optimized this method by aggregating fairness constraints to better estimate the true fairness violation for all data.
Although some fair FL algorithms do not directly access the training data (Chu et al., 2018; Du et al., 2018), faithfully sharing the model/gradients in FL could incur privacy leakage risks. The privacy breach could happen during the training or inference stage. The attack could be carried out by either the server or the clients (Zhou et al., 2017). For example, an honest-but-curious server can lunch a reconstruction attack (Zhou et al., 2017) to recover the private data from the gradients uploaded by the victim client. However, the main challenge to training a fair model in FL is restricted data access, e.g., data never leaving local devices, which is an under-investigated topic in FL literature. In the case of the adversary clients/server in FL, some privacy-preserving techniques, such as DP, can be combined with the aforementioned fair FL approaches to prevent privacy leakage.
### Discussion of Privacy and Fairness Interactions
The complex interactions between privacy and fairness have been thoroughly examined and documented in various studies. These investigations highlight the intricate trade-offs and challenges that arise when attempting to simultaneously address both privacy and fairness objectives (Zhou et al., 2017).
The impact of privacy and fairness on each other is indeed bilateral. In one scenario, privacy measures can degrade fairness. For instance, in widely-used privacy mechanisms like DP-SGD, to protect privacy, the algorithm clips and adds noise to the gradients. However, due to the scarcity of data for certain groups, these modifications can disproportionately affect underrepresented groups, exacerbating unfairness. Therefore, the implementation of DP can inadvertently worsen existing unfairness by disproportionately impacting certain groups.
In another case, fairness can increase privacy risks. To achieve fairness, it may be necessary to collect additional demographic information about users, even if it is irrelevant to the task at hand.
This data collection is aimed at guiding modifications to the model, such as addressing inconsistent responses or removing discrimination in statistical models (Kalalal and Triggs, 2011; Kalal and Triggs, 2012; Kalalal and Triggs, 2013). However, the collection of such sensitive information raises privacy concerns, as it expands the scope of data being collected and potentially increases the risk of privacy breaches.
In the context of Federated Learning (FL), the cooperative game between clients and the server adds complexity to the privacy and fairness challenges. FL introduces new privacy attack surfaces, as discussed in Section 3, where potential malicious participants can actively or passively infer the private data of other clients. Consequently, securing private information in FL requires even stronger privacy protection measures compared to the centralized setting. Merely protecting group membership is insufficient to address the privacy risks in FL. Furthermore, the non-i.i.d. (non-independent and identically distributed) nature of FL poses another challenge. In a typical FL system, clients' data are sampled from different distributions, leading to data heterogeneity. A model that achieves fairness within the local distribution of each client is not guaranteed to perform unbiasedly on a global scale. The non-i.i.d. issue also introduces potential fairness concerns at the client level, as the performance of the model can vary significantly among clients. It is crucial to address this variation and ensure fairness across all participating clients in FL. The challenge lies in training a fair model in FL without violating the data access restrictions imposed by each client. Finding methods to mitigate the fairness issues arising from the non-i.i.d. nature of the data while respecting the privacy and data access constraints in FL remains a challenging task.
## 6. Open research directions
The research community has made fruitful progress in privacy and fairness in FL. However, throughout this survey, we found this field still faces several challenges that need to be solved.
* **Trade-offs between Privacy and Fairness**. The interaction between privacy and fairness is an under-studied topic. Existing works have focused on exploring the two notions in isolation, either focused on privacy-preserving machine learning (Kalalal and Triggs, 2011) or on paradigms that respect fairness (Kalalal and Triggs, 2013). However, as demonstrated by several studies (Kalalal and Triggs, 2013; Kalalal and Triggs, 2013), privacy and fairness may compete with each other. In the realm of FL, challenges and opportunities coexist. On the one hand, restricted information and non-i.i.d distribution complicate the problem settings. On the other hand, the flexibility of the FL paradigm may enable more possible solutions. For instance, the personalized model (Kalalal and Triggs, 2011; Kalalal and Triggs, 2013) has been widely used in FL to address statistical challenges by assigning clients personalized models. We may combine privacy and personalized models to achieve a better trade-off between privacy, fairness, and utility. Thus, we believe it is worth examining the trade-offs between privacy and fairness in FL.
* **The Compatibility of Fairness and DP**. We believe it would be worth investigating techniques that simultaneously accommodate fairness and DP. As pointed out in Dwork et al.'s (Dwork et al., 2012) work, given a carefully designed distance metric, it is possible to achieve individual fairness through \(\epsilon\)-DP. Two characteristics of FL make individual fairness a superior choice over group fairness: 1) Data distribution in FL may vary significantly between clients, and individual fairness is more suitable in such cases. Since it is defined at the sample level, thus, it generates better than group notions when addressing new samples which may be distinct from those in the training set; 2) The restricted access to information in FL lends itself more to individual fairness because individual fairness relies on a Lipschitz continual prediction model and does not require access to demographic data. This perfectly fits the FL setting.
* **How can one satisfy fairness at both the algorithm and client levels in FL?** The majority of studies on fairness in FL focus on promoting fairness at the client level. However, client-level fairness does not necessarily imply algorithmic fairness. Consider a scenario where multiple
companies (clients) collaborate to train a credit card approval model. Consumer demographic compositions vary between each company. Although a federated model trained subject to client-level fairness constraints might handle the different companies fairly, the model could still be biased towards sensitive attributes (such as race or educational background). This raises a question: _How can one satisfy fairness at both the algorithm and the client levels while preserving privacy in FL?_.
## 7. Conclusion
In this article, we conducted a detailed survey of data privacy and model fairness issues in FL. Uniquely, we also documented the interactions between privacy and fairness from the perspective of trade-offs. In terms of privacy in FL, we first reviewed privacy attacks in FL. Then, we presented three kinds of privacy-preserving techniques. Regarding fairness, we first analyzed the possible sources of bias and how bias can be introduced on both the client and server sides. Following a review of the notions of fairness adopted in machine learning and those originating from FL, a discussion of the various fairness-aware FL algorithms is presented. The last part of the survey focused on the interactions between privacy and fairness. We identified three relations in the general context and further listed possible solutions to achieve both fair and private FL.
## Acknowledgments
This paper is supported by the Australian Research Council Discovery DP200100946 and DP230100246, and NSF under grants III-1763325, III-1909323,III-2106758, and SaTC-1930941.
| Federated learning (FL) は近年話題となっています。導入以来、研究者はプライバシーの保護や公平な結果の保証を目的としたFLシステムを開発しようと努めてきました。そのほとんどの研究が1つまたはもう1つを対象としており、このアプローチは、プライバシーと公平性の相互作用を比較的に少ない研究対象として扱われています。しかし、プライバシーと公平性は競合しており、それぞれを孤立して考えることは、他方の利益を損なう可能性があるためです。これらの2つの重要なテーマの広い視点を得るため、プライバシーと公平性の問題に関する詳細な文献的調査を行っており、FLのユニークな課題とフェデレーション環境における解決策を強調しています。さらに、プライバシーと公平性の間の様々な相互作用を体系的に調査し、プライバシーと公平性の相互作用がどのように影響し、公平でプライバシーを考慮したFLにおける新たな研究方向を示しました |
2307.03776 | Double-$Q$ spin chirality stripes in the anomalous Hall antiferromagnet
CoNb$_3$S$_6$ | The metallic antiferromagnet CoNb$_3$S$_6$ exhibits a giant anomalous Hall
effect (AHE) that cannot be explained by a collinear N\'eel order on
intercalated Co ions. Thus, a noncoplanar structure is expected. We carried out
resonant elastic x-ray scattering (REXS) to reexamine the magnetic structure of
CoNb$_3$S$_6$ and found a double-$Q$ ($2Q$) order with a $(\frac{1}{2}00)$
commensurate component and a long-wavelength modulation. Circular dichroism and
linear polarization analysis reveal that the commensurate components on the two
Co sites are noncollinear and the modulation is helical. The resulting magnetic
structure has a staggered scalar spin chirality forming a stripe pattern in
real space. Furthermore, we found that the helical modulation wavevector
exhibits a sample dependence and develops a low-symmetry domain structure. We
propose that quenched-in lattice strain controls the helical domain structure,
accounting for much of the sample dependence. These results provide insight
into the mechanism of the AHE in CoNb$_3$S$_6$ and identifies potential routes
for controlling the Hall response and realizing other unconventional electronic
phenomena in metallic antiferromagnets. | Ben Zager, Raymond Fan, Paul Steadman, Kemp Plumb | 2023-07-07T18:00:12 | http://arxiv.org/abs/2307.03776v1 | # Double-\(Q\) spin chirality stripes in the anomalous Hall antiferromagnet CoNb\({}_{3}\)S\({}_{6}\)
###### Abstract
The metallic antiferromagnet CoNb\({}_{3}\)S\({}_{6}\) exhibits a giant anomalous Hall effect (AHE) that cannot be explained by a collinear Neel order on intercalated Co ions. Thus, a noncoplanar structure is expected. We carried out resonant elastic x-ray scattering (REXS) to reexamine the magnetic structure of CoNb\({}_{3}\)S\({}_{6}\) and found a double-\(Q\) (\(2Q\)) order with a \((\frac{1}{2}00)\) commensurate component and a long-wavelength modulation. Circular dichroism and linear polarization analysis reveal that the commensurate components on the two Co sites are noncollinear and the modulation is helical. The resulting magnetic structure has a staggered scalar spin chirality forming a stripe pattern in real space. Furthermore, we found that the helical modulation wavevector exhibits a sample dependence and develops a low-symmetry domain structure. We propose that quenched-in lattice strain controls the helical domain structure, accounting for much of the sample dependence. These results provide insight into the mechanism of the AHE in CoNb\({}_{3}\)S\({}_{6}\) and identifies potential routes for controlling the Hall response and realizing other unconventional electronic phenomena in metallic antiferromagnets.
Materials with complex magnetic phases beyond traditional ferro- and antiferromagnetism exhibit a diverse range of phenomena that are both fundamentally rich and offer many potential applications as next-generation electronic and spintronic devices. Such phases include noncoplanar, chiral, and topological spin textures [1; 2], altermagnetism [3; 4], multiferroics [5], and multipolar magnetism [6]. In these materials, the intricate magnetic symmetries allow for the coupling between charge, magnetic, and lattice degrees of freedom from which effective composite degrees of freedom emerge and give rise to novel macroscopic response.
Transition metal dichalcogenides intercalated with \(3d\) transition metal ions form a class of materials where such complex magnetic phases are stabilized through an interplay between localized spins on the \(3d\) sites and itinerant electrons in the host layers [7; 8]. Diverse phenomena are possible depending on the host compound, intercalation species, and intercalation ratio. Co-intercalated NbS\({}_{2}\), CoNb\({}_{3}\)S\({}_{6}\), is of particular interest because it exhibits a giant anomalous Hall effect (AHE) that cannot be explained by its reported collinear antiferromagnetic structure [9; 10; 11]. A series of neutron diffraction measurements have found the symmetry-related magnetic propagation vectors \((\frac{1}{2}00)\), \((0\frac{1}{2}0)\), and \((\frac{1}{2}\frac{1}{2}0)\), but disagree on the orientation of the moments and the presence of single-\(Q\) domains or multi-\(Q\) order [10; 11; 12; 13; 14]. Elucidating the precise details of the magnetic structure is an essential step towards understanding the origin of the giant AHE in this antiferromagnet, and potentially tuning the properties to realize new functionalities.
In this letter, we reexamine the magnetic structure of CoNb\({}_{3}\)S\({}_{6}\) using Co \(L_{3}\) edge resonant elastic x-ray scattering (REXS). We find a double-\(Q\) (\(2Q\)) magnetic structure with a commensurate \(\mathbf{Q}_{0}=(\frac{1}{2}00)\) component and incommensurate \(\mathbf{Q}_{0}\pm\mathbf{q}\) modulation giving rise to a staggered scalar spin chirality with a modulated stripe or checkerboard pattern. The commensurate component of the structure is noncollinear and the incommensurate component is helical. The data confirms that \((\frac{1}{2}00)\) and \((0\frac{1}{2}0)\) peaks belong to separate \(2Q\) domains. Finally, we found that the modulation varies between samples and shows an asymmetric domain pattern implicating lattice strains to influence the magnetic structure, and likely the anomalous Hall response of CoNb\({}_{3}\)S\({}_{6}\).
Single crystals were grown using chemical vapor transport [9] with the nominal stoichiometry Co:Nb:S=1:3:6. Four different samples from the same growth were measured. All samples undergo abrupt magnetic transitions at 28.6 K, and exhibit sharp (100)-type Bragg peaks, indicating a well-ordered triangular lattice of intercalated
Figure 1: (a) Magnetic REXS intensity in sample 3 at 16 K using circularly polarized x-rays. (b) Experimental geometry, incident (outgoing) polarization channels \(\sigma\) (\(\sigma^{\prime}\)) and \(\pi\) (\(\pi^{\prime}\)) correspond to \(\alpha\) (\(\eta\)) of 0\({}^{\circ}\) and 90\({}^{\circ}\) respectively. (c) Summary of observed magnetic peaks in the first Brillouin zone across all measured samples. Large white circles are \(\mathbf{Q}_{0}\!=\!(\frac{1}{2}00)\) and \((0\frac{1}{2}0)\), green circles show the magnetic reflections observed in samples 1 and 2 and blue squares show those found in sample 3 as in (a). Dashed lines with empty markers show positions of symmetry allowed peaks that were not observed.
Co ions [15; 16]. REXS experiments were performed at the I10 beamline at Diamond Light Source using the RASOR endstation [17] with the experimental geometry shown in Fig. 1(b). Samples were mounted in the \((HK0)\) scattering plane to access (\(\frac{1}{2}00\)) and (\(0\frac{1}{2}0\)) magnetic wavevectors at \(2\theta\!=\!106.2\)deg at the Co \(L_{3}\) edge (778.5 eV). In this geometry the x-ray beam scatters from a (110) facet probing an effective area of \(20\!\times\!200\)\(\mu\)m, and with a penetration depth of 0.3 \(\mu\)m. Thus, our measurements probe a macroscopic sample volume containing many basal plane layers. Data was collected for four different samples using an area detector and full linear polarization analysis (FLPA) was carried out on a fifth sample using a point detector and multilayer polarization analyzer optimized for the Co \(L_{3}\) edge [18]. Measurements were performed for zero-field-cooling (ZFC) and field-cooling (FC) under the application of a 0.1 T permanent magnet along the (001) direction.
Representative reciprocal space maps of the magnetic scattering are shown in Fig. 1(a). Primary magnetic reflections at \(\mathbf{Q}_{0}\!=\!\big{(}\frac{1}{2}00\big{)}\) and (\(0\frac{1}{2}0\)) were observed in all samples, consistent with previous reports [11]. Our fine resolution measurements also revealed a long-wavelength incommensurate modulation of the magnetic structure through satellite magnetic reflections at \(\mathbf{Q}_{0}\!\pm\!\mathbf{q}\) [Fig. 1]. These satellites showed three distinct behaviors between samples as summarized in Fig. 1(b). In samples 1 and 2, satellites appear at \((\frac{1}{2},\pm\delta,0)\) and \((\pm\delta,\frac{1}{2},0)\), in sample 3 one set of satellites appears at \((\pm\delta,\frac{1}{2},0)\) while the other set appears at \((\frac{1}{2}\mp\delta,\pm 2\delta,0)\), i.e. purely transverse to the main peak. No satellite reflections were observed in sample 4 [15]. At \(T\!=\!14\) K, we find \(\delta\!=\!3.0(3)\!\times\!10^{-3}\) r.l.u. \(=3.7(3)\!\times\!10^{-3}\) A\({}^{-1}\), corresponding to a modulation with 170(15) nm wavelength, or 97(10) nm for sample 3 (\(\frac{1}{2}00\)). These results were consistent across multiple zero-field-cooled (ZFC) and field-cooled (FC) cycles.
The different satellite wavevectors that appear between \((\frac{1}{2}00)\) and \((0\frac{1}{2}0)\) reflections in sample 3 [Fig. 1(a)] break both \(C_{6}\) rotational and mirror symmetry about (110) of the triangular lattice, while the satellite reflections observed in samples 1 and 2 possess mirror symmetry but break the \(C_{6}\) rotational symmetry. Such symmetry reductions indicate that \((\frac{1}{2}00)\) and \((0\frac{1}{2}0)\) belong to distinct domains, not a single multi-\(Q\) domain, as will be further confirmed by the linear polarization analysis presented below. In this case, the particular long wavelength modulation of the magnetic structure realized in each domain of a given sample may be selected through a symmetry-breaking field such as small lattice strains that are quenched in during crystal synthesis.
Fig. 2 shows the temperature-dependent integrated intensities at \(\mathbf{Q}_{0}\!=\!(\frac{1}{2}00)\) and \(\mathbf{Q}_{0}\!+\!(\overline{\delta}\ 2\delta\ 0)\) in sample 3. Both sets of peaks have the same critical temperature of \(T_{N}=28\) K and an intensity that varies smoothly with temperature. We also observed a smooth decrease in the magnitude of the satellite wave vector with decreasing temperature, decreasing towards \(\mathbf{Q}_{0}\) as temperature is decreased [Fig. 2], characteristic of a helical magnetic modulation [19]. The spectral lineshape of incident energy scans across the 778.5 eV resonance is typical for Co\({}^{2+}\)[20; 21] and further verifies the magnetic origin of observed peaks [Fig. 2 inset, and Fig. 3 (b) inset].
Further details of the magnetic structure are revealed through the polarization-dependent resonant x-ray scattering. All magnetic reflections exhibit a finite circular dichroism (CD), [Fig. 3], arising at \(\mathbf{Q}_{0}\) from noncollinearity of the commensurate component, and at \(\mathbf{Q}_{0}\pm\mathbf{q}\) from a helical modulation [15]. The CD at \((\frac{1}{2}00)\) shows a variation along the \((1\bar{2}0)\) direction suggesting the presence of opposite chirality domains that may have slightly different values of \(\delta\) or are spatially separated on a length scale comparable to the beam size. We also find that the CD varies between ZFC and 0.1 T FC measurements, especially for the \((\frac{1}{2}00)\) peaks, which is consistent with a redistribution of chiral domains, as we discuss below.
Precise moment directions were determined from full linear polarization analysis (FLPA) by measuring the intensity at \(\mathbf{Q}_{0}\) as a function of incident polarization angle \(\alpha\) at various fixed analyzer angles \(\eta\) [Fig. 1(b)]. The polarization-dependent intensity shown in Fig. 4(a) is directly sensitive to the real space orientations of the Fourier component \(\mathbf{m}_{\mathbf{Q}_{0},n}\) of the magnetic moments.
To model the polarization-dependent scattering intensity in CoNb\({}_{3}\)S\({}_{6}\), we consider a \(2Q\) magnetic structure with propagation vectors \(\mathbf{Q}_{0}\) and \(\mathbf{Q}_{0}\pm\mathbf{q}\) (See SI [15] for alternate possibilities). The real space form of this
Figure 2: Normalized temperature dependence of main and satellite peak, and parameter \(\delta\) in satellite wavevector \((\delta,-2\delta,0)\) for sample 3 measured on the point detector with \(\pi\) polarized x-rays. The inset shows a fixed-\(Q\) energy scan of the main peak, with the dashed line showing total fluorescence yield (TFY).
structure is
\[\begin{split}\mathbf{m}_{n}(\mathbf{r}_{j})&=\cos(\mathbf{q}\cdot \mathbf{r}_{j}+\psi_{n})\cos(\mathbf{Q}_{0}\cdot\mathbf{r}_{j}+\phi_{n})\hat{\mathbf{u}}_{n}\\ &+\sin(\mathbf{Q}_{0}\cdot\mathbf{r}_{j}+\phi_{n})\hat{\mathbf{v}}_{n},\\ &+\cos(\mathbf{q}\cdot\mathbf{r}_{j}+\psi_{n}+\tfrac{\pi}{2}\chi)\cos(\bm {Q}_{0}\cdot\mathbf{r}_{j}+\phi_{n})\hat{\mathbf{w}}_{n},\end{split} \tag{1}\]
where \(n\!=\!1,2\) labels the sublattices at 2 Co sites in the unit cell, \(\chi\!=\!\pm 1\) is the helix chirality, \(\phi_{n}\) and \(\psi_{n}\) are the phases on sublattice \(n\) for \(\mathbf{Q}_{0}\) and \(\mathbf{Q}_{0}\!\pm\!\mathbf{q}\) respectively, and \(\hat{\mathbf{u}}_{n}\), \(\hat{\mathbf{v}}_{n}\), and \(\hat{\mathbf{w}}_{n}\) are unit vectors, which we assume to be orthogonal to maintain a constant moment size. The Fourier components of this structure are
\[\begin{split}\mathbf{m}_{\mathbf{Q}_{0},n}&=\mathbf{m}^{*}_{ -\mathbf{Q}_{0},n}=-\tfrac{i}{2}e^{i\phi_{n}}\hat{\mathbf{v}}_{n}\\ \mathbf{m}_{\mathbf{Q}_{0}\pm\mathbf{q},n}&=\tfrac{\mathbf{n}^{*}}{ 4}e^{i\phi_{n}\pm i\psi_{n}}(\hat{\mathbf{u}}_{n}\pm i\hat{\mathbf{w}}_{n}).\end{split} \tag{2}\]
We parameterize the magnetic structure in terms of the angle \(\mu_{n}\) of \(\hat{\mathbf{u}}_{n}\) from the lattice vector \(\mathbf{a}_{1}\!=\!a\hat{\mathbf{x}}\) within the \(a\)-\(b\) plane, out-of-plane canting angle \(\nu_{n}\) of \(\mathbf{v}_{n}\), and phases \(\phi_{n}\), \(\psi_{n}\). We fit the commensurate component \(\mathbf{m}_{\mathbf{Q}_{0},n}\) to the measured FLPA shown in Fig. 4(a). \(\nu_{1}\!=\!\nu_{2}\) is ruled out as this always leads to zero \(\pi\)-\(\pi^{\prime}\) intensity, inconsistent with the observed FLPA and CD. For our analysis, we have fixed \(\mu_{1}\!=\!\mu_{2}\!=\!\mu\) and \(\nu_{1}\!=\!-\nu_{2}\!=\!\nu\). We find no improvements in the model by relaxing these constraints.
While the phase variables \(\phi_{n}\) and \(\psi_{n}\) cannot be uniquely determined from our measurements, symmetry constrains \(\Delta\phi\!=\!\phi_{2}\)-\(\phi_{1}\) to either \(0^{*}\) or \(180^{*}\)[22]. We separately consider two cases: \(\Delta\phi\!=\!0^{*}\) or \(\Delta\phi\!=\!180^{*}\) and find for both \(\mu\!=\!109(1)^{*}\) at \((\tfrac{1}{2}00)\) and \(\mu\!=\!12(1)^{*}\) at \((0\tfrac{1}{2}0)\), or nearly \(\pm 80^{*}\) from \(\mathbf{Q}_{0}\). The in-plane angle relative to \(\mathbf{Q}_{0}\) is opposite in each domain, with the same broken symmetry as the modulation wavevectors in samples 1 and 2 [15]. For \(\Delta\phi=0^{*}\) we find \(\nu=37(2)^{*}\) at \((\tfrac{1}{2}00)\) and \(\nu\!=\!24(2)^{*}\) at \((0\tfrac{1}{2}0)\). While for \(\Delta\phi\!=\!180^{*}\), we find \(\nu\!=\!14(2)^{*}\) at \((\tfrac{1}{2}00)\) and \(\nu\!=\!9(2)^{*}\) at \((0\tfrac{1}{2}0)\). Both cases adequately describe the data at \((\tfrac{1}{2}00)\) while neither fully matches the intensity at \(\pi\)-\(\sigma\) for \((0\tfrac{1}{2}0)\). We attribute this discrepancy to an experimental artifact likely due to a slight analyzer misalignment. Furthermore, we cannot rule out contributions from domains with different moment orientations because the x-ray intensity measures an ensemble-average for a given \(\mathbf{Q}\). The results of our fit are summarized in Table 1 and Fig. 4(b) shows a real-space representation of the magnetic structure found in sample 3.
Our measurements reveal that CoNb\({}_{3}\)S\({}_{6}\) hosts a non-coplanar magnetic structure [Fig. 4(d)] that may strongly influence the electronic transport properties. In particular, a nonzero scalar spin chirality, \(\chi_{s}\!=\!\mathbf{m}(\mathbf{r}_{i})\cdot[\mathbf{m}(\mathbf{r}_{j})\times\mathbf{m}(\mathbf{r}_{k})]\) for sites \(i\), \(j\), \(k\) on a triangular plaquette generates an effective magnetic field felt by conduction electrons passing over the plaquette: \(\mathbf{b}_{\alpha}\!\propto\!t_{\alpha}\chi_{s,\alpha}\hat{\mathbf{n}}_{\alpha}\), where \(\hat{\mathbf{n}}_{\alpha}\) is the plaquette normal vector and \(t_{\alpha}\!=\!t_{ij}t_{jk}t_{ki}\) is the hopping integral around the plaquette [13]. We compute the total scalar spin chirality for CoNb\({}_{3}\)S\({}_{6}\) using the real space spin structures found above by considering separate contributions from intra-sublattice plaquettes \(\chi_{s}^{\perp}\), and inter-sublattice plaquettes \(\chi_{s}^{\perp}\), involving two sites from one sublattice and one site from the opposite one [Fig. 4(c)].
We find that the uniform net scalar chirality vanishes \(\chi_{s}(Q\!=\!0)\!=\!0\), but the local scalar chirality is finite. \(\chi_{s}^{\parallel}\) forms stripes with propagation vector \(\mathbf{Q}_{0}\) and no variation along \(\mathbf{q}\). The stripes on each sublattice are \(\pm 90^{*}\) out of phase for \(\Delta\phi=0^{*}\) or \(180^{*}\). \(\chi_{s}^{\perp}\) depends on \(\nu\) and forms complex structures that depend on the choice of the phase variables. We apply the constraint \(\Delta\psi\!=\!0^{*}\) or \(180^{*}\) to consider four different combinations of phases that result in stripe or checkerboard like patterns of chirality [15]. In all cases, the magnitudes of \(\chi_{s}^{\parallel}\) and
\begin{table}
\begin{tabular}{c c c c c} \hline \hline & \multicolumn{2}{c}{\(\Delta\phi=0^{*}\)} & \multicolumn{2}{c}{\(\Delta\phi=180^{*}\)} \\ \hline Peak & \(\mu\) & \(\nu\) & \(\mu\) & \(\nu\) \\ \hline \((\tfrac{1}{2}00)\) & \(109(1)^{*}\) & \(37(2)^{*}\) & \(109(1)^{*}\) & \(14(2)^{*}\) \\ \((0\tfrac{1}{2}0)\) & \(12(1)^{*}\) & \(24(2)^{*}\) & \(12(1)^{*}\) & \(9(2)^{*}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters obtained from FLPA describing the commensurate Fourier component, for the two possible choices of \(\Delta\phi\).
vanish as \(q\to 0\). Thus, the incommensurate modulation is essential for providing a finite local scalar spin chirality in CoNb\({}_{3}\)S\({}_{6}\).
Due to the opposite canting between sublattices, \(\chi_{s}^{\perp}\) is much larger than \(\chi_{s}^{\parallel}\) for the values of \(\nu\) we have found. Although the relative magnitude of the intra- and inter-sublattice hopping integrals are unknown, the effective field produced by \(\chi_{s}^{\perp}\) will dominate for all feasible values and the qualitative behavior is unchanged by \(\chi_{s}^{\parallel}\)[15]. In order to visualize \(\chi_{s}\), we project \(\chi_{s}^{\perp}\) onto the \(z\) component of the plaquette normal vectors, shown in Fig. 4(d) and (e) for two different possibilities of the relative phases. In all four cases, the scalar chirality develops an intricate pattern, modulated along both \(\mathbf{Q}_{0}\) and \(\mathbf{q}\)[15]. Such staggered chirality cannot directly account for the AHE. However, the staggered chirality in the presence of finite spin-orbit coupling may generate a net Berry curvature that can account for the AHE [23]. Alternatively, the structure we have found may play a role in the AHE via the crystal chirality or alternating effects [3; 4; 24; 25]. The finite local chirality should also give rise to nonlinear or nonreciprocal transport [26] in samples with suitably controlled domain structures.
The present findings clarify the microscopic origins of the giant Hall signal in CoNb\({}_{3}\)S\({}_{6}\) and highlight the importance of mesoscale (magnetic domains) in this itinerant frustrated magnet. A noncoplanar magnetic structure may be stabilized through competing exchange interactions [27] or through itinerant frustration in a Kondo lattice model [28; 29; 30; 31; 32]. Given that CoS\({}_{6}\) octahedra in CoNb\({}_{3}\)S\({}_{6}\) are disconnected, with distances of 5.75 A between nearest neighbor Co sites, superexchange is not likely between local Co moments and a Kondo lattice picture is more natural. Such a mechanism is also consistent with angle resolved photoemission experiments [33]. In this model, the long wavelength modulation at \(\mathbf{q}\) is due to biquadratic interactions originating from Fermi surface instabilities and giving rise to stripes of scalar chirality similar to our observations [34; 29; 30]. The specific value of \(\mathbf{q}\) depends on precise details of the Fermi surface and chemical potential that are further influenced by intrinsic material factors such as lattice strains or chemical defects, consistent with the sample dependence we have observed. This intrinsic sensitivity of the magnetic modulation also affects the domain structure that will, in turn, influence electronic transport in macroscopic samples.
Indeed, a striking feature of our results is the broken symmetry of the domains and differences between samples with identical thermodynamic characterization and well-ordered Co triangular lattices [15]. We refer to the three possible commensurate wavevectors as \(Q_{0}\)-domains. For a given \(Q_{0}\)-domain, the symmetry-allowed directions of \(\mathbf{q}\) form \(q\)-domains. For a given \(\mathbf{Q}_{0}\), \(\mathbf{q}\) pair, the helix chirality \(\chi=\pm 1\) and canting direction \(\pm\nu\) give four types of \(\chi\)-domains. In all samples, we only observed a single \(q\)-domain for a given \(Q_{0}\), breaking the expected symmetry between each \(Q_{0}\). The \(q\)-domains in sample 3 [blue squares in Fig. 1(c)] break both \(C_{6}\) rotational and mirror symmetry about (110). The \(q\)-domains in samples 1 and 2, [green circles in Fig. 1(c)] break \(C_{6}\) rotational symmetry, while retaining mirror symmetry about the (110). This same symmetry-breaking is exhibited by the in-plane orientation of the commensurate component measured from FLPA. Although we did not measure reciprocal space maps on that sample, it has the same sur
face normal as samples 1 and 2, so we expect it to show the same modulation types. It is thus likely that the in-plane orientation of the commensurate component and the helical modulation wavevector are pinned.
The appearance of distinct \(q\)-domains between two \(Q_{0}\)-domains in a single sample implicates a symmetry breaking strain field. In helimagnets where competing and anisotropic exchange interactions control the modulation, helical wavevectors often align with the strain direction [35; 36; 37; 38; 39]. Similarly, small strains can modify the Fermi surface and break degeneracies between nesting vectors in itinerant magnets [40; 41]. The domain selection in samples 1 and 2 can thus naturally be explained by a residual strain along the (110) surface normal that favors \(\mathbf{q}\) more closely aligned with (110) [42]. However, in sample 3, the modulations at \((\frac{1}{2}00)\) and \((0\frac{1}{2}0)\) are not along symmetry-equivalent directions and these two peaks cannot be described as separate domains of the same structure. In this sample, the surface normal is tilted away from (110), possibly giving rise to a rotated residual strain that may stabilize the transverse modulation in one \(Q_{0}\)-domain. This same rotation in the other \(Q_{0}\)-domain would place the modulation closer to the longitudinal direction (\(q_{\parallel}\)), which may be unstable.
The two types of \(\chi\)-domains we have identified in CoNb\({}_{3}\)S\({}_{6}\) provide a microscopic picture for the chiral _micro_ and _macro_ domains [10], which respectively require field-cooling with 0.1 T and 8 T to align. We observed a finite CD after zero-field cooling that must arise from an unbalanced population of \(\chi\)-domains in the absence of any external symmetry-breaking perturbation, suggesting that magnetic chirality may be coupled to structural chirality [43; 44; 45]. Field cooling under a 0.1 T field alters the CD at both \(\mathbf{Q}_{0}\) and \(\mathbf{Q}_{0}\pm\mathbf{q}\), consistent with chiral microdomains that arise from spin canting [15].
While the AHE in CoNb\({}_{3}\)S\({}_{6}\) has been shown to vary greatly with both Co [46] and S [47] stoichiometry, the central importance of the domain structure is implicit through the observed order of magnitude enhancement of the anomalous Hall conductivity in mesoscale samples [10]. Our measurements further highlight and provide a microscopic picture for these domains, demonstrating that quenched in strains can act to give distinct magnetic phenomena between samples with identical stoichiometry. Given the dependence of the magnetic structure on this strain, the AHE may also show a pronounced in-plane anisotropy based on the particular domain populations preferred by the sample geometry. Future work might take advantage of this and employ strain directly to control the symmetry-breaking and tune new phases [48; 49].
In summary, we have discovered a double-\(Q\) noncoplanar magnetic structure in CoNb\({}_{3}\)S\({}_{6}\) exhibiting a staggered scalar spin chirality \(\chi_{s}\). The result rules out a uniform scalar chirality as the origin of the anomalous Hall effect, but opens up the possibility for other nontrivial transport phenomena in this itinerant antiferromagnet. Theoretical work is needed to understand the mechanism underlying this magnetic structure and its impact on transport properties. On the other hand, the magnetic domain pattern reveals a potential tunablity of the magnetic modulation through lattice strain that opens up new avenues for controlling the coupling between magnetic order and electronic transport in itinerant antiferromagnets.
## Acknowledgements
We are grateful to Cristian Batista for helpful discussions and comments on this manuscript. Work at Brown University was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0021265. This work was carried out with the support of Diamond Light Source, beamline I10 under proposal numbers MM30765 and MM30768. We thank Mark Sussmuth for technical support at I10.
| 金属反ferro磁性材料CoNb$_3$S$_6$は、コリニアなN\'eel秩序による説明ができない巨大な異様性Hall効果（AHE）を示します。したがって、非コラン式構造が期待されます。私たちは共鳴弾性X線散乱（REXS）を行い、CoNb$_3$S$_6$の磁性構造を再検討し、2Q ($2Q$) 秩序を検出しました。これは、（1/2 00） commensurable 分子が含まれており、長波長変動も含まれています。円偏光および線偏光分析は、両方のコサイトの commensurate 分子が非コランニアであることを示し、変動は螺旋状であることを明らかにしています。これにより、磁性構造は、実空間における縞状のStaggered Scalar Spin Chirality を形成しています。さらに、螺旋変動波ベクトルには |
2310.00868 | RT-GAN: Recurrent Temporal GAN for Adding Lightweight Temporal
Consistency to Frame-Based Domain Translation Approaches | While developing new unsupervised domain translation methods for endoscopy
videos, it is typical to start with approaches that initially work for
individual frames without temporal consistency. Once an individual-frame model
has been finalized, additional contiguous frames are added with a modified deep
learning architecture to train a new model for temporal consistency. This
transition to temporally-consistent deep learning models, however, requires
significantly more computational and memory resources for training. In this
paper, we present a lightweight solution with a tunable temporal parameter,
RT-GAN (Recurrent Temporal GAN), for adding temporal consistency to individual
frame-based approaches that reduces training requirements by a factor of 5. We
demonstrate the effectiveness of our approach on two challenging use cases in
colonoscopy: haustral fold segmentation (indicative of missed surface) and
realistic colonoscopy simulator video generation. The datasets, accompanying
code, and pretrained models will be made available at
\url{https://github.com/nadeemlab/CEP}. | Shawn Mathew, Saad Nadeem, Alvin C. Goh, Arie Kaufman | 2023-10-02T03:13:26 | http://arxiv.org/abs/2310.00868v1 | RT-GAN: Recurrent Temporal GAN for Adding Lightweight Temporal Consistency to Frame-Based Domain Translation Approaches
###### Abstract
While developing new unsupervised domain translation methods for endoscopy videos, it is typical to start with approaches that initially work for individual frames without temporal consistency. Once an individual-frame model has been finalized, additional contiguous frames are added with a modified deep learning architecture to train a new model for temporal consistency. This transition to temporally-consistent deep learning models, however, requires significantly more computational and memory resources for training. In this paper, we present a lightweight solution with a tunable temporal parameter, RT-GAN (Recurrent Temporal GAN), for adding temporal consistency to individual frame-based approaches that reduces training requirements by a factor of 5. We demonstrate the effectiveness of our approach on two challenging use cases in colonoscopy; haustral fold segmentation (indicative of missed surface) and realistic colonoscopy simulator video generation. The datasets, accompanying code, and pretrained models will be made available at [https://github.com/nadecmlab/CEP](https://github.com/nadecmlab/CEP).
Colonoscopy, domain translation, generative-adversarial networks, temporal consistency, video-to-video translation.
## I Introduction
The video modality has become more and more commonplace in medical tasks and procedures as cameras provide eyes for the doctors and other instrument guidance systems. Processing these video sequences in realtime can provide new tools to assist doctors during their procedures and patient care. As more procedures and tools bring cameras into the workflow, new interesting and innovative tasks arise. When tackling new tasks that process video sequences it is quite natural to begin by breaking the problem down into frames. For tasks such as segmentation and depth inference, where a single correct answer can be predicted from the image, individually processed frames can be stitched together to produce adequate results. Realistic video generation, on the other hand, requires certain elements such as textures and geometry to be consistent between frames. Frame-by-frame processing in this case, would produce flickery results. These problems require components to provide temporal consistency between frames. Providing these temporal components require additional overhead in both the development time and resources.
Recently, unsupervised domain translation methods have shown promising results across different endoscopy tasks, but not all have been extended to video. The domain translation models that have been extended to video create new models from scratch to accommodate video sequences. Once a frame-based model has been finalized, one can either try simple post-processing normalization across frames to get "quasi-consistency" [18] or train a new model from scratch with full temporal consistency. The first approach is only possible on very specific tasks, such as depth estimation, where there is one correct result. Tasks such as realistic image generation cannot be concatenated together with simple approaches. The second, more general option however requires significantly more computational and memory resources for training.
Moreover, temporally-consistent unsupervised video-to-video domain translation (RecycleGAN [1] derivatives) typically requires learning both directions of translation when only a single direction may be relevant, for example, colonoscopy to depth, colonoscopy to fold segmentation, synthetic to real colonoscopy simulation, etc. This forward and backward learning with temporal components increases the number of learnable parameters by several orders of magnitude. Even still, the general approaches like RecycleGAN may not utilize domain specific knowledge that can vastly improve results. In this work, we present Recurrent Temporal Generative-Adversarial Network (RT-GAN) for adding lightweight temporal-consistency to unsupervised image-to-image domain translation models (that reduces training requirements by a factor of 5). RT-GAN allows traversal between temporal consistency and fidelity to the frame-based models using a single tunable weight parameter while focusing on a single translation direction. Specifically, RT-GAN uses recurrent information by referencing the previous frame and its result as seen in Figure 1c. A temporal discriminator takes the generator's results for 3 consecutive frames to build temporal consistency. A recent approach, RAVE [7], also uses recurrent generators for video domain translation, but needs to be trained from scratch. Frame-based domain translation models can learn useful representations with task-specific components (priors, losses, etc) but these components need to redesigned or dropped altogether when transitioning to a new unsupervised video-to-video domain translation model such as RAVE. In contrast, RT-GAN builds on the representations learned by any
unsupervised image-to-image domain translation model and adds temporal consistency to these without needing to redesign task-specific components. We demonstrate the effectiveness of RT-GAN in adding lightweight temporal consistency to two frame-based models, FoldIt [16] haustrild segmentation model with some inherent "quasi-consistency" across frames and CLTS-GAN color-lighting-texture-specular reflection augmentation model with no consistency at all across frames.
The contributions of this work are as follows:
1. A method to add temporal consistency to established frame-based models
2. An efficient and less resource-hungry method for video domain translation
3. A tunable way to control the trade off between temporal smoothness and faithfulness to the frame-based model
4. Demonstrate the effectiveness of RT-GAN in two challenging use cases: haustrild segmentation (indicative of missed surface) and real colonoscopy video simulation.
## II Related Works
Generative Adversarial Networks (GANs) [8] proposed image generation using adversarial learning. GANs use two networks, one to generate images and one to learn the differences between the dataset and generated images. Many different variations and use cases of GANs were created to solve a number of different tasks including face generation [10, 13, 29], segmentation [5, 12, 23], future frame prediction [11, 24] and more. The adversarial learning approach has also shown to be useful in video generation tasks [27, 30, 35].
Domain translation is the task of translating a sample from one domain to another. Pix2Pix [9] is a network that provides a solution for supervised image-to-image domain translation. Pix2pix used conditional GANs [19] with an L1 loss to tackle but it requires ground truth/paired information. In contrast, CycleGAN [36] presented a method for unsupervised/unpaired domain translation using a cycle consistency loss for forward and backward translation between two given domains. A number of follow-up works have adopted this cycle consistency loss or variations of it for task-specific frame-based image-to-image domain translation models [11, 16, 34]. The task-specific components in these frame-based models are critical for obtaining best results but these need to be redesigned or dropped altogether when transitioning to a completely new video-to-video domain translation model.
Similar to image-to-image-domain translations, there are video-to-video domain translation that require ground truth annotations [31, 32]. Bashkirova et al. performed a number of experiments using a CycleGAN model that uses 3D convolutions with varying input types such as randomly sorted frames, ordered frames, and frames stacked as a 3D tensor. They found that using the stacking frames into 3D tensors provided the best results at the cost of extra training requirements [2]. Bansal et al. proposed RecycleGAN [1], a network for unsupervised video retargeting, that does not require any task-specific modules and adds temporal consistency components (such as optical flow) on CycleGAN to extend it to videos (Figure 0(b)). Specifically, an additional future frame prediction network is added for temporal consistency. This increases memory requirements especially since two predictor networks are needed, one for each domain. OfGAN [33] predicts optical flow using an architecture similar to the one shown in Figure 0(b) to translate synthetic colonoscopy sequences to real colonoscopy video sequences; OfGAN relies on texture, lighting, and specular reflection information to be embedded in the input videos to generate realistic colored output sequences. MocycleGAN [3], similar to OfGAN, estimates optical flow for unsupervised video domain translation for more general tasks. Chu et al. [4] presented UVT that used an additional network to warp frames for temporal consistency, similar to the next frame predictor network. A more recent work, RAVE [7], provides real-time video domain translation using a recurrent
Fig. 1: Depicting how temporal consistency can be added. X is the input video and Y is the resulting video output. (a) Depicts the case where each frame is processed individually by a frame-based model. These results can be jittery without temporal consistency. (b) Shows one way that temporal consistency is added to models such as RecycleGAN [1] and OfGAN [33]. An optical flow module or deep learning frame predictor is used to predict the next frame from the current frame. These additional modules require more resources for training. (c) Illustrates RT-GAN adversarial approach to temporal consistency. RT-GAN references the previous frame and its output to translate the current frame. Three consecutive output frames from generators are passed into a discriminator to provide temporal consistency. The first frame, \(Y_{0}^{\prime}\) is generated from a fully trained frame-based model. The other two frames are created by RT-GAN to be temporally consistent with \(Y_{0}^{\prime}\).
generator. RAVE's generator applies an adversarial loss to learn temporal consistency similar to our work (Figure 0(c)) but needs to be trained from scratch while requiring modifications of domain-specific components for new problems.
## III Dataset
The OC and VC dataset was created from 10 patients that had VC procedures followed by OC procedures. The OC videos were cropped to a size of 256x256 to remove borders in the frames created by the fish-eye lens in the colonoscope. The videos for VC were created from triangulated meshes of the colon extracted from CT scans as described by Nadeem et al. [20]. A virtual camera flies through the mesh with random rotations and lights at both sides of the camera. To better replicate the conditions of the colonoscopy procedure, the inverse square fall-off property is applied to the lights [15]. The videos for both the VC and OC datasets were split into 300 sets of 3 sequential frames. In total, training dataset is composed of 1500 frame triplets, while the validation and testing datasets were composed of 900 and 600 frame triplets respectively. Haustral fold segmentation data is generated in a similar manner to Mathew et al. [16].
## IV Methods
Typically for unsupervised domain translation, at least 2 generator networks are being updated during training time. One generator learns the translation between the input domain and output domain, while the other learns the inverse direction. Typically, applications only require one generator to provide the domain translation results. RT-GAN only trains one generator reducing resources during training (see Table I). RT-GAN builds off of the results from a fully trained frame-based model, \(F\). The results of the frame-based model can be pre-computed, so it does not affect the required resources for training. The RT-GAN's generator \(G\), translates from the input domain \(X\) to the output domain \(Y\). \(G\) takes 3 images as input to produce the output \(y^{\prime}_{t}\). The first input is the frame, \(x_{t}\) that is to be translated. The next input is the previous frame in the input sequence, \(x_{t-1}\), to give the network context and a better understanding of motion. The last input image for \(G\) is \(y^{\prime}_{t-1}\), the result for \(x_{t-1}\). \(y^{\prime}_{t-1}\) gives the generator context on the previous frame with which the output needs to be temporally consistent with. The input for RT-GAN's generator can be seen in Figure 0(c).
\(G\) is trained using two discriminators, each having its own adversarial loss. The adversarial loss described below:
\[\mathcal{L}_{adv}(G,D,y,y^{\prime})=\text{log}(D(y))+\text{log}(1-D(y^{\prime })), \tag{1}\]
where \(y^{\prime}\) is from the generator and \(y\) is from the training data.
The first discriminator, \(D_{t}\), learns temporal consistency. \(D_{t}\) compares a 3 frame sequence from the output domain to a 3 frame sequence created from the generators. The first frame in the triplet is provided by F, while the next 2 temporally consistent frames are provided by \(G\). \(G\) aligns its results with \(F\) in order to provide temporal consistency, but \(F\)'s results is independent of \(G\). The temporal adversarial loss is described as,
\[\mathcal{L}_{t}(G,F,D_{t},Y,X)=\mathcal{L}_{adv}(G,D_{t},\{y_{0},y_{1},y_{2}\},\{F(x_{0}),y^{\prime}_{1},y^{\prime}_{2}\}), \tag{2}\]
where \(y^{\prime}_{1}\) is \(G(x_{0},x_{1},F(x_{1}))\) and \(y^{\prime}_{2}\) is \(G(x_{1},x_{2},y^{\prime}_{1})\).
A separate discriminator, \(D_{f}\), ensures that \(G\)'s results appear similar to \(F\). It compares the paired input and output frames for \(F\) and \(G\). The adversarial loss for \(D_{f}\) is described as:
\[\mathcal{L}_{f}(G,F,D_{f},X)=\mathcal{L}_{adv}(G,D_{f},\{x_{1},F(x_{1})\},\{x _{1},y^{\prime}_{1}\}) \tag{3}\]
A stationary loss, \(\mathcal{L}_{s}\) is included to add extra stability to the model. When the camera is stationary the output should remain the same. This is simulated in the stationary loss and is defined as:
\[\mathcal{L}_{s}(G,X)=\|y^{\prime}_{1}-G(x_{1},x_{1},y^{\prime}_{1})\|_{1}, \tag{4}\]
where \(\|\cdot\|\) represents the \(\ell 1\) norm.
The complete objective function for the network is:
\[\mathcal{L}_{obj}=\lambda\mathcal{L}_{t}(G,F,D_{t},Y,X)+\mathcal{L}_{f}(G,F,D _{f},X)+\mathcal{L}_{S}(G,X) \tag{5}\]
where \(\lambda\) is a tunable weight to determine the tradeoff between the temporal smoothness and fidelity to the frame-based model. \(D_{t}\) is a PatchGAN discriminator with 3D convolutions to help it learn temporal information while \(D_{f}\) is a PatchGAN discriminator with 2D convolutions to learn spatial information. \(G\) uses a Resnet architecture with 9 blocks.
## V Results
The training time and memory usage of RT-GAN is analyzed in Table I. RT-GAN reduces the number of learnable parameters by a factor of 5 while decreasing the training time when compared with RecycleGAN. Compared the TempCycleGAN, a video domain translation model with a minimal amount of image generators, RT-GAN reduces the number of learnable parameters and training time by a factor of 2. FoldIt, a frame-based model for haustral fold segmentation, uses
fewer resources than RecycleGAN as it deals with individual frames. RT-GAN still requires lesser resources than FoldIt because it only learns one direction of translation while FoldIt learns four [16]. CLTS-GAN [17] only learns two directions of translation, so RT-GAN reduces the learnable parameters in half. When training RT-GAN, the hardware requirements are capped by the frame-based model since RT-GAN requires lesser resources. To get an intuition of the computa
Fig. 2: Comparisons of the results for RT-GAN with stitched images from FoldIt and RecycleGAN. The first row shows the optical colonoscopy frames from [14] while the second row shows FoldIt’s results. FoldIt shows a bit of jitteriness, and in the first sequence the deeper parts of the colon gets smoothed out. The third row shows TempCycleGAN, which is unable to properly handle the depth and the deeper parts of the colon. RecycleGAN is displayed in the fourth row. On the right sequences, the model marks deeper parts of the colon as folds due to the lack of task specific modules. The last row shows RT-GAN results that are smoother and maintain the depth of the colon in its output. Full results are found in the **supplementary video**.
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tional/memory training resources for video domain translation approaches in colonoscopy, OFGAN [33] (the colonoscopy simulator model) used 4 GPUs with 16 GB each for training whereas RT-GAN was trained on a single 24 GB GPU.
To test the effectiveness of RT-GAN in fold segmentation context (indicative of the total missed surface during colonoscopy), we added RT-GAN on top of FoldIt hustral fold frame-based model [16]. In Figure 2, we compare RT-GAN, FoldIt, TempCycleGAN, and RecycleGAN results on public video sequences from Ma et al. [14]. RecycleGAN has many variants, however sifting through all the variants and applying task-specific components requires great effort on part of the end users. We chose RecycleGAN for comparisons since it has all the base temporal components seen in the more advanced variants and is not task-specific. We also compare with TempCycleGAN which has shown results on medical tasks. FoldIt and RecycleGAN both had jitterly results. FoldIt occasionally smooths out the deeper parts of the endolumen. In contrast, RecycleGAN translated these deeper endolumen parts as folds since it does not contain any task-specific modules or losses. RT-GAN utilizes the task-specific modules from FoldIt while providing temporal consistency. TempCycleGAN is more consistent, however, similar to RecycleGAN it doesn't have the task specific additions and it fails to accommodate the deeper portions of the endoluminal view. The complete videos sequences for these results can be seen in the **supplementary video**.
For quantitative analysis, a synthetic colon with ground truth annotations is rendered using two different textures, similar to [16]. Table II shows that RT-GAN's additional temporal consistency provided improvement on the IoU and DICE scores for both textures. RT-GAN is also more consistent than the other models despite different textures. Additionally, the optical flow can be compared between in the input sequences and output sequences as done by Rivoir et al. [25]. The mean difference between the input optical flow and output optical flow on our textured colons for RecycleGAN, FoldIt, and RT-GAN are 2.4788, 0.9021, and 0.8479, respectively. This indicates that RT-GAN is better capable of capturing to motion between frames when compared with other models like RecycleGAN and FoldIt. The synthetic colon results can be found in Figure 3. In Figure 4, the \(\lambda\) parameter to control temporal consistency is shown. When \(\lambda\) is set to a lower value, it tries to be more faithful to FoldIt. As \(\lambda\) is increased, RT-GAN makes the annotations smoother so it looks more temporally consistent.
We evaluated RT-GAN on real colonoscopy video generation/simulation using the frame-based CLTS-GAN model [17]. CLTS-GAN creates colonoscopy frames with different colors, lighting, textures, and specular reflections using noise parameters. For real colonoscopy video generation/simulation, RT-GAN was trained for 200 epochs on 1800 frame triplets
Fig. 4: Results for varying temporal weights (\(\lambda\)). The first row is the input and the second row shows \(\lambda=0.2\). As \(\lambda\) decreases RT-GAN’s is more faithful to FoldIt. The next row shows \(\lambda=1\) where there is a balance between temporal and the frame losses. The last row shows \(\lambda=5\). Here the annotation shapes tend to remain consistent between frames. Full videos are found in the **supplementary material**.
of colonoscopy video and 3D renderings of the colon using virtual colonoscopy from [17]. The results of real colonoscopy video generation from synthetic sequences are shown in Figure 5. The top half shows video generation from virtual colonoscopy renderings. CLTS-GAN's use of noise parameters allows it to generate drastically different output across frames. RT-GAN is much smoother and the specular reflections and textures are consistent; in the **supplementary video**, the overall color and lighting changes over time since RT-GAN only looks at the previous frame (and doesn't have a longer-term memory, an issue we will resolve in the future). The bottom portion of Figure 5 compares (RT-GAN + CLTS-GAN) with OfGAN [33]. OfGAN is confined to creating textures and specular reflections that are embedded in its input video. In contrast, CLTS-GAN adds additional texture and specular reflections but lacks temporal consistency. RT-GAN uses CLTS-GAN's texture and specular information and adds temporal consistency on top of it. Full videos of the colonoscopy video generation results are in the **supplementary video**.
## VI Conclusion and Future work
In this paper, we presented a new method, RT-GAN, to add temporal consistency to established frame-based models. This is an efficient and considerably less resource-hungry method for video domain translation. RT-GAN also provides a tunable way to control the trade off between temporal smoothness and faithfulness to the frame-based model. We demonstrate the effectiveness of RT-GAN in two challenging use cases: haustral fold segmentation (indicative of missed surface) and real colonoscopy video simulation.
There are two general failure cases for RT-GAN that can be seen in Figure 6. The first failure case is the lack of long
Fig. 5: Results for RT-GAN trained on CLTS-GAN. The top portion shows results on rendered mesh frames. CLTS-GAN’s results change drastically over time. RT-GAN builds off CLTS-GAN to provide consistent specular and texture between frames. The bottom half shows results using OfGAN’s input video, which embeds texture and specular information. CLTS-GAN adds more intricate specular reflections and textures, and RT-GAN inherits this property. OfGAN relies on the embedded texture and specular to produce its output. Full videos are in the **supplementary video**.
term memory, which is shown in the top portion of Figure 6. The deeper parts of the colon that are annotated in the first few frames disappear in the later frames. This is due to the fact that RT-GAN only receives information from the previous frame. Information about the deeper parts of the colon from earlier frames get lost. A similar effect is seen in the global drift when RT-GAN generates realisitc colonoscopy video sequences. Long term memory or transformer components from models like [7, 22, 26, 28] could mitigate this issue. Additionally, RT-GAN can inherit some of the limitations of its frame based model. In the bottom portion of Figure 6, one of the failure cases for FoldIt is displayed. FoldIt cannot handle frame occlusion and hence both FoldIt and RT-GAN can hallucinate endoluminal view for occluded frames. Going forward, we plan to try RT-GAN on non-colonoscopy datasets such as the AdaptOR Challenge dataset for mitral valve repair simulation (unsupervised video-to-video domain translation task) to show how it can be applied to other tasks [6].
| endoscopyビデオの新しい非 supervised ドメイン転送手法を開発する過程で、初期段階では、個々のフレームに有効なアプローチを開始することが一般的です。個々のフレームモデルが確定したら、追加の連続したフレームは、変更された深層学習アーキテクチャを使用して、時間的整合性を備えた新しいモデルをトレーニングするために、訓練します。しかし、時間的整合性を備えた深層学習モデルへの移行は、トレーニングに必要な計算資源とメモリリソースを大幅に増大させる必要があります。この論文では、個々のフレームベースのアプローチに時間的整合性を付加するための軽量な解決策である、RT-GAN（時系列GAN）を提案します。これは、トレーニングの要件を5倍削減するのに役立ちます。私たちは、コ loscopy の 2 つのチャレンジングなユースケースで、RT-GAN の有効性を示しました。それは、ヘローフォールドセグメンテーション（表面 |
2303.00950 | Open Problem: Optimal Best Arm Identification with Fixed Budget | Best arm identification or pure exploration problems have received much
attention in the COLT community since Bubeck et al. (2009) and Audibert et al.
(2010). For any bandit instance with a unique best arm, its asymptotic
complexity in the so-called fixed-confidence setting has been completely
characterized in Garivier and Kaufmann (2016) and Chernoff (1959), while little
is known about the asymptotic complexity in its "dual" setting called
fixed-budget setting. This note discusses the open problems and conjectures
about the instance-dependent asymptotic complexity in the fixed-budget setting. | Chao Qin | 2023-03-02T04:07:08 | http://arxiv.org/abs/2303.00950v1 | ###### Abstract
###### Abstract
_Best arm identification_ or _pure exploration_ problems have received much attention in the COLT community since Bubeck et al. (2009) and Audibert et al. (2010). For any bandit instance with a unique best arm, its asymptotic complexity in the so-called _fixed-confidence setting_ has been completely characterized in Garivier and Kaufmann (2016) and Chernoff (1959), while little is known about the asymptotic complexity in its "dual" setting called _fixed-budget setting_. This note discusses the open problems and conjectures about the instance-dependent asymptotic complexity in the fixed-budget setting.
multi-armed bandit, best arm identification, pure exploration, asymptotic complexities]Open Problem: Optimal Best Arm Identification with Fixed Budget Chao Qin [email protected] Columbia University
P. Loh and Maxim Raginsky
## 1 Introduction and problem formulation
We consider the so-called best arm identification (BAI) or pure exploration problems where there is a finite number of arms. An experimenter can sequentially select arms to measure and observes independent noisy observations of their quality. The experimenter's goal is to confidently identify a best arm through allocating measurement effort in an adaptive and intelligent manner. BAI problems have also been studied under different names for several decades, e.g., _ranking and selection_ or _ordinal optimization_ in the literature of statistics and operations research. The literature of machine learning mainly studies BAI problems in two settings. One is called _fixed-confidence setting_ where the objective is minimizing the expected number of collected samples while guaranteeing the probability of incorrect decision after the stopping time less than a pre-specified level, and the other is called _fixed-budget setting_ where the objective is minimizing the probability of incorrect decision after a given budget of samples is used up. For any bandit instance with a unique best arm, its asymptotic complexity in the fixed-confidence setting has been fully characterized. See for example, Garivier and Kaufmann (2016) and Chernoff (1959). Although both settings seem "dual" to each other, the instance-dependent asymptotic complexity in the fixed-budget setting is unclear for a very long time. This note briefly include the existing results in the fixed-confidence setting and discusses the open problems and conjectures about the instance-dependent asymptotic complexity in the fixed-budget setting.
We use bold letters to denote vectors. A bandit instance \(\mathbf{\mu}\) consists of \(k\) unknown distributions or arms \(\mathbf{\mu}=(\mu_{1},\ldots,\mu_{k})\) with respective expectations \(\mathbf{\theta}=(\theta_{1},\ldots,\theta_{k})\). For the ease of exposition, we assume the bandit instance \(\mathbf{\mu}\) has a unique best arm. Denote it by \(I^{*}(\mathbf{\mu})\triangleq\arg\max_{i\in[k]}\theta_{i}\) where \([k]\triangleq\{1,\ldots,k\}\). The bandit instance \(\mathbf{\mu}\) is _unknown_ to an experimenter who wants to confidently identify the best arm \(I^{*}(\mathbf{\mu})\) at the end of the experiment. At each time \(t=1,2,\ldots\), according to the information collected so far, she can choose an arm \(I_{t}\in[k]\) to measure and then observes an independent noisy observation \(Y_{t,I_{t}}\) drawn from distribution \(\mu_{I_{t}}\).
## 2 Fixed-confidence setting and its known results
In the fixed-confidence setting, the experimenter can stop gathering samples at any time and returns an estimate of the identity of the best arm after that. The experimenter's algorithm is then composed of three rules: a sampling rule that determines which arm to sample at each time, a stopping rule that decides whether to stop at each time, and a decision rule that at the stopping time \(\tau\), returns an estimate \(\hat{I}_{\tau}\) of the identity of the best arm based on the first \(\tau\) observations.
Let \(\mathcal{S}\) be the class of bandit instances with a unique best arm. Garivier and Kaufmann (2016) studies algorithms that guarantee a _uniformly_ small probability of incorrect decision (at the stopping time) below a pre-specified level \(\delta>0\), in the sense that
\[\forall\boldsymbol{\mu}\in\mathcal{S},\quad\mathbb{P}_{\boldsymbol{\mu}} \left(\hat{I}_{\tau_{\delta}}\neq I^{*}(\boldsymbol{\mu})\right)\leq\delta \tag{1}\]
where \(\tau_{\delta}\) is an _almost surely finite_ stopping time. The notation \(\mathbb{P}_{\boldsymbol{\mu}}(\cdot)\) indicates that we are evaluating the probability of events when the observations from chosen arms are drawn under the bandit instance \(\boldsymbol{\mu}\). In the learning theory literature, such algorithms are called \(\delta\)_-Probably-Approximately-Correct_ or \(\delta\)_-PAC_. Among such algorithms, we would like to minimize the expected number of collected samples denoted by \(\mathbb{E}_{\boldsymbol{\mu}}[\tau_{\delta}]\). Garivier and Kaufmann (2016) shows that for any \(\delta\)-PAC algorithm,
\[\forall\boldsymbol{\mu}\in\mathcal{S},\quad\liminf_{\delta\to 0}\frac{ \mathbb{E}_{\boldsymbol{\mu}}[\tau_{\delta}]}{\log(1/\delta)}\geq\Gamma_{\rm fc }^{*}(\boldsymbol{\mu}) \tag{2}\]
where
\[\Gamma_{\rm fc}^{*}(\boldsymbol{\mu})=\left(\sup_{\boldsymbol{w}\in\Sigma_{ \boldsymbol{\mu}}}\inf_{\boldsymbol{\nu}\in\mathrm{Alt}(\boldsymbol{\mu})} \sum_{i=1}^{k}w_{i}\mathrm{KL}(\mu_{i}\|\nu_{i})\right)^{-1}. \tag{3}\]
Here \(\Sigma_{k}\) is the probability simplex of dimension \(k-1\); \(\mathrm{Alt}(\boldsymbol{\mu})\triangleq\{\boldsymbol{\nu}\in\mathcal{S}:I^{* }(\boldsymbol{\nu})\neq I^{*}(\boldsymbol{\mu})\}\) is the set of bandit instances whose unique best arm is different from \(\boldsymbol{\mu}\)'s unique best arm; \(\mathrm{KL}(p\|q)\) denotes the Kullback-Leibler (KL) divergence between distributions \(p\) and \(q\). The subscript \(\mathrm{fc}\) in \(\Gamma_{\rm fc}^{*}\) is the acronym of "fixed-confidence". Besides the information-theoretic lower bound in Equation (2), Garivier and Kaufmann (2016) also proposes the so-called Track-and-Stop algorithms that are \(\delta\)-PAC and can guarantee
\[\forall\boldsymbol{\mu}\in\mathcal{S},\quad\limsup_{\delta\to 0}\frac{ \mathbb{E}_{\boldsymbol{\mu}}[\tau_{\delta}]}{\log(1/\delta)}\leq\Gamma_{\rm fc }^{*}(\boldsymbol{\mu}). \tag{4}\]
Since the lower and upper bounds in Equations (2) and (4) are the same, the function \(\Gamma_{\rm fc}^{*}:\mathcal{S}\rightarrow\mathbb{R}\) characterizes the asymptotic complexity in the fixed-confidence setting.
## 3 Fixed-budget setting and its open problems
In the fixed-budget setting, a budget of \(n\) samples is fixed and given. After collecting \(n\) samples, the experimenter needs to decide an estimate of the identity of the best arm denoted by \(\hat{I}_{n}\). An algorithm is then only consists of a sampling rule and a decision rule. The experimenter's objective in the fixed-budget setting is to minimize the probability of incorrect decision defined as
\[p_{\boldsymbol{\mu},n}\triangleq\mathbb{P}_{\boldsymbol{\mu}}\left(\hat{I}_{n }\neq I^{*}(\boldsymbol{\mu})\right).\]
This setting seems "dual" to the fixed-confidence setting in the sense that instead of minimizing the number of samples subject to a uniformly small probability of incorrect decision, here we minimize the probability of incorrect decision subject to a fixed budget of samples. However, little is known about the asymptotic complexity in the fixed-budget setting.
Open problem 1.The first and foremost open problem is whether there are a desirable algorithm class \(\mathcal{A}\) and a well-defined function \(\Gamma^{*}_{\text{fb}}:\mathcal{S}\rightarrow\mathbb{R}\) such that for any algorithm in \(\mathcal{A}\),
\[\forall\boldsymbol{\mu}\in\mathcal{S},\quad\liminf_{n\rightarrow\infty}\frac{ n}{\log(1/p_{\boldsymbol{\mu},n})}\geq\Gamma^{*}_{\text{fb}}(\boldsymbol{\mu})\]
and there is an algorithm that belongs to \(\mathcal{A}\) and guarantees
\[\forall\boldsymbol{\mu}\in\mathcal{S},\quad\limsup_{n\rightarrow\infty}\frac {n}{\log(1/p_{\boldsymbol{\mu},n})}\leq\Gamma^{*}_{\text{fb}}(\boldsymbol{ \mu}).\]
Here the subscript \(\text{fb}\) in \(\Gamma^{*}_{\text{fb}}\) is the acronym of "fixed-budget".
Discussion on potential algorithm class.Kaufmann et al. (2016) studies the so-called _consistent_ algorithms such that for any \(\boldsymbol{\mu}\in\mathcal{S}\), the probability of incorrect decision \(p_{\boldsymbol{\mu},n}\) goes to zero when \(n\) increases to infinity. The class of consistent algorithms is relatively large, and we believe it might not be the right algorithm class for characterizing the asymptotic complexity in the fixed-budget setting. Note that in the fixed-confidence setting, the class of \(\delta\)-PAC algorithms defined in Equation (1) is restrictive in the sense that it requires a uniformly small probability of incorrect decision (at the stopping time) for any bandit instance \(\boldsymbol{\mu}\in\mathcal{S}\). This restriction helps the analysis of the asymptotic complexity in the fixed-confidence setting. We believe it is necessary to come up with a natural but more restrictive algorithm class in the fixed-budget setting. For example, besides the convergence of the probability of incorrect decision to zero, we may also need to control the convergence rate of the algorithms in the class. One potential algorithm class contains all the algorithms that perform uniformly no worse than uniform sampling, i.e., for any algorithm in this class, it achieves a lower or the same value of \(\limsup_{n\rightarrow\infty}\frac{n}{\log(1/p_{\boldsymbol{\mu},n})}\) for any bandit instance \(\boldsymbol{\mu}\in\mathcal{S}\). This leads to the following open problem.
Open problem 2.This open problem is whether there is an algorithm other than uniform sampling itself that performs uniformly no worse than uniform sampling in the fixed-budget setting.
Indeed in the fixed-confidence setting, one can show that for any bandit instance \(\boldsymbol{\mu}\), \(\Gamma^{*}_{\text{fc}}(\boldsymbol{\mu})\) is less than or equal to the value of \(\limsup_{n\rightarrow\infty}\frac{\mathbb{E}_{\boldsymbol{\mu}}[\tau_{ \text{g}}]}{\log(1/\delta)}\) under uniform sampling. This implies those asymptotically optimal algorithms in the fixed-confidence setting perform uniformly no worse than uniform sampling. We tend to believe that those algorithms also have advantages over uniform sampling in the fixed-budget setting, but the answer to this open problem is unclear.
## 4 Conjectures
In this section, we state two existing conjectures in the literature. Unfortunately, neither of them is correct in general.
Conjecture 1.Since the fixed-budget and fixed-confidence settings are "dual" to each other, one conjecture is that \(\Gamma^{*}_{\text{fb}}=\Gamma^{*}_{\text{fc}}\).
Conjecture 2.Another conjecture is that \(\Gamma^{*}_{\mathrm{fb}}=\Gamma^{*}_{\mathrm{na}}\) where \(\Gamma^{*}_{\mathrm{na}}\) defined later is the asymptotic complexity in a _non-adaptive_ version of the fixed-budget setting studied in Glynn and Juneja (2004) (and the subscript \(\mathrm{na}\) in \(\Gamma^{*}_{\mathrm{na}}\) is the acronym of "non-adaptive"). They consider sampling rules that fix the probability vector \(\mathbf{w}\) of selecting \(k\) arms in each time and thus do not adapt to the observations from sequentially selected arms. They show that for any bandit instance \(\mathbf{\mu}\in\mathcal{S}\),
\[\forall\mathbf{w}\in\Sigma_{k},\quad\liminf_{n\to\infty}\frac{n}{\log(1/p_{\mathbf{ \mu},n})}\geq\Gamma^{*}_{\mathrm{na}}(\mathbf{\mu})\]
and
\[\exists\mathbf{w}^{*}(\mathbf{\mu})\in\Sigma_{k},\quad\limsup_{n\to\infty}\frac{n}{ \log(1/p_{\mathbf{\mu},n})}\leq\Gamma^{*}_{\mathrm{na}}(\mathbf{\mu})\]
where
\[\Gamma^{*}_{\mathrm{na}}(\mathbf{\mu})=\left(\sup_{\mathbf{w}\in\Sigma_{k}}\inf_{\mathbf{ w}\in\operatorname{Alt}(\mathbf{\mu})}\sum_{i=1}^{k}w_{i}\mathrm{KL}(\nu_{i}\|\mu_{i}) \right)^{-1}. \tag{5}\]
At the first glance, the complexity term \(\Gamma^{*}_{\mathrm{na}}(\mathbf{\mu})\) in Equation (5) looks the same as \(\Gamma^{*}_{\mathrm{fc}}(\mathbf{\mu})\) in Equation (3). Indeed they are different since KL divergence is not symmetrical in general, but for Gaussian distributions, \(\Gamma^{*}_{\mathrm{na}}(\mathbf{\mu})=\Gamma^{*}_{\mathrm{fc}}(\mathbf{\mu})\). Note that the optimal sampling vector \(\mathbf{w}^{*}(\mathbf{\mu})\) depends on the knowledge of unknown bandit instance \(\mathbf{\mu}\), so it is unknown a priori. Hence, the sampling rule that always fixes the optimal sampling vector \(\mathbf{w}^{*}(\mathbf{\mu})\) for each bandit instance \(\mathbf{\mu}\) is not a valid choice for the adaptive fixed-budget setting of our interest.
Neither conjecture is correct.The results in Ariu et al. (2021) imply neither conjecture is correct in general for Bernoulli bandits. Inspired by the construction in Carpentier and Locatelli (2016), Ariu et al. (2021) constructs a set of bandit instances with large number of arms and shows that neither conjecture can hold for all the instances. We believe that one can also show similar negative results for Gaussian bandits.
## 5 Known results for two-armed bandits
Though neither conjecture is correct in general, Kaufmann et al. (2016) shows that both conjectures hold for two-armed Gaussian bandits with known variances, i.e., \(\Gamma^{*}_{\mathrm{fb}}(\mathbf{\mu})=\Gamma^{*}_{\mathrm{fc}}(\mathbf{\mu})=\Gamma^ {*}_{\mathrm{na}}(\mathbf{\mu})\) for any such bandit instance \(\mathbf{\mu}\). It further proves that the optimal sampling rule is non-adaptive, which fixes the sampling vector \((\frac{\sigma_{1}}{\sigma_{1}+\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}+ \sigma_{2}})\) where \(\sigma_{1}\) and \(\sigma_{2}\) are the known variances of the two arms. Recently, Kato et al. (2022) shows that when the gap between the unknown means of the two arms goes to zero, even the variances are also unknown, the upper bound of the proposed algorithm matches the instance-dependent lower bound in Kaufmann et al. (2016). Adusumilli (2022) studies the diffusion regime of two-armed Gaussian bandits and proves that the same sampling vector \((\frac{\sigma_{1}}{\sigma_{1}+\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}+ \sigma_{2}})\) is also minimax optimal. However, for two-armed Bernoulli bandits, Kaufmann et al. (2016) shows that though the optimal sampling vector exists, it requires the knowledge of unknown means of the arms, which is unknown a priori. It is unclear whether there is an algorithm can achieve the asymptotic optimality without such a requirement.
## Acknowledgments
We thank Kaito Ariu, Remy Degenne, Sandeep Juneja, Masahiro Kato, Junpei Komiyama, Wouter M. Koolen, Pierre Menard, Daniel Russo and Assaf Zeevi for fruitful discussions. | Bestの腕の識別または純粋な探索問題がCOLTコミュニティで、Bubeck et al. (2009)とAudibert et al. (2010)以降、注目を集めています。 すべての banditインスタンスに、唯一の最適な腕がある場合は、その有望な安定化設定における漸近的な複雑さは、Garivier and Kaufmann (2016)とChernoff (1959)で完全に特徴付けられており、一方、その“対”設定である固定予算設定における漸近的な複雑さは、あまり知られていません。このノートは、固定予算設定におけるインスタンス依存性の漸近的な複雑さのオープンな問題と仮説について議論します。 |
2308.15949 | Latency-aware Unified Dynamic Networks for Efficient Image Recognition | Dynamic computation has emerged as a promising avenue to enhance the
inference efficiency of deep networks. It allows selective activation of
computational units, leading to a reduction in unnecessary computations for
each input sample. However, the actual efficiency of these dynamic models can
deviate from theoretical predictions. This mismatch arises from: 1) the lack of
a unified approach due to fragmented research; 2) the focus on algorithm design
over critical scheduling strategies, especially in CUDA-enabled GPU contexts;
and 3) challenges in measuring practical latency, given that most libraries
cater to static operations. Addressing these issues, we unveil the
Latency-Aware Unified Dynamic Networks (LAUDNet), a framework that integrates
three primary dynamic paradigms-spatially adaptive computation, dynamic layer
skipping, and dynamic channel skipping. To bridge the theoretical and practical
efficiency gap, LAUDNet merges algorithmic design with scheduling optimization,
guided by a latency predictor that accurately gauges dynamic operator latency.
We've tested LAUDNet across multiple vision tasks, demonstrating its capacity
to notably reduce the latency of models like ResNet-101 by over 50% on
platforms such as V100, RTX3090, and TX2 GPUs. Notably, LAUDNet stands out in
balancing accuracy and efficiency. Code is available at:
https://www.github.com/LeapLabTHU/LAUDNet. | Yizeng Han, Zeyu Liu, Zhihang Yuan, Yifan Pu, Chaofei Wang, Shiji Song, Gao Huang | 2023-08-30T10:57:41 | http://arxiv.org/abs/2308.15949v3 | # Latency-aware Unified Dynamic Networks for Efficient Image Recognition
###### Abstract
Dynamic computation has emerged as a promising strategy to improve the inference efficiency of deep networks. It allows selective activation of various computing units, such as layers or convolution channels, or adaptive allocation of computation to highly informative spatial regions in image features, thus significantly reducing unnecessary computations conditioned on each input sample. However, the practical efficiency of dynamic models does not always correspond to theoretical outcomes. This discrepancy stems from three key challenges: 1) The absence of a _unified formulation_ for various dynamic inference paradigms, owing to the fragmented research landscape; 2) The undue emphasis on algorithm design while neglecting _scheduling strategies_, which are critical for optimizing computational performance and resource utilization in CUDA-enabled GPU settings; and 3) The cumbersome process of evaluating practical latency, as most existing libraries are tailored for static operators. To address these issues, we introduce **Latency-Aware Unified Dynamic Networks (LAUDNet)**, a comprehensive framework that amalgamates three cornerstone dynamic paradigms--spatially-adaptive computation, dynamic layer skipping, and dynamic channel skipping--under a unified formulation. To reconcile theoretical and practical efficiency, LAUDNet integrates algorithmic design with scheduling optimization, assisted by a latency predictor that accurately and efficiently gauges the inference latency of dynamic operators. This latency predictor harmonizes considerations of algorithms, scheduling strategies, and hardware attributes. We empirically validate various dynamic paradigms within the LAUDNet framework across a range of vision tasks, including image classification, object detection, and instance segmentation. Our experiments confirm that LAUDNet effectively narrows the gap between theoretical and real-world efficiency. For example, LAUDNet can reduce the practical latency of its static counterpart, ResNet-101, by over 50% on hardware platforms such as V100, RTX3090, and TX2 GPUs. Furthermore, LAUDNet surpasses competing methods in the trade-off between accuracy and efficiency. Code is available at: [https://www.github.com/LeapLabTHU/LAUDNet](https://www.github.com/LeapLabTHU/LAUDNet).
Dynamic networks, Efficient inference, Convolutional neural networks.
## 1 Introduction
Deep neural networks have demonstrated exceptional capabilities in various domains such as computer vision [1, 2, 3, 4, 5], natural language processing [6, 7, 8, 9], and multi-modal understanding/generation [10]. Despite their stellar performance, the intensive computational requirements of these deep networks often limit their deployment on resource-constrained platforms, like mobile phones and IoT devices, highlighting the need for more efficient deep learning models.
Unlike traditional static networks [2, 3, 4] which process all inputs uniformly, dynamic models [11] adaptively allocate computation in a data-dependent fashion. This adaptivity involves bypassing certain network layers [12, 13, 14, 15] or convolution channels [16, 17] conditionally, and executing spatially adaptive inference that concentrates computational effort on the most informative regions of an image [18, 19, 20, 21, 22, 23]. As the field evolves and various dynamic models show promise, it begs the question: _How can we design a dynamic network for practical use?_
Addressing this question is challenging due to difficulties in fairly comparing different dynamic-computation paradigms. These challenges fall into three categories: 1) The lack of a unified framework to encompass different paradigms, as research in this area is often fragmented; 2) The focus on algorithm design, which often results in the mismatch between practical efficiency and their theoretical computational potential, due to the significant impact of scheduling strategies1 and hardware properties on real-world latency; 3) The laborious task of evaluating a dynamic model's latency on different hardware platforms, as common libraries (_e.g._ cuDNN) are not built to accelerate many dynamic operators.
Footnote 1: Scheduling strategies are essential for practical efficiency because they optimize the use of GPU threads and memory with CUDA codes.
In response, we introduce a **Latency-Aware Unified Dynamic Network (LAUDNet)**, a framework that unifies three representative dynamic-inference paradigms. Specifically, we examine the algorithmic design of layer skipping, channel skipping, and spatially dynamic convolution, integrating them through a "mask-and-compute" scheme (Fig. 1 (a)).
Next, we delve into the challenges of translating theoretical efficiency into tangible speedup, especially on multi-core processors such as GPUs. Traditional literature commonly adopts hardware-agnostic FLOPs (floating-point operations) as a crude efficiency measure, failing to provide latency-aware guidance for algorithm design. In dynamic networks,
adaptive computation coupled with sub-optimal scheduling strategies intensifies the gap between FLOPs and latency. Moreover, most existing methods execute adaptive inference at the finest granularity. For instance, in spatial-wise dynamic inference, the decision to compute each feature pixel is made independently [19, 20, 21]. This fine-grained flexibility results in non-contiguous memory access [21], necessitating specialized scheduling strategies (Fig. 1 (b)).
Given that dynamic operators exhibit unique memory access patterns and scheduling strategies, libraries designed for static models, like cuDNN, fail to optimize dynamic models effectively. The lack of library support implies that each dynamic operator requires individualized scheduling optimization, code refinement, compilation, and deployment, making network latency evaluation across hardware platforms labor-intensive. To address this, we propose a novel latency prediction model that efficiently estimates network latency by taking into account algorithm design, scheduling strategies, and hardware properties. Compared to hardware-agnostic FLOPs, our predicted latency offers a more realistic representation of dynamic model efficiency.
Guided by the latency prediction model, we tackle the aforementioned challenges within our latency-aware unified dynamic network (LAUDNet) framework. For a given hardware device, we use the predicted latency as the guiding metric for algorithm design and scheduling optimization, as opposed to the conventionally used FLOPs (Fig. 1 (c)). In this context, we propose coarse-grained dynamic networks where "whether-to-compute" decisions are made at the patch/group level rather than individual pixels/channels. Though less flexible than pixel/channel-level adaptability in prior works [16, 17, 19, 20, 21], this approach encourages contiguous memory access, enhancing real-world speedup on hardware. Our improved scheduling strategies further permit batching inference. We investigate dynamic inference paradigms, focusing on the accuracy-latency trade-off. Notably, previous research has established a correlation between latency and FLOPs on CPUs [21, 23], hence in this paper, we primarily target the GPU platform, a more challenging but less explored environment.
The LAUDNet is designed as a general framework in two ways: 1) Multiple adaptive inference paradigms can be easily implemented in various CNN backbones, like ResNets [2] and RegNets [24]; and 2) The latency predictor functions as an off-the-shelf tool that can be readily applied to diverse computing platforms, such as server-end GPUs (Tesla V100, RTX3090), desktop-level GPU (RTX3060) and edge devices (Jetson TX2, Nvidia Nano).
We evaluate LAUDNet's performance across multiple CNN architectures for image classification, object detection, and instance segmentation. Our results show that LAUDNet significantly improves the efficiency of deep CNNs, both in theory and practice. For instance, the inference latency of ResNet-101 on ImageNet [1] is reduced by \(>\)50% on different types of GPUs (_e.g._, V100, RTX3090 and TX2), without compromising accuracy. Moreover, our method outperforms various lightweight networks in low-FLOPs scenarios.
Although parts of this work were initially published in a conference version [25], this paper significantly expands our previous efforts in several key areas:
* A unified dynamic-inference framework is proposed. While the preliminary paper [25] predominantly focused on spatially adaptive computation, this paper delves deeper into two additional and important dynamic paradigms, specifically, dynamic layer skipping and channel skipping (Fig. 1 and Sec.3.1). Furthermore, we integrate these paradigms into a unified framework, and provide more thorough study on architecture design and complexity analysis (Sec.3.2).
* The latency predictor has been enhanced to support an expanded set of dynamic operators, including layer skipping and channel skipping (Sec. 3.3). Moreover, we adopt Nvidia Cutlass [26] to optimize the scheduling strategies. Hardware evaluations demonstrate that our latency predictor can accurately predict the latency on real hardware (Fig. 5).
* For the first time, we incorporate batching inference for our dynamic operators (Sec. 3.4). This innovation leads to more consistent prediction outcomes and an enhanced speedup ratio on GPU platforms (Fig. 8, 12).
* We undertake an exhaustive analysis of various dynamic granularities (Fig. 9) and paradigms
Fig. 1: An overview of our method. (a) illustrates three representative adaptive inference _algorithms_ (_i.e._ spatial-wise dynamic convolution, channel skipping, and layer skipping); (b) is an example of the _scheduling_ strategy for spatial-wise dynamic convolution; and (c) presents our key idea of using the latency to _guide_ both algorithm design and scheduling optimization.
(Fig. 10,11,13, Tab. 2,3), spanning different vision tasks and platforms, with added evaluations on contemporary GPUs like RTX3060 and RTX3090. We are confident that our results will offer valuable insights to both researchers and practitioners.
## 2 Related works
**Efficient deep learning** has garnered substantial interest. Traditional solutions involve lightweight model design [27, 28, 29, 30], network pruning [31, 32, 33, 34], weight quantization [35, 36, 37, 38], and knowledge distillation [39, 40]. However, these _static_ methods have sub-optimal inference strategy, leading to intrinsic redundancy since they process all inputs with equal computation.
**Dynamic networks**[11, 12, 18, 41] propose an appealing alternative to static models by enabling input-conditional _dynamic inference_. This adaptive approach has yielded superior results across various domains. In visual recognition, prevalent dynamic paradigms include early exiting [12, 13, 42], layer skipping [14, 15, 41], channel skipping [16, 17, 43], and spatial-wise dynamic computation [19, 20, 21]. This paper primarily targets the latter three paradigms, as they can be readily applied to arbitrary visual backbones, thereby offering a generality advantage. Layer skipping and channel skipping explore _structural_ redundancy within deep networks by selectively activating computation units, such as layers or convolution channels when processing different inputs. Spatial-wise dynamic models alleviate spatial redundancy in image features and selectively assign computation to the regions most pertinent to the task at hand.
Despite their effectiveness, previous studies often fail to recognize the shared underlying formulation across different dynamic paradigms. In contrast, we introduce a unified framework that encompasses all three paradigms, facilitating a thorough exploration of dynamic networks. Additionally, existing methods primarily concentrate on algorithm design, which often results in a significant disparity between theoretical and practical efficiency. In our latency-aware co-design framework, we bridge this gap by utilizing latency directly from our latency predictor to guide both algorithm design and scheduling optimization. This approach results in improved latency performance across diverse platforms.
**Hardware-aware network design.** Researchers have acknowledged the necessity to bridge the gap between theoretical and practical efficiency of deep models by considering actual latency during network design. Two primary approaches have emerged: the first entails conducting speed tests on targeted devices and deriving guidelines to facilitate _hand-designing_ lightweight models [30], and the second involves performing speed tests for various types of _static_ operators and modeling the latency predictor as a small trainable model [44, 45, 46, 47]. Neural architecture search (NAS) techniques [48, 49] are then used to _search_ for hardware-friendly models.
Our work distinguishes itself from these approaches in two significant ways: 1) while existing works predominantly focus on constructing _static_ models that inherently exhibit computational redundancy by treating all inputs uniformly, our goal is to design latency-aware _dynamic_ models that adjust their computation based on inputs; 2) conducting speed tests for dynamic operators across various hardware devices can be laborious and impractical. To circumvent this, we propose a latency prediction model that efficiently estimates the inference latency of dynamic operators on any given computing platform. This model accounts for algorithm design, scheduling strategies, and hardware properties simultaneously, providing valuable insights without the need for extensive speed testing.
## 3 Method
This section begins by providing an introduction to the foundational concepts underlying three dynamic inference paradigms (Sec. 3.1). We then present the architecture design of our LAUDNet framework, which unifies these paradigms under a cohesive _mask-and-compute_ formulation (Sec. 3.2). Next, we explain the latency prediction model (Sec. 3.3), which guides the determination of granularity settings and scheduling optimization (Sec. 3.4). Finally, we describe the training strategies for our LAUDNet (Sec. 3.5).
### _Preliminaries_
**Spatially adaptive computation.** Existing spatial-wise dynamic networks typically incorporate a masker \(\mathcal{M}^{\text{s}}\) within each convolutional block of a CNN backbone. Given an input \(\mathbf{x}\in\mathbb{R}^{H\times W\times C}\) to a block, where \(H\) and \(W\) represent the feature height and width, and \(C\) denotes the channel number. Assuming a convolution stride of \(1\), the masker \(\mathcal{M}^{\text{s}}\) takes \(\mathbf{x}\) as input and generates a binary-valued spatial mask \(\mathbf{M}^{\text{s}}=\mathcal{M}^{\text{s}}(\mathbf{x})\in\{0,1\}^{H\times W}\). Each element in \(\mathbf{M}^{\text{s}}\) determines whether to perform convolution operations at the corresponding output location. Unselected regions are populated with values from skip connection [19, 20].
During inference, the current scheduling strategy for spatial-wise dynamic convolutions generally involve three steps [50] (Fig. 1 (b)): 1) _gathering_, which re-organizes the selected pixels (if the convolution kernel size is greater than \(1\times 1\), the neighbors are also required) along the _batch_ dimension; 2) _computation_, which performs convolution on the gathered input; and 3) _scattering_, which fills the computed pixels on their corresponding locations of the output feature. Compared to performing convolutions on the entire feature map, this scheduling strategy reduces computation at the cost of overhead from mask generation and non-contiguous _memory access_. As a result, the overall latency could even be increased, particularly when the _granularity_ of dynamic convolution is at the pixel level (Fig. 6).
**Dynamic layer skipping**[14, 15, 51] adaptively determines whether to execute each layer or block, leveraging the structural redundancy of deep models to achieve data-dependent network _depth_. The implementation of dynamic layer skipping is similar to spatially adaptive inference, but with a scalar \(0,1\) decision variable \(\mathbf{M}^{\text{l}}\) instead of a spatial \(H\times W\) mask. Compared to spatially adaptive inference, layer skipping provides less flexibility but more regular computation patterns. Moreover, it generally does not require special scheduling strategies, as the original convolution operators remain unmodified.
**Dynamic channel skipping**[16, 17, 52] takes a more conservative approach to dynamic architecture versus full
layer skipping. It uses a \(C\)-dimensional vector \(\mathbf{M}^{\mathrm{c}}\in\{0,1\}^{C}\) to adaptively determine the runtime _width_ of a convolution layer with \(C\) output channels. For instance, the \(i\)-th (\(1\leq i\leq C\)) channel is computed only if \(\mathbf{M}^{\mathrm{c}}_{i}=1\). The scheduling of dynamic channel skipping usually requires gathering convolution kernels instead of feature pixels as in spatially dynamic computation (compare Fig. 2 (b) and (c)).
### _LAUDNet architecture_
**Overview**. Our analysis in Sec.3.1 reveals that the three dynamic inference paradigms share a common _"mask-and-compute"_ scheme, with the key difference being the _mask shapes_. Leveraging this insight, we propose a unified framework (Fig.2) where lightweight modules generate the channel mask \(\mathbf{M}^{\mathrm{c}}\) and the spatial/layer mask \(\mathbf{M}^{\mathrm{s}/1}\), respectively. Notably, layer skipping can be treated as a special case of spatially adaptive inference by introducing the concept of _granularity_ in dynamic computation as follows.
**Dynamic granularity**. As mentioned in Sec. 3.1, using _pixel-level dynamic convolutions_[19, 20, 21] poses substantial challenges for realistic speedup on multi-core processors due to non-contiguous memory access. To address this, we propose to optimize the _granularity_ of dynamic computation. For spatially adaptive inference, instead of producing an \(H\times W\) mask directly, we first generate a low-resolution mask \(\mathbf{M}^{\mathrm{s}}_{\mathrm{coarse}}\in\{0,1\}^{\frac{H}{2}\times\frac{W} {2}}\), where \(S\) is the _spatial granularity_. Each element in \(\mathbf{M}^{\mathrm{s}}_{\mathrm{coarse}}\) determines computation for a corresponding \(S\times S\) feature patch. For instance, the first ResNet stage2 deal with \(56\times 56\) features. Then the valid
Fig. 3: The architecture design of two types of maskers. The spatial/layer masker (a) is composed of a an adaptive pooling layer and a \(1\times 1\) convolution. The channel makser (b) consists of a global average pooling and a 2-layer MLP. The argmax operation is directly applied to obtain the discrete decisions during inference, while Gumbel Softmax [53, 54] is utilized for end-to-end training (Sec. 3.5).
Fig. 2: Our proposed LAUDNet block. (a) we first use a lightweight module to generate the channel mask \(\mathbf{M}^{\mathrm{c}}\) or the spatial/layer mask \(\mathbf{M}^{\mathrm{s}}/\mathbf{M}^{\mathrm{l}}\). The granularity of dynamic inference is controlled by \(G\) (for channel skipping) and \(S\) (for spatially adaptive computation). During training, the channel mask is multiplied with the input and output of the \(3\times 3\) convolution, and the spatial mask is applied on the final output of the block. Layer skipping could be easily implemented by setting \(S\) equal to the feature resolution. The scheduling strategies in inference ((b) for spatial-wise dynamic convolution and (c) for channel skipping) is performed to decrease memory access and facilitate parallel computation (Sec. 3.4). Note that we omit layer skipping here due to its simplicity: the whole block will be executed if the layer masker produces a value of 1.
choices for \(S\) are \(\{1,2,4,7,8,14,28,56\}\). The mask \(\mathbf{M}_{\mathrm{coarse}}^{\mathrm{s}}\) is then upsampled to the size of \(H\times W\). Notably, \(S=1\) corresponds to pixel-level granularity [19, 20, 21], while \(S\!=\!56\) naturally implements layer skipping.
Similarly, we introduce channel granularity \(G\) for channel skipping. Each element in \(\mathbf{M}_{\mathrm{coarse}}^{\mathrm{c}}\in\{0,1\}^{\frac{G}{G}}\) determines computation for \(G\) feature channels. The choice of the spatial granularity \(S\) and the channel granularity \(G\) for each block will be guided by our latency predictor (Sec. 3.3) for balancing flexibility and efficiency. Note that we apply the channel mask at the first two convolution layers within a block. This design is compatible with various backbone architectures, including those with arbitrary bottleneck ratios or group convolutions [24].
**Masker design.** We design different structures for spatial (layer) and channel-wise dynamic computation. As shown in Fig. 3 (a), the spatial masker uses an adaptive pooling layer to downsample the input \(\mathbf{x}\) to the size of \(\frac{H}{S}\times\frac{W}{S}\times C\), followed by a \(1\times 1\) convolution layer producing the soft logits \(\widehat{\mathbf{M}}_{\mathrm{coarse}}^{\mathrm{s}}\in\mathbb{R}^{d\times \frac{W}{S}\times 2}\). For the channel masker, we use a 2-layer MLP (Fig. 3 (b)) to produce channel-skipping decisions. Given input channels \(C\) the target mask dimension \(D\!=\!C/G\), we set the hidden units in the MLP as \(\max\{\lfloor D/16\rfloor,16\}\), where \(\lfloor\cdot\rfloor\) denotes a round-down operation. Appendix C.1 shows this design effectively reduces the latency of channel maskers, especially in late stages with more channels.
**Computational complexity.** We first point out that the masker FLOPs are negligible compared to the backbone convolutions. Therefore, we mainly analyse the complexity of standard convolution blocks here.
For spatially adaptive computation, We define the _activation ratio_\(r^{\mathrm{s}}=\frac{\sum_{i,j}\mathbf{M}_{i,j}^{\mathrm{s}}\in[0,1]}{H\times W} \in[0,1]\) to denote the fraction of computed pixels. Following [20], we further compute \(r_{\mathrm{dil}}^{\mathrm{s}}\) of a dilated spatial mask to represent the activation ratio of the first convolution in a block. It is observed in our experiments that \(r_{\mathrm{dil}}^{\mathrm{s}}\) is generally close to \(r^{\mathrm{s}}\). With FLOPs \(F_{1},F_{2},F_{3}\) for the three convolution layers, the _theoretical_ speedup is \(\frac{r_{\mathrm{dil}}F_{1}+r^{\mathrm{s}}F_{2}+r^{\mathrm{s}}F_{3}}{F_{1}+F_{ 2}+F_{3}}\!\approx\!r^{\mathrm{s}}\).
For channel skipping, the _activation ratio_ is \(r^{\mathrm{c}}=\frac{\sum_{j}\mathbf{M}_{i}^{\mathrm{c}}}{C}\in[0,1]\). Apply the mask before and after the \(3\times 3\) convolution makes its complexity quadratic with respect to \(r^{\mathrm{c}}\). The overall speedup is \(\frac{r^{\mathrm{c}}F_{1}+(r^{\mathrm{c}})^{2}F_{2}+r^{\mathrm{c}}F_{3}}{F_{ 1}+F_{2}+F_{3}}\leq r^{\mathrm{c}}\).
### _Latency predictor_
As stated before, it is laborious to evaluate the latency of dynamic operators on different hardware platforms. To efficiently seek preferable dynamic paradigms and granularity settings on any target device, we propose a latency prediction model \(\mathcal{G}\). Given hardware properties \(\mathbf{H}\), layer parameters \(\mathbf{P}\), dynamic paradigm \(\mathbf{D}\), spatial/channel granularity \(S/C\), and activation rates \(r^{\mathrm{s}}/r^{\mathrm{c}}\), \(\mathcal{G}\) directly _predicts_ block execution latency \(\ell=\mathcal{G}(\mathbf{H},\mathbf{P},\mathbf{D},S,C,r^{\mathrm{s}},r^{ \mathrm{c}})\).
**Hardware modeling.** We model a device with multiple processing engines (PEs) for parallel computation (Fig. 4). The memory system has three levels [55]: 1) off-chip memory, 2) on-chip global memory, and 3) memory in PE. In practice, the latency mainly comes from two processes: _data movement_ and _parallel computation_:
\[\ell=\ell_{\mathrm{data}}+\ell_{\mathrm{computation}}+\ell_{\mathrm{Const}}, \tag{1}\]
where \(\ell_{\mathrm{Const}}\) is a hardware-specific constant. This model accurately predicts both \(\ell_{\mathrm{data}}\) and \(\ell_{\mathrm{computation}}\), enabling more practical efficiency measurement than FLOPs.
**Latency prediction.** Given hardware properties and model parameters, adopting a proper _scheduling strategy_ is key to maximizing resource utilization through increased parallelism and reduced memory access. We use Nvidia Cutlass [26] to _search_ for the optimal scheduling (tiling and in-PE parallelism configurations) of dynamic operations. The data movement latency can then be easily obtained from data shapes and target device bandwidth. Furthermore, the computation latency is derived from hardware properties. Please refer to Appendix A for more details.
**Empirical validation.** We evaluate the performance of our latency predictor with a ResNet-101 block on an RTX3090 GPU, varying the activation rate \(r\). The blue curves represent the predictions, and the scattered dots are obtained via _searching_ for a proper scheduling strategy (implemented with custom CUDA code) using Nvidia Cutlass [26]. All the three dynamic paradigms are tested. Fig. 5 compares predictions to real GPU testing latency, showing accurate estimates across a wide range of activation rates.
Fig. 4: Our hardware model.
Fig. 5: Comparison between the real and predicted latency of a dynamic block in LAUD-ResNet-101.
### _Scheduling optimization_
We use general optimization methods like fusing activation functions and batch normalization (BN) layers into convolution layers. We also optimize our dynamic convolution blocks as follows.
**Operator fusion for spatial maskers**. As mentioned in Sec. 3.2, spatial maskers have negligible computation but take the full feature map as input, making them _memory-bounded_ (latency is dominated by memory access). Since the masker shares its input with the first \(1\times 1\) conv (Masker-Conv1\(\times\)1 in Figure 2 (b)), fusing them avoids repeated input reads. However, this makes the convolution spatially static, potentially increasing computation. For simplicity, we adopt such operator fusion in all tested models. In practice, we find that operator fusion improves efficiency in most scenarios.
**Fusing gather and dynamic convolution**. Traditional approaches first gather the input pixels of the first dynamic convolution in a block. The gather operation is also a _memory-bounded_ operation. Furthermore, when the kernel size exceeds 1\(\times\)1, input patches overlap, leading to repeated loads/stores. We fuse gathering into dynamic convolution to reduce the memory access (Gather-Conv3x3 in Fig. 2 (b)).
Note that for dynamic channel skipping (Fig. 2 (c)), gathering is conducted on convolution kernels rather than features. The weight gather operations is also fused with convolution by our scheduling optimization.
**Fusing scatter and add.** Conventional methods scatter the final convolution outputs before the element-wise addition. We fuse these two operators (Scatter-Add in Fig. 2 (b)) to reduce memory access costs. The ablation study in Sec. 4.2 validates the effectiveness of the proposed fusing methods.
**Batching inference** is enabled by recording patch, location, and sample correspondences during gathering and scattering (Fig. 2 (b, c)). Inference with a larger batch size facilitates parallel computation, making latency more dependent on computation versus kernel launching or memory access. See Appendix C.1 for empirical analysis.
### _Training_
**Optimization of non-differentiable maskers.** The masker modules produce binary variables for discrete decisions, and cannot be directly optimized with back-propagation. Following [20, 21, 23], we adopt straight-through Gumbel Softmax [53, 54] for end-to-end training. Take spatial-wise dynamic inference as an example, let \(\bar{\mathbf{M}}^{\mathrm{c}}\!\in\!\mathbb{R}^{H\times W\times 2}\) denote the output of the spatial mask generator \(\mathcal{M}^{\mathrm{s}}\). The decisions are obtained with the argmax function during inference. Training uses a differentiable Softmax approximation:
\[\bar{\mathbf{M}}^{\mathrm{c}}=\frac{\exp\left\{\left(\log\left(\bar{\mathbf{M }}^{\mathrm{c}}_{i:,0}\right)+\mathbf{G}_{i:,0}\right)/\tau\right\}}{\sum_{k= 0}^{1}\exp\left\{\left(\log\left(\bar{\mathbf{M}}^{\mathrm{c}}_{i:,k}\right)+ \mathbf{G}_{i:,k}\right)/\tau\right\}}\in[0,1]^{H\times W}, \tag{2}\]
where \(\tau\) is the Softmax temperature. Similarly, a channel masker \(\mathcal{M}^{\mathrm{c}}\) produces a \(2C\)-dimensional vector \(\bar{\mathbf{M}}^{\mathrm{c}}\in\mathbb{R}^{2C}\), where \(C\) is the channel number of the \(3\times 3\) convolution in a block. We first reshape \(\bar{\mathbf{M}}^{\mathrm{c}}\) into the size of \(C\times 2\), and apply Gumbel Softmax along the second dimension to produce \(\bar{\mathbf{M}}^{\mathrm{c}}\in[0,1]^{C}\). Following [20, 23], we let \(\tau\) decay exponentially from 5.0 to 0.1 in training to facilitate the optimization of maskers.
**Training objective**. As analyzed in Sec. 3.2, the FLOPs of each dynamic convolution block can be calculated based on our defined activation rate \(r^{\mathrm{s}}\) (or \(r^{\mathrm{c}}\)). Let \(F_{\mathrm{dyn}}\) and \(F_{\mathrm{stat}}\) denote the overall dynamic and static network FLOPs. We optimize their ratio to approximate a target \(0<t<1\): \(L_{\mathrm{FLOPs}}=(\frac{F_{\mathrm{dyn}}}{F_{\mathrm{stat}}}-t)^{2}\). In addition, we define \(L_{\mathrm{bounds}}\) as in [20] to constrain the upper/lower bounds in early training epochs.
We further propose to leverage the static counterparts of our dynamic networks as "teachers" to guide the optimization procedure. Let \(\mathbf{y}\) and \(\mathbf{y}^{\prime}\) denote the output logits of a dynamic "student" model and its static "teacher", respectively. Our final loss can be written as
\[L=L_{\mathrm{task}}+\alpha(L_{\mathrm{FLOPs}}+L_{\mathrm{bounds}})+\beta T^{2 }\cdot\mathrm{KL}(\sigma(\mathbf{y}/T)||\sigma(\mathbf{y}^{\prime}/T)), \tag{3}\]
where \(L_{\mathrm{task}}\) represents the task-related loss, _e.g._, cross-entropy loss in classification. \(\mathrm{KL}(\cdot||\cdot)\) denotes the Kullback-Leibler divergence, and \(\alpha,\beta\) are the coefficients balancing these items. We use \(\sigma\) to denote the log-Softmax function, and \(T\) is the temperature for computing KL-divergence.
## 4 Experiments
In this section, we first introduce the experiment settings in Sec. 4.1. Then the latency of different granularity settings are analyzed in Sec. 4.2. The performance of our LAUDNet on ImageNet is further evaluated in Sec. 4.3, followed by the visualization results in Sec. 4.4. We finally validate our method on the object detection and instance segmentation tasks (Sec. 4.5). For simplicity, we add "\(\mathrm{LAUD}^{\mathrm{s}/c/1}\)-" as a prefix before model names to denote our LAUDNet with different dynamic paradigms (s for spatial, c for channel and 1 for layer), _e.g._, \(\mathrm{LAUD}^{\mathrm{s}}\)-ResNet-50.
### _Experiment setup_
**Image classification** experiments are conducted on the ImageNet [1] dataset. We implement our LAUDNet on four representative architectures extending up to a broad spectrum of computational costs: ResNet-50, ResNet-101 [2], RegNetY-400M, and RegNetY-800M [24]. As per the established methodology in [20], we initialize the backbone parameter from a torchvision pre-trained checkpoint ([https://pytorch.org/vision/stable/models.html](https://pytorch.org/vision/stable/models.html)), and fine-tune the whole network for 100 epochs employing the loss function in Eq. (3). We fix \(\alpha\!=\!10,\beta\!=\!0.5\) and \(T\!=\!4.0\) for all dynamic models. Note that we adopt the pretrain-fintune paradigm mainly to reduce the training cost, as Gumbel Softmax usually requires longer training for convergence.
**Latency prediction.** We evaluate our LAUDNet on various types of hardware platforms, including two server GPUs (Tesla V100 and RTX3090), a desktop GPU (RTX3060) and two edge devices (_e.g._, Jetson TX2 and Nvidia Nano). The major properties considered by our latency prediction model include the number of processing engines (#PE), the floating-point computation in a processing engine (#FP32), the frequency and the bandwidth. It can be observed from Tab. 4 that server GPUs generally have a larger #PE than IoT devices. If not stated otherwise, the batch size is set as 128 for V100, RTX3090 and RTX3060 GPUs. On edge devices TX2 and Nano, testing batch size is fixed as 1.
More details are provided in Appendix B.
### _Latency prediction results_
This subsection presents the latency prediction results of dynamic convolutional blocks using two distinct backbones: ResNet-50 [2] (on V100) and RegNetY-800MF [24] (on TX2). Each block features a bottleneck structure with varying channel numbers and convolution groups, and the RegNetY employs Squeeze-and-Excitation (SE) [56] modules. We define \(\ell_{\mathrm{dyn}}\) as the latency of a dynamic convolutional block and \(\ell_{\mathrm{stat}}\) as the latency of a static block. The ratio of the two is denoted as \(r_{\ell}=\frac{\ell_{\mathrm{dyn}}}{\ell_{\mathrm{stat}}}\), with a realistic speedup being achieved when \(r_{\ell}<1\).
**Effect of spatial granularity.** The primary objective here is to investigate how the _granularity_ of dynamic computation impacts the latency ratio \(r_{\ell}\). We explore the correlation between \(r_{\ell}\) and the activation rate \(r^{\mathrm{s}}\) (refer to Sec. 3.2) for varying _granularity_ settings. The results in Fig. 5(a) (ResNet on V100) and Fig. 5(c) (RegNetY-800M on TX2) demonstrate that:
* Despite the implementation of our optimized scheduling strategies, pixel-level dynamic convolution (\(S\)=1) does not consistently enhance practical efficiency. This approach to fine-grained adaptive inference has been adopted in previous works [20, 21, 57]. Our findings help elucidate why these studies only managed to achieve realistic speedup on less potent CPUs [21] or specialized devices [57];
* By contrast, a coarse granularity setting (\(S>1\)) significantly mitigates this issue across both devices. Realistic speedup (\(r_{\ell}<1\)) is attainable with larger activation values (\(r^{\mathrm{s}}\)) when \(S>1\).
The latency prediction results are further used to determine preferable spatial granularity settings for the first 3 stages. Note that for the final stage where the feature resolution is \(7\!\times\!7\), \(S\!=\!1\) and \(S\!=\!7\) correspond to two distinct dynamic paradigms (spatially adaptive inference and layer skipping). The relationship curves between \(r_{\ell}\) and \(S\) depicted in Fig. 5(b) (ResNet on V100) and Fig. 5(d) (RegNetY-800M on TX2) reveal the following:
* The latency ratio \(r_{\ell}\) generally decreases as \(S\) increases for a given \(r\) on V100;
* An excessively large \(S\) (indicating less flexible adaptive inference) provides negligible improvement on both devices. In particular, increasing \(S\) from 7 to 14 in the second stage of LAUD-RegNetY-800MF on TX2 detrimentally impacts efficiency. This is hypothesized to be due to the oversized patch size causing additional memory access costs on this device, which has fewer processing engines (PEs);
\begin{table}
\begin{tabular}{c c c c} \hline \hline Masker-Conv & Gather-Conv & Scatter-Add & Latency (\(\mathrm{\SIUnitSymbolMicro s}\)) \\ \hline ✗ & ✗ & ✗ & 162.4 \\ ✓ & ✗ & ✗ & 135.1 \\ ✓ & ✓ & ✗ & 131.7 \\ ✓ & ✓ & ✓ & **118.3** \\ \hline \hline \end{tabular}
\end{table} TABLE I: Ablation studies on operator fusion.
Fig. 6: Latency prediction results for LAUD\({}^{\text{s}/1}\)-ResNet blocks on the Nvidia Tesla V100 GPU (a, b) and LAUD\({}^{\text{s}/1}\)-RegNetY-800MF blocks on the Nvidia Jetson TX2 GPU (c, d). The circle markers (\(\bullet\)) represent spatial-wise dynamic computation, and the star markers (\(\star\)) denote layer skipping, which is implemented via the largest granularity \(S\) in each stage.
* Layer skipping (marked by \(\star\)) consistently outperforms spatial-wise dynamic computation (marked by \(\bullet\)). We will analyze their performance across various vision tasks in Sec. 4.3 and Sec. 4.5.
Based on these results, we can strike a balance between flexibility and efficiency by choosing suitable \(S\) for different models and devices. For instance, we can simply set \(S^{\mathrm{net}}\)=4-4-2-13 in a LAUD\({}^{\mathrm{e}}\)-ResNet-50 to achieve realistic speedup.
Footnote 3: We use this form to represent the \(S\) settings for the 4 network stages.
**Effect of channel granularity.** We further investigate how the channel granularity \(G\) influences the realistic latency of channel-skipping dynamic models. Using \(\mathrm{LAUD}^{\mathrm{e}}\)-ResNet as an example, results presented in Fig. 7 show that the performance of channel skipping is less sensitive to the channel granularity \(G\). Setting \(G=2\) improves efficiency only in deeper stages, while extending \(G\) beyond 2 offers diminishing benefits. This aligns with our understanding that channel skipping requires more regular operations compared to spatially sparse convolution, implying that \(G=1\) can already employ impactful speedup. Moreover, the curves in Fig. 7a are generally convex, since the computation of the \(3\times 3\) convolution is quadratic in relation to \(r^{c}\) (Sec. 3.2).
**Ablation study of operator fusion.** We delve further into the effect of our operator fusion described in Sec. 3.4. Using a convolutional block from the first stage of a LAUD\({}^{\mathrm{e}}\)-ResNet-50 (\(S\)=4, \(r^{\mathrm{e}}\)=0.6) as a case study, it can be observed from the results in Table I that each operator fusion stage contributes to reducing the practical latency of a block by effectively reducing memory access overhead. Particularly, the fusion of the masker operation and the first convolution stands out as an essential contributor to latency reduction.
**Ablation study of batch size.** To establish a suitable testing batch size, we graph the relationship between latency per image and batch size for LAUD-ResNet-50 in Fig. 8. Two server-end GPUs (V100 and RTX3090) are tested. The results highlight that latency diminishes with an increase in batch size, eventually reaching a stable plateau when the batch size exceeds 128 on both platforms. This is comprehensible since a larger batch size favors enhanced computation parallelism, resulting in latency becoming more dependent on theoretical computation (FLOPs). The results on the desktop-level GPU, RTX3060 (Fig. 12 in Appendix C.1), show a similar phenomenon. Based on these observations, we report the latency on server-end and desktop-level GPUs with a batch size of 128 henceforth.
### _ImageNet classification_
#### 4.3.1 Comparison of spatial/channel granularities
We begin by comparing different granularities for spatial and channel-wise dynamic computation. Based on the analysis in Sec. 4.2, the candidates for spatial and channel granularities are \(S^{\mathrm{net}}\in\)1-1-1-1, 4-4-2-1, 8-4-7-1 and \(G^{\mathrm{net}}\in\)1-1-1, 2-2-2-2 respectively. We select ResNet-50 and RegNetY-800M as backbones, and compare various settings on TX2 and V100. The results in Fig. 9 reveal that:
Fig. 8: Relationship between the latency per image and batch size of LAUD-ResNet-50 on V100 (a) and 3090 (b) GPUs.
Fig. 7: Latency prediction results for LAUD\({}^{\mathrm{e}}\)-ResNet blocks on the Nvidia Tesla V100 GPU.
Fig. 9: Comparison of different granularities (\(S\) and \(G\)) in LAUD-ResNet-50 (a) and LAUD-RegNetY-800M (b). The latency on TX2 (left) and V100 (right) are presented.
* Regarding spatially dynamic computation, the optimal granularity \(S^{\mathrm{net}}\) is contingent on both network structures and hardware devices. For instance, \(S^{\mathrm{net}}\)=8-4-7-1 achieves a preferable performance on V100 for both models, yet incurs substantial inefficiency on TX2. This corresponds to our results in Fig. 6.
* Elevating the channel granularity \(G\) from 1 to 2 does yield sort of speedup for ResNet-50, but renders comparable performance in the case of RegNetY-800M. We hypothesize that a larger \(G\) is only beneficial for models with more extensive channel numbers, which also aligns with observations from Fig. 7.
#### 4.3.2 Comparison of dynamic paradigms
Having decided on the optimal granularities, we submit different dynamic paradigms to a more detailed comparison. Additionally, our LAUDNet is compared to various competitive baselines. The findings are illustrated in Fig. 10. **Standard baseline comparison: ResNet_s**. The compared baselines include various types of dynamic inference approaches: 1) layer skipping (SkipNet [14] and Conv-AIG [15]); 2) channel skipping (BAS [17]); and 3) pixel-level spatial-wise dynamic network (DynConv [20]). For our LAUDNet, we select the best granularity settings for spatial-wise and channel-wise dynamic inference. Layer skipping implemented in our framework is also included. We set training targets (cf. Sec. 3.5) \(t\in\{0,4,\cdots,0.8\}\) for our dynamic models to evaluate their performance across different sparsity regimes. We apply scheduling optimization (Sec. 3.4) uniformly across all models [15, 20] for a fair comparison.
The results are exhibited in Fig. 10 (a). On the left we plot the relationship between accuracy and FLOPs. It becomes obvious that our LAUD-ResNets, with various granularity settings, considerably outperform competing dynamic networks. Moreover, on ResNet-101, the three paradigms seem fairly comparable, whereas, on ResNet-50, layer skipping falls behind, especially when the training target is small. This is understandable because layer skipping might be overly aggressive for more shallow models.
Interestingly, the scenario alters as we explore real latency (middle on TX2 and right on V100). On the less potent TX2, latency generally exhibits a stronger correlation with theoretical FLOPs, given that it is _computation-bounded_ (that means, the latency is primarily focused around computation) on such IoT devices. However, different dynamic paradigms yield varying acceleration impacts on server-end GPU, V100, as latency could be impacted by the memory access cost. For instance, layer skipping takes precedence over the other
Fig. 11: Visualization results of activation rates \(r^{s/c/1}\) and selected patches by LAUD”-ResNet-101.
Fig. 10: Main results of LAUDNet implemented on ResNet (a) and RegNetY (b).
two paradigms on the deeper ResNet-101. With the target activation rate \(t=0.4\), our LAUD\({}^{\text{l}}\)-ResNet-101 reduces the inference latency of its static counterpart by \(\sim\)53%. On the shallower ResNet-50, channel skipping keeps pace with layer skipping on some low-FLOPs models. Although our proposed course-grained spatially adaptive inference trails behind the other two schemes, it significantly outclasses the previous work using pixel-level dynamic computation [20]. The additional results in Appendix C.2 also demonstrate the preferable efficiency of layer skipping on RTX3060 and RTX3090. Channel skipping outperforms the other two paradigms only on the edge device, Nvidia Nano.
**Lightweight baseline comparison: RegNets.** We further evaluate our LAUDNet in lightweight CNN architectures, _i.e._ RegNets-Y [24]. Two different sized models are tested: RegNetY-400MF and RegNetY-800MF. Compared baselines include other types of efficient models, _e.g._, MobileNets-v2 [28], ShuffletNets-v2 [30] and CondenseNets [33].
The results are presented in Fig. 10 (b). We observe that while channel skipping surpasses the other two paradigms substantially in the accuracy-FLOPs trade-off, it is less efficient than layer skipping on most models except RegNet-Y-800M. Remarkably, layer skipping emerges as the most dominant paradigm. We theorize that this is due to the model width (number of channels) of RegNet-Y being limited, and the inference latency still being bounded by memory access. Moreover, layer skipping enables skipping the memory-bounded SE operation [56]. The results on desktop-level and server-end GPUs (Appendix C.2) further showcase the superiority of layer skipping.
### _Visualization and interpretability_
We present visualization results of LAUDNet to delve into its interpretability from the perspectives of networks' structural redundancy and images' spatial redundancy.
**Activation rate.** Fig. 11 (a) illustrates the average activation rates \(r^{\text{s/c/l}}\) of each block in LAUD\({}^{\text{s/c/l}}\)-ResNet-101 (\(t\)=0.5) on the ImageNet validation set. The results uncover that
* The activation rate patterns for spatially dynamic convolution and layer skipping are similar. The activation rates \(r^{\text{s}}\) and \(r^{\text{l}}\) seem more binarized (close to 0 or 1) in stages 1, 2, and 4. The dynamic region/layer selection predominantly occurs in stage 3;
* These two paradigms tend to maintain the entire feature map (\(r^{\text{s/l}}\)=1.0) at the first block of stages 2, 3, and 4, where the convolutional stride is 1. This aligns with the settings in [52, 15], where the training targets for these blocks are manually set to 1. Notably, we train our LAUDNet to meet an overall computational target, rather than confining the targets for different blocks as done in [52, 15].
* Channel skipping results in activation rates that are more centered around 0.5 throughout the network.
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline Detection & \multirow{2}{*}{Backbone} & \multirow{2}{*}{Backbone} & \multicolumn{5}{c}{Backbone Latency (ms)} & \multirow{2}{*}{mAP (\%)} \\ \cline{3-3} Framework & & FLOPs (G) & & V100 & 3090 & 3060 & TX2 & Nano \\ \hline \multirow{8}{*}{Faster R-CNN} & ResNet-101 (Baseline) & 141.2 & 33.9 & 29.8 & 44.8 & 586.4 & 1600.4 & 39.4 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.6) & 90.7 & 32.4 & 36.8 & 40.4 & 402.2 & 1082.4 & **40.3** \\ & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(S^{\text{s/c}}\)=4-4-7-1, \(t\)=0.5) & 79.5 & 30.4 & 29.4 & 38.2 & 390.4 & 1050.7 & 40.0 \\ & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(S^{\text{s/c}}\)=4-4-7-1, \(t\)=0.4) & **67.9** & 27.4 & 26.2 & 34.5 & 340.0 & 911.4 & 39.5 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.8) & 112.37 & 30.6 & 30.0 & 42.0 & 471.6 & 1264.3 & 40.2 \\ & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.7) & 96.42 & 27.9 & 27.3 & 37.8 & 400.4 & 1065.4 & 40.0 \\ & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.6) & 80.73 & 23.9 & 24.6 & 33.9 & 335.7 & **884.0** & 39.7 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(t\)=0.5) & 97.97 & 24.2 & 22.1 & 32.2 & 409.2 & 1114.1 & 40.2 \\ & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(t\)=0.4) & 86.71 & **19.8** & **18.2** & **26.5** & **331.2** & 899.9 & 39.5 \\ \hline \multirow{8}{*}{RetinaNet} & ResNet-101 (Baseline) & 141.2 & 33.9 & 29.8 & 44.8 & 586.4 & 1600.4 & 38.5 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(S^{\text{s/c}}\)=4-4-2-1, \(t\)=0.5) & 77.8 & 29.0 & 32.7 & 36.7 & 350.1 & 937.1 & 39.3 \\ \cline{1-1} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(S^{\text{s/c}}\)=4-4-7-1, \(t\)=0.4) & **66.4** & 28.1 & 26.0 & 35.2 & 335.0 & 897.1 & 38.9 \\ \cline{1-1} \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.6) & 79.6 & 23.7 & 24.4 & 33.7 & 331.2 & 871.4 & 39.3 \\ \cline{1-1} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.5) & 65.5 & 20.9 & 22.1 & 30.4 & **278.7** & **724.6** & 38.5 \\ \cline{1-1} \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(t\)=0.5) & 95.1 & 23.6 & 21.5 & 31.4 & 397.7 & 1082.5 & **39.4** \\ \cline{1-1} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(t\)=0.3) & 74.4 & **18.7** & **17.3** & **25.0** & 311.4 & 846.3 & 38.6 \\ \hline \hline \end{tabular}
\end{table} TABLE II: Object detection results on the COCO dataset.
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{Segmentation} & \multirow{2}{*}{Backbone} & \multirow{2}{*}{Backbone} & \multicolumn{5}{c}{Backbone Latency (ms)} & \multirow{2}{*}{mAP (\%)} \\ \cline{3-3} Framework & & FLOPs (G) & V100 & 3090 & 3060 & TX2 & Nano & \\ \hline \multirow{8}{*}{Mask R-CNN} & ResNet-101 (Baseline) & 141.2 & 33.9 & 29.8 & 44.8 & 586.4 & 1600.4 & 36.1 & 40.0 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(S^{\text{s/c}}\)=4-2-1, \(t\)=0.5) & 80.5 & 29.7 & 33.5 & 37.5 & 361.9 & 969.9 & **37.0** & **41.0** \\ & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(S^{\text{s/c}}\)=4-2-1, \(t\)=0.4) & **69.2** & 26.4 & 29.6 & 33.7 & **314.3** & **838.8** & 36.1 & 40.0 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.8) & 112.7 & 30.7 & 30.0 & 42.1 & 473.1 & 1269.3 & 36.9 & 40.9 \\ \cline{2-8} & LAUD\({}^{\text{s/c}}\)-ResNet-101 (\(C^{\text{s/c}}\)=2-2-2, \(t\)=0.7) & 95.9 & 27.1 & 27.2 & 37.7
**Dynamic patch selection.** We visualize the spatial masks generated by our third block of a LAUD*-ResNet-101 (\(S^{\mathrm{net}}\)=4-4-2-1) in Fig. 11 (b). The highlighted areas denote the locations of \(1\) elements in a mask, while computations in the dimmed regions are skipped by our dynamic model. It becomes evident that the masker is adept at pinpointing the most task-related areas, even minutiae such as the tiny aircraft at the corner, thereby trimming unnecessary computations in background zones. Such findings imply that, a granularity of \(S\)=4 is aptly flexible for identifying crucial regions, paving the way for a harmonious balance between accuracy and efficiency. Intriguingly, the masker is able to pick out objects which are _not labeled_ for that particular sample - for instance, the flower next to the hummingbird or the person clutching the camera. This signals that our spatially dynamic networks inherently discern regions imbued with semantic significance, and their prowess isn* blackled by mere classification labels. Such a trait is invaluable for a slew of downstream tasks, like object detection and instance segmentation (Sec. 4.5), tasks which necessitate the identification of various classes and objects within an image. For a broader range of visualization results, readers can refer to Appendix C.3.
### _Dense prediction tasks_
Our LAUDNet is further put to test on downstream tasks, _i.e._ COCO [58] object detection (as seen in Table II) and instance segmentation (presented in Table III). For object detection, the mean average precision (mAP) stands as the barometer for network efficacy. For instance segmentation, the APmask does deeper to gauge the nuance of dense prediction. The average backbone FLOPs, and the average backbone latency on the validation set are used to measure the network efficiency. We test two prevalent detection frameworks: Faster R-CNN [59] with Feature Pyramid Network [60] and RetinaNet [61]. For the instance segmentation task, we employ Mask R-CNN [62]. Owing to the universality of our method, we can effortlessly substitute the backbones with ours which has been pre-trained on ImageNet, and fine-tune the entire models on COCO under the standard setting for 12 epochs (for detailed setup, please refer to Appendix B.3). The input images are resized to a short side of 800 with a long side not exceeding 1333. The results of our LAUD-ResNet-101 with various dynamic paradigms are displayed in Table II. The results clearly show that our LAUDNet can consistently enhance both the mAP and efficiency. Furthermore, channel skipping might realize more noticeable latency improvement on less capable devices, while layer skipping outperforms on server-grade V100 GPUs.
## 5 Conclusion
In this paper, we propose to build _latency-aware_ unified dynamic networks (LAUDNet) under the guidance of a _latency prediction model_. By collectively considering the algorithm, scheduling strategy, and hardware properties, we can accurately estimate the practical latency of different dynamic operators on any computing platforms. Based on an empirical analysis of the correlation between latency and the _granularity_ of spatial-wise and channel-wise adaptive inference, the algorithm and scheduling strategies are optimized to attain realistic speedup on a range of multi-core processors, such as Tesla V100 and Jetson TX2. Our experiments on image classification, object detection, and instance segmentation tasks affirm that the proposed method markedly boosts the practical efficiency of deep CNNs and surpasses numerous competing approaches. We believe our research brings useful insights into the design of dynamic networks. Future works include explorations on more types of model architectures (_e.g._ Transformers, large language models) and tasks (_e.g._ low-level vision tasks and vision-language tasks).
## Acknowledgments
This work is supported in part by the National Key R&D Program of China under Grant 2021ZD0140407, the National Natural Science Foundation of China under Grants 62022048 and 62276150, Guoqiang Institute of Tsinghua University and Beijing Academy of Artificial Intelligence.
| 動的計算は、深層ネットワークの推論効率を向上させる有望な道として登場しています。それは、計算ユニットの選択的な活性化を可能にし、入力サンプルごとに不要な計算を削減するのに役立ちます。しかし、これらの動的モデルの実効性は、理論的な予測とは異なる場合があります。このずれは、以下の3点に起因します。1) さまざまな研究の分断により、統一されたアプローチが欠如していること、2) CUDAを有効にしたGPU環境におけるアルゴリズム設計の優先度が高すぎること、3) 静的オペレータに対応する多くのライブラリの存在により、実用的なレイテンシを測定するのが難しいこと。これらの問題に対処するため、私たちは、レイテンシを意識した統一的な動的ネットワーク（LAUDNet）というフレームワークを開発しました。このフレームワークには、空間的に適応可能な計算、動的層スキップ、動的チャ |
2305.02211 | Influence zones for continuous beam systems | Unlike influence lines, the concept of influence zones is remarkably absent
within the field of structural engineering, despite its existence in the
closely related domain of geotechnics. This paper proposes the novel concept of
a structural influence zone in relation to continuous beam systems and explores
its size numerically with various design constraints applicable to steel framed
buildings. The key challenge involves explicitly defining the critical load
arrangements, and is tackled by using the novel concepts of polarity sequences
and polarity zones. These lead to the identification of flexural and (discovery
of) shear load arrangements, with an equation demarcating when the latter
arises. After developing algorithms that help identify both types of critical
load arrangements, design data sets are generated and the influence zone values
are extracted. The results indicate that the influence zone under ultimate
state considerations is typically less than 3, rising to a maximum size of 5
adjacent members for any given continuous beam. Additional insights from the
influence zone concept, specifically in comparison to influence lines, are
highlighted, and the avenues for future research, such as in relation to the
newly identified shear load arrangements, are discussed. | Adrien Gallet, Andrew Liew, Iman Hajirasouliha, Danny Smyl | 2023-02-24T11:14:15 | http://arxiv.org/abs/2305.02211v1 | # Influence zones for continuous beam systems
###### Abstract
Unlike influence lines, the concept of influence zones is remarkably absent within the field of structural engineering, despite its existence in the closely related domain of geotechnics. This paper proposes the novel concept of a structural influence zone in relation to continuous beam systems and explores its size numerically with various design constraints applicable to steel framed buildings. The key challenge involves explicitly defining the critical load arrangements, and is tackled by using the novel concepts of polarity sequences and polarity zones. These lead to the identification of flexural and (discovery of) shear load arrangements, with an equation demarcating when the latter arises. After developing algorithms that help identify both types of critical load arrangements, design data sets are generated and the influence zone values are extracted. The results indicate that the influence zone under ultimate state considerations is typically less than 3, rising to a maximum size of 5 adjacent members for any given continuous beam. Additional insights from the influence zone concept, specifically in comparison to influence lines, are highlighted, and the avenues for future research, such as in relation to the newly identified shear load arrangements, are discussed.
_Keywords_: influence zones, influence lines, load arrangements, continuous beams, structural design, pattern loads, polarity zones.
## 1 Introduction
Influence lines, which derive from Betti's theorem established in 1872 [1], are a well-established tool in structural engineering to identify the worst-case load placement on structural systems [2, 3, 4], and are widely applied in research related to continuous beam systems [5, 6], rigid frames [7], bridge engineering [8] and structural health monitoring [9, 10]. Influence zones, on the other hand, also known as zones of influence, are an established concept within the field of geotechnical engineering, helping to identify the area
of engineering soils likely to be affected by loading due to sub- and superstructure construction [11], providing geotechnical engineers valuable design insight in deep foundation design [12, 13], settlement estimations [14] and preserving groundwater supplies [15].
Despite the obvious discipline link between geotechnical and structural engineering, the equivalent use of an influence zone in structural engineering does not exist in literature. Here, the term _structural influence zone_ would refer to the zone in which applied forces, stiffness provisions and support conditions, or changes thereof, impact the design of the surrounding structural system.
The dearth of literature on such an _influence zone_ is surprising. For instance, the concept of influence zones also exists outside of geotechnical literature. Some examples are available in research related to the study of saltwater-freshwater interfaces [16], harmful emission concentrations at traffic intersections [17], reverse \(k\)-nearest neighbour algorithms [18, 19], propagation path of surfaces waves [20] and ecological studies on below-ground plant competition [21].
Furthermore, one can readily identify situations where knowledge of the _influence zone_ could be beneficial in design. For example, the size of the influence zone could allow an engineer to avoid the need to model an entire structure for the design of a single element whilst being confident that structural information outside the influence zone is irrelevant, with direct applications in multi-disciplinary projects [22]. The impact of late design changes (due to changes in loading or structural provisions), which are known to cause significant time lags until the associated engineering analysis is completed [23], could be more effectively addressed by knowing immediately the selection of members impacted by the said design change. Similarly, engineers are typically required to verify assumptions made in preliminary design [24]. In such cases, the use of an influence zone-based approach could guide what information to incorporate when building an independent model of the design problem. In all of these scenarios, there is valuable design insight to be gained from the _influence zone_.
This article aims to address the above mentioned knowledge gap by numerically introducing the concept of influence zones in relation to continuous beam systems. First, the theory and methodology for evaluating the influence zone will be introduced in section 2, followed by a systematic analysis of critical load arrangements in section 3. The explicit formulations of critical load arrangements allow for the efficient generation of design data sets and the evaluation of their respective influence zones in section 4, the results of which are discussed in section 5. In addition to the _influence zone_, this paper proposes other novel concepts such as _polarity zones_, identifies an entirely new set of critical pattern loads named _shear load arrangements_, and proposes efficient _load arrangement algorithms_ for continuous beam systems of arbitrary member size.
## 2 Methodology
### Overview
Consider a continuous beam system, as shown in Figure 1, consisting out of \(m\) members, indexed by \(i\), which is subjected to \(w_{i}\) uniformly distributed loads (UDLs) from vector \(\mathbf{w}\), with each member having span length \(L_{i}\) from vector \(\mathbf{L}\). When designing this
system to identify the minimum required structural properties of the members (size optimisation) denoted \(I_{i}\) to form vector \(\mathbf{I}\), it will need to be designed against the worst-case load arrangement (also known as pattern load) from the set of load arrangements \(\mathbf{J}\) of size \(p\). The over-restrained nature of this structural system (a function of the support fixity and structural connectivity) renders the continuous beam indeterminate. This means that the performance of the system is a function of the structural properties which need to be evaluated, and generally makes the design process iterative. Literature has well established formulations to design such indeterminate systems [25].
Figure 1: An exemplary continuous beam system with \(m=5\) members, subjected to UDLs \(\mathbf{w}\), spans \(\mathbf{L}\) and with designed cross-sectional properties \(\mathbf{I}\), all indexed by \(i\). The system’s indeterminacy requires an iterative design process against various load arrangements \(\mathbf{J}\) of size \(p\) indexed by \(j\).
### Influence zone formulations
Suppose a member within a continuous beam system is designated as the _design beam_ by index \(d\), and a discrete integer \(k\in\mathbf{Z}\) indicates the index position of a member relative to the design beam at \(d\). As shown in Figure 2, if \(\mathbf{K}\) refers to the list of members identified in terms of \(k\) that fall within the influence zone, then the size of the influence zone is denoted by \(k_{\max}=\max(|\mathbf{K}|)\) with \(k_{\max}\in\mathbf{N}^{0}\), representing the set of all positive integers and including \(0\). Two different formulations have been identified to evaluate the influence zone:
* The _local formulation_ identifies the value of \(k_{\max}\) based on whether the design information at the design beam \(d\) significantly influences the surrounding members of indices \(d-k_{\max}\leq i\leq d+k_{\max}\).
* The _global formulation_ identifies the value of \(k_{\max}\) based on when the design information at members with indices \(i<d-k_{\max}\) and \(i>d+k_{\max}\) becomes inconsequential for the design beam at \(d\).
Figure 2: An example demonstrating how influence lines relate to influence zones, and what an influence zone of size \(k_{\max}=2\) corresponds to in relation to a given design beam (here \(d=3\), highlighted in yellow).
For the continuous beam system established in Figure 1, the "design information" include the UDLs \(w_{i}\) and spans \(L_{i}\). Although the terms "significantly influences" and "becomes inconsequential" are currently undefined, they refer to an error threshold that will be explained later. Whilst the _local_ and _global_ formulations differ in terms of where the design information impact is measured from (locally at the design beam for the _local formulation_ or outside the influence zone for the _global formulation_), as long as the design constraints are identical, the size of the influence zone \(k_{\max}\) each formulation identifies will be identical.
There are various methodologies one could employ to establish the influence zone using either formulation. For example, _analytical_ approaches making use of concepts such as perturbation theories based on the relationship between force vectors and stiffness matrices may be viable. On the other hand, influence zones could be approached experimentally with the use of physical models or numerically with the use of finite element methods.
Each methodology has its own disadvantages. It is not intuitive how one would evaluate the size of the influence zone using a perturbation based approach if large design perturbations are required with multiple load arrangements. Experimental procedures would be limited by the number of design scenarios that can be tested, whilst numerical approaches would make mechanical assumptions on the material and structural behaviour of the system.
A numerical approach was preferred since it allows a multitude of design scenarios to be investigated and for statistical conclusions to be determined. Both the local and global formulations were attempted with the use of a numerical model, yet only the latter formulation was fully developed. This was because with the global formulation, the influence zone \(k_{\max}\) could be measured in relation to the utilisation ratio of the design beam \(d\) directly, which made evaluating and reporting the influence zone easier. In the local formulation, the utilisation ratio of all surroundings members outside the influence zone would have to be monitored.
### Mathematical formulation
Mathematically, the global formulation can be expressed as follows. For a given continuous beam system as depicted in Figure 1, and the design constraints expressed in Equation 1,
\[\begin{array}{l}w_{\min}\;<w_{i}<w_{\max}\\ L_{\min}\;<L_{i}\;<L_{\max}\\ I_{\min}\;<I_{i}\;<I_{\max}\end{array} \tag{1}\]
the size of the influence zone of a given design beam \(d\) is found when the value of \(k_{\max}\in\mathbf{N}^{0}:k_{\max}\in[0,m]\)**and** all values larger than \(k_{\max}\) fulfil the following condition:
\[\begin{array}{l}\left|\;1-\frac{u_{d,\mathrm{cap}}}{u_{d, \mathrm{true}}}\;\right|\leq\epsilon_{\max}\\ \\ u_{d,\mathrm{cap}}=\max\left(\;\sum_{i=-k_{\max}}^{k_{\max}}\mathbf{u}_{d,i, j}(\mathbf{w},\mathbf{L},\mathbf{I},\mathbf{J})\;\right)\end{array} \tag{2}\]
where \(\epsilon_{\max}\) represents the maximum error threshold for the difference between \(u_{d,\text{cap}}\), the captured utilisation ratio of the design beam \(d\) for a given value of \(k_{\max}\), and \(u_{d,\text{true}}\), the true utilisation ratio of the design beam \(d\) if the contribution of all members of the continuous beam system had been considered. \(\mathbf{u}_{d,i,j}\) is the utilisation ratio contribution function towards the design beam \(d\) by member \(i\) based on the UDLs \(\mathbf{w}\), spans \(\mathbf{L}\), structural properties \(\mathbf{I}\) and load arrangements \(\mathbf{J}\) indexed by \(j\).
The global formulation as written in Equation 2 measures the point at which the contributions outside of \(k_{\max}\) "becomes inconsequential" by minimising the difference between \(u_{d,cap}\) and \(u_{d,true}\) based on \(\epsilon_{\max}\). As \(k_{\max}\) increases, the ratio \(u_{d,cap}/u_{d,true}\) will approach unity, attaining unity if all structural members (\(k_{\max}=m\)) are considered within the influence zone. If the error threshold \(\epsilon_{\max}\) is relaxed, an influence zone less than the total number of beam members \(m\) can be found. The influence zone is therefore a heuristic measure based on an acceptable maximum error threshold \(\epsilon_{\max}\).
The importance of the design constraints as specified by Equation 1 is that they allow for the statistical estimation of the maximum influence zone size based on the diversity of design information variation that can arise. The maximum influence zone value for a type of structural system should always be understood with explicit reference to the design constraints it was evaluated by.
### Design constraints and assumptions
The design constraints considered in this investigation were chosen for their relevance in the design of continuous steel framed buildings, which is reflected by the range of UDLs and spans of the design data sets. Four individual design scenarios are considered to study the influence zone in depth, with each set featuring an increasing variation in span lengths and applied loads, summarised in Table 1. Length and UDL values are discretized in \(0.5\,\,\mathrm{m}\) and \(5\,\,\mathrm{kN/m}\) increments respectively, and are drawn from a random uniform distribution.
\begin{table}
\begin{tabular}{c c c c} \hline \hline Data set & \(G_{k,i}=\) & \(Q_{k,i}\in\) & \(L_{i}\in\) \\ \hline Set 1 & \(3.0\,\mathrm{kN/m}+\) & \(a\) for all \(i\), with \(a\in\) & \(b\) for all \(i\), with \\ _Zero variation_ & self-weight & \([0\,\mathrm{kN/m},60\,\mathrm{kN/m}]\) & \(b\in[1\,\mathrm{m},12\,\mathrm{m}]\) \\ Set 2 & \(3.0\,\mathrm{kN/m}+\) & \([20\,\mathrm{kN/m},40\,\mathrm{kN/m}]\) & \([4\,\mathrm{m},8\,\mathrm{m}]\) \\ _Low variation_ & self-weight & \([10\,\mathrm{kN/m},50\,\mathrm{kN/m}]\) & \([2\,\mathrm{m},10\,\mathrm{m}]\) \\ Set 4 & \(3.0\,\mathrm{kN/m}+\) & \([0\,\mathrm{kN/m},60\,\mathrm{kN/m}]\) & \([1\,\mathrm{m},12\,\mathrm{m}]\) \\ _High variation_ & self-weight & \([0\,\mathrm{kN/m},60\,\mathrm{kN/m}]\) & \([1\,\mathrm{m},12\,\mathrm{m}]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design constraints for various design scenarios used in this investigation based on Eurocode terminology, with \(G_{k}\) and \(Q_{k}\) being the characteristic permanent and variable actions. Increasing set numbers correspond to increasing design variation, a proxy for design complexity. Span and UDL values are discretized in \(0.5\,\,\mathrm{m}\) and \(5\,\,\mathrm{kN/m}\) increments respectively, and will be drawn from a random uniform distribution.
Further design and modelling constraints/assumptions include restricting the cross-sectional properties to prismatic BS EN 10365:2017 UKB I-sections, designing for S355 steel with perfectly linear elastic behaviour using Timoshenko-Ehrenfest beam theory (yet the design was conducted using plastic section properties as allowed by EN 1993-1-1 5.4.2(2) [26]). It was assumed that all spans are laterally restrained (and hence not susceptible to lateral instability), with elements designed against EC3 ULS checks (and notably not SLS requirements) with EN 1990 Eq. 6.10 load combination factors [27].
### The key challenge
The most important aspect for evaluating the influence zone using equation 2 is the member-based utilisation ratio contribution function \(\mathbf{u}_{\mathrm{d,i,j}}\). Whilst the UDLs \(\mathbf{w}\) and spans \(\mathbf{L}\) are given, the critical load arrangement from set \(\mathbf{J}\) which will determine the required structural properties \(\mathbf{I}\) is unknown. Furthermore, the critical load arrangement for a design beam could also differ based on the assumed value of the influence zone size \(k_{\mathrm{max}}\).
One approach would be to use a naive, brute-force procedure to trial every possible load arrangement to create the set \(\mathbf{J}_{\mathrm{naive}}\) with a corresponding set size of \(p_{naive}=2^{m}\) for \(\mathbf{J}\) in equation 2. This is not an issue for systems with few members, but if larger systems with \(m>10\) members need to be modelled to study the influence zone in depth, a brute-force approach becomes computationally expensive. The issue of computational cost in relation to critical load arrangements of large-scale systems is well acknowledged in literature, and various methodologies have been employed using probability [28] and possibility theories [29, 30]. Among the latter, fuzzy sets using interval finite-element methods have been shown to be efficient and accurate [31, 32].
However, whilst these interval-based methods are effective at evaluating the bounds (the worst case force/moment value) of the critical load arrangement, they do not in fact reveal what this load arrangement looks like. This is problematic for the evaluation of the influence zone, since Equation 2 relies on being able to identify this set \(\mathbf{J}\) explicitly. Another approach would be to use the load arrangements prescribed by design manuals, yet these consist out of a heuristic set of load arrangements that are known to be non-conservative [32].
Due to these limitations, a rigorous study is conducted to identify and validate the set of critical load arrangements _a priori_ for the design problem identified in Section 2.1, labelled \(\mathbf{J}_{\mathrm{crit}}\). This will not only reduce the computational cost of both designing the members and evaluating their influence zone with Equation 2; it will also highlight the relationship between influence lines and influence zones, providing an intuition on the size of the latter. This study is followed by the numerical generation of randomly distributed data sets based on the design constraints identified in Section 2.4, allowing the evaluation and statistical analysis of the influence zone for various continuous beam systems.
## 3 Critical load arrangement investigation
### Polarity sequences
Influence lines can be used to identify the critical load arrangements for a given continuous beam system. The design problem is restricted to positive UDL values only (no
uplift) which can be activated on or off (1 or 0 as activation factors). Therefore, by integrating the influence line for each individual beam \(i\), one can evaluate the net contribution (positive or negative) a given beam causes in terms of bending moments and shear forces at the influence line (IL) location when subjected to a positive, unit UDL. The net-contribution of each beam can be either positive or negative at the IL location, that is "hogging or sagging" for bending moments and "clockwise or anti-clockwise" for shear forces, respectively, which is termed as the _polarity_ of that particular beam. This procedure is shown in Figure 3.
The last row of Figure 3 therefore reflects a particular _polarity sequence_ for a given IL location, which can be directly used to identify the critical load arrangement for that IL location. When all beams of positive polarity are loaded, then the maximum positive internal forces are generated at the IL location, and vice-versa, loading the negative polarity members leads to the maximum negative internal forces.
Figure 3: An exemplary process of arriving from influence line plots (top row) to polarity sequences (bottom row) via integrated influence lines (middle row) for a) major axis bending moment \(M_{y}\) and b) major axis shear force \(V_{z}\) about the specified influence line (IL) location.
### Polarity zones
A rigorous qualitative study of the polarity sequences for different IL locations and design scenarios revealed 5 unique polarity sequences that occur along specific segments of a given beam span termed _polarity zones_, which are illustrated in Figure 4 for the central beam highlighted in red.
These 5 polarity zones are common to all members of both homogeneous (equal spans and cross-sections) as well as heterogeneous continuous beam systems, although the exact boundaries between one zone varied depending on the relative magnitude of spans and cross-section properties. The sequences identified in Figure 4 also apply to larger beam systems with the polarity direction alternating at each successive beam. For example, if the 5-member system was extended by an additional member on either side of the system (to give a 7 member system), the left-most member of the Type I polarity sequence would have a positive polarity, and similarly, the right-most member would have a negative polarity. The same logic extends to the other four sequences.
Each polarity sequence is indicative of two critical load arrangements that maximise the positive or negative internal member forces respectively. The maximum positive load
Figure 4: Polarity zones that occur along various span segments of a \(m=5\) homogeneous beam system of equal span and cross-sectional properties. The same zones and sequences, although at different boundaries, occur in heterogeneous (varying UDL and span) systems.
arrangement for Type I is also equal to the maximum negative load arrangement for Type IV, since these sequences are polar opposites of each other, which is also true for the Type II and Type IV sequences. Consequently, these 5 polarity zones correspond to 6 unique critical load arrangements for a given beam, namely positive Type I, II and III along with their (negative) polar opposites. The only exceptions occur for the beams at either end of the spans, named end-span beams, in which the Type I and Type IV sequences collapse into the Type III sequence (or its polar opposite) at the left end, and similarly for the Type II and Type V sequences at the right end, resulting in four unique load arrangements for end-span beams.
Whilst each non-end-span beam has 6 unique critical load arrangements, it does not mean that the beam system has \(6m\) unique load arrangements (\(m\) is the number of members in the beam system). This is because, as shown in Figure 5, the maximum positive Type V load arrangement for one beam is identical to the maximum positive Type I load arrangement of the beam immediately adjacent to (the right of) it. A similar overlap exists between Type II and Type IV sequences, and the two Type III load arrangements (for maximum and negative internal forces) are identical for all beams. Through a process of elimination, it is possible to simplify the actual total number of potential critical load arrangements to \(p_{\text{flex}}=2m\). This set will be termed the _flexural load arrangements_ set \(\mathbf{J}_{\text{flex}}\), and can be evaluated using Algorithm 1 provided in A, with an example output for a \(m=5\) system shown in Figure 6, grouped in alternating and adjacently loaded arrangements.
The load arrangement set \(\mathbf{J}_{\text{flex}}\) of size \(p_{\text{flex}}=2m\) identified here is a literal exponential improvement to the brute-force approach of analysing and designing against \(p=2^{m}\) load arrangements and for evaluating the influence zone with Equation 2. It needs to be shown though that all critical load arrangements \(\mathbf{J}_{\text{crit}}\) fall within \(\mathbf{J}_{\text{flex}}\) (i.e. \(\mathbf{J}_{\text{crit}}\in\mathbf{J}_{\text{flex}}\)).
Figure 5: Polarity sequences are identical for adjacently lying beams (highlighted in red) for Type I and Type V sequences as shown by Figure a) and Figure c), as well as Type II and Type IV sequences, as shown by Figure b) and Figure d).
### Shear beams and the impact on critical load arrangements
To check if the load arrangement set \(\mathbf{J}_{\text{flex}}\) contains all critical pattern loads, various continuous beam systems of size up to \(m=10\) (to facilitate computational feasibility) were numerically generated, using randomly distributed UDLs and span lengths based on the design constraints identified in Table 1. By calculating all \(2^{m}\) load arrangements, evaluating the utilisation ratio (based on the moment and shear force combinations) of each, the load arrangement that caused the worst-case utilisation ratio could be identified. This set of critical load arrangements was then compared against the \(\mathbf{J}_{\text{flex}}\) set identified by Algorithm 1 provided in A.
Although \(\mathbf{J}_{\text{flex}}\) tended to cause the critical utilisation ratio in the majority of cases, there were instances were other load arrangements not within \(\mathbf{J}_{\text{flex}}\) controlled the design. This unexpected behaviour occurred in cases where short spanning, deep beams were included. Analysing these special cases in detail indicated that the exact conditions under which the previously unidentified load arrangements occur were generally related to the following \(L_{\text{shear}}\) span limit quantified by
\[\sqrt{\frac{6EI_{yy}}{GA_{z}}}<L_{\text{shear}} \tag{3}\]
where \(E\) and \(G\) are the Young's and shear modulus of the material respectively, and \(I_{yy}\) and \(A_{z}\) are the major second moment of area and shear area of the prismatic beam, respectively.
Although the \(L_{\text{shear}}\) span limit appears to be related to shear beams, this is the first time that shear beams have been reported in literature to cause novel critical load arrangements. As shown in Figure 7, shear beams appear to flip the polarity of the immediately adjacent member when measured outwardly from a given IL location, with all subsequent members alternating the polarity direction as before.
When shear beams (as defined by the \(L_{\text{shear}}\) limit) occur, they introduce new critical load arrangements not found within \(\mathbf{J}_{\text{flex}}\). The increase in terms of the final utilisation factor of the beams was typically in the range of 4-5%, although larger increases were also observed. Whilst a thorough analysis of the increase in utilisation ratio caused by these newly identified load arrangements would be of interest, it falls outside the scope of this study. Instead, an algorithm will be presented capable of identifying these new load arrangements _a priori_, which is the main objective of this investigation as explained in section 2.5.
Figure 6: The critical load arrangements set \(\mathbf{J}_{\text{flex}}\) of size \(p=2m\) for a 5-member continuous beam system (\(p=10\)) grouped in alternating and adjacently loaded arrangements.
### Determining shear beam induced critical load arrangements _a priori_
The principal issue when evaluating the shear beam induced critical load arrangements _a priori_, hereafter referred to as the _shear load arrangements_\(\mathbf{J}_{\mathrm{shear}}\), is the fact that the final material and cross-sectional properties to evaluate the \(L_{\mathrm{shear}}\) limit in Equation 3 are not known until the beam is designed. This creates a causality dilemma and hence needs to be addressed.
In clear opposition to the \(\mathbf{J}_{\mathrm{flex}}\) set, which does not depend on the continuous beam system properties, the shear load arrangements cannot be established _in universum_ without some system knowledge. However, by taking advantage of the design constraints set by Equation 1, one can identify _a priori_ what members are potentially susceptible to cause shear load arrangements by re-writing Equation 3 as:
Figure 7: A schematic demonstrating the impact of a shear beam (highlighted in yellow) on a standard polarity sequence of a continuous beam system when spans shorter than the shear span limit \(L_{\mathrm{shear}}\) (as identified by Equation 3) occur. Note the flipped polarity directions of the members on the right-hand side of the system.
\[\sqrt{6\left(\frac{E}{G}\right)_{\max}\left(\frac{I_{yy}}{A_{z}}\right)_{\max}}<L_{ \mathrm{shear,max}} \tag{4}\]
The above equation groups the maximum material and cross-sectional property ratios together. By limited the design space to S355 steel and UKB section sizes as specified in Section 2.4, the maximum material ratio (\((E/G)_{\max}=2.600\)) and cross-sectional property ratio (\((I_{yy}/A_{z})_{\max}=0.397m^{2}\)) can be evaluated. Consequently, beams shorter than the shear span limit are susceptible to cause shear load arrangements (in this case \(L_{\mathrm{shear,max}}=2.49\)m). In identifying these susceptible members _a priori_, it is possible to evaluate the shear load arrangements using Algorithm 2 provided in B.
Algorithm 2 transforms the flexural load arrangement from set \(\mathbf{J}_{\mathrm{flex}}\) based on a list of susceptible shear beams identified by Equation 4. This is achieved by flipping the on/off activation factor of the load arrangement if a shear beam is encountered whilst travelling outwardly in both the left (-1) and right (1) direction from a start beam index. This operation transforms the flexural load arrangement based on the behaviour identified visually in Figure 7, and needs to check four individual case conditions to account for continuous beam systems that have multiple, potentially adjacently lying, shear beams.
Since every beam system is of size \(m\), the time complexity of a single pass of Algorithm 2 is \(O(m)\). However, since every flexural load arrangement (\(2m\)), and every combination of \(n\) potential shear beams (\(2^{n}-1\) combinations, as the zero set is already considered in \(\mathbf{J}_{\mathrm{flex}}\) by default), and every possible start-index (\(m\)) needs to be computed, the time complexity to evaluate the shear set \(\mathbf{J}_{\mathrm{shear}}\) would be \(O(m^{3}\,2^{n})\). It should be noted that this process is computationally expensive.
It was observed that passing every possible start index generated either duplicate shear load arrangements, or occasionally existing flexural load arrangements. For example, for a given singular potential shear beam location, the algorithm would result in the same transformed shear load arrangement for all start-indices starting on the left and right hand-side of that susceptible shear beam location. Similarly, the two alternating arrangements from \(\mathbf{J}_{\mathrm{flex}}\) would result in an already existing adjacent arrangement from \(\mathbf{J}_{\mathrm{flex}}\) if only a singular susceptible shear beam exists.
Using such logic, it is sufficient to pass only adjacent arrangements from \(\mathbf{J}_{\mathrm{flex}}\) along with the left-hand (or right-hand) index of the adjacently loaded spans as the start index for Algorithm 2 to yield an effective set of potential shear load arrangements. By not having to evaluate Algorithm 2 for every possible start index of each load arrangement, the computational complexity reduces to \(O(m^{2}\,2^{n})\). From this, it also follows that since the alternating load arrangement is never transformed (which leaves only \(2(m-1)\) load arrangements to be passed to the algorithm) and since \(2^{n}-1\) possible shear beam combinations can exist, the maximum number of unique critical shear load arrangements should be of size \(p_{\mathrm{shear}}=2(m-1)(2^{n}-1)\).
### Validating flexural and shear load arrangement algorithms
A design data set consisting of 32 UDL and 32 span values sampled from a random uniform distribution for a \(m=10\) beam system was generated based on the high-variation design scenario identified in Section 2.4. Significantly higher variable UDLs
(\(Q_{k,i}\in[200\,\mathrm{kN/m},400\,\mathrm{kN/m}]\)) were applied to increase the likelihood of deep beams and thereby critical shear load arrangements, allowing the performance of the algorithm to be stress-tested. This resulted in \(10\times 32\times 32=10240\) individual beam design examples, for which the critical load arrangement \(J_{\mathrm{crit}}\) could be identified.
The results of this validation exercise are illustrated in Figure 8, which plots the critical load arrangement index for each design beam example. Every load arrangement index corresponds to a unique load arrangement out of the naive set \(\mathbf{J}_{\mathrm{naive}}\) of size \(p_{\mathrm{naive}}=2^{m}=1024\). The set \(\mathbf{J}_{\mathrm{naive}}\) was ordered so that the load arrangements for set \(\mathbf{J}_{\mathrm{flex}}\) are first, followed by those of set \(\mathbf{J}_{\mathrm{shear}}\), and subsequently all others. The design examples themselves were sorted twice: first in ascending number of shear beam occurrences, and subsequently in ascending load arrangement indices. This results in the gradual increase of the \(J_{\mathrm{crit}}\) indices as seen in Figure 8.
Figure 8: Load arrangement index for each design beam example ordered in increasing number of shear beam occurrences and critical load arrangement indices. This confirms visually that the critical load arrangement \(J_{\mathrm{crit}}\) for each design beam example from the generated data set falls within either \(\mathbf{J}_{\mathrm{flex}}\) or \(\mathbf{J}_{\mathrm{shear}}\) and are significantly smaller than \(\mathbf{J}_{\mathrm{naive}}\). Figure b) is an enlarged view of Figure a).
Figure 8 sheds insight on a number of important points. The first is that the critical load arrangement \(J_{\text{crit}}\) for every single beam example from the \(10240\) data set occurred within the \(\mathbf{J}_{\text{flex}}\) or \(\mathbf{J}_{\text{shear}}\) sets, validating the qualitative analysis based on the polarity zones and sequences identified previously. This also emphasises the validity of Algorithm 1 and 2. Furthermore, the set size predictions \(p_{flex}=2m\) and \(p_{shear}=2(m-1)(2^{n}-1)\) are also confirmed. For the \(m=10\) member system designed here, \(p_{flex}=20\), and depending on the number of shear beam occurrences of each system, which varied from \(n=\{0,1,2,3,4\}\), the number of shear load arrangements varied from \(p_{shear}=\{0,18,54,126,270\}\). This corresponded to \(p_{total}=\{20,38,74,146,290\}\) respectively, as indicated by the \(y\)-axis of Figure 8 b).
Figure 8 a) also emphasises how much smaller sets \(\mathbf{J}_{\text{flex}}\) and \(\mathbf{J}_{\text{shear}}\) are in comparison to \(\mathbf{J}_{\text{naive}}\). This will greatly reduce the number of load-arrangements that need to be analysed, reducing the computational cost of both optimally designing the continuous beam system and evaluating the influence zone for system lengths of \(m>10\). Further insights generated by Figure 8 are discussed in Section 5.
### Summary of critical load arrangements of continuous beams
By adding the set of critical flexural and shear load arrangements together, it is possible to explicitly define _a priori_ the set of critical load arrangements for any continuous beam system under defined design constraints. For the purpose of the influence zone concept and Equation 2, it is assumed that: \(\mathbf{J}\rightarrow\mathbf{J}_{\text{crit}}\in\mathbf{J}_{\text{flex}} \cup\mathbf{J}_{\text{shear}}\). The results from this systematic critical load arrangement investigation are summarised in Table 2.
\begin{table}
\begin{tabular}{c c c} \hline \hline Set & Set Size & Algorithm Complexity \\ \hline Critical load arrangements & 6 & \(O(1)\) \\ per internal beam & & \\ Critical load arrangements & 4 & \(O(1)\) \\ per end-span beam & & \\ \(\mathbf{J}_{\text{flex}}\) - Critical flexural arrangements per beam & \(2m\) & \(O(m)\) \\ system & & \\ \(\mathbf{J}_{\text{shear}}\) - Critical shear arrangements per beam & \(2(m-1)(2^{n}-1)\) & \(O(m^{2}\,2^{n})\) \\ system & & \\ \(\mathbf{J}_{\text{naive}}\) - Naive load arrangements & \(2^{m}\) & \(O(2^{m})\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Load arrangements set summary for \(m\) dimensional beam systems containing \(n\) shear beams with associated algorithm complexities.
## 4 Influence zone evaluation
### Explicitly defining the utilisation ratio contribution function
By taking advantage of the concept of integrated influence lines and polarity zones from Section 3.1, and by having explicitly defined the critical load arrangement set \(\mathbf{J}\rightarrow\mathbf{J}_{\mathrm{crit}}\) in Section 3.6, it is possible to define the utilisation ratio contribution function \(\mathbf{u}_{d,i,\,j}\) as:
\[\begin{split}\mathbf{u}_{d,i,\,j}&\rightarrow \mathbf{D}_{\mathrm{ULS}}(I_{d},M_{d,i,j},V_{d,i,j})\\ M_{d,i,j}&=w_{i}\ J_{i,j}\int_{i}\mathbf{M}_{ \mathrm{IL},d}\\ V_{d,i,j}&=w_{i}\ J_{i,j}\int_{i}\mathbf{V}_{ \mathrm{IL},d}\end{split} \tag{5}\]
\(\mathbf{D}_{\mathrm{ULS}}\) represents the ULS steel cross-section design checks based on Eurocode EN 1993-1-1 6.2 [26], \(I_{d}\) represents the cross-sectional properties, \(M_{d,i,j}\) denotes the major axis moment while \(V_{d,i,j}\) is the major axis shear force of the design beam \(d\), \(w_{i}\) is the UDL, and \(J_{i,j}\) is the activation factor of the load arrangement \(j\) from the set \(\mathbf{J}_{\mathrm{crit}}\) for beam \(i\). Integrals \(\int_{i}\mathbf{M}_{\mathrm{IL},\mathrm{d}}\) and \(\int_{i}\mathbf{V}_{\mathrm{IL},\mathrm{d}}\) are the integrated influence line values across beam \(i\) for a particular influence line location within the design beam \(d\) as introduced in Figure 3.
The influence line locations within the design beam need to correspond with the worst-case internal force locations. Whilst engineering experience would dictate those to occur over the supports, they can in fact arise anywhere along the design beam depending on the exact distribution of UDLs, spans and cross-sectional properties, since each segment of the design beam has its own critical load arrangement as highlighted by Figure 4. In this study, a total of 11 influence line locations were sampled, one at either support and another 9 equidistantly distributed between the supports.
For a given design beam and \(k_{\mathrm{max}}\) value, Equation 5 will therefore result in \(11p\) utilisation ratios (recall that \(p\) is the set size of \(\mathbf{J}\)), from which the critical utilisation ratio (the maximum one) is evaluated in Equation 2 to check if the \(\epsilon_{\mathrm{max}}\) threshold has been attained. Note that as \(k_{\mathrm{max}}\) increases, the critical influence line location and the critical load arrangement can vary, yet as \(k_{\mathrm{max}}\) approaches \(m\), they will equate to the location and load arrangement that governed the design for that particular beam.
### Design data set generation
The size of the continuous beam system \(m\) to be modelled needs to be at least double the maximum influence size \(k_{\mathrm{max}}\). This is because the highest influence zone measurable for the middle span of a continuous beam is by design half the system length \(m\). Therefore, size \(m\) needs to be chosen such that \(\max(\mathbf{k}_{\mathrm{max}})<m/2\), where \(\mathbf{k}_{\mathrm{max}}\) is the list of all influence values \(k_{\mathrm{max}}\) of the continuous beam system. Since evaluating \(k_{\mathrm{max}}\) is the main aim of this investigation, a sufficiently large value for \(m\) needs to be assumed; \(m=15\) was used for this purpose.
Individual design data sets consisting of 32 UDL and 32 span values sampled from a random uniform distribution for a \(m=15\) beam system were created based on the design constraints identified in Section 2.4 for Sets 2, 3, and 4, each containing \(32\times 32\times 15=15360\) beam designs. For Set 1, the difference within the beam systems only varied in terms of
the identical span \(L\) and UDLs \(Q_{k}\) of the beams, which were also sampled in 0.5 m and 5 kN/m increments respectively. Given that this results in 23 span and 13 UDL increments for Set 1 respectively, Set 1 contained \(23\times 13\times 15=4485\) design examples.
For the design optimisation of the continuous beam systems, a coupled analysis and design approach was taken, optimising for minimum structural depth. Design sensitivity analysis was avoided by an implicit ordering of the UKB section list based on structural capacity. The influence zone values were extracted using Equation 2 and Equation 5.
### Influence zone results
The influence zone results are shown in Figure 9 for a max error threshold \(\epsilon_{\text{max}}=0.005\) and various design data sets defined in Section 2.4. For all sets investigated, the most common influence zone value (the mode) was \(k_{\text{max}}=3\), and the majority of influence zone values were at \(k_{\text{max}}\leq 3\), meaning the span and applied loading information of a given beam along with that of the three adjacent spans on either side captured the correct utilisation ratio of the design beam with less than a \(\pm 0.5\%\) error in the majority of cases.
However, the various sets reveal differences in the maximum and distribution of the influence zone. The maximum influence zone value for Set 1 was \(k_{\text{max}}=4\), whereas it was
Figure 9: Influence zone results for various design constraints with a max error threshold \(\epsilon_{\text{max}}=0.005\) indicating the percentage frequency distributions of the influence zone values \(k_{\text{max}}\) and minimum utilisation factors captured for each \(k_{\text{max}}\) value for a given design beam \(d\).
\(k_{\max}=5\) for Set 2, 3 and 4. Furthermore, as the set number increases, which corresponds with an increase in variation of the design information in terms of spans and UDLs, the influence zone value distribution appears to flatten and widen. For example, it was the high-variation Set 4 which actually contained the most influence zone values \(k_{\max}=0\) for 1.6% of the design examples, whereas the zero variation Set 1 only had 0.4% of its design examples exhibit an influence zone of \(k_{\max}=0\). The minimum utilisation curve (red curve with point markers in Figure 9) captured by each influence zone value suggests that, in general, increasing design variation leads to greater maximum influence zone values.
The average and maximum influence zone values were also calculated for various error thresholds as shown in Table 3. Note that the maximum influence zone value of \(k_{\max}=7\) for Set 4 with the highest error threshold confirms that the \(m=15\) member-size assumption was sufficient for the purpose of this study. Together Figure 9 and Table 3 provide evidence for the following conclusions:
* A decrease in the acceptable error threshold correlates with an increase in both the average and maximum influence zone range.
* An increase in design variation correlates with an increase in the maximum influence zone range.
* An increase in design variation, however, correlates with a decrease in average influence zone range in most instances where the acceptable error threshold is relatively tight (\(\epsilon_{\max}\leq 10\%\)). At higher error thresholds the trend is less discernible.
It should be noted that an error threshold of less than 0.5% is relatively small in comparison to uncertainties that exist in structural design. These uncertainties include, for example, material yield strength and imposed UDL values (consider that variable UDL values \(Q_{k}\) are increased 50% with a load combination factor of 1.5 within the Eurocodes [27]). Furthermore, the design constraints of design set 4 represent the top end of design variation which may occur in typical continuous beam systems. Consequentially, it is reasonable to suggest that for continuous beam systems with design constraints specified
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{Error \(\epsilon_{\max}\) [\%]} & \multicolumn{4}{c}{Average \(k_{\max}\)} & \multicolumn{4}{c}{Maximum \(k_{\max}\)} \\ \cline{2-10} & Set 1 & Set 2 & Set 3 & Set 4 & Set 1 & Set 2 & Set 3 & Set 4 \\ \hline
0.1 & 4.60 & 4.46 & 3.84 & 3.37 & 5 & 5 & 6 & 7 \\
0.5 & 2.89 & 2.86 & 2.69 & 2.38 & 4 & 5 & 5 & 5 \\
1 & 2.76 & 2.75 & 2.35 & 2.06 & 3 & 3 & 5 & 5 \\
5 & 1.52 & 1.39 & 1.29 & 1.17 & 2 & 3 & 3 & 4 \\
10 & 0.98 & 0.98 & 0.89 & 0.83 & 2 & 2 & 3 & 4 \\
20 & 0.76 & 0.84 & 0.73 & 0.67 & 1 & 1 & 2 & 3 \\
50 & 0.00 & 0.30 & 0.41 & 0.43 & 0 & 1 & 1 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Influence zone results for various maximum error thresholds \(\epsilon_{\max}\) for each design data set, evaluating average and maximum influence zone values \(k_{\max}\). Note that increasing set numbers corresponds with increasing design variation, a proxy for design complexity, see Table 1 for details.
in section 2.4 the influence zone values are on average \(k_{\max}<3\), and in the most extreme case \(k_{\max}=5\).
## 5 Discussion
The results along with the methodology of the influence zone investigation has led to a number of important findings. These include introducing the novel concept of the _structural influence zone_ and a numerical methodology of evaluating it, discovering novel _shear load arrangements_ with the help of _polarity zones_ and _polarity sequences_, and introducing _load arrangement algorithms_ to explicitly identify critical load arrangements of continuous beam systems of any arbitrary member size, which were a necessary prerequisite for the influence zone study. Each of these findings are discussed in detail and contextualised with relevant existing literature.
### Influence zone insights
The influence zone results confirm that the impact of loading, and by extension of any design information, drops off sharply the further away one moves from the influence line location. This behaviour can be identified across all influence line diagrams found within this paper, such as Figures 3 and 7. This investigation has formulated this concept as the _influence zone_, shown how it applied to continuous beam systems, and rigorously studied the influence zone distributions under various design assumptions and error thresholds.
An important element of the influence zone definition in Section 2.3 should be brought to light. The influence zone value \(k_{\max}\) is only found when two important conditions are met, notably when both the smallest value of \(k_{\max}\) and all values larger than this value of \(k_{\max}\) conform to Equation 2. This was a necessary constraint to account for the fact that the ratio \(u_{d,cap}/u_{d,true}\) sometimes converged towards unity by oscillation.
The cause of this behaviour results from adjacent load arrangements (see Figure 6) which were sometimes more critical when only a segment of (as opposed to the entire) load arrangement was considered. For example, the maximum \(u_{d,cap}/u_{d,true}\) ratio calculated for Set 4 when assuming an influence zone value of \(k_{\max}=1\) was 1.89, only for the ratio to drop to 0.942 and 0.998 for influence zone values \(k_{\max}=2\) and \(k_{\max}=3\), respectively. Future research on influence zones should keep this non-intuitive behaviour in mind, since a simple \(u_{d,cap}/u_{d,true}<r_{cap}\) threshold, in which \(r_{cap}\) represents a minimum threshold of captured utilisation would lead to an underestimation of the influence zone distribution.
### Demarcating influence zones from influence lines
Although there is a proximal relationship between the concept of influence zones and influence lines, mostly evidenced by Equation 5 where integrated influence lines play an important role for the evaluation of influence zones, these two concepts differentiate themselves in important ways. This distinction also applies to the two-dimensional application of influence lines known as influence surfaces [33, 34, 35, 36].
Whilst influence lines/surfaces are exact analytical tools that define the mechanical response of a known structural system about a particular point, influence zones are a heuristic design tool that offer insight on what information is relevant to the design of
the structural system to begin with based on certain analytical assumptions. The value of influence lines/surfaces arise during analysis on a system-by-system basis, whereas the value of influence zones arise during design after having studied them in their statistical aggregate.
This distinction could be considered further evidence supporting the demarcation between design and analysis in structural engineering. Previous literature has highlighted the difference between _knowledge-that_ explains fundamental facts about systems (such as influence lines) versus _knowledge-how_ something can be designed or solved (such as influence zones) [37, 38]. Recent literature has suggested that the processes of analysis and design solve related, albeit oppositely posed problems known as forward and inverse problems respectively [39]. Influence lines can be seen as a tool that solves the former, whereas influence zones solve the latter.
As a matter of fact, the influence zone concept was developed whilst developing a design model for continuous beam systems from an inverse problem perspective, and allows the _a priori_ knowledge of what span and loading information is relevant for design of a particular continuous beam. It is possible that the influence zone concept could serve as an important heuristic tool in the design of continuous structural systems, supporting the view that the application of heuristics is a cornerstone for engineering design [40]. Further novel ideas might be uncovered when approaching engineering design from an inverse problem perspective.
### Flexural load arrangements
An important contribution of this investigation was presenting the flexural load arrangements clearly through the use of _polarity sequences_. Notably the _polarity zones_ highlight which load arrangement is critical for specific segments of a beam, which could be useful in the design of tapered (non-prismatic) continuous beam systems [41, 42].
The influence zone study allows the contextualisation of simplified load arrangement provisions. For example, whilst Annex AB.2 from EN 1993-1-1 [26] covers alternating flexural load arrangements in full, it specifies that for the adjacent flexural load arrangement type, the two adjacently loaded spans are the only spans required to factor the variable load (\(Q_{k}\)). In essence, the variable load information on all other spans aside from the beam under consideration and the two directly adjacent spans are ignored, which is the technical equivalent of assuming an influence zone to \(k_{\max}=1\).
With help of Table 3, it is possible to infer that an influence zone value \(k_{\max}=1\) is likely to introduce an error between \(5-10\%\) in terms of the true utilisation for design scenarios with no UDL or span variation (the average \(k_{\max}\) value for \(\epsilon_{\max}=5\%\) and \(\epsilon_{\max}=10\%\) is \(1.52\) and \(0.98\) for Set 1 respectively). The simplified Eurocode provisions are therefore, on average, a reasonable simplification to capture the impact of variable load arrangements. However, the maximum influence zone value of Set 1 with \(k_{\max}=1\) corresponds to an error of \(\epsilon_{\max}=20\%\), and when considering non-heterogeneous continuous beam systems (reflected by Set 2, 3 and 4), this error can increase up to \(\epsilon_{\max}=50\%\) and more. This is further evidence, as already pointed out in literature, that the load arrangement provisions from building codes can be non-conservative and hence lead to unsafe designs [32].
The simplified provisions within the Eurocodes, which also exist within EN 1992-1-1
5.1.3 [43] and other codes [44], need to be understood in context of the \(1.5Q_{k}\) load factors and the dead load contribution \(G_{k}\), which invariably will lessen the underestimation made by the provisions. Nonetheless, the validity of the design code recommendations for flexural load arrangements could be investigated further, especially for highly irregular beam and floor arrangements [45].
### Shear load arrangements
Unlike flexural load arrangements, which have been identified in literature and building codes, the shear load arrangements are a novel discovery. To the authors' knowledge, this is the first time that deep beams have been identified to cause new critical load arrangements in literature. Although shear load arrangements sometimes resulted in identical utilisation ratios to that of flexural ones, initial analyses pointed to an average increase in utilisation ratio of 4-5%, while larger deviations were occasionally observed. Figure 8 also highlights that these shear load arrangements were relatively prevalent within the design scenarios considered.
Confirmation and validation of these shear load arrangements by future research is encouraged. Of particular interest is why Equation 4 defines the exact point when these critical load arrangements arise. One notable difference in the mechanical assumption in this investigation of load arrangements as to that of previous studies was the use of Timoshenko-Ehrenfest rather than Euler-Bernoulli beam theory. For example, the two seminal works on establishing the bounds of critical load arrangements using fuzzy set based finite-element methods used Bernoulli-Euler beam theory [31, 32]. A re-investigation with deep beams as defined by Equation 3 and Timoshenko-Ehrenfest beam theory should reveal more critical bounds of load arrangements than previously identified with interval-finite-element methods. The extent to which these shear load arrangements require special provisions within building codes will require further exploration.
### Critical load arrangement algorithms
The critical _load arrangement algorithms_ provided in A and B, along with a study of their computational complexity, were key for the evaluation of the influence zone. Limiting the design space to a fraction of the naive \(J_{naive}\) load arrangement set without making heuristic simplifications was crucial in both the data set generation and influence zone evaluation steps.
It is likely that there is further room for improvement for Algorithm 2 for evaluating the shear load arrangements for a known list of susceptible shear beams. The current formulation, as explained in Section 3.4, still leads to either pre-existing flexural load arrangements, or creates duplicate shear load arrangements. On average, 74.7% of the outputs obtained from Algorithm B were unique, with a best-case efficiency of 88.8% and a worst-case efficiency of 12.7%. This suggests that an algorithm with a lesser computational complexity than \(O(m^{2}\,2^{n})\) might be achievable through further investigation.
### Future investigations and application of the influence zone concept
This investigation will hopefully serve as a starting point for future studies related to the influence zone. There were several limitations within this study, notably not accounting for serviceability checks and limiting the design space to positively loaded UDLs. Furthermore, only 11 equidistant points were sampled about each beam during design and extraction of influence zone values. A more efficient approach could sample specific points against the critical load arrangement that apply to that particular polarity zone.
Further studies could be conducted for different material and design information assumptions, while studies could also be expanded to 2D continuous frames and shells, with fixed and semi-rigid connections. As previously explained, such numerical studies could be validated with either analytical or experimental approaches, along with using local, as opposed to global, influence zone formulations as discussed in section 2.2.
The influence zone concept and associated results could be a helpful piece of information when teaching the design of large-scale, steel-framed continuous beam systems, may have applications in other research areas such as reliability engineering, and help in the future development of generalised design models [39].
## 6 Conclusions
A novel concept termed the _influence zone_ was proposed in relation to continuous beam systems. The investigation developed a local and global formulation, of which the latter one was explored numerically with design constraints applicable to steel framed buildings. The key challenge was the explicit definition of critical load arrangements to allow the computational feasible generation of design data sets and evaluation of their respective influence zones. The investigation led to three important outcomes:
* The development of polarity sequences and polarity zones which led to the demarcation between previously known flexural load arrangements and the newly discovered shear load arrangements, with an explicit span limit equation for when these novel load arrangements occur.
* Two algorithms capable of finding these two types of load arrangements, and providing evidence that they encompass all critical permutations in comparison to the naive, brute-force approach.
* The generation of design data sets from which the influence zone values for various degrees of design complexities and error thresholds could be rigorously studied. For error thresholds deemed acceptable in structural design, the influence zone for continuous beams within steel framed building under ultimate state considerations is on average less than 3, going to a maximum influence zone value of 5.
The influence zone is a heuristic design tool that differentiates itself from influence lines (and influence surfaces) and demonstrates the value of the inverse problem perspective through which it was evaluated by. This study opens the scope for future research, notably in the evaluation of influence zones for various materials and structural systems, validating and explicating the existence of shear load arrangements, and encouraging research on improving the existing algorithm that identifies them.
## Acknowledgements
**Authors' contributions:** Adrien Gallet: Conceptualization, Methodology, Investigation, Software, Formal analysis, Validation, Visualization, Writing - Original Draft Andrew Liew: Writing - Review & Editing Iman Hajirasouliha: Writing - Review & Editing Danny Smyl: Supervision, Writing - Review & Editing
**Data statement:** Data used in this article are available from the authors upon request.
**Competing interests:** The authors declare that they have no competing interests.
| structural influence zone |
2307.13565 | Decision-Focused Learning: Foundations, State of the Art, Benchmark and
Future Opportunities | Decision-focused learning (DFL) is an emerging paradigm that integrates
machine learning (ML) and constrained optimization to enhance decision quality
by training ML models in an end-to-end system. This approach shows significant
potential to revolutionize combinatorial decision-making in real-world
applications that operate under uncertainty, where estimating unknown
parameters within decision models is a major challenge. This paper presents a
comprehensive review of DFL, providing an in-depth analysis of both
gradient-based and gradient-free techniques used to combine ML and constrained
optimization. It evaluates the strengths and limitations of these techniques
and includes an extensive empirical evaluation of eleven methods across seven
problems. The survey also offers insights into recent advancements and future
research directions in DFL.
Code and benchmark: https://github.com/PredOpt/predopt-benchmarks | Jayanta Mandi, James Kotary, Senne Berden, Maxime Mulamba, Victor Bucarey, Tias Guns, Ferdinando Fioretto | 2023-07-25T15:17:31 | http://arxiv.org/abs/2307.13565v4 | # Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities
###### Abstract
Decision-focused learning (DFL) is an emerging paradigm in machine learning which trains a model to optimize decisions, integrating prediction and optimization in an end-to-end system. This paradigm holds the promise to revolutionize decision-making in many real-world applications which operate under uncertainty, where the estimation of unknown parameters within these decision models often becomes a substantial roadblock. This paper presents a comprehensive review of DFL. It provides an in-depth analysis of the various techniques devised to integrate machine learning and optimization models, introduces a taxonomy of DFL methods distinguished by their unique characteristics, and conducts an extensive empirical evaluation of these methods proposing suitable benchmark dataset and tasks for DFL. Finally, the study provides valuable insights into current and potential future avenues in DFL research.
## 1 Introduction
Real-world applications frequently confront the task of decision-making under uncertainty, such as planning the shortest route in a city, determining optimal power generation schedules, or managing investment portfolios (Sahinidis, 2004; Liu & Liu, 2009; Kim, Lewis, & White, 2005; Hu, Wang, & Gooi, 2016; Delage & Ye, 2010; Garlappi, Uppal, & Wang, 2006). In such scenarios, estimating unknown parameters often poses a significant challenge.
Machine Learning (ML) and Constrained Optimization (CO) serve as two key tools for these complex problems. ML models estimate uncertain quantities, while CO models optimize objectives within constrained spaces. This sequential process, commonly referred
to as _predictive_ and _prescriptive_ modeling, as illustrated in Figure 1, is prevalent in fields like operations research and business analytics (den Hertog and Postek, 2016). For instance, in portfolio management, the prediction stage forecasts asset returns, while the prescriptive phase optimizes returns based on these predictions.
A commonly adopted approach involves handling these two stages--prediction and optimization--separately and independently. This "two-stage" process first involves training an ML model to create a mapping between observed features and the relevant parameters of a CO problem. Subsequently, and independently, a specialized optimization algorithm is used to solve the decision problem, which is specified by the predicted problem parameters. The underlying assumption in this methodology is that superior predictions would lead to precise models and consequently, high-quality decisions. Indeed, if the predictions of parameters were perfectly accurate, they would enable the correct specification of CO models which can be solved to yield fully optimal decisions. However, ML models often fall short of perfect accuracy, leading to suboptimal decisions due to propagated prediction errors. Thus, in many applications, the predictive and prescriptive modelings are not isolated but rather, deeply interconnected, and hence should ideally be modeled jointly.
This is the goal of the **decision-focused learning** (DFL) paradigm, which directly trains the ML model to make predictions that lead to good decisions. In other words, DFL integrates prediction and optimization in an end-to-end system trained to optimize a criterion (i.e., a loss function) that is based on the resulting decisions.
Since many ML models, including neural networks (NNs), are trained via gradient-based optimization, the gradients of the loss must be backpropagated through each constituent operation of the model. In DFL, the loss function is dependent on the solution of an optimization model, thus the optimization solver is _embedded_ as a component of the ML model. In this integration of prediction and optimization, a key challenge is _differentiating through the optimization problem_. An additional challenge arises from decision models operating on discrete variables, which produce discontinuous mappings and hinder gradient-based learning. Hence, examining _smooth_ surrogate models for these discrete mappings, along with their differentiation, becomes crucial. These two challenges are the core emphasis and central focal points in DFL.
Figure 1: Decision-making under uncertainty involves both predictive and prescriptive analytics. In the predictive stage, the uncertain parameters are predicted from the feature variables using an ML model. In the prescriptive stage, a decision is prescribed by solving a CO problem using the predicted parameters.
This manuscript presents a comprehensive survey of decision-focused learning and makes several contributions. First, to navigate the complex methodologies developed in recent years, the paper proposes the first categorization of DFL methods into four distinct classes: **(1)** analytical differentiation of optimization mappings, **(2)** analytical smoothing of optimization mappings, **(3)** smoothing by random perturbations, and **(4)** differentiation of surrogate loss functions. This categorization, as illustrated in Figure 4, serves as a framework for comprehending and organizing various DFL methodologies. Next, the paper compiles a selection of problem-specific DFL models, making them publicly available to facilitate broader access and usage. An integral part of this paper involves benchmarking the performance of various available methodologies on _seven_ distinct problems. This provides an opportunity for comparative understanding and assists in identifying the relative strengths and weaknesses of each approach. The code and data used in the benchmarking are accessible through [https://github.com/PredOpt/predopt-benchmarks](https://github.com/PredOpt/predopt-benchmarks). Finally, this survey addresses the critical need to look forward, by discussing the outstanding challenges and offering an outlook on potential future directions in the field of DFL.
### Paper organization.
Following this introduction, the paper is structured as follows. Preliminary concepts are discussed in Section 2, which introduces the problem setting and explicates the challenges in implementing DFL. The subsequent Section 3 offers a comprehensive review of recently proposed methodologies for handling these challenges, neatly organized into broad classes of related techniques. Secion 4 presents interesting real-world examples of DFL applications. Section 5 brings forth seven benchmark DFL tasks from public datasets, with a comparative evaluation of eight DFL methodologies presented in the following section. The manuscript concludes by providing a discourse on the current challenges and possible future directions in DFL research.
## 2 Preliminaries
This section presents an overview of the problem setting, along with preliminary concepts and essential terminology. Then, the central modeling challenges are discussed, setting the stage for a review of current methodologies in the design and implementation of DFL solutions. Throughout the manuscript, vectors are denoted by boldface lowercase letters, such as \(\mathbf{x}\), while scalar components within the vector \(\mathbf{x}\) are represented with a subscript \(i\), denoting the \(i^{\text{th}}\) item within \(\mathbf{x}\) as \(x_{i}\). Similarly, the vectors \(\mathbf{1}\) and \(\mathbf{0}\) symbolize the vector of all-ones and all-zeros, respectively.
### Problem Setting
In operations research and business analytics, decisions are often quantitatively modeled using CO problems. These problems model various decision-making scenarios, but may not be efficiently solvable and often demand specialized solution algorithms that are tailored to their specific form. In many real-world applications, some parameters of the CO problems are uncertain and must be inferred from contextual data (hereafter referred to as _features_). The settings considered in this manuscript involve estimating those parameters through
predictive inferences made by ML models, and subsequently, the final decisions are modeled as the solution to the CO problems based on those inferences.
In this setting, the decision-making processes can be described by _parametric_ CO problems, defined as,
\[\mathbf{x}^{\star}(\mathbf{c})=\underset{\mathbf{x}}{\operatorname{ argmin}}\;\;f(\mathbf{x},\mathbf{c}) \tag{1a}\] \[\mathtt{s.t.}\;\;\boldsymbol{g}(\mathbf{x},\mathbf{c}) \leq\mathbf{0}\] (1b) \[\boldsymbol{h}(\mathbf{x},\mathbf{c}) =\mathbf{0}. \tag{1c}\]
The goal of the optimization problem above is to find \(a\) solution \(\mathbf{x}^{\star}(\mathbf{c})\in\mathbb{R}^{n}\), a minimizer of the objective function \(f\), satisfying a set \(\boldsymbol{g}\) of inequality and a set \(\boldsymbol{h}\) of equality constraints. The _parametric_ problem formulation defines \(\mathbf{x}^{\star}(\mathbf{c})\) as a function of the parameters \(\mathbf{c}\in\mathbb{R}^{k}\). In the present setting, this function can naturally be interpreted as part of an overall composite function that encompasses ML inference and decision-making, and returns optimal decisions given feature variables as input.
CO problems can be categorized in terms of the forms taken by the functions defining their objectives (1a) and constraints (1b-1c). These forms also determine important properties of the optimization mapping \(\mathbf{c}\to\mathbf{x}^{\star}(\mathbf{c})\) when viewed as a function from problem parameters to optimal solutions, such as its continuity, differentiability, and injectivity.
In this manuscript, it is assumed that the constraints are fully known prior to solving, i.e., \(\boldsymbol{h}(\mathbf{x},\mathbf{c})=\boldsymbol{h}(\mathbf{x})\) and \(\boldsymbol{g}(\mathbf{x},\mathbf{c})=\boldsymbol{g}(\mathbf{x})\), and restrict the dependence on \(\mathbf{c}\) to the objective function only. This is the setting considered by almost all existing works surveyed. While it is also possible to consider uncertainty in the constraints, this leads to the possibility of predicting parameters that lead to solutions that are infeasible with respect to the ground-truth parameters. The learning problem has not yet been well-defined in this setting (unless a recourse action to correct infeasible solutions is used (Hu, Lee, & Lee, 2022, 2023a)). For this reason, in the following sections, only \(f\) is assumed to depend on \(\mathbf{c}\), so that \(\boldsymbol{g}(\mathbf{x})\leq\mathbf{0}\) and \(\boldsymbol{h}(\mathbf{x})=\mathbf{0}\) are satisfied for all outputs of the decision model. For notational convenience, the feasible region of the CO problem in (1), will be denoted by \(\mathcal{F}\) (i.e., \(\mathbf{x}\in\mathcal{F}\) if and only if \(\boldsymbol{g}(\mathbf{x})\leq\mathbf{0}\) and \(\boldsymbol{h}(\mathbf{x})=\mathbf{0}\)).
If the true parameters \(\mathbf{c}\) are known exactly, the corresponding 'true' optimal decisions may be computed by solving (1). In such scenarios, \(\mathbf{x}^{\star}(\mathbf{c})\) will referred to as the _full-information optimal decisions_(Bertsimas & Kallus, 2020). This paper, instead, considers problems where the parameters \(\mathbf{c}\) are unknown but can be estimated as a function of empirically observed features \(\mathbf{z}\). The problem of estimating \(\mathbf{c}\) falls under the category of supervised machine learning problems. In this setting, a set of past observation pairs \(\{(\mathbf{z}_{i},\mathbf{c}_{i})\}_{i=1}^{N}\) is available and used to train a ML model \(m_{\boldsymbol{\omega}}\) (with trainable parameters \(\boldsymbol{\omega}\)), so that parameter predictions take the form \(\mathbf{\hat{c}}=m_{\boldsymbol{\omega}}(\mathbf{z})\). Then, a decision \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) can be made based on the predicted parameters. \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) is referred to as a _prescriptive decision_. The overall learning goal is to optimize the set of prescriptive decisions made over a distribution of feature variables \(\mathbf{z}\sim\mathcal{Z}\), with respect to some evaluation criterion on those decisions. Thus, while the machine learning model \(m_{\boldsymbol{\omega}}\) is trained to predict \(\mathbf{\hat{c}}\), its performance is evaluated on the basis of the corresponding optimal solutions \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\). This paper uses the terminology _Predict-Then-Optimize_ problem to refer to the problem of predicting \(\mathbf{\hat{c}}\), to improve the evaluation of \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\).
### Learning Paradigms
The defining challenge of the Predict-Then-Optimize problem setting is the gap in modeling between the prediction and the optimization components: while \(m_{\mathbf{\omega}}\) is trained to predict \(\mathbf{\hat{c}}\), it is evaluated based on the subsequently computed \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\). Using standard ML approaches, learning of the predictions \(\mathbf{\hat{c}}=m_{\mathbf{\omega}}(\mathbf{z})\) can only be supervised by the ground-truth \(\mathbf{c}\) under standard loss functions \(\mathcal{L}\), such as mean squared error or cross-entropy. In principle, it is favorable to train \(m_{\mathbf{\omega}}\) to make predictions \(\mathbf{\hat{c}}\) that optimize the evaluation criterion on \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) directly. This distinction motivates the definition of two alternative learning paradigms for Predict-Then-Optimize problems.
Prediction-focused learning (PFL).A straightforward approach to this supervised ML problem is to train the model to generate accurate parameter predictions \(\mathbf{\hat{c}}\) with respect to ground-truth values \(\mathbf{c}\). This paper introduces the term _prediction-focused learning_ to refer to this approach (also called two-stage learning (Wilder, Dilkina, & Tambe, 2019a)) because the model is trained with a focus on the accuracy of the parameter predictions preceding the decision model. Here, the training is agnostic of the downstream optimization problem. At the time of making the decision, the pre-trained model's predictions \(\mathbf{\hat{c}}\) are passed to optimization routines which solve (1) to return \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\). Typical ML losses, such as the mean squared error (MSE) or binary cross entropy (BCE), are used to train the prediction model in this case.
\[MSE(\mathbf{\hat{c}},\mathbf{c})=\frac{1}{N}\|\mathbf{c}-\mathbf{\hat{c}}\|^{2} \tag{2}\]
Such loss functions, like Eq. (2), which measure the prediction error of \(\mathbf{\hat{c}}\) with respect to \(\mathbf{c}\), are referred to as _prediction losses_. Algorithm 1 illustrates prediction-focused learning using the MSE loss.
Decision-focused learning (DFL).By contrast, in _decision-focused_ learning, the ML model is trained to optimize the evaluation criteria which measure the quality of the resulting decisions. As the decisions are realized after the optimization stage, this requires the integration of prediction and optimization components, into a composite model which produces full decisions. From this point of view, generating the predicted parameters \(\mathbf{\hat{c}}\) is an intermediary step of the integrated approach, and the accuracy of \(\mathbf{\hat{c}}\) is not the primary focus in training. The focus, rather, is on the error incurred after optimization. A measure of error with respect to the integrated model's prescriptive decisions, when used as a loss function for training, is henceforth referred to as a _task loss_. The essential difference from the aforementioned prediction loss is that it measures the error in \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\), rather than in \(\mathbf{\hat{c}}\).
The objective value achieved by using the predicted \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) is generally suboptimal with respect to the true objective parameters \(\mathbf{c}\). Often, the end goal is to generate predictions \(\mathbf{\hat{c}}\) with an optimal solution \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) whose objective value in practice (i.e., \(f(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{c})\)) comes close to the full-information optimal value \(f(\mathbf{x}^{\star}(\mathbf{c}),\mathbf{c})\). In such cases, a salient notion of task loss is the _regret_, defined as the difference between the full-information optimal objective value and the objective value realized by the prescriptive decision. Equivalently, it is the magnitude of suboptimality of the decision \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) with respect to the optimal solution \(\mathbf{x}^{\star}(\mathbf{c})\) under ground-truth parameters \(\mathbf{c}\):
\[\textit{Regret}(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{c})=f(\mathbf{x} ^{\star}(\mathbf{\hat{c}}),\mathbf{c})-f(\mathbf{x}^{\star}(\mathbf{c}), \mathbf{c}) \tag{3}\]
Note that minimizing regret is equivalent to minimizing the value of \(f(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{c})\), since the term \(f(\mathbf{x}^{\star}(\mathbf{c}),\mathbf{c})\) is constant with respect to the prediction model. While regret may be considered the quintessential example of a task loss, other task losses can arise in practice. For example, when the ground-truth target data are observed in terms of decision values \(\mathbf{x}\), rather than parameter values \(\mathbf{c}\), they may be targeted using the typical training loss functions such as \(MSE(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{x})\).
Relationship between prediction and task losses.As previously mentioned, an ML model is trained without considering the downstream CO problem in prediction-focused learning for Predict-Then-Optimize tasks; still the ML model is evaluated at test time on the basis of its resulting CO problem solutions. This is based on an underlying assumption that generating accurate predictions with respect to a standard prediction loss will result in good prescriptive decisions. Note that zero prediction loss always implies zero task loss, since \(\mathbf{\hat{c}}=\mathbf{c}\) implies \(\mathbf{x}^{\star}(\mathbf{\hat{c}})=\mathbf{x}^{\star}(\mathbf{c})\). However, in practice, it is impossible to learn a model that makes no prediction error on any sample. The model error can only be minimized in one metric, and the minimization of the prediction error and the resulting decision error do not in general coincide (Wilder et al., 2019). Furthermore, the prediction loss and the task loss are, in general, not continuously related. These principles are illustrated by the following example:
Example.The shortcomings of training with respect to prediction errors can be illustrated with a relatively simple CO problem. For this illustration, consider a knapsack problem (Pisinger & Toth, 1998). The objective of the knapsack problem is to select a subset of maximal value from an overall set of items, each having its own value and unit weight, subject to a capacity constraint. The capacity constraint imposes that the sum of the weights of the selected items cannot be higher than the capacity \(C\). This knapsack problem with unit weights can be formulated as follows:
\[\mathbf{x}^{\star}(\mathbf{c})=\operatorname*{argmax}_{\mathbf{x}\in\{0,1\}} \mathbf{c}^{\top}\mathbf{x}\ \ \texttt{s.t.}\sum_{i}x_{i}\leq\text{Capacity} \tag{4}\]
In a Predict-Then-Optimize variant of this knapsack problem, the item weights and knapsack capacity are known, but the item values are unknown and must be predicted using observed features. The ground-truth item value \(\mathbf{c}\) implies the ground-truth solution \(\mathbf{x}^{\star}(\mathbf{c})\). Overestimating the values of the items that are chosen in \(\mathbf{x}^{\star}(\mathbf{c})\) (or underestimating the values of the items that are not chosen) increases the prediction error. Note that these kind of prediction errors, even if they are high, do not affect the solution, and thus do not affect the task loss either. On the other hand, even low prediction errors for some item values may change the solution, affecting the task loss. That is why after a certain point, reducing prediction errors does not decrease task loss, and sometimes may increase it. DFL aims to address this shortcoming of PFL: by minimizing the task loss directly, prediction errors are implicitly traded off on the basis of how they affect the resulting decision errors.
The discrepancy between the prediction loss and the task loss has been exemplified in Figure 2 for a very simple knapsack problem with only two items. For this illustration, assume that both the items are of unit weights and the capacity of the knapsack is one, i.e., only one of the two items can be selected. The true values of the first and second items are 2.5 and 3 respectively. The point \((2.5,3)\), marked with \(\clubsuit\), represents the true item values. In
this case the true solution is \((0,1)\), which corresponds to selecting only the second item. It is evident that any prediction in the blue shaded region leads to this solution. For instance, the point \((1.5,3)\), marked with \(\clubsuit\), corresponds to predicting \(1.5\) and \(3\) as values of the two items respectively and this results in selecting the second item. On the other hand, the point \((2.5,2)\), marked with \(\clubsuit\), triggers the wrong solution \((1,0)\), although the squared error values of \(\clubsuit\) and \(\clubsuit\) are identical. Also, note that overestimating the value of the second item does not change the solution. For instance, the point \((1.5,4)\), marked with \(\clubsuit\), corresponds to overestimating the value of the second item to \(4\) while keeping the value of the first item the same as the point in \(\clubsuit\). This point is positioned directly above the point in \(\clubsuit\) and still stays in the blue-shaded region. Similarly, the point \((0.5,3)\), marked with \(\clubsuit\), results from underestimating the value of the first item and is in the blue shaded region too. Although these two points have higher values of squared error than the point marked with \(\clubsuit\), they trigger the right solution, resulting in zero regret.
Empirical risk minimization and bilevel form of DFL.The minimization of either the prediction loss in PFL or the task loss in DFL, can be expressed as an _empirical risk minimization_ (ERM) (Vapnik, 1999) problem over a training dataset containing feature variables and their corresponding parameters \(\mathcal{D}\equiv\{(\mathbf{z_{i}},\mathbf{c_{i}})\}_{i=1}^{N}\). For concreteness, the respective ERM problems below assume the use of the MSE and regret loss functions, but the principles described here hold for a wide range of alternative loss functions.
Figure 2: An illustrative numerical example with a knapsack problem with two items to exemplify the discrepancy between prediction error and regret. The figure illustrates that two points can have the same prediction error but different regret. Furthermore, it demonstrates that overestimating the values of the selected items or underestimating the values of the items that are left out does not change the solution, and thus does not increase the regret, even though the prediction error does increase.
PFL, by minimizing the prediction error with respect to the ground-truth parameters directly, takes the form of a standard regression problem:
\[\min_{\mathbf{\omega}}\frac{1}{N}\sum_{i=1}^{N}\|m_{\mathbf{\omega}}(\mathbf{z_{i}})- \mathbf{c_{i}}\|^{2}, \tag{5}\]
which is an instance of unconstrained optimization. In the case of DFL, it is natural to view the ERM as a bilevel optimization problem:
\[\min_{\mathbf{\omega}}\frac{1}{N}\sum_{i=1}^{N}\left(f(\mathbf{x^{*}} (\mathbf{\hat{c}_{i}}),\mathbf{c_{i}})-f(\mathbf{x^{*}}(\mathbf{c_{i}}), \mathbf{c_{i}})\right) \tag{6a}\] \[\mathtt{s.t.}\ \ \mathbf{\hat{c}_{i}}=m_{\mathbf{\omega}}(\mathbf{z_{i}}); \ \mathbf{x^{*}}(\mathbf{\hat{c}_{i}})=\operatorname*{argmin}_{\mathbf{x}\in \mathcal{F}}\mathbf{\hat{c}_{i}}^{\top}\mathbf{x}. \tag{6b}\]
The outer-level problem (6a) minimizes task loss on the training set while the inner-level problem (6b) computes the mapping \(\mathbf{c}\rightarrow\mathbf{x^{*}}(\mathbf{c})\). Solving (6) is computationally more challenging than solving (5) in the prediction-focused paradigm. In both cases, optimization by stochastic gradient descent (SGD) is the preferred solution method for training neural networks.
Algorithms 1 and 2 compare the gradient descent training schemes for each of these problems. Algorithm 1 is a standard application of gradient descent, in which the derivatives of Line 6 are generally well-defined and can be computed straightforwardly (typically by automatic differentiation). Line 7 of Algorithm 2 shows that direct differentiation of the mapping \(\mathbf{c}\rightarrow\mathbf{x^{*}}(\mathbf{c})\) can be used to form the overall task loss gradient \(\frac{d\mathcal{L}}{d\mathbf{\omega}}\), by providing the required chain rule term \(\frac{d\mathbf{x^{*}}(\mathbf{\hat{c}})}{d\mathbf{\hat{c}}}\). However, this differentiation is nontrivial as the mapping itself lacks a closed-form representation. Further, many interesting and practical optimization problems are inherently nondifferentiable and even discontinuous as functions of their parameters, precluding the direct application of Algorithm 2 to optimize (6) by gradient descent. The following subsections review the main challenges of implementing Algorithm 2.
```
0: training data D\(\equiv\{(\mathbf{z_{i}},\mathbf{c_{i}})\}_{i=1}^{N}\) Hyperparams: \(\alpha\)- learning rate
1: Initialize \(\mathbf{\omega}\).
2:for each epoch do
3:for each instance \((\mathbf{z},\mathbf{c})\) do
4:\(\mathbf{\hat{c}}=m_{\mathbf{\omega}}(\mathbf{z})\)
5:\(\mathcal{L}=(\mathbf{\hat{c}}-\mathbf{c})^{2}\)
6:\(\mathbf{\omega}\leftarrow\mathbf{\omega}-\alpha\frac{d\mathcal{L}}{d\mathbf{\hat{c}} }\frac{d\mathbf{\hat{c}}}{d\mathbf{\omega}}\)
7:endfor
8:endfor
```
**Algorithm 1** Gradient-descent in prediction-focused learning
```
0:\(\mathcal{F}\), training data D\(\equiv\{(\mathbf{z_{i}},\mathbf{c_{i}},\mathbf{x^{*}}(\mathbf{c_{i}})\}_{i=1}^{N}\); Hyperparams: \(\alpha\)- learning rate
1:Initialize \(\mathbf{\omega}\).
2:for each epoch do
3:for each instance \((\mathbf{z},\mathbf{c},\mathbf{x^{*}}(\mathbf{c}))\) do
4:\(\mathbf{\hat{c}}=m_{\mathbf{\omega}}(\mathbf{z})\)
5:\(\mathbf{x^{*}}(\mathbf{\hat{c}})=\operatorname*{argmin}_{\mathbf{x}\in\mathcal{ F}}f(\mathbf{x},\mathbf{\hat{c}})\)
6:\(\mathcal{L}=f(\mathbf{x^{*}}(\mathbf{\hat{c}}),\mathbf{c})-f(\mathbf{x^{*}}( \mathbf{c}),\mathbf{c})\)
7:\(\mathbf{\omega}\leftarrow\mathbf{\omega}-\alpha\frac{d\mathcal{L}}{d\mathbf{x^{*}}( \mathbf{\hat{c}})}\frac{d\mathbf{x^{*}}(\mathbf{\hat{c}})}{d\mathbf{\hat{c} }}\frac{d\mathbf{\hat{c}}}{d\mathbf{\omega}}\)
8:endfor
9:endfor
```
**Algorithm 2** Gradient-descent in decision-focused learning with regret as task loss
### Challenges to Implement DFL
Differentiation of CO mappings.To minimize a task loss by gradient descent training, its partial derivatives with respect to the prediction model parameters \(\mathbf{\omega}\) must be computed to carry out at the parameter update at Line 7 of Algorithm 2. Since the task loss \(\mathcal{L}\) is a function of \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\), the gradient of \(\mathcal{L}\) with respect to \(\mathbf{\omega}\) can be expressed in the following terms by using the chain rule of differentiation:
\[\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{c})}{d\mathbf{ \omega}}=\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{c})}{ d\mathbf{x}^{\star}(\mathbf{\hat{c}})}\frac{d\mathbf{x}^{\star}(\mathbf{\hat{c}})}{d \mathbf{\hat{c}}}\frac{d\mathbf{\hat{c}}}{d\mathbf{\omega}} \tag{7}\]
The first term in the right side of (7), can be computed directly as \(\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{c})\) is typically a differentiable function of \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\). A deep learning library (such as TensorFlow (Abadi, Agarwal, Barham, Brevdo, Chen, Citro, Corrado, Davis, Dean, Devin, Ghemawat, Goodfellow, Harp, Irving, Isard, Jia, Jozefowicz, Kaiser, Kudlur, Levenberg, Mane, Monga, Moore, Murray, Olah, Schuster, Shlens, Steiner, Sutskever, Talwar, Tucker, Vanhoucke, Vasudevan, Viegas, Vinyals, Warden, Wattenberg, Wicke, Yu, & Zheng, 2015), PyTorch (Paszke, Gross, Massa, Lerer, Bradbury, Chanan, Killeen, Lin, Gimelshein, Antiga, et al., 2019)) computes the last term by representing the neural network as a computational graph and applying automatic differentiation (autodiff) in the reverse mode (Baydin, Pearlmutter, Radul, & Siskind, 2018). However, the second term, \(\frac{d\mathbf{x}^{\star}(\mathbf{\hat{c}})}{d\mathbf{\hat{c}}}\), may be nontrivial to compute given the presence of two major challenges: **(1)** The mapping \(\mathbf{\hat{c}}\rightarrow\mathbf{x}^{\star}(\mathbf{\hat{c}})\), as defined by the solution to an optimization problem, _lacks a closed form_ which can be differentiated directly, and **(2)** for many interesting and useful optimization models, the mapping is _nondifferentiable_ in some points, and has zero-valued gradients in others, precluding the straightforward use of gradient descent. As shown in the next subsection, even the class of linear programming problems, widely used in decision modeling, is affected by both issues. Section 3 details the various existing approaches aimed at overcoming these challenges.
Computational costAnother major challenge in decision-focused learning is the computational resources required to train the integrated prediction and optimization model. Note that Line 5 in Algorithm 2 evaluates \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\). This requires solving and differentiating the underlying optimization problem for each observed data sample, in each epoch. This imposes a significant computational cost even when dealing with small-scale and efficiently solvable problems, but can become an impediment in the case of large and (NP-)hard optimization problems. Section 3 reviews the techniques proposed thus far for reducing the computational demands of DFL and improving scalability.
### Optimization Problem Forms
The effectiveness of solving an optimization problem depends on the specific forms of the objective and constraint functions. Considerable effort has been made to developing efficient algorithms for certain optimization forms. Below, the readers are provided an overview of the key and widely utilized types of optimization problem formulations.
#### 2.4.1 Convex optimization
In _convex_ optimization problems, a convex objective function is to be optimized over a convex feasible space. This class of problems is distinguished by the guarantee that any locally optimal is also globally optimal (Boyd, Boyd, & Vandenberghe, 2004). Since many optimization methods converge provably to local minima, convex problems are considered to be reliably and efficiently solvable relative to _nonconvex_ problems. Despite this, convex optimization mappings still impose significant computational overhead on Algorithm 2 since they must be solved for each data sample in each epoch, and most convex optimizations are orders of magnitude more complex than conventional neural network layers (Amos & Kolter, 2017). Like all parametric optimization problems, convex ones are implicitly defined mappings from parameters to optimal solutions, lacking a closed form that can be differentiated directly. However as detailed in Section 3.1, they can be canonicalized to a standard form, which facilitates automation of their solution and backpropagation by a single standardized procedure (Agrawal, Amos, Barratt, Boyd, Diamond, & Kolter, 2019).
The class of convex problems is broad enough to include some which yield mappings \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) that are differentiable everywhere, and some which do not. The _linear programs_, which are convex and form nondifferentiable mappings with respect to their objective parameters, are notable examples of the latter case and are discussed next. The portfolio optimization problem (44), which contains both linear and quadratic constraints, provides an example of a parametric convex problem which admits useful gradients over some regions of its parameter space and not others. Where the (quadratic) variance constraint (44b) is not active, it behaves as a linear program. Elsewhere, the optimal solution is a smooth function of its parameters.
#### 2.4.2 Linear programming
_Linear programs_ (LPs) are convex optimization problems whose objective and constraints are composed of affine functions. These programs are predominant as decision models in operations research, and have endless industrial applications since the allocation and transfer of resources is typically modeled by linear relationships between variables (Bazaraa, Jarvis, & Sherali, 2008). The parametric LPs considered in this manuscript take the following
Figure 3: In decision-focused learning, the neural network model is trained to minimize the task loss
form:
\[\mathbf{x}^{\star}(\mathbf{c}) =\operatorname*{argmin}_{\mathbf{x}}\mathbf{c}^{\top}\mathbf{x}\] (8a) s.t. \[A\mathbf{x} =\mathbf{b} \tag{8b}\] \[\mathbf{x} \geq\mathbf{0} \tag{8c}\]
Compared to other classes of convex problems, LPs admit efficient solution methods, even for large-scale problems (Bazaraa et al., 2008; Ignizio and Cavalier, 1994). From a DFL standpoint, however, LPs pose a challenge, because the mapping \(\mathbf{c}\to\mathbf{x}^{\star}(\mathbf{c})\) is nondifferentiable. Although the derivatives of mapping (8) are defined almost everywhere, they provide no useful information for gradient descent training. To see this, first note the well-known fact that a linear program always takes its optimal value at a vertex of its feasible set (Bazaraa et al., 2008). Since the number of vertices in any such set is finite, (8) maps a continuous parameter space to a discrete set of solutions. As such, it is a piecewise constant mapping. Therefore its derivatives are zero almost everywhere, and undefined elsewhere. Prevalent strategies for incorporating linear programs in decision-focused learning thus typically rely on differentiating smooth approximations to the LP, as detailed in Section 3.2.
Many operations research problems, such as the allocation and planning of resources, can be modeled as LPs. Also many prototypical problems in algorithm design (e.g., sorting and top-\(k\) selection) can be formulated as LPs with continuous variables, despite admitting only discrete integer solutions, by relying on the total unimodularity of the constraint matrices (Bazaraa et al., 2008). In what follows, some examples of machine learning models of LPs and how they might occur in a Predict-Then-Optimize context are given.
**Shortest paths.**: Given a directed graph with a given start and end node, the goal in the shortest path problem is to find a sequence of arcs of minimal total length that connects the start end the end node. The decision variables are binary indicators of each edge's inclusion in the path. The linear constraints ensure \([0,1]\) bounds on each indicator, as well as flow balance through each node. These flow balance constraints capture that, except for the start and end node, each node has as many incoming selected arcs as outgoing selected arcs. For the start node, there is one additional outgoing selected arc, and for the end node, there is one more incoming selected arc. The parameters in the linear objective represent the arc lengths. In many realistic settings--as well as in several common DFL benchmarks (Elmachtoub and Grigas, 2022; Pogancic et al., 2020)--these are unknown, requiring them to be predicted from features before a shortest path can be computed. This motivating example captures the realistic setting in which the shortest route between two locations has to be computed, but in which the road traversal times are uncertain (due to unknown traffic conditions, for example), and have to be predicted from known features (such as day of the week, time of day and weather conditions).
**Bipartite matching.**: Given is a graph consisting of two sets of nodes, and arcs connecting each node of the first set to each node of the second. The arcs are weighted but the weights are unknown and must be predicted. The optimization task is to choose a subset of arcs such that each node from each set is involved in a selected arc at most once, and the total weight of the selected arcs is maximized. The variables lie in
\([0,1]\) and indicate the inclusion of each edge. The constraints ensure that each node is involved at most once in a selected arc. The objective parameters represent arc weights. With a complete bipartite graph, matchings can be construed as permutations, and are presented a permutation matrices, which can be employed in tasks such as learning to rank (Kotary, Fioretto, Van Hentenryck, & Zhu, 2022).
**Sorting and Ranking.**: The sorting of any list of predicted values can be posed as a linear program over a feasible region whose vertices correspond to all of the possible permutations of the list. The related ranking, or argsort problem assigns to any length-\(n\) list a permutation of sequential integers \([n]\) which sorts the list. By smoothing the linear program, these basic operations can be differentiated and backpropagated (Blondel, Teboul, Berthet, & Djolonga, 2020).
**Top-\(k\) selection.**: Given a set of items and item values that must be predicted, the task is to choose the subset of size \(k\) with the largest total value in selected items. In addition to \([0,1]\) bounds on the indicator variables, a single linear constraint ensures that the selected item indicators sum to \(k\). A prevalent example can be found in multilabel classification (Amos, Koltun, & Kolter, 2019; Martins & Astudillo, 2016).
**Computing the maximum.**: This is a special case of top-\(k\) selection where \(k=1\). When the LP's objective is regularized with the entropy term \(H(\mathbf{x})=\mathbf{x}\cdot\log\mathbf{x}\), the mapping from predicted values to optimal solutions is equivalent to a softmax function (Agrawal et al., 2019).
**Max-flow/ min-cut.**: Given a network with predefined source and sink nodes, and predicted flow capacities on each arc, the task is to find the maximum flow rate that can be channeled from source to sink. Here the predicted flow capacities occupy the right-hand side of the linear constraints, which is not in line with the DFL problem description given in subsection 2.1. However, in the related min-cut problem--which is equivalent to the dual linear program of the max-flow problem--the flow capacities are the parameters in the objective function. The max-flow problem can thus be cast as an equivalent min-cut problem and DFL can be used to learn to predict the flow capacities.
#### 2.4.3 Integer linear programming
Integer Linear Programs (ILPs) are another mainstay in operations research and computer science. ILPs differ from LPs in that the decision variables \(\mathbf{x}\) are restricted to integer values, i.e., \(\mathbf{x}\in\mathbb{Z}^{k}\) where \(\mathbb{Z}^{k}\) is the set of integral vectors of appropriate dimensions. Like LPs, ILPs are challenging to use in DFL because they yield discontinuous, nondifferentiable mappings. Computationally however, they are more challenging due to their NP-hard complexity, which may preclude the exact computation of the mapping \(\mathbf{\hat{c}}\rightarrow\mathbf{x}^{\star}(\mathbf{\hat{c}})\) at each step of Algorithm 2. Their differentiation is also significantly more challenging, since the discontinuity of their feasible regions prevents many smoothing techniques that can be applied in DFL with LPs.
In the following, examples of how ILPs may occur in a Predict-Then-Optimize setting are provided.
**Knapsack.**: The knapsack problem has been used as a benchmark in several papers about DFL (Mandi, Demirovic, Stuckey, & Guns, 2020; Mandi & Guns, 2020; Demirovic, Stuckey, Bailey, Chan, Leckie, Ramamohanarao, & Guns, 2019). Given are weights of a set of items, as well as a capacity. The items also have associated values, which have to be predicted from features. The optimization task involves selecting a subset of the items that maximizes the sum of the weights associated with the selected items, whilst ensuring that the sum of the associated weights does not exceed the capacity.
**Travelling salesperson problem**: In the travelling salesperson problem, the list of cities, and the distances between each pair of cities, is given. The goal is to find a path of minimal length that visits each city exactly once. In the Predict-Then-Optimize setting, the distances between the cities first have to be predicted (Pogancic et al., 2020) from observable empirical data.
**Combinatorial portfolio optimization.**: Portfolio optimization involves making optimal investment decisions across a range of financial assets. In the combinatorial Predict-Then-Optimize variant, the decisions are discrete, and must be made on the basis of the predicted next period's increase in the value of several assets (Ferber, Wilder, Dilkina, & Tambe, 2020).
**Diverse bipartite matching.**: Diverse bipartite matching problems are similar to the bipartite matching problems described in 2.4.2, but are subject to additional diversity constraints (Ferber et al., 2020; Mulamba, Mandi, Diligenti, Lombardi, Bucarey, & Guns, 2021; Mandi, Bucarey, Tchomba, & Guns, 2022) In this variant, edges have additional properties. The diversity constraints enforce lower and upper bounds on the proportion of edges selected with a certain property. This precludes the LP formulation, and makes the use of ILP more interesting.
**Energy-cost aware scheduling.**: Energy-cost aware scheduling involves scheduling a set of tasks across a set of machines in a way that minimizes the overall energy cost involved. As future energy costs are unknown, they first have to be predicted (Mulamba et al., 2021; Mandi et al., 2020, 2022; Mandi & Guns, 2020).
#### 2.4.4 Integer nonlinear programming
In integer nonlinear programming, the objective function and/or the constraints are nonlinear. Performing DFL on integer nonlinear programs faces the same challenges as performing DFL on ILPs: integer nonlinear programs are computationally expensive to solve, are implicit mappings with zero-valued gradients almost everywhere, and have discontinuous feasible regions, hindering the use of the smoothing techniques that can be applied in DFL with LPs. Additionally, because of their nonlinear nature, many of the techniques developed for DFL with ILPs, which assume linearity, do not work on integer nonlinear programs (Elmachtoub & Grigas, 2022; Pogancic et al., 2020). To the best of our knowledge, no DFL method has specifically been developed for or tested on integer nonlinear programs. The most closely related work is (Ferber, Huang, Zha, Schubert, Steiner, Dilkina, & Tian, 2022), which learns an approximate ILP surrogates for integer nonlinear programs, which could then in turn be used in a DFL loop.
## 3 Review of Decision-focused Learning Methodologies
To the best of our knowledge, this manuscript is the first to provide a comprehensive review of methods developed for and suitable for DFL in gradient-based training. Concurrently with this paper, Sadana, Chenreddy, Delage, Forel, Frejinger, and Vidal (2023) survey recently proposed approaches to address Predict-Then-Optimize problems (referred to as contextual optimization within it). While Sadana et al. (2023) also cover some of the works to be surveyed later, this manuscript goes beyond by presenting an extensive review solely focused on DFL methods and proposing the first categorization of existing DFL methods.
This section will describe several methodologies which address the challenge of differentiating an optimization mapping for DFL in gradient-based training. In essence, different approaches propose different smoothed surrogate approximations of \(\frac{d\mathcal{X}^{\star}(\mathbbm{e})}{d\mathbbm{e}}\) or \(\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbbm{e}))}{d\mathbbm{e}}\), which is used for backpropagation. This paper proposes the first categorization of existing DFL approaches into the following four distinct classes:
**Analytical Differentiation of Optimization Mappings:**: Methodologies under this category aim to compute exact derivative for backpropagation by differentiating the optimality conditions for certain optimization problem forms, for which the derivative exists and non-zero.
**Analytical Smoothing of Optimization Mappings:**: These approaches deal with combinatorial optimization problems (for which the analytical derivatives are zero almost everywhere) by performing smoothing of combinatorial optimization problems, which results in approximate problems that can be differentiated analytically.
**Smoothing by Random Perturbations:**: Methodologies under this category utilize implicit regularization through perturbations, constructing smooth approximations of optimization mappings.
**Differentiation of Surrogate Loss Functions:**: Methodologies under this category propose convex surrogate loss functions of specific task loss such as regret.
**Decision-Focused Learning without Optimization in the Loop:**: These methodologies bypass the need for computing \(\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbbm{e}))}{d\mathbbm{e}}\) by utilizing surrogate losses, which reflect the quality of the decisions, but do not require computing the solution of the optimization problem for differentiation.
Figure 4 presents key characteristics of these four methodology classes, highlighting the types of problems that can be addressed within each class. Next, each category is thoroughly described.
### Analytical Differentiation of Optimization Mappings
As discussed before, differentiating through parametric CO problems comes with two main challenges. First, since CO problems are complex, implicitly defined mappings from parameters to solutions, computing the derivatives is not straightforward. Second, since some
CO problems result in piecewise-constant mappings, their derivatives are zero almost everywhere, and do not exist elsewhere.
This subsection pertains to CO problems for which the second challenge does not apply, i.e., problems that are smooth mappings. For these problems, all that is required to implement DFL is direct differentiation of the mapping in Eq. (1).
Differentiating unconstrained relaxations.An early work discussing differentiation through constrained argmin problems in the context of machine learning is (Gould, Fernando, Cherian, Anderson, Cruz, & Guo, 2016). It first proposes a technique to differentiate the argmin of a smooth, _unconstrained_ convex function. When \(V(\mathbf{c})=\operatorname*{argmin}_{\mathbf{x}}f(\mathbf{c},\mathbf{x})\), it can be shown that when all second derivatives of \(f\) exist,
\[\frac{dV(\mathbf{c})}{d\mathbf{c}}=-\frac{f_{\mathbf{ex}}(\mathbf{c},V( \mathbf{c}))}{f_{\mathbf{xx}}(\mathbf{c},V(\mathbf{c}))} \tag{9}\]
where \(f_{\mathbf{ex}}\) is the second partial derivative of \(f\) with respect to \(\mathbf{c}\) followed by \(\mathbf{x}\). This follows from implicit differentiation of the first-order optimality conditions
\[\frac{df}{d\mathbf{x}}(\mathbf{c},V(\mathbf{c}))=0 \tag{10}\]
with respect to \(\mathbf{c}\), and rearranging terms. Here the variables \(\mathbf{c}\) are the optimization problem's defining parameters, and the variables \(\mathbf{x}\) are the decision variables.
Figure 4: An overview of the categorization of DFL methodologies in four classes.
This technique is then extended to find approximate derivatives to _constrained_ problems with inequality constraints \(g_{i}(\mathbf{c},\mathbf{x})\leq 0\), \(1\leq i\leq m\), by first relaxing the problem to an unconstrained problem, by means of the log-barrier function
\[F(\mathbf{c},\mathbf{x})=f(\mathbf{c},\mathbf{x})-\mu\sum_{i}\log(-g_{i}( \mathbf{c},\mathbf{x})) \tag{11}\]
and then differentiating \(\operatorname*{argmin}_{\mathbf{x}}F(\mathbf{c},\mathbf{x})\) with respect to \(\mathbf{c}\) for some choice of the scaling factor \(\mu\). Since this approach relies on approximations and requires hyperparameter tuning for the factor \(\mu\), subsequent works focus on differentiating constrained optimization problems directly via their own global conditions for optimality, as discussed next.
Differentiating KKT conditions of quadratic programs.More recent approaches are based on differentiating the optimality conditions of a CO problem directly, i.e., without first converting it to an unconstrained problem. Consider an optimization problem and its optimal solution:
\[\mathbf{x}^{\star}=\operatorname*{argmax}_{\mathbf{x}} f(\mathbf{x}) \tag{12a}\] \[\mathtt{s.t.} \boldsymbol{g}(\mathbf{x})\leq 0\] (12b) \[\boldsymbol{h}(\mathbf{x})=0 \tag{12c}\]
and assume that \(f\), \(g\) and \(h\) are differentiable functions of \(\mathbf{x}\). The Karush-Kuhn-Tucker (KKT) conditions are a set of equations expressing optimality conditions for a solution \(\mathbf{x}^{\star}\) of problem (12) (Boyd et al., 2004):
\[\nabla f(\mathbf{x}^{\star})+\sum_{i}w_{i}\nabla h_{i}(\mathbf{x }^{\star})+\sum_{j}u_{j}\nabla g_{j}(\mathbf{x}^{\star})=0 \tag{13a}\] \[g_{j}(\mathbf{x}^{\star})\leq 0 \ \forall j\] (13b) \[h_{i}(\mathbf{x}^{\star})=0 \ \forall i\] (13c) \[u_{j}\geq 0 \ \forall j\] (13d) \[u_{j}g_{j}(\mathbf{x}^{\star})=0 \ \forall j \tag{13e}\]
OptNet is a framework developed by Amos and Kolter (2017) to differentiate through optimization mappings that are convex quadratic programs (QPs) by differentiating through these KKT conditions. In convex quadratic programs, the objective \(f\) is a convex quadratic function and the constraint functions \(g,h\) are linear over a continuous domain. In the most general case, each of \(f\), \(g\) and \(h\) are dependent on a distinct set of parameters, in addition to the optimization variable \(\mathbf{x}\):
\[f(\mathbf{c},Q,\mathbf{x})=\frac{1}{2}\mathbf{x}^{\top}Q\mathbf{ x}+\mathbf{c}^{\top}\mathbf{x} \tag{14a}\] \[g(A,\mathbf{b},\mathbf{x})=R\mathbf{x}-\mathbf{s}\] (14b) \[h(R,\mathbf{s},\mathbf{x})=A\mathbf{x}-\mathbf{b} \tag{14c}\]
When \(\mathbf{x}\in\mathbb{R}^{k}\) and the number of equality constraints is \(M_{in}\) and \(M_{eq}\), respectively, a QP problem is specified by parameters \(Q\in\mathbb{R}^{k\times k}\), \(\mathbf{c}\in\mathbb{R}^{k}\), \(R\in\mathbb{R}^{k\times M_{in}}\), \(\mathbf{s}\in\mathbb{R}^{M_{in}}\), \(A\in\mathbb{R}^{k\times M_{eq}}\)
and \(\mathbf{b}\in\mathbb{R}^{M_{eq}}\). Note that for this problem to be convex, \(Q\) must be positive-semidefinite always, which can be ensured by learning instead parameters \(\mathbf{q}\in\mathbb{R}^{k}\) and taking \(Q=\mathbf{q}^{\top}\mathbf{q}\).
A defining characteristic of quadratic programs such as (14) is their straightforward parameterization. This is due to the fact that any linear or quadratic function can be fully specified by a square matrix or a vector of parameters, respectively. Here, problem (12) is viewed as a mapping \((Q,\mathbf{c},R,\mathbf{s},A,\mathbf{b})\to\mathbf{x}^{\star}(Q,\mathbf{c},R, \mathbf{s},A,\mathbf{b})\), which parameterizes the space of all possible quadratic programs and their solutions. The presence of such a canonical form allows for separation of a problem's inherent structure from its parameters (Grant and Boyd, 2008), and is key to creating a differentiable mapping from parameters to optimal solutions in an automated way, without necessitating additional analytical transformations.
The gradients are sought with respect to each of the parameters in \((Q,\mathbf{c},R,\mathbf{s},A,\mathbf{b})\). For this purpose, Amos and Kolter (2017) argue that the inequalities (13b) and (13d) can be dropped, resulting in a system of equalities representing optimality conditions for \(\mathbf{x}^{\star}\).
Exact gradients \(\frac{d\mathbf{x}^{\star}}{dP}\) for any \(P\in\{Q,\mathbf{c},R,\mathbf{s},A,\mathbf{b}\}\) can then be retrieved by solving the differential KKT conditions:
\[\begin{bmatrix}Q&R^{\top}&A^{\top}\\ D(\mathbf{w}^{*})R&D(R\mathbf{x}^{*}-\mathbf{s})&0\\ A&0&0\end{bmatrix}\begin{bmatrix}d\mathbf{x}\\ d\mathbf{w}\\ d\mathbf{u}\end{bmatrix}=-\begin{bmatrix}dQ\mathbf{x}^{*}+d\mathbf{c}+dR^{\top}\mathbf{ w}^{*}+dA^{\top}\mathbf{u}^{*}\\ D(\mathbf{w}^{*})dR\mathbf{x}^{*}-D(\mathbf{w}^{*})d\mathbf{s}\\ dA\mathbf{x}^{*}-d\mathbf{b}\end{bmatrix} \tag{15}\]
where the shorthand \(d\) stands for the derivative \(\frac{d}{dP}\). This is another example of implicit differentiation, and requires solving a linear system of equations. Later, Konishi and Fukunaga (2021) extended the method of Amos and Kolter (2017), where they compute the second order derivative of the solution. This allows to train gradient boosting models, which require the gradient as well as the Hessian matrix of the loss.
In summary, the techniques in this category compute the derivatives of the solution with respect to the parameters (if they exist) by leveraging implicit differentiation of the KKT conditions.
Differentiating optimality conditions of conic programs.Another class of problems with a parametric canonical form are the conic programs, which take the form:
\[\mathbf{x}^{\star}(A,\mathbf{b},\mathbf{c})=\operatorname*{argmax }_{\mathbf{x}}\ \ \mathbf{c}^{\top}\mathbf{x} \tag{16a}\] \[\mathtt{s.t.}\ \ A\mathbf{x}-\mathbf{b}\in\mathcal{K} \tag{16b}\]
where \(\mathcal{K}\) is a nonempty, closed, convex cone.
A framework for differentiating the mapping (16) for any \(\mathcal{K}\) is proposed in (Agrawal, Barratt, Boyd, Busseti, & Moursi, 2019c), which starts by forming the homogeneous self-dual embedding of (16), whose parameters form askew-symmetric block matrix composed of \(A\), \(\mathbf{b}\), and \(\mathbf{c}\). Following (Busseti, Moursi, & Boyd, 2019), the solution to this embedding is expressed as the problem of finding a zero of a mapping containing a skew-symmetric linear function and projections onto the cone \(\mathcal{K}\) and its dual. The zero-value of this function is implicitly differentiated, in a similar manner to the KKT conditions of a quadratic program as in (Amos and Kolter, 2017). The overall mapping (16) is viewed as the composition of function that maps \((A,\mathbf{b},\mathbf{c})\) onto the skew-symmetric parameter space of the self-dual embedding, the rootfinding problem that produces a solution to the embedding, and a
transformation back to a solution of the primal and dual problems. The overall derivative is found by a chain rule applied over this composition.
Subsequent work (Agrawal et al., 2019a) leverages the above-described differentiation of cone programs to develop a more general differentiable convex optimization solver--Cvxpylayers. It is well known that conic programs of the form (16) can provide canonical representations of convex programs (Nemirovski, 2007). The approach described by Agrawal et al. (2019a) is based on this principle; that a large class of _parametric_ convex optimization problems can be recast as equivalent parametric cone programs, with an appropriate choice of the cone \(\mathcal{K}\). A major benefit of this representation is that it allows a convex program to be separated with respect to its defining parameters \((A,\mathbf{b},\mathbf{c})\) and its structure \(\mathcal{K}\), allowing a generic procedure to be applied for solving and differentiating the transformed problem with respect to \(A\), \(\mathbf{b}\) and \(\mathbf{c}\).
The framework for transforming convex programs to cone programs of the form (16) is drawn from (Grant & Boyd, 2008), which is based on two related concepts. First is the notion of _disciplined convex programming_, which assists the automation of cone transforms by imposing a set of rules or conventions on how convex programs can be represented. Second is the notion of _graph implementations_, which represent functions as optimization problems over their epigraphs, for the purpose of generically representing optimization problems and assisting conversion between equivalent forms. The associated software system called cvx allows for disciplined convex programs to be converted to cone programs via their graph implementations. Subsequently, the transformed problem is solved using conic optimization algorithms, and its optimal solution is converted to a solution of the original disciplined convex program. Differentiation is performed through each operation and combined by the chain rule. The transformation of parameters between respective problem forms, and the solution recovery step, are differentiable by virtue of being affine mappings (Agrawal et al., 2019a). The intermediate conic program is differentiated via the methods of (Agrawal et al., 2019c).
Solver unrolling and fixed-point differentiation.While the methods described above for differentiation through CO problems are generic and applicable to broad classes of problems, other practical techniques have been proven effective and even advantageous in some cases. A common strategy is that of solver _unrolling_, in which the solution to (1) is found by executing an optimization algorithm in the computational graph of the predictive model. Then, the mapping (1) is backpropagated simply by automatic differentiation or 'unrolling' through each step of the algorithm, thus avoiding the need to explicitly model \(\frac{d\mathbf{x}^{*}(\mathbf{c})}{d\mathbf{c}}\)(Domke, 2012). While this approach leads to accurate backpropagation in many cases, it suffers disadvantages in efficiency due to the memory and computational resources required to store and apply backpropagation over the entire computational graph of an algorithm that requires many iterations (Amos & Kolter, 2017). Additionally, it has been observed that unrolling over many solver iterations can leads to vanishing gradient issues reminiscent of recurrent neural networks (Monga, Li, & Eldar, 2021). On the other hand, unrolling allows for the learning of unspecified algorithm parameters, such as gradient descent step sizes or weights in an augmented lagrangian, which can be exploited to accelerate the forward-pass convergence of the optimization solver. A comprehensive survey of algorithm unrolling for image processing applications is provided in (Monga et al., 2021).
Another way in which a specific solution algorithm may provide gradients though a corresponding optimization mapping, is by implicit differentiation of its fixed-point conditions. Suppose that the solver iterations
\[\mathbf{x}_{k+1}(\mathbf{c})=\mathcal{U}(\mathbf{x}_{k}(\mathbf{c}),\ \mathbf{c}) \tag{17}\]
converge as \(k\rightarrow\infty\) to a solution \(\mathbf{x}^{\star}(\mathbf{c})\) of the problem (1), then the fixed-point conditions
\[\mathbf{x}^{\star}(\mathbf{c})=\mathcal{U}(\mathbf{x}^{\star}(\mathbf{c}),\ \mathbf{c}) \tag{18}\]
are satisfied. Assuming the existence of all derivatives on an open set containing \(\mathbf{c}\) to satisfy the implicit function theorem, it follows by implicit differentiation with respect to \(\mathbf{c}\) that
\[(I-\Phi)\frac{d\mathbf{x}^{\star}}{d\mathbf{c}}=\Psi, \tag{19}\]
which is a linear system to be solved for \(\frac{d\mathbf{x}^{\star}}{d\mathbf{c}}\), in terms of \(\Phi=\frac{d\mathcal{U}}{d\mathbf{x}^{\star}}(\mathbf{x}^{\star}(\mathbf{c}),\ c)\) and \(\Psi=\frac{d\mathcal{U}}{\mathbf{c}}(\mathbf{x}^{\star}(\mathbf{c}),\ \mathbf{c})\).
The relationship between unrolling and differentiation of the fixed-point conditions is studied by Kotary, Dinh, and Fioretto (2023), which shows that backpropagation of (1) by unrolling (17) is equivalent to solving the linear system (19) by fixed-point iteration. As such, the convergence rate of the backward pass in unrolling is determined by the convergence rate of the equivalent linear system solve, and can be calculated in terms of the spectral radius of \(\Phi\).
Discussion.In contrast to most other differentiable optimization methods surveyed in this article, the analytical approaches in this subsection allow for backpropagation of coefficients that specify the constraints as well the objective function. For example, Amos and Kolter (2017) propose parametric quadratic programming layers whose linear objective parameters are predicted by previous layers, and whose constraints are learned through the layer's own embedded parameters. This is distinct from most cases of DFL, in which the optimization problems have fixed constraints and no trainable parameters of their own.
Furthermore, the techniques surveyed in this subsection are aimed at computing exact gradients of parametric optimization mappings. However, many applications of DFL contain optimization mappings that are discontinuous and piecewise-constant. Such mappings, including parametric linear programs (8), have gradients that are zero almost everywhere and thus do not supply useful descent directions for SGD training. Therefore, the techniques of this subsection are often applied after regularizing the problem analytically with smooth functions, as detailed in the next subsection.
### Analytical Smoothing of Optimization Mappings
To differentiate through combinatorial optimization problems, the optimization mapping first has to be smoothed. While techniques such as noise-based gradient estimation (surveyed in Section 3.3) provide smoothing and differentiation simultaneously, analytical differentiation first incorporates smooth analytical terms in the optimization problem's formulation, and then analytically differentiates the resulting optimization problem using the techniques discussed in Section 3.1.
Analytical smoothing of linear programs.Note that while an LP problem is convex and has continuous variables, only a finite number of its feasible solutions can potentially be optimal. These points coincide with the vertices of its feasible polytope (Bazaraa et al., 2008). Therefore the mapping \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) in (8), as a function of \(\mathbf{\hat{c}}\), is discontinuous and piecewise constant, and thus requires smoothing before it can be differentiated through. An approach to do so was presented in Wilder et al. (2019), which proposes to augment the linear LP objective function with the Euclidean norm of its decision variables, so that the new objective takes the following form
\[\mathbf{x}^{\star}(\mathbf{c}) =\operatorname*{argmax}_{\mathbf{x}}\mathbf{c}^{\top}\mathbf{x}- \mu\|\mathbf{x}\|_{2}^{2} \tag{20a}\] \[=\operatorname*{argmin}_{\mathbf{x}}\|\mathbf{x}-\frac{\mathbf{c }}{\mu}\|_{2}^{2} \tag{20b}\]
where the above equality follows from expanding the square and cancelling constant terms, which do not affect the argmax. This provides an intuition as to the effect of such a quadratic regularization: it converts a LP problem into that of projecting the point \(\frac{\mathbf{c}}{\mu}\) onto the feasible polytope, which results in a continuous mapping \(\mathbf{\hat{c}}\to\mathbf{x}^{\star}(\mathbf{\hat{c}})\). Wilder et al. (2019) then train decision-focused models by solving and backpropagating the respective quadratic programming problem using the framework of (Amos and Kolter, 2017), in order to learn to predict objective parameters with minimal regret. At test time, the quadratic smoothing term is removed. This article refers to such regret-based DFL with quadratically regularized linear programs as the _Quadratic Programming Task Loss_ method (QPTL).
Other forms of analytical smoothing for linear programs can be applied by adding different regularization functions to the objective function. Some common regularization terms for LPs include the entropy function \(H(\mathbf{x})=\sum_{i}x_{i}\log x_{i}\) and the binary entropy function \(H_{b}(\mathbf{x})=H(\mathbf{x})+H(\mathbf{1}-\mathbf{x})\). To differentiate the resulting smoothed optimization problems, the framework of Agrawal et al. (2019) can be used. Alternatively, problem-specific approaches that do not employ (Agrawal et al., 2019) have also been proposed. For example, (Blondel et al., 2020) proposes a method for problems where \(H\) smooths an LP for differentiable sorting and ranking, and (Amos et al., 2019) proposes a way to differentiate through problems where \(H_{b}\) is used in a multilabel classification problem. Both works propose fast implementations for both the forward and backward passes of their respective optimization problems.
In a related approach, Mandi and Guns (2020) propose a general, differentiable LP solver based on log-barrier regularization. For a parametrized LP of standard form (8), gradients are computed for the regularized form in which the constraints \(\mathbf{x}\geq\mathbf{0}\) are replaced with log-barrier approximations:
\[\mathbf{x}^{\star}(\mathbf{c})=\operatorname*{argmin}_{\mathbf{x}} \mathbf{c}^{\top}\mathbf{x}+\lambda\sum_{i}x_{i} \tag{21a}\] \[\text{s.t. }A\mathbf{x}=\mathbf{b} \tag{21b}\]
While similar in this sense to (Gould et al., 2016), this method exploits several efficiencies specific to linear programming, in which the log-barrier term serves a dual purpose of rendering (21) differentiable and also aiding its solution. Rather than forming and solving
this regularized LP problem directly, the solver uses an interior point method to produce a sequence of log-barrier approximations to the LP's homogenous self-dual (HSD) embedding. Early stopping is applied in the interior point method, producing a solution to (21) for some \(\lambda\), which serves as a smooth surrogate problem for differentiation. A major advantage of this technique is that it only requires optimization of a linear program, making it in general more efficient than direct solution of a regularized problem as in the approaches described above.
Analytical smoothing of integer linear programs.To differentiate through ILPs, Wilder et al. (2019) propose to simply drop the integrality constraints, and to then smooth and differentiate through the resulting LP relaxation, which is observed to give satisfactory performance in some cases. Ferber et al. (2020) later extended this work by using a more systematic approach to generate the LP relaxation of the ILP problem. They use the method of cutting planes to discover an LP problem that admits the same solution as the ILP. Subsequently, the method of (Wilder et al., 2019) is applied to approximate the LP mapping's derivatives. Although this results in enhanced performance with respect to regret, there are some practical scalability concerns, since the cut generation process is time consuming but also must be repeated for each instance in each training epoch.
### Smoothing by Random Perturbations
A central challenge in DFL is the need for smoothing operations of non-smooth optimization mappings. Techniques that perform the smoothing operation by adding explicit regularization functions to the optimization problems' objective function have been surveyed in Section 3.2. This section instead surveys techniques, which use implicit regularization via perturbations. These techniques construct smooth approximations of the optimization mappings by adopting a probabilistic point of view. To introduce this point of view, the CO problem in this section is not viewed as a mapping from \(\mathbf{c}\) to \(\mathbf{x}^{\star}(\mathbf{c})\). Rather, it is viewed as a function that maps \(\mathbf{c}\) onto a probability distribution over the feasible region \(\mathcal{F}\). From this perspective, \(\mathbf{x}^{\star}(\mathbf{c})\) can be viewed as a random variable, conditionally dependent on \(\mathbf{c}\). The motivation behind representing \(\mathbf{x}^{\star}(\mathbf{c})\) as a random variable is that the rich literature of likelihood maximization with latent variables, in fields such as Probabilisic Graphical Models (PGMs) (Koller and Friedman, 2009), can be exploited.
Implicit differentiation by perturbation.One seminal work in the field of PGMs is by Domke (2010). This work contains an important proposition, which deals with a setup where a variable \(\boldsymbol{\theta}_{1}\) is conditionally dependent on another variable \(\boldsymbol{\theta}_{2}\) and the final loss \(\mathcal{L}\) is defined on the variable \(\boldsymbol{\theta}_{1}\). Let \(p(\boldsymbol{\theta}_{1}|\boldsymbol{\theta}_{2})\) and \(\mathbb{E}[\boldsymbol{\theta}_{1}|\boldsymbol{\theta}_{2}]\) be the conditional distribution and the conditional mean of \(\boldsymbol{\theta}_{1}\). The loss \(\mathcal{L}\) is measured on the conditional mean \(\mathbb{E}[\boldsymbol{\theta}_{1}|\boldsymbol{\theta}_{2}]\) and the goal is to compute the derivative of \(\mathcal{L}\) with respect to \(\boldsymbol{\theta}_{2}\). Domke (2010) proposes that the derivative of \(\mathcal{L}\) with respect to \(\boldsymbol{\theta}_{2}\) can be approximated by the following finite difference method:
\[\frac{dL}{d\boldsymbol{\theta}_{2}}\approx\frac{1}{\delta}\Bigg{(}\mathbb{E}[ \boldsymbol{\theta}_{1}|\big{(}\boldsymbol{\theta}_{2}+\delta\frac{d}{d \boldsymbol{\theta}_{1}}\big{(}\mathcal{L}(\mathbb{E}[\boldsymbol{\theta}_{1} |\boldsymbol{\theta}_{2}]\big{)}\big{)}]-\mathbb{E}[\boldsymbol{\theta}_{1}| \boldsymbol{\theta}_{2}]\Bigg{)} \tag{22}\]
where \(\frac{d}{d\boldsymbol{\theta}_{1}}[\mathcal{L}(\mathbb{E}[\boldsymbol{\theta} _{1}])]\) is the derivative \(\mathcal{L}\) with respect to \(\boldsymbol{\theta}_{1}\) at \(\mathbb{E}[\boldsymbol{\theta}_{1}]\). Notice that the first term in (22) is the conditional mean after perturbing the parameter \(\boldsymbol{\theta}_{2}\) where magnitude of
the perturbation is modulated by the derivative of \(\mathcal{L}\) with respect to \(\mathbf{\theta}_{1}\). Taking inspiration from this proposition, by defining a conditional distribution \(p(\mathbf{x}^{\star}(\mathbf{\hat{c}})|\hat{c})\), one can compute the derivative of the regret with respect to \(\mathbf{\hat{c}}\) in the context of DFL.
To perfectly represent the deterministic mapping \(\mathbf{c}\rightarrow\mathbf{x}^{\star}(\mathbf{c})\), the straightforward choice is to define a Dirac mass distribution, which assigns all probability mass to the optimal point and none to other points, i.e.,
\[p(\mathbf{x}|\mathbf{c})=\begin{cases}1&\mathbf{x}=\mathbf{x}^{\star}(\mathbf{ c})\\ 0&\text{otherwise}\end{cases} \tag{23}\]
Differentiation of blackbox combinatorial solvers.Note that with the distribution in (23) \(\mathbb{E}_{\mathbf{x}\sim p(\mathbf{x}|\mathbf{c})}[x|c]=\mathbf{x}^{\star}( \mathbf{c})\). Hence, using conditional probability in the proposition in (22), \(\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))}{d\mathbf{\hat{c}}}\) can be computed in the following way:
\[\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))}{d\mathbf{\hat{c}}} \approx\nabla^{(DBB)}\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))=\left( \mathbf{x}^{\star}\Big{(}\mathbf{\hat{c}}+\delta\frac{d\mathcal{L}(\mathbf{x} ^{\star}(\mathbf{\hat{c}}))}{d\mathbf{x}^{\star}(\mathbf{\hat{c}})}\Big{)}- \mathbf{x}^{\star}\Big{(}\mathbf{\hat{c}}\Big{)}\right) \tag{24}\]
The gradient computation methodology proposed by Pogancic et al. (2020) takes the form of (24). They interpret it as substituting the jump-discontinuous optimization mapping with a piece-wise linear interpolation. It is a linear interpolation of the mapping \(\mathbf{\hat{c}}\rightarrow\mathbf{x}^{\star}(\mathbf{\hat{c}})\) between the points \(\mathbf{\hat{c}}\) and \(\mathbf{\hat{c}}+\delta\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}) )}{d\mathbf{x}}|_{\mathbf{x}=\mathbf{x}^{\star}(\mathbf{\hat{c}})}\). Pogancic et al. (2020) call this 'differentiation of blackbox' (DBB) solvers, because this approach considers the CO solver as a blackbox oracle, i.e., it does not take cognizance of how the solver works internally.
In a subsequent work, Sahoo, Paulus, Vlastelica, Musil, Kuleshov, and Martius (2023) propose to treat \(\frac{d\mathbf{x}^{\star}(\mathbf{\hat{c}})}{d\mathbf{\hat{c}}}\) as a negative identity matrix while backpropagating the loss. However, they notice that such an approach might run into unstable learning for scale-invariant optimization problems such as LPs and ILPs. To negate this effect, they suggest multiplying the cost vector with the matrix of the invariant transformation. In case of LPs and ILPs this can be achieved by normalizing the cost vector through projection onto the unit sphere.
Perturb-and-MAP.However, at this point it is worth mentioning that Domke (2010) assumes, in his proposition, that the distribution \(p(\theta_{1}|\theta_{2})\) in (22) belongs to the exponential family of distributions (Barndorff-Nielsen, 1978). Note that the distribution defined in (23) is not a distribution of the exponential family. Nevertheless, a tempered softmax distribution belonging to exponential family can be defined to express the mapping in the following way:
\[p_{\tau}(\mathbf{x}|\mathbf{c})=\begin{cases}\frac{\exp(-f(\mathbf{x},\mathbf{ c})/\tau)}{\sum_{\mathbf{x}^{\prime}\in\mathcal{F}}\exp(-f(\mathbf{x}^{\prime}, \mathbf{c})/\tau)}&\mathbf{x}\in\mathcal{F}\\ 0&\text{otherwise}\end{cases} \tag{25}\]
In this case, the log unnormalized probability mass at each \(\mathbf{x}\in\mathcal{F}\) is proportional to \(\exp(-f(\mathbf{x},\mathbf{c})/\tau)\), the exponential of the negative of the tempered objective value. The idea behind (25) is to assign a probability to each feasible solution such that solutions with a better objective value have a larger probability. The parameter \(\tau\) affects the way in which objective values map to probabilities. When \(\tau\to 0\), the distribution becomes the argmax distribution in (23), when \(\tau\rightarrow\infty\), the distribution becomes uniform. In other words,
the value of \(\tau\) determines how drastically the probability changes because of a change in objective value. Good values for \(\tau\) are problem-dependent, and thus tuning \(\tau\) is advised.
Note that (22) deals with conditional expectation. As in the case of tempered softmax distribution, the conditional expectation is not always equal to the solution to the CO problem, it must be computed first to use the finite difference method in (22). However, computing the probability distribution function in (25) is not tractable, as the denominator (also called the partition function) requires iterating over all feasible points in \(\mathcal{F}\). Instead, Papandreou and Yuille (2011) propose a novel approach, known as _perturb-and-MAP_, to estimate the probability using perturbations. It states that the distribution of the maximizer after perturbing the log unnormalized probability mass by i.i.d. \(\text{Gumbel}(0,\epsilon)\) noise has the same exponential distribution as (25). To make it more explicit, if \(\tilde{\mathbf{c}}=\mathbf{c}+\boldsymbol{\eta}\), where the perturbation vector \(\boldsymbol{\eta}\stackrel{{\text{i.i.d.}}}{{\sim}}\text{Gumbel}(0,\epsilon)\),
\[\mathbb{P}[\mathbf{x}=\underset{\mathbf{x}^{\prime}}{\text{ argmax}}-f(\mathbf{x}^{\prime},\tilde{\mathbf{c}})]=p_{\epsilon}(\mathbf{x}| \mathbf{c}) \tag{26}\]
The perturb-and-MAP framework can be viewed as a method of stochastic smoothing (Abernethy, Lee, & Tewari, 2016). A smoothed approximation of the optimization mapping is created by considering the average value of the solutions of a set of _nearby perturbed_ points. With the help of (26), the conditional distribution and hence the conditional mean can be approximated by Monte Carlo simulation.
Differentiable perturbed optimizers.Berthet, Blondel, Teboul, Cuturi, Vert, and Bach (2020) propose another approach for perturbation-based differentiation. They name it differentiable perturbed optimizers (DPO). They make use of the perturb-and-MAP framework to draw samples from the conditional distribution \(p(\mathbf{x}|\mathbf{c})\). In particular, they use the reparameterization trick (Kingma & Welling, 2014; Rezende, Mohamed, & Wierstra, 2014) to generate samples from \(p(\mathbf{x}|\mathbf{c})\). The reparameterization trick uses a change of variables to rewrite \(\mathbf{x}\) as a _deterministic function_ of \(\mathbf{c}\) and a random variable \(\boldsymbol{\eta}\). In this reformulation, \(\mathbf{x}\) is still a random variable, but the randomness comes from the variable \(\boldsymbol{\eta}\). They consider \(\boldsymbol{\eta}\) to be a random variable having a density proportional to \(\exp(-\nu(\boldsymbol{\eta}))\) for a twice-differentiable function \(\nu\). Moreover, they propose to multiply the random variable \(\boldsymbol{\eta}\) with a temperature parameter \(\epsilon>0\), which controls the strength of perturbing \(\mathbf{c}\) by the random variable \(\boldsymbol{\eta}\). In summary, first \(\mathbf{c}\) is perturbed with random perturbation vector \(\epsilon\boldsymbol{\eta}\), where \(\boldsymbol{\eta}\) is sampled from the aforementioned density function, and then the maximizer of the perturbed vector \(c+\epsilon\boldsymbol{\eta}\) is viewed as a sample from the conditional distribution for given values of \(\mathbf{c}\) and \(\epsilon\), i.e., \(\mathbf{x}_{\epsilon}^{\star}(\mathbf{c})=\mathbf{x}^{\star}(\mathbf{c}+ \epsilon\boldsymbol{\eta})\) is considered as a sample drawn from \(p(\mathbf{x}|\mathbf{c})\) for a given \(\epsilon\). They call \(\mathbf{x}_{\epsilon}^{\star}(\mathbf{c})\) a _perturbed optimizer_. Note that, for \(\epsilon\to 0\), \(\mathbf{x}_{\epsilon}^{\star}(\mathbf{c})\rightarrow\mathbf{x}^{\star}( \mathbf{c})\). Like before, \(\mathbf{x}_{\epsilon}^{\star}(c)\) can be estimated by Monte Carlo simulation by sampling i.i.d. random noise \(\boldsymbol{\eta}^{(m)}\) from the aforementioned density function. The advantage is that the Monte Carlo estimate is _continuously_ differentiable with respect to \(\mathbf{c}\). This Monte Carlo estimate \(\bar{\mathbf{x}}_{\epsilon}^{\star}(\mathbf{c})\) can be expressed as:
\[\bar{\mathbf{x}}_{\epsilon}^{\star}(\mathbf{c})=\frac{1}{M}\sum_{m=1}^{M} \mathbf{x}^{\star}\Big{(}\mathbf{c}+\epsilon\boldsymbol{\eta}^{(m)}\Big{)} \tag{27}\]
Moreover, its derivative can be estimated by Monte Carlo simulation too
\[\frac{d\bar{\mathbf{x}}_{\epsilon}^{\star}(\mathbf{c})}{d\mathbf{c}}=\frac{1}{ \epsilon}\frac{1}{M}\sum_{m=1}^{M}\mathbf{x}^{\star}(\mathbf{c}+\epsilon\mathbf{ \eta}^{(m)})\nu^{\prime}(\mathbf{\eta}^{(m)})^{\top} \tag{28}\]
where \(\nu^{\prime}\) is the first order derivative of \(\nu\). They can approximate \(\frac{d\mathbf{x}^{\star}(\mathbf{c})}{d\mathbf{c}}\) by \(\frac{d\bar{\mathbf{x}}_{\epsilon}^{\star}(\mathbf{c})}{d\mathbf{c}}\) to implement the backward pass. As mentioned before, if \(\epsilon\to 0\), the estimation will be an unbiased estimate of \(\mathbf{x}^{\star}(\mathbf{c})\). However, in practice, for low values of \(\epsilon\), the variance of the Monte-Carlo estimator will increase, leading to unstable and noisy gradients. This is in line with the smoothing-versus-accuracy trade-off mentioned before. Berthet et al. (2020) use this DPO framework to differentiate any optimization problem with linear objective. For a CO problem with discrete feasible space, they consider the convex hull of the discrete feasible region. Furthermore, Berthet et al. (2020) construct the Fenchel-Young loss function and show for Fenchel-Young loss function, the gradient can be approximated in the following way:
\[\nabla\mathcal{L}^{FY}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))=-\big{(}\bar{ \mathbf{x}}_{\epsilon}^{\star}(\mathbf{\hat{c}})-\mathbf{x}^{\star}(\mathbf{c })\big{)} \tag{29}\]
In a later work, Dalle, Baty, Bouvier, and Parmentier (2022) extend the perturbation approach, where they consider multiplicative perturbation. This is useful when the cost parameter vector is restricted to be non-negative, such as in the applications of shortest path problem variants. The work of Paulus, Choi, Tarlow, Krause, and Maddison (2020) can also be viewed as an extension of the DPO framework. They introduce stochastic softmax tricks (SST), a framework of Gumbel-softmax distribution, where they propose differentiable methods by sampling from more complex categorical distributions.
Implicit maximum likelihood estimation (I-MLE).Niepert, Minervini, and Franceschi (2021) also use the _perturb-and-MAP_ framework. However, they do not sample noise from the Gumbel distribution, rather they report better results when the noise \(\mathbf{\eta}^{\gamma}\) is sampled from a _Sum-of-Gamma_ distribution with hyperparameter \(\gamma\). Combining the finite difference approximation (22) with the perturb-and-MAP framework, the gradient takes the following form:
\[\frac{d\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))}{d\mathbf{\hat{c}}} \approx\nabla^{(IMLE)}\mathcal{L}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))= \left(\mathbf{x}^{\star}\Big{(}\mathbf{\hat{c}}+\delta\frac{d\mathcal{L}( \mathbf{x}^{\star}(\mathbf{\hat{c}}))}{d\mathbf{x}^{\star}(\mathbf{\hat{c}})}+ \epsilon\mathbf{\eta}^{\gamma}\Big{)}-\mathbf{x}^{\star}\Big{(}\mathbf{\hat{c}}+ \epsilon\mathbf{\eta}^{\gamma}\Big{)}\right) \tag{30}\]
where \(\epsilon>0\) is a temperature parameter, which controls the strength of noise perturbation. Clearly, (30) turns into (24) when there is no noise perturbation, i.e., if \(\mathbf{\eta}^{\gamma}=0\).
Discussion.One major advantage of the methodologies explained in this subsection is that for gradient computation they call the optimization solver as a blackbox oracle and only use the solution returned by it for gradient computation. In essence, these techniques are not concerned with _how_ the CO problem is solved. The users can utilize any techniques of their choice--constraint programming (CP) (Rossi, van Beek, & Walsh, 2006), Boolean satisfiability (SAT) (Gomes, Kautz, Sabharwal, & Selman, 2008) or linear programming (LP) and integer linear programming (ILP) to solve the CO problem.
### Differentiation of Surrogate Loss Functions
The methodologies explained in the preceding subsections can be viewed as implementations of differentiable optimization layers, which solve the CO problem in the forward pass and return useful approximations of \(\frac{d\mathbf{x}^{\star}(\mathbf{c})}{d\mathbf{c}}\) in the backward pass. Consequently, those methodologies can be used to introduce optimization layers _anywhere in a neural network architecture_, and can be combined with arbitrary loss functions. In contrast, the methodologies that will be introduced next can only be used to differentiate regret (3)--a specific task loss. Hence, models can only be trained in an end-to-end fashion using these techniques when the CO problem occurs in the _final_ stage of the pipeline, as in the case of Predict-Then-Optimize problems. Also note that the computation of the regret requires both the ground-truth cost vector \(\mathbf{c}\); as well as ground-truth solution \(\mathbf{x}^{\star}(\mathbf{c})\). If \(\mathbf{c}\) is observed, \(\mathbf{x}^{\star}(\mathbf{c})\) can be computed. However, if only \(\mathbf{x}^{\star}(\mathbf{c})\) is observed, \(\mathbf{c}\) cannot directly be recovered. Hence, the techniques that will be discussed next are not suitable when the true cost vectors \(\mathbf{c}\) are not observed in the training data.
Smart "Predict, Then Optimize".Elmachtoub and Grigas (2022) developed Smart "Predict, Then Optimize" (SPO), a seminal work in DFL. As the gradient of the regret with respect to cost vector \(\mathbf{\hat{c}}\) is zero almost everywhere, SPO instead uses a surrogate loss function that has subgradients which _are_ useful in training. They start by proposing a convex surrogate upper bound of regret, which they call the SPO+ loss.
\[\mathcal{L}_{SPO+}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))=2\mathbf{\hat{c}}^{ \top}\mathbf{x}^{\star}(\mathbf{c})-\mathbf{c}^{\top}\mathbf{x}^{\star}( \mathbf{c})+\max_{\mathbf{x}\in\mathcal{F}}\{\mathbf{c}^{\top}\mathbf{x}-2 \mathbf{\hat{c}}^{\top}\mathbf{x}\} \tag{31}\]
Then, they derive the following useful subgradient of \(\mathcal{L}_{SPO+}(\mathbf{x}^{\star}(\mathbf{\hat{c}}))\):
\[\mathbf{x}^{\star}(\mathbf{c})-\mathbf{x}^{\star}(2\mathbf{\hat{c}}-\mathbf{ c})\in\partial\mathcal{L}_{SPO+} \tag{32}\]
This subgradient is used in place of to update the model parameters in the backward pass.
From a theoretical point of view, the SPO+ loss has the Fisher consistency property with respect to the regret, under certain distributional assumptions. A surrogate loss function satisfies the Fisher consistency property if the function that minimizes the surrogate loss also minimizes the true loss in expectation (Zou, Zhu, & Hastie, 2008). Concretely, this means that minimizing the SPO+ loss corresponds to minimizing the regret in expectation. While training ML models with a finite dataset, an important property of considerable interest would be _risk bounds_(Massart & Nedelec, 2006). Liu and Grigas (2021) develop risk bounds for SPO+ loss and show that low excess SPO+ loss risk translates to low excess regret risk. Furthermore, El Balghiti, Elmachtoub, Grigas, and Tewari (2019) develop worst-case generalization bounds of the SPO loss.
The SPO framework is applicable not only to LPs, but to _any CO problems where the cost parameters appear linearly in the objective function_. This includes QPs, ILPs and MILPs. Mandi et al. (2020) empirically investigated how the framework performs on ILP problems. However, as these problems are much more computationally expensive to solve than the ones considered by Elmachtoub and Grigas (2022), they compared the regular SPO methodology with a variant in which, it is significantly cheaper to solve the CO problem during training. To be specific, they consider LP relaxations of the ILPs These LP relaxations are obtained by considering the continuous relaxation of the ILPs, i.e., they are variants of the ILPs
in which the integrality constraints are dropped. Using the LP relaxations significantly expedite training, without any cost: Mandi et al. (2020) did not observe a significant difference in the final achieved regret between these two approaches, with both of them performing better than the prediction-focused approach. However, one should be cautious to generalize this result across different problems, as it might be dependent on the integrality gap between the ILP and its LP relaxation.
Next, within this category, a different type of DFL technique is being surveyed. In these DFL techniques, the surrogate loss functions are supposed to reflect the decision quality, but their computations do _not_ involve solving the CO problems, thereby avoiding the zero-gradient problem.
Noise contrastive estimation.One such approach is introduced by Mulamba et al. (2021). Although their aim is still to minimize regret, computation of \(\nabla_{\mathbf{\hat{c}}}\text{{Regret}}(\mathbf{x}^{\star}(\mathbf{\hat{c}}), \mathbf{c})\) has been avoided by using a surrogate loss function. In their work, the CO problem is viewed from a probabilistic perspective, as in (25). However, instead of maximum likelihood estimation, the noise contrastive estimation (NCE) (Gutmann & Hyvarinen, 2010) method is adopted. NCE has been extensively applied in many applications such as language modeling (Mnih & Teh, 2012), information retrieval (Huang, He, Gao, Deng, Acero, & Heck, 2013) and entity linking (Gillick, Kulkarni, Lansing, Presta, Baldridge, Ie, & Garcia-Olano, 2019). Its basic idea is to learn to discriminate between data coming from the true underlying distribution and data coming from a noise distribution. In the context of DFL, this involves contrasting the likelihood of ground-truth solution \(\mathbf{x}^{\star}(\mathbf{c})\) and a set of negative examples \(S\). In other words, the following ratio is maximized:
\[\max_{\mathbf{\hat{c}}}\sum_{\mathbf{x}^{\prime}\in S}\frac{p_{\tau}(\mathbf{ x}^{\star}(\mathbf{c})|\mathbf{\hat{c}})}{p_{\tau}(\mathbf{x}^{\prime}|\mathbf{ \hat{c}})} \tag{33}\]
where \(\mathbf{x}^{\prime}\in S\) is a negative example. Because the probability \(p_{\tau}(\mathbf{x}^{\star}(\mathbf{c})|\mathbf{\hat{c}})\) is defined as in (25), when \(\tau=1\), maximizing (33) corresponds to minimizing the following loss:
\[\mathcal{L}_{NCE}(\mathbf{\hat{c}},\mathbf{c})=\sum_{\mathbf{x}^{\prime}\in S }f(\mathbf{x}^{\star}(\mathbf{c}),\mathbf{\hat{c}})-f(\mathbf{x}^{\prime}, \mathbf{\hat{c}}) \tag{34}\]
In other words, this approach learns to predict a \(\mathbf{\hat{c}}\) for which ground-truth solution \(\mathbf{x}^{\star}(\mathbf{c})\) achieves a good objective value, and for which other feasible solutions \(\mathbf{x}^{\prime}\) achieve worse objective values. Note that when \(f(\mathbf{x}^{\star}(\mathbf{c}),\mathbf{\hat{c}})\leq f(\mathbf{x}^{\prime}, \mathbf{\hat{c}})\) for all \(\mathbf{x}^{\prime}\in\mathcal{F}\), it holds that \(\mathbf{x}^{\star}(\mathbf{c})=\mathbf{x}^{\star}(\mathbf{\hat{c}})\), and thus the regret is zero. Also note that computing \(\mathcal{L}_{NCE}(\mathbf{\hat{c}},\mathbf{c})\) does not involve computing \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\), circumventing the zero-gradient problem.
As an alternative to NCE, Mulamba et al. (2021) also introduce a maximum a posteriori (MAP) approximation, in which they only contrast the ground-truth solution with the most probable negative example from \(S\) according to the current model:
\[\mathcal{L}_{MAP}(\mathbf{\hat{c}},\mathbf{c}) =\max_{\mathbf{x}^{\prime}\in S}f(\mathbf{x}^{\star}(\mathbf{c}), \mathbf{\hat{c}})-f(\mathbf{x}^{\prime},\mathbf{\hat{c}})\] \[=f(\mathbf{x}^{\star}(\mathbf{c}),\mathbf{\hat{c}})-f(\mathbf{x}^ {\prime},\mathbf{\hat{c}})\text{ where }\mathbf{x}^{\prime}=\operatorname*{argmin}_{ \mathbf{x}\in S}f(\mathbf{x},\mathbf{\hat{c}}) \tag{35}\]
Note that whenever \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\in S\), it holds that \(\mathcal{L}_{MAP}(\mathbf{\hat{c}},\mathbf{c})=f(\mathbf{x}^{\star}(\mathbf{ c}),\mathbf{\hat{c}})-f(\mathbf{x}^{\star}(\mathbf{\hat{c}}),\mathbf{\hat{c}})\). This is also known as _self-contrastive estimation_ (SCE) (Goodfellow, 2015) since the ground-truth is contrasted with the most likely output of the current model itself.
Also note that for optimization problems with a linear objective, the losses are \(\mathcal{L}_{NCE}(\mathbf{\hat{c}},\mathbf{c})=\sum_{\mathbf{x}^{\prime}\in S} \mathbf{\hat{c}}^{\top}(\mathbf{x}^{\star}(\mathbf{c})-\mathbf{x}^{\prime})\) and \(\mathcal{L}_{MAP}(\mathbf{\hat{c}},\mathbf{c})=\mathbf{\hat{c}}^{\top}( \mathbf{x}^{\star}(\mathbf{c})-\mathbf{x}^{\prime})\), where \(\mathbf{x}^{\prime}=\operatorname*{argmin}_{\mathbf{x}\in S}f(\mathbf{x}, \mathbf{\hat{c}})\). In order to prevent the model from simply learning to predict \(\mathbf{\hat{c}}=\mathbf{0}\), the following alternate loss functions are proposed for these kinds of problems:
\[\mathcal{L}_{NCE}^{(\mathbf{\hat{c}}-\mathbf{c})}(\mathbf{\hat{c}},\mathbf{c} )=\sum_{\mathbf{x}^{\prime}\in S}(\mathbf{\hat{c}}-\mathbf{c})^{\top}( \mathbf{x}^{\star}(\mathbf{c})-\mathbf{x}^{\prime}) \tag{36}\]
\[\mathcal{L}_{MAP}^{(\mathbf{\hat{c}}-\mathbf{c})}(\mathbf{\hat{c}},\mathbf{c} )=\max_{\mathbf{x}^{\prime}\in S}(\mathbf{\hat{c}}-\mathbf{c})^{\top}( \mathbf{x}^{\star}(\mathbf{c})-\mathbf{x}^{\prime}) \tag{37}\]
Construction of \(S\).Forming \(S\) by sampling points from the feasible region \(\mathcal{F}\) is a crucial part of using the contrastive loss functions. To this end, Mulamba et al. (2021) proposes to construct \(S\) by caching all the optimal solutions in the training data. That is why they name \(S\) as'solution cache'. While training, more feasible points are gradually added to \(S\) by solving for some of the predicted cost vectors. However, in order to avoid computational cost, the solver call is not made for each predicted cost during training. Whether to solve for a predicted cost vector is decided by pure random sampling, i.e., is based on a biased coin toss with probability \(p_{solve}\). Intuitively, the \(p_{solve}\) hyperparameter determines the proportion of instances for which the CO problem is solved during training. Experimentally, it has been reported that \(p_{solve}=5\%\) of the time is often adequate, which translates to solving for only \(5\%\) the predicted instances. This translates to reducing the computational cost by approximately \(95\%\), since solving the CO problems represents the major bottleneck in terms of computation time in DFL training.
Approximation of a solver by solution-cache.Furthermore, Mulamba et al. (2021) propose a solver-free training variant for any methodology that treats the optimization solver as a blackbox oracle. Such methodologies include the aforementioned I-MLE, DBB, SPO. In this solver-free implementation, solving the optimization problem is substituted with a cache lookup strategy, where the minimizer within the cache \(S\subset\mathcal{F}\) is considered as a proxy for the solution to the optimization problem (i.e., the minimizer within \(\mathcal{F}\)). This significantly reduces the computational cost as solving an optimization problem is replaced by a linear search within a limited cache. Such an approximation can be useful in case the optimization problem takes long to solve.
DFL as a learning to rank (LTR) problem.In a later work, Mandi et al. (2022) observe that \(\mathcal{L}_{NCE}\) (34) can be derived by formulating DFL as a _pairwise learning to rank_ task (Joachims, 2002). The learning to rank task consists of learning the implicit order over the solutions in \(S\) invoked by the objective function values achieved by the solutions with respect to \(\mathbf{c}\). In other words, it involves learning to predict a \(\mathbf{\hat{c}}\) that ranks the solutions in \(S\) similarly to how \(\mathbf{c}\) ranks them. In the _pairwise_ approach, \(\mathbf{x}^{\star}(\mathbf{c})\) and any \(\mathbf{x}^{\prime}\in S\) are treated as a pair and the model is trained to predict \(\mathbf{\hat{c}}\) such that the ordering of each pair is the same for \(\mathbf{c}\) and \(\mathbf{\hat{c}}\). The loss is considered to be zero if \(\mathbf{\hat{c}}^{\top}\mathbf{x}^{\star}(\mathbf{c})\) is smaller than \(\mathbf{\hat{c}}^{\top}\mathbf{x}^{\prime}\) by at least a margin of \(\Theta>0\). The pairwise loss is formally defined in the following form:
\[\mathcal{L}_{\text{Pairwise}}(\mathbf{\hat{c}},\mathbf{c})=\sum_{\mathbf{x}^ {\prime}\in S}\max\big{(}0,\Theta+(f(\mathbf{x}^{\star}(\mathbf{c}),\mathbf{ \hat{c}})-f(\mathbf{x}^{\prime},\mathbf{\hat{c}}))\big{)} \tag{38}\]
Another loss function is formulated by considering the difference in differences between the objective values at the true optimal \(\mathbf{x}^{\star}(\mathbf{c})\) and non-optimal \(\mathbf{x}^{\prime}\) with \(\mathbf{c}\) and \(\mathbf{\hat{c}}\) as the parameters.
\[\mathcal{L}_{\text{PairwiseDifference}}(\mathbf{\hat{c}},\mathbf{c})=\sum_{ \mathbf{x}^{\prime}\in S}\bigg{(}\big{(}f(\mathbf{x}^{\star}(\mathbf{c}), \mathbf{\hat{c}})-f(\mathbf{x}^{\prime},\mathbf{\hat{c}})\big{)}-\big{(}f( \mathbf{x}^{\star}(\mathbf{c}),\mathbf{c})-f(\mathbf{x}^{\prime},\mathbf{c}) \big{)}\bigg{)}^{2} \tag{39}\]
Further, motivated by _listwise learning to rank task_(Cao, Qin, Liu, Tsai, & Li, 2007), a loss function is proposed by Mandi et al. (2022) where the ordering of all the items in \(S\) is considered, rather than the ordering of pairs of items. Cao et al. (2007) define this listwise loss based on a _top-one probability_ measure. The top-one probability of an item is the probability of it being the best of the set. Note that such probabilistic interpretation in the context of DFL is already defined in Section 3.3. Mandi et al. (2022) make use of the tempered softmax probability defined in (25). Recall that this \(p_{\tau}(\mathbf{x}|\mathbf{c})\) can be interpreted as a probability measure of \(\mathbf{x}\in\mathcal{F}\) being the minimizer of \(f(\mathbf{x},\mathbf{c})\) in \(\mathcal{F}\) for a given \(\mathbf{c}\). However, as mentioned before, direct computation of \(p_{\tau}(\mathbf{x}|\mathbf{c})\) requires iterating over all feasible points in \(\mathcal{F}\), which is intractable. Therefore Mandi et al. (2022) compute the probability with respect to \(S\subset\mathcal{F}\). This probability measure finally is used to define a listwise loss--the cross-entropy loss between \(p_{\tau}(\mathbf{x}|\mathbf{c})\) and \(p_{\tau}(\mathbf{x}|\mathbf{\hat{c}})\), the distributions obtained for ground-truth \(\mathbf{c}\) and predicted \(\mathbf{\hat{c}}\). This can be written in the following form:
\[\mathcal{L}_{\text{Listwise}}(\mathbf{\hat{c}},\mathbf{c})=\bigg{(}-\frac{1} {|S|}\sum_{\mathbf{x}^{\prime}\in S}p_{\tau}(\mathbf{x}^{\prime}|\mathbf{c}) \log p_{\tau}(\mathbf{x}^{\prime}|\mathbf{\hat{c}})\bigg{)} \tag{40}\]
The main advantage of (34), (35), (38), (39) and (40) is that they are differentiable and can be computed directly by any neural network library via automatic differentiation. Also note that the computation and differentiation of the loss functions are solver-free, i.e., they need not solve the optimization problem to compute the loss or its derivative.
Learning efficient surrogate solvers.Another research direction without optimization in the loop is based on reducing the computational cost associated with repeatedly solving optimization problems, by learning efficiently computable and differentiable surrogate losses that approximate and replace the true task loss. Shah, Wang, Wilder, Perrault, and Tambe (2022) propose to learn a surrogate of the regret function by parametric local losses. Due to the difficulty of learning a single convex surrogate function to estimate regret, a convex local surrogate is learned for each data sample in training. By design, the surrogate losses are automatically differentiable, and thus they eliminate the need for a differentiable optimization solver.
### Discussion
So far, in this section, an extensive overview of different DFL methodologies have been provided. For the ease of the readers, a summary of some of the key DFL techniques, discussed so far, have been provided in Table 1. The second column of Table 1 highlights the form of the CO problem applicable to the technique. Note that although some techniques are generally applicable to any optimization problem forms, most techniques have been evaluated so far using CO problems with linear objective functions. The third column
\begin{table}
\begin{tabular}{c c c c} \hline \hline Methodologies & CO Problem Forms & Computation of Gradient & Optimization \\ & & & Layer \\ \hline OptNet & & Implicit differentiation of & \\ (Amos \& Kolter, 2017) & Convex QPs & KKT conditions & \\ Cvxpylayers & & Implicit differentiation of & \\ (Agrawal et al., 2019a) & Convex problems & & \\ Fold-opt & Convex and nonconvex & & \\ (Kotary et al., 2023) & problems & & \\ \hline QPTL & & Implicit differentiation & \\ (Wilder et al., 2019a) & LPs, ILPs & after transforming into QPs & \\ & & by adding regularizer & \\ & & Implicit differentiation of & \\ & & HSD of (relaxed) LPs by & \\ (Mandi \& Guns, 2020) & & adding log-barrier relaxation & \\ & & & Conversion of ILPs into LPs & \\ & & & by method of cutting planes & \\ & & & before applying QPTL & \\ \hline DBB & Optimization problems & & Differentiation of \\ (Pogancic et al., 2020) & with a linear objective & & linear interpolation & \\ & & & of optimization mapping & \\ & & & Treating the CO solver as & \\ & & with a linear objective & \\ & & & negative identity mapping & \\ & & & \\ & & & Finite difference approximation & \\ & & with a linear objective & \\ & & & with perturb-and-MAP & \\ & & & \\ & & & Differentiation of & \\ & & & perturbed optimizer & \\ & & & \\ & & & Differentiation of perturbed & \\ & & & Fenchel-Young loss & \\ & & & \\ & & & Differentiation of & \\ & & & surrogate SPO+ loss & \\ & & & \\ & & & \\ & & & Differentiation of & \\ & & & surrogate contrastive loss & \\ & & & \\ & & & Differentiation of & \\ & & & surrogate LTR loss & \\ & & & \\ & & & Differentiation of a & \\ & & & \\ & & & learned convex local surrogate loss & \\ \hline \hline \end{tabular}
\end{table}
Table 1: A concise overview of gradient modeling techniques in key DFL methodologies that use gradient-based learning.
summarizes the gradient computation technique. The fourth column indicates whether that particular technique is compatible with any generic task loss. Techniques, termed as implementations of _differentiable optimization layers_, can be embedded in any stage of an NN architecture. The other techniques are applicable where optimization is the final stage of the pipeline (such as in Predict-Then-Optimize problem formulations) and a particular loss (most often regret) is used as the task loss.
### Other Aspects of Decision-Focused Learning
In the following, some aspects related to DFL, that have not yet been discussed in this manuscript, will be highlighted. To begin with, it should be noted that certain CO problems may have _multiple non-unique optimal solutions_ for a given cost vector. This can occur when the cost vector of an LP is parallel to one of the faces of the feasible polyhedron. Moreover, problems involving symmetric graphs often exhibit multiple optimal solutions, especially when the problems' solution can be transformed into other solutions through automorphisms (Weisstein, 2000). It is important to note that if the predicted cost vector has multiple non-unique optimal solutions, each of these solutions may have different value of regret. In such scenarios, Elmachtoub and Grigas (2022) propose to consider the worst-case regret. To do so, the set of optimal solutions of \(\hat{\mathbf{c}}\) can be represented by \(\mathcal{X}^{\star}(\hat{\mathbf{c}})\). And then the worst-case regret can be defined in the following form:
\[\textit{Regret}(\mathbf{x}^{\star}(\hat{\mathbf{c}}),\mathbf{c})=\max_{ \mathbf{x}^{\star}(\hat{\mathbf{c}})\in\mathcal{X}^{\star}(\hat{\mathbf{c}})} f(\mathbf{x}^{\star}(\hat{\mathbf{c}}),\mathbf{c})-f(\mathbf{x}^{\star}( \mathbf{c}),\mathbf{c}) \tag{41}\]
Having addressed the possibility of the presence of multiple non-unique optimal solutions in a CO problem, the focus now turns to other important facets of DFL.
#### 3.6.1 Prediction-focused vs. Decision-focused learning
DFL methodologies are expected to deliver lower regret than a PFL approach in Predict-Then-Optimize problems, as the ML model is directly trained to achieve low regret. However, as discussed before, the implementation of DFL poses significant challenges. In fact, practitioners may be tempted to resort to a PFL approach to circumvent the computational costs associated with DFL, when dealing with real-world Predict-Then-Optimize problems. To encourage practitioners to adopt DFL methodologies, it is crucial to investigate scenarios where DFL methodologies outperform the PFL approach. To this end, Elmachtoub, Lam, Zhang, and Zhao (2023) conduct a theoretical comparison of the limiting distributions of the optimality gaps between the two approaches in the context of stochastic optimization. They show the PFL approach that does not consider optimization while training the model asymptotically outperforms the integrated prediction and optimization approach, employed by DFL methodologies, _if_ the underlying prediction model is well-specified. This is intuitive, as a well-specified model tends to produce highly accurate predictions, which can contribute to the success of the PFL approach. In such cases, the DFL methodologies might perform worse than PFL since training in DFL involves _approximate_ gradients (because the true gradient is zero almost everywhere), whereas the gradient is well-defined for a PFL approach. On the other hand, they show that if the model is not well-specified, a PFL approach perform suboptimally compared to the DFL approach. Hence, it is recommended to
use DFL when there exists aleatoric or epistemic uncertainty. As most real-world settings include various sorts of uncertainty--both aleatoric and epistemic--DFL methodologies are expected to outperform the PFL approach. In a separate work, Cameron, Hartford, Lundy, and Leyton-Brown (2022) show that the suboptimality of the PFL becomes more pronounced in the presence of correlations between the predicted parameters.
#### 3.6.2 Alternatives to gradient-based decision-focused learning
The methodologies explained so far implement DFL by gradient descent training, which is the go-to approach for training neural networks. However, note that there exist other machine learning frameworks, such as tree-based methods, which do not require gradient-based training. To avoid the problem of zero-valued gradients altogether, several works have considered alternatives to gradient-based learning instead.
In SPO Trees (SPOTs) (Elmachtoub, Liang, & McNellis, 2020), the predictive model is a decision tree or ensemble of decision trees. Such models can be learned by recursive partitioning with respect to the regret directly, and thus do not require the use of the SPO+ surrogate loss function introduced by Elmachtoub and Grigas (2022). Alternatively, the tree learning problem can be posed as a MILP and be solved by an off-the-shelf solver, in the same spirit as Jeong, Jaggi, Butler, and Sanner (2022). Jeong et al. (2022) formulate the problem of minimizing regret as a mixed-integer linear program (MILP), when the predictive model is linear. They start from the bilevel optimization formulation, introduced in (6a) and (6b). First, the transition points where the solution of the lower level program (6b) changes are identified, and then the solution space is exhaustively partitioned, and for each partition the solution is annotated. This paves the way to construct a MILP formulation of the outer program (6a). This MILP problem is solved to learn the parameters \(\mathbf{\omega}\) of the linear predictive model. The resulting model is guaranteed to be globally optimal, which is not the case for gradient-based methods that might get stuck in a local optimum. However, their method is limited to ML models that are linear and optimization problems that are binary MILPs.
Demirovic, Stuckey, Guns, Bailey, Leckie, Ramamohanarao, and Chan (2020) consider linear ML models and represent the objective function of the CO problem as a piece-wise linear function of the ML parameters. In this proposed technique, the ML parameters are updated via coordinate descent algorithm, where each component of the cost vector is updated at a time to minimize the regret keeping other components fixed. This technique requires identifying the _transition points_, where regret changes, as function of each component of the cost parameter. Demirovic et al. (2020) consider CO problems that can be solved by dynamic programming and identify the transition points using dynamic programming. In a later work, Guler, Demirovic, Chan, Bailey, Leckie, and Stuckey (2022) extend this technique by employing a 'divide-and-conquer' algorithm to identify the transition points for CO problems whose objective function is a bilinear function of the decision variables and the predicted parameters. This development generalizes the previous work (Demirovic et al., 2020) to cover much broader class of CO problems and offers a substantial speed improvement. The 'branch & learn' approach proposed by HU, Lee, Lee, and Zhong (2022), which consider CO problems that can be solved by recursion, also extends this technique.
#### 3.6.3 Predicting parameters in the constraints
The majority of the works in DFL aim to predict parameters in the objective function and assume that the feasible space is precisely known. However, in many applications the unknown parameters occur in the constraints as well as in the objectives. When the parameters in the constraints are predicted and prescribed decisions are made using the predicted parameters, one major issue is that the prescribed decisions might turn out to be infeasible with respect to the true parameters. In this case, the task loss should not only minimize the suboptimality of the prescribed decisions, but it should also penalize if the prescribed decisions become infeasible. Hence designing DFL algorithms suitable for such problems entails a few additional considerations. The first consideration deals with quantifying the extent of infeasibility when the prescribed decisions become infeasible with respect to the true parameters. In this regard, Garcia, Street, Homem-de Mello, and Munoz (2021) propose to add artificial slack variables with high penalty costs in the objective function to penalize infeasible decisions. In a recent work, Hu et al. (2022) introduce the notion of _post-hoc_ regret, wherein a non-negative penalty is added to regret to account for the conversion of infeasible solutions into feasible ones. This idea of a penalty function shares a fundamental resemblance to the concept of recourse action in stochastic programming (Ruszczynski and Shapiro, 2003). In a later work, Hu, Lee, and Lee (2023b) apply the 'branch & learn' (HU et al., 2022) to minimize _post-hoc_ regret in CO problems, solvable by recursion.
The second consideration is formulating a task loss that strikes a balance between trading off suboptimality and the measure of infeasibility. The next consideration is computing the gradients of this task loss with respect to the parameters in the constraints. Some of the techniques discussed in Section 3.1 can be utilized for this purpose. For example, the gradient can be obtained by solver unrolling. Tan, Delong, and Terekhov (2019) compute the gradient by unrolling a LP. As the parameters in the constraints are also present in the the KKT conditions (13), it is possible to compute the gradients for optimization problems, with differentiable constraint functions by differentiating the KKT conditions using the techniques discussed in Section 3.1. Hu et al. (2022) shows how the gradient can be computed by differentiating the KKT conditions for packing and covering LPs. For an LP, Tan, Terekhov, and Delong (2020) provide a empirical risk minimization formulation considering both the suboptimlaity of the prescribed decision and the feasibility of the true optimal decisions. This formulation takes the form of a non-linear optimization program and they propose to compute the derivative by considering its sequential quadratic programming (SQP) approximation.
The task of computing the gradients of the task loss with respect to the parameters in the constraints is particularly challenging for combinatorial optimization problems, which often involves discrete feasible space. For combinatorial optimization problems, it might happen that no constraints are active at the optimal point. So, slight changes of the parameters in the constraints do not change the optimal solution leading towards the problem of zero gradients. Hence coming up with meaningful gradients for back-propagation is a big challenge for combinatorial optimization problems. Paulus, Rolinek, Musil, Amos, and Martius (2021) develop a differentiable optimization layer for ILPs, which considers the downstream gradient of the solution as an input and returns the directions of the updating
the parameters in the backward pass. They update the parameters along the directions so that the Euclidean distance between the solution of the updated parameter and the updated solution with the downstream gradient is minimized. For ILPs, Nandwani, Ranjan, Mausam, and Singla (2023) view the task of constraint learning from the lens of learning hyperplanes, which is common in classification tasks. Such an approach requires negative samples. However, the negative samples in this setting must also include infeasible points, which is different from the framework proposed by Mulamba et al. (2021).
#### 3.6.4 Model robustness in decision-focused learning
The issue of model robustness arises often in deep learning. As has been shown in many works, it is often possible for malicious actors to craft inputs to a neural network in such a way that the output is manipulated (evasion attacks) (Goodfellow, Shlens, & Szegedy, 2014), or to generate training data which cause adverse effects on the performance of the trained model (poisoning attacks). As a subset of machine learning, some adversarial settings also apply in DFL.
Evasion attacks, despite being the most commonly studied adversarial attacks, do not generalize straightforwardly to DFL since they inherently pertain to classification models with finite output spaces. On the other hand, it is shown by Kinsey, Tuck, Sinha, and Nguyen (2023) that effective poisoning attacks can be made against DFL models. The paper shows that while such attacks can be effective, they are computationally expensive due to the optimization which must be repeatedly evaluated to form the attacks. On the other hand, it is also demonstrated that poisoning attacks designed against two-stage models can be transferred to fully integrated DFL models.
Separately, Johnson-Yu, Wang, Finocchiaro, Taneja, and Tambe (2023) study robustness of decision-focused learning under label noise. The paper provides bounds on the degradation of regret when test-set labels are corrupted by noise relative to those of the training set. An adversarial training scheme is also proposed to mitigate this effect. The robust training problem is equivalent to finding the equilibrium solution to a Stackelberg game, in which a figurative adversary applies label noise that is optimized to raise regret, while the main player seeks model parameters that minimize regret.
#### 3.6.5 Stochastic optimization
Settings in decision-focused learning based on stochastic optimization models are studied by Donti, Kolter, and Amos (2017). In contrast to more typical settings, the downstream decision model is considered to be a stochastic optimization problem. In this formulation, it is only possible to predict parameters of a random distribution that models the parameters of an optimization problem. For instance, the mean and variance of load demands in a power scheduling problem could be modeled as parameters of the optimization problem. Their work shows how such problems can be converted to DFL with deterministic decision models and solved using the techniques described in this article. To this end, it also introduces an effective technique for approximating the derivatives through arbitrary convex optimization problems, by forming and differentiating their quadratic programming approximations, as computed by sequential quadratic programming.
#### 3.6.6 Problems other than optimization problems
Furthermore, we believe that DFL can be further extended to encompass problems beyond optimization problems, thereby broadening its applicability. For instance, to integrate symbolic reasoning into neural network architectures, Wang, Donti, Wilder, and Kolter (2019) make use of MAXSAT solvers and perform end-to-end training of the neural network by differentiating through the semidefinite program (SDP) relaxations of the MAXSAT problems. Wilder, Ewing, Dilkina, and Tambe (2019b) consider the K-means clustering in a graph as the optimization problem, i.e., the optimization problem in their case is to cluster the nodes of a given graph into \(K\) segments. They embed the \(K\)-means clustering as a layer in a neural network architecture by differentiate through the clustering layer. Wang, Shah, Chen, Perrault, Doshi-Velez, and Tambe (2021) further extend DFL, for sequential decision making problems, where the decision making problems have been formulated as Markov decision processes (MDPs). In such cases, the DFL problem deals with the challenge of predicting the unknown parameters in the MDPs.
#### 3.6.7 Active learning algorithm for DFL
Active learning concerns with ML problems where labeled data are scarce or expensive to obtain. To address the challenge of limited training data, active learning algorithms choose the most informative instances for labeling (Settles, 2009). Liu, Grigas, Liu, and Shen (2023) study active learning in DFL paradigm. To choose datapoints for which to ask for a label, they propose to use notion of 'distance to degeneracy' (El Balghiti et al., 2019). Distance to degeneracy measures how far the predicted cost vector is from the set of cost vectors that have multiple optimal solutions. They argue that if distance to degeneracy is higher at a datapoint, there is more certainty regarding the solution (of the CO problem); hence they propose to acquire the label of a datapoint if its distance to degeneracy is lower than a threshold.
#### 3.6.8 Multi-task decision-focused learning
In most DFL works, a single task is considered. For instance, in the shortest path benchmark considered by Elmachtoub and Grigas (2022), the grid structure and the start and end nodes are the same in all instances. However, one often has to deal with multiple tasks at once, in which it would be convenient to make decision-focused predictions, without having to train a separate model for each task. A first step in this direction was recently taken in (Tang and Khalil, 2023a). This paper proposes a way of training a model in a decision-focused way with respect to multiple tasks at once. They consider two kinds of architectures. The first is a regular multi-layer perceptron that outputs a single vector \(\mathbf{\hat{c}}\) which is used in the different tasks. The different resulting task losses then get aggregated to inform the update to weights \(\boldsymbol{\omega}\), i.e., the weights \(\boldsymbol{\omega}\) are trained to produce a \(\mathbf{\hat{c}}\) that generally works well for the different tasks considered. The second architecture is a multi-headed one, consisting of one or more shared first layers, followed by a dedicated head for every task. This means that a different vector \(\hat{c}_{i}\) is produced for every task. Their results show that they can train a model that can make effective decision-focused predictions for multiple tasks at once, and that this is particularly beneficial when not that many training data are available. However,
a remaining limitation is that the model can still not be trained with the aim of _generalizing_ to new tasks.
## 4 Applications of Decision-Focused Learning
The Predict-Then-Optimize problem occurs in many real-world applications, as optimal decisions can be found by solving CO problems and due to the presence of uncertainty, some parameters of the CO problems must be estimated. Having seen the development of DFL for Predict-Then-Optimize problems in the preceding section, practical uses of DFL in various application domains will be presented below. As DFL techniques, which predict cost parameters have been reviewed in Section 3, applications, presented below, focus on the task of predicting only the cost parameters.
Computer vision.The DBB framework (Pogancic et al., 2020) (reviewed in Section 3.3) has been used by Rolinek, Musil, Paulus, Vlastelica, Michaelis, and Martius (2020a) for differentiating rank-based metrics such as precision and recall and by Rolinek, Swoboda, Zietlow, Paulus, Musil, and Martius (2020b) and Kainmueller, Jug, Rother, and Myers (2014) for differentiating bipartite matching in deep graph and multi-graph matching problems respectively in the application of semantic keypoint matching of images.
Fair Learning to Rank.In learning to rank (LTR), a machine learning model must produce rankings of documents in response to users' search queries, in which those most relevant to a given query are placed in the highest ranking positions. In this setting, the relevance of documents to queries is often measured empirically by historical user click rates (Cao et al., 2007). In fair learning to rank (FLTR), this relevance-based matching must be performed subject to strict constraints on the relative exposure between predefined groups. Due to the difficulty of enforcing such constraints on the outputs of a machine learning model, many FLTR frameworks resort to a two-stage approach in which prediction of query-document relevance scores is learned by a typical LTR model without constraints on fairness of exposure. At test time, the predicted relevance scores inform the objective of a separate fair ranking optimization program (Singh & Joachims, 2018). Kotary, Fioretto, Van Hentenryck, and Zhu (2021) use DFL to unify the prediction of relevance scores with the subsequent optimization of fair rankings, in an end-to-end model trained by SPO which learns to map user queries directly to the fair ranking policies that optimize user relevance. The result is a FLTR model which outperforms previous penalty-based models in terms of both user relevance and fairness, with the ability to directly control their trade-offs by modifying the fairness constraints of the optimization layer.
Route optimization.Ferber, Griffin, Dilkina, Keskin, and Gore (2023) present an interesting application, where DFL is used to combat the challenge of wildlife trafficking. They consider the problem of predicting the flight trajectory of traffickers based on a given pair of source and destination airports. It is framed as a shortest path problem in a graph, where each node is an airport. In the prediction stage, the probability of using a directed edge \((i,j)\) to leave the node \(i\) is predicted. In the optimization stage, the most likely path from the source to the destination is found by solving a shortest path problem where the negative log probabilities are used as edge weights. In this Predict-Then-Optimize formulation, the probabilities are predicted via DFL, using the DBB framework for gradient computation.
Solving a shortest path problem by considering the negative log probabilities as edge weights has also been explored by Mandi, Canoy, Bucarey, and Guns (2021). In (Mandi et al., 2021), the objective is to prescribe most preferred routing in capacitated vehicle routing problem (CVRP) (Toth and Vigo, 2015) for last-mile delivery applications. A high probability value for the edge \((i,j)\) indicates that it is the preferred edge to leave the node \(i\). However, they do not observe any advantage of DFL paradigm over PFL paradigm and attribute this to the lack of training data instances (fewer than 200 instances). DFL is used for last-mile delivery applications by Chu, Zhang, Bai, and Chen (2021) too. However, there the objective is to minimize total travel time. In the prediction stage, the travel times of all the edges are predicted and in the optimization stage, the CVRP is solved to minimize the total travel time. The underlying model is trained using the SPO framework to directly minimize the total travel time.
#### Maritime transportation.
The inspection of ships by port state control has been framed as a Predict-Then-Optimize problem by Yang, Yan, and Wang (2022). Due to limited number of available personnel, the aim is to identify non-compliant ships with high attention risk beforehand and select those ships for inspection. A ship can be found to be non-compliant by port state control in multiple categories. If a ship is found to be non-compliant for a category, the deficiency number for that category will be recorded as one. In the prediction stage, a linear model is built to identify deficiency numbers of the ships in all the categories and in the optimization stage, a CO problem is solved to select ships maximizing the total number of deficiencies. Due to the nature of the large-scale optimization problem, training in the SPO framework is not practical. Therefore, they employ pairwise-comparison based loss function, similar to Eq. (38) to implement DFL. Ship maintenance activities by ship owners have been framed as Predict-Then-Optimize problems by Tian, Yan, Liu, and Wang (2023). The ship owners have to schedule regular maintenance activities to remain compliant. However, as maintenance activities are expensive, the objective of identifying categories that may warrant immediate detentions has been considered. To do so, in the prediction stage, a random forest model is built to predict the deficiency number (likelihood of non-compliance) for each category. In the optimization stage, a CO problem is formulated considering maintenance cost and detention cost to determine whether maintenance activity should be scheduled for each category. The random forest models are trained to directly minimize regret using SPOTs (Elmachtoub et al., 2020).
#### Planning and Scheduling.
Wahdany, Schmitt, and Cremer (2023) provide a use-case of DFL in renewable power system application. In their work, the prediction stage involves the task of generating wind power forecasts. As these forecasts are further used in power system energy scheduling, the task of minimizing power system operating costs has been considered. Cvxpylayers(Agrawal, Amos, Barratt, Boyd, Diamond, & Kolter, 2019) has been used to directly train the model with the objective of minimizing power system operating costs. DFL is applied in power system application by Sang, Xu, Long, Hu, and Sun (2022) also. In the prediction stage electricity prices are predicted and the optimization stage deals with optimal energy storage system scheduling to maximize arbitrage benefits. Lower values of regret have been reported when the prices are predicted using the SPO framework.
#### Communication technology.
DFL is applied in mobile wireless communication technology application by Chai, Wong, Tong, Chen, and Zhang (2022). Fluid antenna system
(Wong, Tong, Zhang, & Zhongbin, 2020) is one of the recent development in mobile wireless communication technology. However, its effectiveness depends on the position of the radiating element, known as the port. Chai et al. (2022) frame the port selection problem as a Predict-Then-Optimize problem, where in the prediction stage signal-to-noise ratio for each position of the port is predicted and then the optimal position of the port is decided in the optimization stage. They use LSTM as the predictive model and report the SPO framework is very effective for such port selection applications.
Solving non-linear combinatorial optimization problems.Ferber et al. (2022) study the problem of learning a linear surrogate optimizer to solve non-linear optimization problems. The objective is to learn a surrogate linear optimizer whose optimal solution is the same as the solution to the non-linear optimization problem. Learning the parameters of the surrogate linear optimizer entails backpropagating through the optimizer, which is implemented using Cvxpylayers (Agrawal et al., 2019b).
Interested readers are referred to (Qi & Shen, 2022) for more applications of Predict-Then-Optimize problems in various areas within operations management.
## 5 Experimental Evaluation on Benchmark Problemsets
DFL recently has received increasing attention. The methodologies discussed in Section 3 have been tested so far on several different datasets. Because a common benchmark for the field has not yet been set up, comparisons among methodologies are sometimes inconsistent. In this section, an effort is made to propose several benchmark test problems for evaluating DFL methodologies. 1 Then some of the methodologies explained in Section 3 are compared on these test problems.
Footnote 1: During the course of writing this manuscript, we have become aware of the PyEPO project (Tang & Khalil, 2023b), which develops an interface for benchmarking DFL methodologies. However, it is important to emphasize that our work differs significantly from PyEPO. While PyEPO focuses on providing an interface for implementing DFL methodologies, our paper serves as a comprehensive survey that goes beyond benchmarking.
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Methodology** & **Controlling functions** & **Position variables** & **GO-SQu** & **Prediction Model** \\ \hline Shortest path problem & & Linear & Continuous & OR-Tools & Linear \\ on a \(5\times 5\) grid & & & & \\ Portfolio optimization & Quadratic & Continuous & Gurobi & Linear \\ Warcraft shortest path & & Linear & Continuous & Customized python & CNN \\ Energy-cost aware & & & & \\ scheduling & & & & \\ Knapsack problem & & & & \\ Diverse bipartite matching & & & & \\ Subset selections & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Brief overview of the test problems considered for experimental evaluation. The **objective functions are linear** for all the optimization problem.
### Problem Descriptions
All the test problems, which are selected for benchmarking, have been previously used in the DFL literature and their datasets are publicly available. Needless to say, all these problems encompass the two stages--prediction and optimization. Table 2 provides an overview of the experimental setups associated with each test problem, including the specification of the CO problem and the type of predictive model. Next, these test problems are described in detail.
#### 5.1.1 Shortest path problem on a \(5\times 5\) grid
This experiment is adopted from the work of Elmachtoub and Grigas (2022). It is a shortest path problem on a \(5\times 5\) grid, with the objective of going from the southwest corner of the grid to the northeast corner where the edges can go either north or east. This grid consists of 25 nodes and 40 edges.
Formulation of the optimization problem.The shortest path problem on a graph with a set \(V\) of vertices and a set \(E\) of edges can be formulated as an LP problem in the following form:
\[\min_{\mathbf{x}}\ \ \mathbf{c}^{\top}\mathbf{x} \tag{42a}\] \[\mathtt{s.t.}A\mathbf{x}=\mathbf{b}\] (42b) \[\mathbf{x}\geq\mathbf{0} \tag{42c}\]
Where \(A\in\mathbb{R}^{|V|\times|E|}\) is the incidence matrix of the graph. The decision variable \(\mathbf{x}\in\mathbb{R}^{|E|}\) is a binary vector whose entries are 1 only if the corresponding edge is selected for traversal. \(\mathbf{b}\in\mathbb{R}^{|V|}\) is the vector whose entry corresponding to the source and sink nodes are 1 and \(-1\) respectively; all other entries are 0. The constraint (42b) must be satisfied to ensure the path will go from the source to the sink node. The objective is to minimize the cost of the path with respect to the (predicted) cost vector \(\mathbf{c}\in\mathbb{R}^{|E|}\).
Synthetic data generation process.In this problem, the prediction task is to predict the cost vector \(\mathbf{c}\) from the feature vector \(\mathbf{z}\). The feature and cost vectors are generated according to the data generation process defined by Elmachtoub and Grigas (2022). For the sake of completeness, the data generation process is described below.2 Each problem instance has cost vector of dimension \(|E|=40\) and feature vector of dimension \(p=5\). The training data consists of \(\{(\mathbf{z_{i}},\mathbf{c_{i}})\}_{i=1}^{N}\), which are generated synthetically. The feature vectors are sampled from a multivariate Gaussian distribution with zero mean and unit variance, i.e., \(\mathbf{z_{i}}\sim\mathbf{N}(0,I_{p})\) To generate the cost vector, first a matrix \(B\in\mathbb{R}^{|E|\times p}\) is generated, which represents the true underlying model. The cost vectors are then generated according to the following formula:
Footnote 2: The generator in [https://github.com/paulgrigas/SmartPredictThenOptimize](https://github.com/paulgrigas/SmartPredictThenOptimize) is used to generate the dataset.
\[c_{ij}=\bigg{[}\bigg{(}\frac{1}{\sqrt{p}}\big{(}B\mathbf{z_{i}}\big{)}+3 \bigg{)}^{\text{Deg}}+1\bigg{]}\xi_{i}^{j} \tag{43}\]
where \(c_{ij}\) is the \(j^{\text{th}}\) component of cost vector \(\mathbf{c_{i}}\). The _Deg_ parameter specifies the extent of model misspecification, because a linear model is used as a predictive model in the
experiment. The higher the value of _Deg_, the more the true relation between the features and objective parameters deviates from a linear one and the larger the prediction errors will be. Finally, \(\xi_{i}^{j}\) is a multiplicative noise term sampled randomly from the uniform distribution \([1-\vartheta,1+\vartheta]\). The experimental evaluation involves five values of the parameter _Deg_, which are \(1,2,4,6\) and \(8\), and the noise-halfwidth parameter \(\vartheta\) being \(0.5\). Furthermore, for each setting, a different training set of of size \(1000\) is used. In each case, the final performance of the model is evaluated on a test set of size \(10,000\).
Predictive model.In each setting, the underlying predictive model is a one-layer feed-forward neural network without any hidden layer, i.e., a linear model. Basically the input to the model is a \(p\) dimensional vector, and output is a \(|E|\) dimensional vector. Note that a multi-layer neural network model can be used to to improve the accuracy of the predictive model. The intuition behind using a simple predictive model is to test the efficacy of the DFL methods when the predictions are not \(100\%\) accurate. The DFL methods are trained to minimize the regret and the prediction-focused model is trained by minimizing the MSE loss between the true and predicted cost vector.
#### 5.1.2 Portfolio optimization problem
A classic problem that combines prediction and optimization is the Markowitz portfolio optimization problem, in which asset prices are predicted by a model based on empirical data, and then subsequently, a risk-constrained optimization problem is solved for a portfolio which maximizes expected return. This experiment is also adopted from the work of Elmachtoub and Grigas (2022).
Formulation of the optimization problem.In portfolio optimization problem, the objective is to choose a portfolio of assets having highest return subject to a constraint on the total risk of the portfolio. The problem is formulated in the following form:
\[\max_{\mathbf{x}} \ \mathbf{c}^{\top}\mathbf{x}\] (44a) \[\mathtt{s.t.} \ \ \mathbf{x}^{\top}\Sigma\mathbf{x}\leq\gamma\] (44b) \[\
whose entries are drawn uniformly over \([-0.0025\vartheta,0.0025\vartheta]\) is generated. Asset returns are calculated first in terms of their conditional mean \(\bar{c}_{ij}\) as
\[\bar{c}_{ij}\coloneqq(\frac{0.05}{\sqrt{\bar{p}}}(B\mathbf{z_{i}})_{j}+(0.1)^{ \frac{1}{\mathrm{Deg}}})^{\mathrm{Deg}} \tag{45}\]
Then the observed return vectors \(\mathbf{c_{i}}\) are defined as \(c_{ij}\coloneqq\bar{r}_{i}+Lf+0.01\vartheta\xi\), where \(f\sim\mathbf{N}(0,I_{4})\) and noise \(\xi\sim\mathbf{N}(0,I_{d})\) This causes the \(c_{ij}\) to obey the covariance matrix \(\Sigma\coloneqq LL^{\top}+(0.01\zeta)^{2}I\), which is also used to form the constraint (44b), along with a bound on risk, defined as \(\gamma\coloneqq 2.25\ e^{\top}\Sigma e\) where \(e\) is the equal-allocation solution (a constant vector). Four values of the parameter _Deg_--\(1,4,8,16\) have been used in the experimental evaluation. The value of noise magnitude parameter \(\vartheta\) is set to \(1\). It is assumed that the covariance matrix of the asset returns does not depend on the features. The values of \(\Sigma\) and \(\gamma\) are constant, and randomly generated for each setting.
Predictive modelLike the previous experiment, the underlying predictive model is a linear model, whose input is a feature vector \(\mathbf{z}\in\mathbb{R}^{p}\) and output is the return vector \(\mathbf{c}\in\mathbb{R}^{d}\).
#### 5.1.3 Warcraft shortest path problem
This experiment was adopted from the work of Pogancic et al. (2020). Each instance in this problem is an image of a terrain map using the Warcraft II tileset (Guyomarch, 2017). Each image represents a grid of dimension \(d\times d\). Each of the \(d^{2}\) pixels has a fixed underlying cost, which is unknown and to be predicted. The objective is to identify the minimum cost path from the top-left pixel to the bottom-right pixel. From one pixel, one can go in eight neighboring pixels--up, down, front, back, as well as four diagonal ones. Hence, it is a shortest path problem on a graph with \(d^{2}\) vertices and \(\mathcal{O}(d^{2})\) edges.
Formulation of the optimization problem.Note that this is a node-weighted shortest path problem, where each node (pixel) in the grid is assigned a cost value; whereas in the previous shortest path problem, each edge is assigned a cost value. However, this problem can be easily reduced to the more familiar edge weighted shortest path problem by 'node splitting'. 'Node splitting' splits each node into two separate nodes--entry and exit nodes and adds an edge, that has a weight equal to the node weight, from the entry node to the exit node. For each original edge, an edge, with null weight, from the exit node of the source node to the entry node of the sink node, is constructed.
Predictive model.The prediction task is to predict the cost associated with each pixel. The actual cost ranges from from \(0.8\) to \(9.2\) and is dependent on visible characteristics of the pixel. For instance, cost changes depending on whether the pixel represents a water-body, land or wood. The predictive model used in this case is a convolutional neural network (CNN), which predicts the cost of each node (pixel). The model takes the \(d\times d\) image as an input and outputs costs of the \(d^{2}\) pixels. The ResNet18 (He, Zhang, Ren, & Sun, 2016) architecture is slightly modified to form the ML model. The first five layers of ResNet18 are followed by a max-pooling operation to predict the underlying cost of each pixel. Furthermore, a Relu activation function (Agarap, 2019) is used to ensure the predicted cost remains positive, thereby avoiding the existence of negative cycles in the shortest path edge weights.
#### 5.1.4 Energy-cost aware scheduling
This experiment setup was adopted from the work of Mandi et al. (2020). This is a resource-constrained day-ahead job scheduling problem (Simonis, O'Sullivan, Mehta, Hurley, & Cauwer, 1999) with the objective of minimizing energy cost. Tasks must be assigned to a given number of machines, where each task has a duration, an earliest start, a latest end, a resource requirement and a power usage. Each machine has a resource capacity constraint. Also, tasks cannot be interrupted once started, nor migrated to another machine and must be completed before midnight. The scheduling is done in one-day advance. So, the prediction task is to predict the energy prices of the next day.
Formulation of the optimization problem.The scheduling problem is formulated as an ILP. Let \(J\) be the set of tasks to be scheduled on a set of machines \(I\) while maintaining resource requirement of \(W\) resources. The tasks must be scheduled over \(T\) number of time slots. Each task \(j\) is specified by its duration \(\zeta_{j}\), earliest start time \(\zeta_{j}^{(1)}\), latest end time \(\zeta_{j}^{(2)}\), power usage \(\phi_{j}\). Let \(\rho_{jw}\) be the resource usage of task \(j\) for resource \(w\) and \(q_{iw}\) is the capacity of machine \(i\) for resource \(w\). Let \(x_{jit}\) be a binary variable which possesses 1 only if task \(j\) starts at time \(t\) on machine \(i\). The objective of minimizing energy cost while satisfying the required constraints can be expressed by the following ILP:
\[\min_{x_{jit}}\sum_{j\in J}\sum_{i\in I}\sum_{t\in T} x_{jit}\Big{(}\sum_{t\leq t\prime<t+\zeta_{j}}\phi_{j}c_{t\prime} \Big{)} \tag{46a}\] \[\texttt{s.t.}\quad\sum_{i\in I}\sum_{t\in T}x_{jit} =1\,\forall_{j\in J}\] (46b) \[x_{jit} =0\ \ \forall_{j\in J}\forall_{i\in I}\forall_{t<\zeta_{j}^{(1)}}\] (46c) \[x_{jit} =0\ \ \forall_{j\in J}\forall_{i\in I}\forall_{t+\zeta_{j}> }\zeta_{j}^{(2)}\] (46d) \[\sum_{j\in J}\sum_{t-\zeta_{j}<t^{\prime}\leq t}x_{jit}\rho_{jw} \leq q_{iw},\forall_{i\in I}\forall_{w\in W}\forall_{t\in T}\] (46e) \[x_{jit} \in\{0,1\}\forall_{j\in J}\forall_{i\in I}\forall_{t\in T} \tag{46f}\]
The (46b) constraint ensures each task is scheduled once and only once. The constraints in (46c) and (46d) ensure that the task scheduling abides by earliest start time and latest end time constraints. (46e) imposes the constraints of resource requirement.
Data description.The prediction task is to predict the energy prices one day advance. The energy price dataset comes from the Irish Single Electricity Market Operator (SEMO) (Ifrim, O'Sullivan, & Simonis, 2012). This dataset consists of historical energy price data at 30-minute intervals starting from midnight on the 1st of November, 2011 until the 31st of December, 2013. In this setup, each day forms an optimization instance, which comprises of 48 time slots, corresponding to 48 half-hour slots. Each half-hour instance of the data has calendar attributes, day-ahead estimates of weather characteristics, SEMO day-ahead forecasted energy-load, wind-energy production and prices, actual wind-speed, temperature and \(CO_{2}\) intensity, which are used as features. So, the dimension of feature vector is 8. Note that, in this dataset, each \(c_{t}\) in the cost vector is associated with an eight dimensional feature vector, i.e., \(\mathbf{c}\in\mathbb{R}^{48}\) and \(\mathbf{z}\in\mathbb{R}^{48\times 8}\).
Predictive model.As energy prices of each half-hour slot is associated with 8 features, the input to the predictive model is a feature vector of dimension 8 and output is a scalar. In this case also, the predictive model is a linear model, i.e., a feed forward neural network without any hidden layer.
#### 5.1.5 Knapsack problem
This problem setup was also adopted from the work of Mandi et al. (2020). The objective of the knapsack problem is to choose a maximal value subset from a given set of items, subject to a capacity constraint. In this case, the weights of all items and the knapsack capacity are known. What are unknown are the values of the items. Hence, the prediction task is to predict item values of each item.
Formulation of the optimization problem.The formulation of the knapsack optimization problem with unit weights has already been provided in Eq. (4). However, in general the weights of all items are not equal. So, a general knapsack optimization problem can be formulated as follows:
\[\max_{\mathbf{x}}\ \ \mathbf{c}^{\top}\mathbf{x}\] (47a) \[\mathtt{s.t.}\ \mathbf{w}^{\top}\mathbf{x}\leq\text{Capacity}\] (47b) \[\
Getoor, Galligher, & Eliassi-Rad, 2008), where a node represent a publication and an edge represent a citation. So the matching problem is to identify the citation between the two sets of publications. Furthermore, the matching must obey diversity constraints, as described later.
Note that this problem falls under the category of structured output prediction tasks (Nowozin & Lampert, 2011), which requires capturing dependencies and relationships between different parts of the output. In this matching problem, each edge does not have an associated cost in the true sense. Therefore, in the prediction-focused approach the model is trained by directly predicting the presence or absence of each edge. On the other hand, the DFL approaches consider the likelihood of the existence of each edge as the edge weights and then determine which edges should be present while ensuring all the constraints are satisfied.
Optimization problem formulation.Let \(S_{1}\) and \(S_{2}\) denote the two sets. The matching must satisfy the following diversity constraints: a minimum \(\rho_{1}\%\) and \(\rho_{2}\%\) of the suggested pairings should belong to same and distinct fields of study respectively. Let \(c_{ij}\) be the likelihood of an edge existing between article \(i\) and \(j\), \(\forall i\in S_{1},j\in S_{2}\).
With this likelihood value, the matching can be performed by solving the following ILP, which ensures the diversity constraints:
\[\max_{\mathbf{x}} \sum_{i,j}c_{ij}x_{ij} \tag{48a}\] \[\mathtt{s.t.} \sum_{j}x_{ij}\leq 1 \forall i\in S_{1}\] (48b) \[\sum_{i}x_{ij}\leq 1 \forall j\in S_{2}\] (48c) \[\sum_{i,j}\phi_{i,j}x_{ij}\geq\rho_{1}\sum_{i,j}x_{ij}\] (48d) \[\sum_{i,j}(1-\phi_{ij})x_{ij}\geq\rho_{2}\sum_{i,j}x_{ij}\] (48e) \[x_{ij}\in\{0,1\} \forall i\in S_{1},j\in S_{2} \tag{48f}\]
where \(\phi_{ij}\) is an indicator variable, which takes the value 1 only if article \(i\) and \(j\) are of same field, and 0 if they belong to two different fields.
Data description.The network is divided into 27 disjoint topologies, each containing 100 nodes. Each of the instant form an optimization instance. In each instance, the 100 nodes are split into two sets of 50 nodes \(S_{1}\) and \(S_{2}\); so each instance forms a bipartite matching problem between two sets of cardinality 50. Each publication (node) has 1433 bag-of-words features. The feature of an edge is formed by concatenating features of the two corresponding nodes. The prediction task is to estimate \(c_{ij}\) values. In this problem, each individual \(c_{ij}\) is associated with a feature vector of length 2866.
Predictive model.The predictive model is a neural network model. The input to the neural network is a 2866 dimensional vector and final output is a scalar between 0 and 1.
The neural network has one hidden layer and uses a sigmoid activation function on the output.
#### 5.1.7 Subset selections
This experiment is a structured prediction task, in which the object is to learn a mapping from feature vectors to binary vectors which represent subset selections. Unlike the other experiments above, the ground-truth data take the form of optimal solutions to an optimization problem, rather than its corresponding problem parameters. Thus, the regret loss is not suitable for training a prediction model. Instead, a task loss based on the error of the predicted solutions with respect to ground-truth solutions is used in this experiment.
Optimization problem formulation.For any \(\mathbf{c}\in\mathbb{R}^{n}\), the objective of the optimization problem is to output a binary vector in \(\mathbb{R}^{n}\), where the non-zero values correspond to the top-\(k\) values of \(\mathbf{c}\). This can be formulated as an LP problem in the following form:
\[\underset{\mathbf{x}}{\operatorname{argmax}}\ \ \mathbf{c}^{\top} \mathbf{x}\] (49a) \[\mathtt{s.t.}\ \ 1^{\top}\mathbf{x}=k\] (49b) \[\
Pairwise LTR loss (38) [Pairwise], 10. Pairwise difference LTR loss (39) [Pairwise(diff)], 11. Maximum a posteriori contrastive loss (37) [MAP]. The reason behind including prediction-focused approach is that it is considered as a benchmark. Note that among these methodologies, Listwise, Pairwise, Pairwise(diff), and MAP make use of a solution cache. The solution cache is implemented using the procedure proposed by Mulamba et al. (2021). In this approach, the solution cache is initialized by caching all the solutions in the training data and the cache is later expanded by employing a \(p_{solve}\) parameter value greater than zero. As in (Mulamba et al., 2021; Mandi et al., 2022) it is reported that \(p_{solve}=5\%\) is adequate for most applications, the value of \(p_{solve}\) is set to \(5\%\). Next, the procedure systematically followed for the empirical evaluations is explained.
Experimental setup and procedures.The performance of a methodology is sensitive to the choice of the methodology specific hyperparameters as well as some other fundamental hyperparameters, common in any neural network training such as learning rate. These are called hyperparameters because they cannot be estimated by training the model, rather they must be selected before training begins. Tuning hyperparameters is the process of identifying the set of hyperparameter values that are expected to produce the best model outcome. In the experimental evaluations, hyperparameter tuning is performed via grid search. In the grid search, each of the hyperparameters is tried for a set of values. The set of values to be tested on for each hyperparameter is predetermined. Grid search suffers from the curse of dimensionality in the hyperparameter space, as the number of combinations grows exponentially with the number of hyperparameters. However, it is possible to train the different models for different combination of hyperparameter in parallel as the combinations are independent.
The hyperparameter of each model for each experiment is selected based on performance on the validation dataset. For each hyperparameter a range of values as defined in Table 3 is considered. The hyperparameter combination which produces the lowest average regret on the validation dataset is considered to be the 'optimal' one. For both validation and testing, 10 trials are run where in every trial the network weights are initialized with a different seed. To be specific, values of seed from 0 to 9 have been considered. Each model for each setup is trained using Pytorch(Paszke et al., 2019) and PyTorch-Lightning(Falcon et al., 2019) with Adam optimizer(Kingma & Ba, 2014) and 'ReduceLROn
\begin{table}
\begin{tabular}{c c c} \hline \hline \begin{tabular}{c} **Hyperparameter** \\ \end{tabular} & \begin{tabular}{c} **Methodologies Utilizing** \\ **the Hyperparameter** \\ \end{tabular} & \begin{tabular}{c} **Range** \\ \end{tabular} \\ \hline
\begin{tabular}{c} learning rate \\ \end{tabular} & All & \(\{5\times 10^{-4},1\times 10^{-3},5\times 10^{-3},0.01,0.05,0.1,0.5,1.0\}\) \\ \(\lambda\) & I-MLE, DBB & \(\{0.1,1,10,100\}\) \\ \(\epsilon\) & I-MLE, FY & \(\{0.05,0.1,0.5,1,2,5\}\) \\ \(\kappa\) & I-MLE & \(\{5,10,50\}\) \\ \(\tau\) & Listwise & \(\{0.05,0.1,0.5,1,2,5\}\) \\ \(\Theta\) & Pairwise & \(\{0.01,0.05,0.1,1.,10.,50.\}\) \\ \(\mu\) & QPTL, HSD & \(\{0.01,0.1,1.,10.,\}\) \\ damping & HSD & \(\{1\times 10^{-4},0.01,0.1,1.,10\}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The range of hyperparameters for hyperparameter tuning by grid search.
Plateau'(PyTorch, 2017) learning rate scheduler. As mentioned before, the learning rate of Adam optimizer is treated as a hyperparameter. For QPTL, the QP problems are solved using Cvxpylayers (Agrawal et al., 2019). For other methodologies, which treat the CO solver as a blackbox solver, Gurobi(Gurobi Optimization, 2021) or OR-Tools(Perron and Furnon, 2020) is used as the solver. For MAP and LTR losses, the experiments are run with \(p_{solve}\) being 5%.
Evaluation metric.After selecting the 'optimal' hyperparameter combination for each test problem, **10** trials of all the methodologies with the 'optimal' hyperparameter combination are run on test dataset. Unless otherwise mentioned the comparative evaluation is made based on relative regret on the test dataset. The **relative regret** is defined as follows:
\[\frac{1}{N_{test}}\sum_{i=1}^{N_{test}}\frac{\mathbf{c_{i}}^{\top}(\mathbf{x}^ {\star}(\mathbf{\hat{c}_{i}})-\mathbf{x}^{\star}(\mathbf{c_{i}}))}{\mathbf{c_{ i}}^{\top}\mathbf{x}^{\star}(\mathbf{c_{i}})}. \tag{50}\]
In practice, \(\mathbf{c}\) (or \(\mathbf{\hat{c}}\)) can have non-unique optimal solutions. However, note that if all the entries in \(\mathbf{c}\) are continuous, it is very unlikely that \(\mathbf{c}\) will have non-unique solutions. For instance, in the case of an LP, the only circumstance in which the LP can have multiple solutions is when \(\mathbf{c}\) is parallel to one of the faces of the LP polyhedron. Nevertheless, if the cost vector is predicted by an ML model, a pathological case might occur, especially at the beginning of model training, when all the cost parameters are zero. This results in all feasible solutions being optimal with zero cost. However, to avoid this complexity in the experiments, it is assumed that the solution \(\mathbf{x}^{\star}(\mathbf{\hat{c}})\) is obtained by calling an optimization oracle and that if there exist non-unique solutions, the oracle returns a single optimal solution by breaking ties in a pre-specified manner. This is true if a commercial solver such as Gurobi is used to solve the CO problem.
#### 5.2.1 Comparative Evaluations
Next, the performances of the 11 methodologies in the 7 problems are presented with insights.
Shortest path problem on a \(5\times 5\) grid.The comparative evaluation for the synthetic shortest path problem in is shown in Figure 5 with the aid of box plots. To conserve space, boxplots for two values of Deg are shown in Figure 5. The boxplots for all the five degrees are shown in Figure A1 the Appendix. In Figure 5, the value of \(\vartheta\), the noise- halfwidth parameter is 0.5 for all the experiments and the training set for each Deg contains of 1000 instances. The predictive model is a simple linear model implemented as a neural network model with no hidden layers. For Deg 1, the linear predictive model perfectly captures the data generation process. Consequently the PF approach is very accurate and it results in the lowest regret. SPO has slightly higher regret than the PF approach. All the other models have considerable higher regrets. It is followed by MAP and FY. For Deg 8, FY has the lowest regret, closely followed by I-MLE. Then comes Listwise and Pairwise ranking losses followed by QPTL and DBB. In this case, SPO performs poorer than them. MAP and HSD have very high regret but still lower than the PF approach. The relative regret worsens for the PF approach, as the value of Deg parameter is increased. For Deg 2, both PF and SPO have lowest regret. However, their differences with other models reduce in
this case. FY, MAP and I-MLE come at the next three places respectively. For Deg 4, the PF model starts to result in high regret. In this case, I-MLE has the lowest regret, closely followed by FY and SPO. The next three spots are taken by MAP, Listwise and Pairwise respectively. DBB and HSD perform worse than the PF approach. For Deg 6, the best one FY, although its test regret is not very different from SPO, I-MLE and QPTL. Listwise, DBB, Pairwise and MAP come next.
Overall, FY and I-MLE are the top two best-performing approaches, for \(Deg>2\). For \(Deg\) values of 1 and 2, the PF approach has the lowest regret. Note that the performance of SPO is very consistent too. It performs considerably worse than I-MLE and FY only for Deg 8. On the other hand, HSD exhibits higher regret than the other DFL approaches. In fact it does better than the PF approach only for Deg 6 and 8. It also exhibits higher variances.
Portfolio optimization problem.Note that this is an optimization problem with continuous decision variables having quadratic constraints and a linear objective function. Hence, the HSD approach is not applicable for this problem, as it cannot handle non-linear constraints. The boxplots of test regrets for noise magnitude parameter \(\vartheta\) being 1 are shown in Figure 6.
In this problem, in some problem instances, all the return values are negative, which makes a portfolio with zero return to be the optimal portfolio. In such cases, relative regret turns infinite as the denominator is zero in Eq. (50). Hence, for this problem set, the **absolute regret** instead of relative regret is reported in Figure 6. The boxplots for Deg
Figure 5: Comparative evaluations on the synthetic shortest path problem with noise-halfwidth parameter \(\vartheta\) = 0.5. The boxplots show the distributions of relative regrets.
values of 1 and 16 are shown in The boxplots for all the four degrees are shown in Figure A2 the Appendix.
Apparently the PF approach performs very well in this problem; but SPO manages to outperform PF slightly in all cases except for Deg 1. It is evident in Figure 6 that DBB, I-MLE, FY and QPTL perform miserably as they generate regret even higher than the PF approach. All these methodologies were proposed considering problems with linear constraints. Hence the concerns arise that these methodologies be suitable in the presence of quadratic constraints. On the other hand, LTR losses--Pairwise and Pairwise(diff) and contrastive loss function, MAP, perform even better than SPO for Deg 16. For Deg 16, again Pairwise is the best performing model, followed by Listwise, Pairwise(diff), MAP and SPO, in that order. For Deg 1, PF is the best followed by MAP and SPO. For Deg 4 and 8, Pairwise loss function has the lowest regret, closely followed by Pairwise(diff), MAP and SPO. For Deg 1, PF is the best one, followed by MAP, SPO, Pairwise and Pairwise(diff), in that order. The Listwise loss function exhibits high variance for Deg 1 and for Deg values of 4 and 8 it generates high regret for few instances. For Deg 16, it generates average test regret lower than SPO. In general, Figure 6 reveals DBB, I-MLE, FY and QPTL perform poorly in this problem, whereas, SPO, MAP Pairwise and Pairwise(diff) seem to be suitable methodologies for this problem.
Warcraft shortest path problem.Recall that this a shortest path problem in an image with dimension \(d\times d\). The optimization problem can be efficiently solved using Dijkstra's algorithm (Dijkstra, 1959), as underlying costs of all the pixel values are non-negative. Hence the shortest path problem is solved using Dijkstra's algorithm for the methodologies which
Figure 6: Comparative evaluations on the synthetic portfolio optimization problem with noise magnitude \(\vartheta=1\). The boxplots show the distributions of **absolute** regrets.
view the CO solver as a blackbox oracle. However, HSD and QPTL require the problem to be formulated as an LP and require a primal-dual solver. Note in this experiment, the predictive ML model is a CNN, which predicts the cost of each pixel. In this case, training of the ML model is challenging due to the large number of parameters. Hence combining this ML model with computation-intensive modules such as interior point optimizer poses significant challenges. We could not run the experiments with HSD and QPTL because of this computational burden.
The dataset contains four values of \(d\): \(12,18,24,30\). Clearly, as the value of \(d\) increases, the optimization problem contains more number of parameters. The boxplots of comparative evaluations are summarized in Figure 7. The boxplots of other two values of \(d\) can be found in Figure 10 in the Appendix. First note that the PF approach, which is trained by minimizing mse loss between the predicted cost and true cost performs significantly worse than the DFL methodologies. In fact, the performance of the PF approach deteriorates as the image size increases. As the size of the image increases, the same level of prediction error induces greater inaccuracies in the solution. This is because an increase in the area of the image involves dealing with a greater number of decision variables in the CO problem. When the level of prediction error remains constant, the probability of the error in prediction changing at least one of the decision variables also increases. Consequently, there is a higher likelihood of error in the final solution. As the regret of the PF approach is significantly higher, note that the scale of the y-axis is changed to fit it into the plot.
Among the DFL methodologies, Listwise performs best for sizes 12, 18, and 30 and SPO performs best for size 30. In fact, for sizes 12, 18, and 24, there are not many
Figure 7: Comparative evaluations on the Warcraft shortest path problem instances. The boxplots show the distributions of relative regrets.
variations between SPO, Listwise, and MAP. After them, the next three best-performing methodologies are Pairwise (diff), I-MLE and DBB. However, for size 30, DBB comes third after Listwise and MAP, followed by Pairwise (diff), SPO, and I-MLE in that order. FY and Pairwise perform slightly worse than the other DFL methodologies. In general, this set of experiments shows the advantage of the DFL approaches as all of them outperform the PF approach.
Energy-cost aware scheduling.There are three instances of this scheduling problem. All the instances have 3 machines. The first, second, and third instances contain 10, 15, and 20 tasks, respectively. In this problem, the underlying ML model is a simple linear model implemented as a neural network model with no hidden layers. The boxplot of comparative evaluations for the first instance is presented in Figure 8. The boxplots of the other instances can be found in Figure A4 in the Appendix.
Note that the scheduling problem is an ILP problem. For HSD and QPTL, the LPs obtained by relaxing the integrality constraints have been considered. For the first instance, MAP and SPO result in the lowest average regret, closely followed by I-MLE. DBB, FY, and Pairwise(diff) perform better than the PF approach. The performances of the Listwise and Pairwise rankings are worse than the PF. QPTL and HSD also perform poorly in all three instances, probably because in this case the LP obtained by relaxing by removing the integrality is not a proper representation of the ILP. In fact, QPT fails to learn in this problem instance. In the second instance, FY, SPO, and I-MLE are the best three performing models. Then comes MAP and DBB, followed by Pairwise(diff). Again, performances of the Listwise and Pairwise rankings are worse than the PF. In the third instance, again, MAP and SPO deliver the lowest average regret. Then comes I-MLE and FY. The test regret of these two models is very similar. In this case, the performance of Pairwise(diff) is slightly worse than the PF approach, whereas, like before, performances of Listwise and Pairwise ranking are significantly worse. In general, across the three problem instances, it is possible to identify some common patterns. The first one is relaxing the integrality constraints fails to capture the essence of the combinatorial nature of the LP. Consequently, HSD and QPTL perform poorly. Secondly, Listwise and Pairwise ranking performances are significantly worse than the PF approaches. The learning curve suggests (refer to B), these models fail to converge in these problem instances, although in some epochs, they are
Figure 8: Comparative evaluations on the energy-cost aware scheduling problem instances. This boxplot shows the distributions of relative regrets.
able to perform significantly better than the PF approach, their pheromones never plateau. Lastly, SPO, MAP, FY, and I-MLE perform consistently better than the other models.
Knapsack problem.Three instantiations of the knapsack problem are considered for the experiment--each instantiation with a different capacity. The three capacity values are--60, 120 and 180. The boxplot corresponding to capacity value 60 is presented in Figure 9. The boxplots of the other two capacities can be found in Figure A5 in the Appendix. With a capacity of 60, the best three models are QPTL, DBB, and I-MLE, in that order. HSD, SPO, and MAP come next and perform better than the PF approach. FY and LTR losses perform worse than the PF approach. With a capacity of 120, the top three models are DBB, I-MLE, and QPTL. Then comes SPO, HSD and MAP. The Pairwise(diff) model performs slightly better than the PF approach, but the other two LTR losses and FY perform worse. With a capacity of 180, the best three models are DBB, I-MLE and SPO. HSD and QPTL perform better than the PF approach, but MAP, LTR losses, and FY perform worse. In general, for this problem, DBB and I-MLE are the best-performing models across the three capacity values. QPTL, SPO, HSD also consistently perform better than the PF approach in all three cases. However, FY and the LTR losses perform poorly in this problem.
Figure 10: Comparative evaluations on the diverse bipartite matching problem instances. This boxplot shows the distributions of relative regrets.
Figure 9: Comparative evaluations on the knapsack problem instances. This boxplot shows the distributions of relative regrets
Diverse bipartite matching.Three instantiations of the diverse bipartite matching problem are formed by changing the values of \(\rho_{1}\) and \(\rho_{2}\). The values of \((\rho_{1},\rho_{2})\) for the three instantiations are \((10\%,10\%)\), \((25\%,25\%)\), \((50\%,50\%)\) respectively. The boxplot of comparative evaluations for \((\rho_{1},\rho_{2})\) being \((50\%,50\%)\), is presented in Figure 10. As mentioned before, in this problem, each edge is not associated with an edge weight in the true sense. Hence, the PF approach is trained by directly learning to predict whether an edge exists. So the loss used for supervised learning for the PF approach is BCE loss. The DFL approaches consider the predicted probability of each edge as the edge weight and then aim to minimize regret.
In this problem instance QPTL is the best-performing model. FY, Pairwise(diff) and Listwise take the next three places. MAP, Pairiwse, I-MLE and SPO also perform better than the PF approach. The performances of HSD and DBB are similar to that of the PF approach. Also note that the relative regrets of all the models are very high (higher than \(80\%\)) for all three instances. With \(\rho_{1}\) and \(\rho_{2}\) being \(10\%\), I-MLE performs considerably better than all the other models. Then comes HSD, FY, Pairwise and Pairwise(diff) followed by SPO, MAP, DBB and Listwise. When \(\rho_{1}\) and \(\rho_{2}\) take the value of \(25\%\), QPTL, I-MLE and HSD are the top there models, with significantly lower regret than the rest. In this instance, the regrets of Listwise, Pairwise, SPO, FY and MAP are higher than the PF approach. Across the instances, the performances of I-MLE and QPTL are consistently better than the PF approach. In the first two instances, other than I-MLE and QPTL, other DFL models do not significantly better than the PF approach. DFL approaches such as FY, Listwise, Pairwise and MAP perform considerably better than the PF approach only in the third instances. On the other hand, the test regret of DBB is similar to the PF approach across the instances.
Learning subset selections.Subset selection problems of three dimensions: \(n=25\), \(n=50\), and \(n=100\) are considered for evaluation. In each case, the subset size \(k\) is chosen to be \(\frac{n}{5}\). The error of any predicted subset \(\hat{x}\), with respect to ground truth \(x\), is considered to be the fraction of items which are selected in \(x\) but not in \(\hat{x}\). Such occurrences are referred to as mismatches.
Figure 11 shows the average mismatch rates over the size \(n=25\) instances that were achieved by each DFL methodology listed in Table 1, excluding those which assume ground-truth data in the form of problem parameters. Here, the ground-truth data are optimal solutions of (49) representing subset selections. For each assessed method, a distribution of results is shown, corresponding to \(10\) different randomly generated training datsets. Figure A7 shows similar results over the larger problem instances.
Note that it is suggested in (Amos et al., 2019) that the entropy function \(H(\mathbf{x})=\sum_{i}x_{i}\log x_{i}\) is particularly well-suited as a regularizer of the objective in (49), for the purpose of multilabel classification, which is identical to the task in terms of its optimization component and the form of its target data. Hence a Cvxpylayers implementation of this model is included and referred to as ENT.
Figure A7 shows that most of the assessed methods perform similarly, with DBB performing worst regardless of the problem's dimension. HSD is most sensitive with respect to the randomly generated training set; the rest show consistent performance across datasets. QPTL and IMLE each show a marginal advantage over the other methods, but DPO and
ENT are also competitive. Across all methods, variation in performance over the randomly generated datasets tends to diminish as problem size increases.
#### 5.2.2 Comparison on Runtime
While coming up with a useful gradient is considered to be the primary challenge of DFL, as mentioned in Section 2.3, computational cost associated with repeatedly solving CO problems gives rise to the second challenge. DFL methodologies with low computational cost are essential for scalability for implementing DFL for real-world large-scale Predict-Then-Optimize problems. The importance of scalability and low computational cost becomes significant while dealing with large-scale CO problems, especially NP-hard combinatorial optimization problems. Note that while the shortest path and the knapsack problems are relatively easy to solve; the energy-cost aware scheduling problem is much more challenging and can be considered an example real-world large-scale NP-hard combinatorial optimization problems. That is why the scheduling problem is considered to compare the computational costs of the DFL methodologies.
The median training time of an epoch during training of each methodology for two instances of the scheduling problem are shown Figure 12. Recall that the first, second and third instances contain 10, 15 and 20 tasks respectively. So, the first one is the easiest of the three and the third one is the hardest one. The complexity of the scheduling problem is evident from the fact that a single instance of the the knapsack problem takes 0.001 seconds to solve, while solving the most difficult instance of the scheduling problem takes 0.1 seconds, both using Gurobi MIP solver. The readers are cautioned against placing excessive emphasis on the absolute values of training times in Figure 12, as they are subject to system overhead. However, some general conclusions can be drawn from the relative ordering of the training times. It is not surprising that the training time of the PF approach is the lowest, as it does not require solving the CO problem for model training. Training times of SPO, DBB, I-MLE and FY are almost 100 times higher than the PF approach. Although QPTL and HSD consider the relaxed LP problem, it is not always the case that they have lower training times. Recall that QPTL and HSD solve and differentiate the optimization problem using primal-dual solver, which involves matrix factorization. On the other hand, SPO, DBB, I-MLE and FY can leverage faster commercial optimization solvers, as they only require the optimal solution. However, for Instance 3, it seems solving the ILP
Figure 11: Comparative evaluations on the subset selection problem instances of Size 25. This boxplot shows the distributions of mismatch rates.
problem is more computationally expensive than solving differentiating the underlying QP problem using Cvxpylayers.
On the other hand, Listwise, Pairwise, Pairwise(diff) and MAP, all of which are run with \(p_{solve}=5\%\), exhibit significantly lower training time than the other DFL methodologies. In fact, the training time of these methodologies are comparable to the PF approach. From this perspective, these methodologies can be viewed as bridging the gap between between PF and DFL approaches. The same conclusion generally holds true for other experiments as well. However, for relatively easier CO problems, the system overhead time sometime dominates over model training time, which might disrupt the ordering of the model training time.
#### 5.2.3 Discussion
The experimental evaluations reveal that no single methodology performs the best across all experiments. Certain methodologies excel on specific test problems, while others perform better on different test problems. Nevertheless, certain interesting characteristics emerge from the experimental evaluations. Firstly, **the performance of SPO is consistently robust across the test problems**, even though it may not outperform other techniques in every experiment. Secondly, **MAP demonstrates consistent performance across most test problems too**; it only exhibits low quality performance specifically in the knapsack problem for Capacity=180 and in the bipartite matching problem when \(\rho_{1}\) and \(\rho_{2}\) are 25%. Additionally, among the LTR losses, Listwise and Pairwise often exhibit high variances, especially in the scheduling and the knapsack problems. The performance of
Figure 12: Comparative evaluations of per epoch training time of different DFL methodologies on the energy-cost aware scheduling problem.
Pairwise(diff) stands out among the LTR losses due to its lower variance. Its performance is comparable to or slightly worse than MAP for most problems other than the synthetic shortest path problem with high values of Deg, i.e., when the underlying predictive model is completely misspecified. Surprisingly, I-MLE, FY, DBB and QPTL perform worse than the PF approach for the portfolio optimization problem, where a quadratic constraint is present. Across the remaining problems, **the performance of I-MLE is comparable to that of SPO and MAP**. DBB performs considerably worse than I-MLE only in the bipartite matching problem. On the other hand, FY performs well in certain cases, but it is more susceptible to higher variance compared to I-MLE. This is particularly evident in the knapsack problem. Moreover, **QPTL demonstrates robust performance in most experiments**. In fact, QPTL outperforms other models by a substantial margin in the bipartite matching problem. However, QPTL performs poorly compared to others in the scheduling problem, which is an ILP. In this case, the poor performance may be attributed to the fact that QPTL considers a relaxation of the ILP. In this problem the LP solution might differ significantly from the true ILP solution. This is not the case for the knapsack problem, because the solution of the relaxed LP does not deviate significantly from the ILP solution for the knapsack problem. HSD also considers relaxeds for ILP problems. However, it performs worse than QPTL for all but the scheduling problem, where it performs considerably better than QPTL. Finally, due to the limitation of computational resources, we were unable to run QPTL and HSD on the Warcraft shortest path problem. This highlights the advantage of DFL methodologies which can make use of any blackbox combinatorial solver (Dijkstra's shortest path solver for instance) to solve the CO problem. Continuing on this topic of computational cost, MAP and the LTR losses are considerably faster and less computationally intensive when they are run with low values of \(p_{solve}\). As MAP tends to have regret as low as SPO for most test problem, it may be considered a _favorable DFL technique for tackling large-scale real-world Predict-Then-Optimize problems_.
## 6 Future Research Directions
While there is increasing interest in decision-focused learning research, it still need to evolve to incorporate new characteristics to tackle real-world problems. This section aims to summarize the wide range of challenges that remain open. A few promising research directions for future investigations that could be exploited in the upcoming days, are presented next.
DFL for related tasks/ Task generalization.In the current DFL framework, the ML model is tailored to a particular optimization task. However, in many applications the CO problem might slightly differ in different instantiations. For example, in the recent MIT-Amazon Last Mile Routing Challenge (Merchan, Arora, Pachon, Konduri, Winkenbach, Parks, & Noszek, 2022), a TSP problem is solved every day for deciding the routing of last mile package delivery, but nodes of the TSPs change every day as the delivery locations vary. An interesting research direction would be to investigate how a model, which is trained to minimize regret of one optimization problem, would perform if evaluated on a similar but different optimization problems. Future work need to advance the approach proposed by Tang and Khalil (2023a) by training the ML model with the aim of _generalizing_ to new tasks.
Noise contrastive loss functions to learn parameters in the constraints.One key advantage of the noise contrastive loss functions (called MAP in the experimental evaluations) proposed by Mulamba et al. (2021) is that it is differentiable. They view the DFL problem by learning to contrast the likelihood of ground-truth solution and a set of negative examples. However, this work does not consider the case of predicting parameters in the constraints. In future studies, there is potential to extend noise contrastive estimation approach by considering the prediction of parameters within the constraints. This can be achieved by learning to contrast the likelihood of feasible points with that of infeasible ones. However, the efficacy of such an approach may rely on how the infeasible points are selected and that is why an empirical investigation into this aspect would provide valuable insights.
Robust decision-focused learning framework to learn parameters in the constraints.While predicting parameters in the constraints of an optimization problem, the prescribed optimal decision might not be feasible with respect to the true parameters. In such scenarios, an interesting direction would be to recommend a solution which is feasible under extreme distributional variations of the parameters. We believe a framework for optimizing average performance and minimizing worst-case constraint violations could reveal new tracks for theoretical research as well as practical applications. The research in this regard can take inspiration from the well-established field of robust optimization (Ben-Tal, El Ghaoui, & Nemirovski, 2009).
Surrogate loss functions in the absence of ground-truth cost parameters.In may real-world applications, the true cost parameters of the objective function might be latent variables. In such cases the parameters are not observed, only the solutions are observed. So, the parameters would not be available for supervised learning, which entails the use of a task loss other than regret. DFL frameworks, which implement differentiable optimization layer, such as DBB, QPTL or I-MLE are compatible with any task loss. However, the SPO approach, which comes with a theoretical proof of convergence, requires the ground-truth cost vector for gradient computation. This is also true for noise contrastive and LTR losses, whose computation and differentiation do not involve solving the CO problem. Development of surrogate loss functions, which neither require solving nor the true cost vector, would be a valuable contribution with potentials in real-world applications.
Decision-focused learning by score function gradient estimation.Most of the DFL techniques focus on computing the derivative \(\frac{d\mathbf{x}^{*}(\mathbf{\bar{c}})}{d\mathbf{\bar{c}}}\) analytically or construct a surrogate task that provides useful gradients. However, there exists another alternative way to estimate the gradient--zeroth-order estimation of the gradient. A widely used approach to zeroth-order optimization is the score function gradient estimation (Williams, 1992). In order to apply score function gradient estimation in DFL, one has to assume the predicted parameter follows a distribution and then compute Monte Carlo estimate of regret by sampling cost vector from that distribution. The score function gradient estimation would return a gradient that moves the parameters of the distribution in directions that facilitate sampling the cost vector with low values of regret (or task loss in general). Although score function gradient estimation provides unbiased gradient; a major challenge of using this technique is it suffers from high variances, which might destabilize the learning. Hence, conducting further research to examine the potential application of score function gradient estimation in DFL would be a valuable contribution.
Non-linear objective function.Most of the works in DFL, consider optimization problems with linear objectives. This is the primary reason such problems have been considered for experimental evaluations in this work. Any convex optimization problem with nonlinear objective function can be differentiated through CVxypylayers (Agrawal et al., 2019a). However, no DFL technique considers nonlinear objectives with discrete decision variables. As many real-world problems in OR are combinatorial optimization problems with discrete decision variables, developing ML techniques for such problems in the future could be beneficial in real life. For examples, the problem of optimally locating substations in an electrical network to minimize the costs of distribution is formulated as nonlinear programming (Lakhera, Shanbhag, & McInerney, 2011). Another classic OR problem, which does not have a linear objective function is the minimization of makespan in flowshop scheduling. Most of the methodologies discussed in this paper are not applicable to handle such problems.
Bilevel optimization techniques for DFL.As mentioned in Section 2.2, the empirical regret minimization problem can be cast as a pessimistic bilevel optimization problem. We believe that by understanding the mathematical object behind the learning process can lead to better algorithms for DFL, leaving a door open to the bilevel optimization community to tackle this problem.
Optimization as an intermediate layer within neural networks.In a Predict-Then-Optimize problem, the final task is to make a decision by solving a CO problem. However, in many other applications the optimization task may appear as an intermediate task. For instance, consider the task of selecting relevant patches in high resolution images, where the patches are being used for a downstream image recognition task. In (Cordonnier, Mahendran, Dosovitskiy, Weissenborn, Uszkoreit, & Unterthiner, 2021) the patch selection task is modeled as a Top-\(k\) selection CO problem. Note that the Top-\(k\) selection is embedded as an intermediate layer between two neural networks; where the upstream neural network assign score to each patch and the downstream neural network performs the recognition task. Techniques such as I-MLE, DBB, QPTL, DPO, which are implementation of differentiable optimization layer can be applied to tackle problems like this. Although the existence of downstream layer after the CO problem may give rise to novel challenges, embedding the CO problem as an intermediate layer could find extensive use across various domains.
Construction of solution cache.The loss functions which utilize solution cache are very effective to address the computational cost of DFL and promising for large NP-hard real-world Predict-Then-Optimize problems. However, we believe there is a space for research to study the trade-off between the solution cache size and solution quality.
## 7 Conclusion
The survey article begins by underscoring the significance of Predict-Then-Optimize problem formulations, wherein an ML model is followed by a CO problem. The Predict-Then-Optimize problem has emerged as a powerful driving force in numerous real-world applications of artificial intelligence, operations research and business analytics. The key challenge in Predict-Then-Optimize problems is predicting the unknown CO problem parameters in a manner that yields high-quality _solutions_, in comparison to the retrospective solutions
obtained when using the groundtruth parameters. To address this challenge, the DFL paradigm has been proposed, wherein the ML models are directly trained considering the CO problems using task losses that capture the error encountered after the CO problems. However to date, there is no comprehensive survey on DFL. This survey provides a comprehensive overview of DFL, highlighting recent technological advancements, applications and identifying potential future research directions. In Section 2, the problem description has been laid out with examples and then the fundamental challenges in decision-focused learning have been presented. Afterward, Section 3 has presented a categorization with four categories of DFL techniques, which have been thoroughly explained highlighting the trade-offs among these four categories. Then, in Section 4, some examples of applications of DFL techniques to address real-world Predict-Then-Optimize problems, across different domains have been provided. Furthermore, extensive comparative evaluations on different problem sets between 11 DFL techniques have been provided in Section 5. Finally, a discussion of some of the open problems in DFL and an outline of potential research directions have been presented in Section 6. While there has been significant recent progress in DFL, there remain challenges that need to be addressed. For instance, the development of DFL techniques, which can handle uncertain parameters occurring anywhere within a generic CO problem, will have a significant impact on various industrial applications.
We hope this survey article will assist readers to understand the paradigm of decision-focused learning and grasp the fundamental challenges of implementing it in many real-world applications. We aspire this survey to potentially act as a catalyst, inspiring the application of decision-focused learning in diverse domains and contexts as well as stimulating further methodological research and advancements.
## Acknowledgments
This project received partial funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under Grant no. 101070149 and Grant No. 101002802 (CHAT-Opt) and the FWO Flanders project G070521N. This research is also partially supported by NSF grants 2242931, 2232054, 2007164, and NSF CAREER award 2143706. Victor Bucarey was funded by the ANID Fondecyt Iniciacion Grant no. 11220864.
| 決意に焦点を当てた学習 (DFL) は、機械学習 (ML) と制約された最適化を統合して、決断の質を向上させるパラダイムです。DFL は、ML モデルをエンドツーエンドシステムで訓練することで、このアプローチは、不確実な現実世界アプリケーションでの組合せ決断を革命的に変える可能性を示しています。このアプローチでは、決断モデル内の未知のパラメータを推定することは、主要な課題です。この論文では、DFL について包括的なレビューを提供し、そのために、ML と制約された最適化を組み合わせるために使用されている勾配ベースと勾配なしの技術について、詳細な分析を行っています。これらの技術の強みと弱みを評価し、11 つの方法を 7 つの問題で広く評価しています。この調査は、DFL の最近の進歩と今後の研究方向に関する洞察を提供しています。コードとベンチマーク: |
2303.01496 | Robustness Measures for Molecular Detections using High-Resolution
Transmission Spectroscopy of Exoplanets | Ground-based high-resolution transmission spectroscopy has emerged as a
promising technique for detecting chemicals in transiting exoplanetary
atmospheres. Despite chemical inferences in several exoplanets and previous
robustness studies, a robust and consistent detrending method to remove
telluric and stellar features from transmission spectra has yet to be agreed
upon. In this work we investigate the robustness of metrics used to optimise
PCA-based detrending for high-resolution transmission spectra of exoplanets in
the near-infrared. As a case study, we consider observations of the hot Jupiter
HD 189733 b obtained using the CARMENES spectrograph on the 3.5 m CAHA
telescope. We confirm that optimising the detrending parameters to maximise the
S/N of a cross-correlation signal in the presence of noise has the potential to
bias the detection significance at the planetary velocity of optimisation.
However, we find that optimisation using the difference between a
signal-injected cross-correlation function and the direct cross-correlation
function (CCF) is more robust against over-optimisation of noise and spurious
signals. We additionally examine the robustness of weighting the contribution
of each order to the final CCF, and of S/N calculations. Using a prescribed
robust methodology, we confirm H2O in the atmosphere of HD 189733 b (S/N =
6.1). We then investigate two further case studies, of exoplanets HD 209458 b
and WASP-76 b, confirming OH in the atmosphere of WASP-76 b (S/N = 4.7), and
demonstrating how non-robust methods may induce false positive or inflated
detections. Our findings pave the way towards a robust framework for
homogeneous characterisation of exoplanetary atmospheres using high-resolution
transmission spectroscopy in the near-infrared. | Connor J. Cheverall, Nikku Madhusudhan, Måns Holmberg | 2023-03-02T18:56:47 | http://arxiv.org/abs/2303.01496v1 | Robustness Measures for Molecular Detections using High-Resolution Transmission Spectroscopy of Exoplanets
###### Abstract
Ground-based high-resolution transmission spectroscopy has emerged as a promising technique for detecting chemicals in transiting exoplanetary atmospheres. Despite chemical inferences in several exoplanets and previous robustness studies, a robust and consistent detrending method to remove telluric and stellar features from transmission spectra has yet to be agreed upon. In this work we investigate the robustness of metrics used to optimise PCA-based detrending for high-resolution transmission spectra of exoplanets in the near-infrared. As a case study, we consider observations of the hot Jupiter HD 189733 b obtained using the CARMENES spectrograph on the 3.5 m CAHA telescope. We confirm that optimising the detrending parameters to maximise the S/N of a cross-correlation signal in the presence of noise has the potential to bias the detection significance at the planetary velocity of optimisation. However, we find that optimisation using the difference between a signal-injected cross-correlation function and the direct cross-correlation function (CCF) is more robust against over-optimisation of noise and spurious signals. We additionally examine the robustness of weighting the contribution of each order to the final CCF, and of S/N calculations. Using a prescribed robust methodology, we confirm H\({}_{2}\)O in the atmosphere of HD 189733 b (S/N = 6.1). We then investigate two further case studies, of exoplanets HD 209458 b and WASP-76 b, confirming OH in the atmosphere of WASP-76 b (S/N = 4.7), and demonstrating how non-robust methods may induce false positive or inflated detections. Our findings pave the way towards a robust framework for homogeneous characterisation of exoplanetary atmospheres using high-resolution transmission spectroscopy in the near-infrared.
keywords: methods: data analysis - techniques: spectroscopic - planets and satellites: atmospheres.
## 1 Introduction
Thousands of exoplanets have been discovered to date. The first confirmed exoplanet orbiting a Sun-like star was observed in 1995 with the discovery of the hot Jupiter 51 Peg b (Mayor and Queloz, 1995). Since then over 5000 confirmed exoplanets have been discovered1. We now know that the occurrence frequency of planets is high, approaching one per star (Fressin et al., 2013; Fulton et al., 2017). Our accelerating ability to detect and subsequently characterise exoplanets is due to the rapidly improving technology, instrumentation and analysis capabilities available, and this will continue into the future. The field of exoplanets is therefore one of the most active and fast-paced frontiers in astrophysics.
Footnote 1: [https://exoplanets.nasa.gov/](https://exoplanets.nasa.gov/)
Exoplanets are hugely diverse in terms of their orbital parameters, bulk parameters (masses, radii, equilibrium temperatures), internal structures, formation conditions and evolution histories. Their atmospheres span a wide range of chemical compositions and temperature profiles, with various chemical and physical processes at play (e.g. Madhusudhan et al., 2014; Madhusudhan, 2019; Zhang, 2020; Fortney et al., 2021). Characterizing the atmospheres of exoplanets via the spectral signatures of the chemical species present allows us to constrain their diverse properties, contributing to an understanding of their physical processes and how they form. This in turn will enable us to learn more about the planets in our own solar system and their formation history.
Common molecules in exoplanetary atmospheres such as H\({}_{2}\)O, CO, CH\({}_{4}\), HCN, CO\({}_{2}\) and TiO, and atoms such as Na and K, have strong absorption features in the optical and/or near-infrared (NIR) which may be seen in the transmission spectrum (Seager and Sasselov, 2000; Sing et al., 2016; Madhusudhan, 2019). Analysis of the transmission spectrum can therefore constrain the composition of the atmosphere at the day-night terminator region. Charbonneau et al. (2002) were the first to use transmission spectroscopy to characterise an exoplanetary atmosphere, using the Hubble Space Telescope (HST) to identify Na in the atmosphere of the hot Jupiter HD 209458 b. It was not until 2008 that ground-based observations were first used to detect a chemical signature in an exoplanetary atmosphere, when Redfield et al. (2008) and Snellen et al. (2008) made detections of Na in the atmospheres of the hot Jupiters HD 189733 b and HD 209458 b, respectively.
In recent years, high-resolution transmission spectroscopy has emerged as one of the most successful techniques for detecting chem
icals in transiting exoplanetary atmospheres (e.g. Snellen et al., 2010; Wittenbach et al., 2015; Brogi et al., 2016, 2018; Hoeijmakers et al., 2018; Alonso-Floriano et al., 2019; Sanchez-Lopez et al., 2019; Giacobbe et al., 2021). Whereas in low-resolution the molecular lines of different species may overlap, at high-resolution the signatures are more easily separated, giving more confident detections. High-resolution typically implies \(R\) between \(10^{4}\) and \(10^{5}\)(Brogi et al., 2016; van Sluijs et al., 2022), which is achieved by various NIR spectrographs on large (4-8m) ground-based telescopes. For a broadband absorber with lines of comparable strength, the signal-to-noise ratio (S/N) of the planetary signal increases with the square root of the number of lines observed, \(\sqrt{N_{\rm lines}}\), so it is ideal to have a spectrograph of very high resolution and with a wide spectral coverage (Birkby, 2018).
This work focuses on the NIR wavelength range which contains strong spectral features of prominent molecules such as H\({}_{2}\)O, CO, CH\({}_{4}\), and HCN, which are expected to be abundant in H\({}_{2}\)-rich atmospheres (Moses et al., 2013; Madhusudhan et al., 2016). There are a number of high-resolution spectrographs currently in use which cover the NIR, including CARMENES (Quirrenbach et al., 2016; Quirrenbach et al., 2018), CRIRES (Kaeuf et al., 2004; Dorn et al., 2014), GIANO (Oliva et al., 2006; Origlia et al., 2014), SPIRou (The SPIRou Team et al., 2018; Donati et al., 2020), GRINUS (Yuk et al., 2010; Park et al., 2014) and HDS/Subaru (Noguchi et al., 2002). The first detections achieved via high-resolution spectroscopy required spectrographs mounted on 8 m class telescopes, such as CRIRES (Snellen et al., 2010; Brogi et al., 2012; Birkby et al., 2013). Snellen et al. (2010) were the first, using CRIRES to identify CO in the transmission spectrum of HD 209458 b. More recently however, atmospheric characterisation of transiting exoplanets has been possible using spectrographs mounted on 4 m class telescopes such as CARMENES (Alonso-Floriano et al., 2019; Sanchez-Lopez et al., 2019), GIANO (Brogi et al., 2018; Giacobbe et al., 2021) and SPIRou (Boucher et al., 2021). New instruments are continuously becoming available, increasing our capabilities even further and enabling the characterisation of more diverse exoplanetary atmospheres.
Close-in, and therefore strongly irradiated, gas giants called hot Jupiters are the most easily characterised, and most commonly observed, type of exoplanet. They are therefore the most common targets for high-resolution spectroscopy. Various chemical species such as CO (Snellen et al., 2010; Brogi et al., 2012), H\({}_{2}\)O (Birkby et al., 2013; Alonso-Floriano et al., 2019), TiO (Nugroho et al., 2017), HCN (Hawker et al., 2018; Cabot et al., 2019), CH\({}_{4}\) (Guilluy et al., 2019), NH\({}_{3}\), C\({}_{2}\)H\({}_{2}\) (Giacobbe et al., 2021; Guilluy et al., 2022; Carleo et al., 2022), Fe, Ti and Ti* (Hoeijmakers et al., 2018) have been inferred in their atmospheres in both transmission and emission.
When observed using ground-based telescopes, NIR spectral lines produced by molecular species in the exoplanet's atmosphere are buried in stellar features and telluric contamination from the Earth's own atmosphere, both of which are orders of magnitude stronger (Sanchez-Lopez et al., 2019). In order to access the planetary signal, the telluric and stellar lines first have to be removed in a process called detrending. This method typically makes use of the changing Doppler shift of the exoplanet's atmospheric spectrum as it transits in front of the host star. For a hot Jupiter, the orbital velocity of the planet is significantly greater than that of its host star. The planetary spectral lines are thus Doppler-shifted with a much greater amplitude than the stellar lines. Over a sufficient observation period, the planetary spectral lines will be subject to large Doppler shifts, whereas the telluric and stellar lines will remain comparatively stationary (Birkby, 2018). This allows us to separate the planetary spectral lines from those of the host star and the telluric absorption. Once the stellar and telluric lines have been removed, the planetary signal must be extracted from the noise. At high resolution, molecular features are resolved into a dense and unique collection of individual and separate lines, each with a very low S/N. Molecules can be detected by cross-correlating the observed high-resolution spectra, after detrending, with model atmospheric spectra of the planet (Snellen et al., 2010; Brogi et al., 2012; Birkby et al., 2013).
Detrending has proven to be the most challenging step in high-resolution spectroscopy. Cabot et al. (2019) previously investigated the robustness of a common detrending procedure in the context of NIR emission spectroscopy, and found that detrending parameters can potentially be overfit by optimising the detection significance at a single point in planetary velocity space. Doing so can lead to amplified or spurious detections at the expected planetary velocity. Whilst tests were proposed to aid in identifying such false positives, a robust detrending method to avoid them has yet to be established and agreed upon. Inhomogeneous methodologies across the literature can lead to inconsistencies in the quoted significance and robustness of detections (Spring et al., 2022), thereby hindering our ability to place tighter constraints on the compositional diversity of exoplanetary atmospheres. Consistent, robust and reproducible methods are therefore desirable.
In this work we investigate the robustness of molecular detections made using high-resolution transmission spectroscopy in the NIR. Our goal is not an exhaustive exploration of the model space aimed at detecting molecular species. Instead we are focused on assessing the relative robustness of molecular detections using different optimisations of a given detrending procedure for the same model template. In doing so, we aim to identify a robust recipe for detrending. Despite the greater transit depth in the NIR increasing the S/N of transmission spectra, telluric absorption is more severe at redder wavelengths meaning that detrending is more difficult.
The paper is organised as follows. In Section 2 we introduce the general methodology by which a planetary signal can be extracted from the spectra, using observations of HD 189733 b as a case study. In Section 3 we examine the robustness of different detrending optimisations from across the literature (Birkby et al., 2017; Alonso-Floriano et al., 2019; Cabot et al., 2019; Sanchez-Lopez et al., 2019; Giacobbe et al., 2021; Spring et al., 2022; Holmberg and Madhusudhan, 2022). Order weighting and other contributing factors in the determination of the detection S/N are discussed in Section 4. Robust methods to achieve high confidence chemical detections in the atmospheres of exoplanets are used to analyse other datasets in Section 5. Potentially spurious and inflated detections resulting from non-robust methods are also demonstrated. We summarise and discuss our results in Section 6.
## 2 Methods
In this section we describe the main steps involved in analysing high-resolution spectroscopic observations of exoplanetary transmission spectra using the cross-correlation method. As a case study, we here focus on the hot Jupiter HD 189733 b and discuss the observations and the general approach to infer a chemical signature in its atmosphere.
### Observations
In order to demonstrate our methods, as a test case we consider archival CARMENES observations of a transit of the hot Jupiter HD 189733 b on the night of 7th September 2017 (Alonso-Floriano et al.,
2019). HD 189733 b is an extensively studied hot Jupiter orbiting a bright K star (\(V\) = 7.7 mag) (Bouchy et al., 2005). H\({}_{2}\)O has previously been detected in its atmosphere using the same CARMENES observations as we use here (Alonso-Floriano et al., 2019), as well as in other high-resolution transmission spectroscopy studies (Brogi et al., 2016, 2018). Brogi et al. (2016) additionally found CO in the atmosphere of this planet using CRIRES over a spectral range around 2.3 \(\mu\)m. CARMENES is mounted on the 3.5 m telescope at the Calar Alto Observatory and consists of two fiber-fed high-resolution spectrograph channels (VIS and NIR). In this work we only use the NIR channel, which observes a wavelength range of 960-1710 nm at a resolution of \(R\) = 80400 over 28 spectral orders. Each channel is fed by two fibres: fibre A positioned on the target and fibre B on the sky to identify sky emission lines. The data consists of 46 observations (spanning planetary orbital phases -0.0348 \(<\phi<\) 0.0359; \(\phi\) = 0 corresponds to mid-transit), of which 24 are in transit. A median S/N of 134 was observed across the pixels. All observations were obtained with a constant exposure time of 198 s. The airmass increased from a minimum of 1.03 to a maximum of 1.32 over the course of the observing night. The system properties for HD 189733 b are given in Table 1.
### Data Cleaning, Normalisation and Calibration
The pre-processed CARMENES data is publicly available, having been automatically reduced after observation using the dedicated pipeline CARACAL v2.10 (Zechmeister et al., 2014; Caballero et al., 2016). Throughout the analysis each spectral order is treated independently, until they are combined at the end. The first step is to remove bad pixels and outliers from the spectra. After cleaning each spectrum of poor quality pixels, each spectrum is first rescaled such that it's ninetieth percentile flux is unity. 5\(\sigma\) outliers are then iteratively clipped from the time-series of each wavelength channel, with clipped pixels replaced using linear interpolation within that wavelength channel. The flux in each spectral order is then normalised by fitting a quadratic polynomial to the pseudo-continuum (Sanchez-Lopez et al., 2019). Pixels identified by CARMENES fibre B as corresponding to sky emission lines are excluded from the normalisation fit. We remove orders 45-41 and 55-53 from this dataset due to their low S/N, and find that the CARMENES spectra require no further wavelength calibration.
### Detrending
The cleaned and normalised spectra are still dominated by telluric and stellar lines. In order to access the planetary signal, orders of magnitude weaker than such contaminants, they need to be removed (detrending). Principle component analysis (PCA) has been used to do this in several past studies (de Kok et al., 2013; Giacobbe et al., 2021; Holmberg and Madhusudhan, 2022; van Sluijs et al., 2022). PCA finds and removes the common modes in the time-variation of each wavelength channel. As such, the quasi-static telluric and stellar features are removed by this process; during the course of the observations they vary only in depth, not significantly in wavelength, and so they produce common time-variations between wavelength channels. On the other hand, the planetary signal, with its changing Doppler shift, should mostly remain since it moves across different wavelength channels over the observing night. The planetary signal should therefore induce a minimal common time-variation between wavelength channels. An alternative algorithm known as SYSREM (Tamuz et al., 2005; Mazeh et al., 2007) has also often been used instead. SYSREM allows for unequal uncertainties between pixels, and has been successfully used in a number of previous works (e.g. Birkby et al., 2017; Nugroho et al., 2017; Hawker et al., 2018; Alonso-Floriano et al., 2019; Sanchez-Lopez et al., 2019; Cabot et al., 2019; Spring et al., 2022). We find minimal difference between residuals when detrending with each of PCA and SYSREM, and use PCA in this work.
The number of PCA iterations applied refers to the number of principle components removed from the spectra. PCA iterations are applied to the spectra until, in principle, we are left with the continuum-normalized planetary spectrum embedded only in white noise (Birkby, 2018). Sufficient iterations of the detrending algorithm must be applied to remove the contaminants completely. As noted above, the planet signal should remain mostly intact assuming its change in radial velocity is sufficiently large. However, in reality detrending also erodes the planetary signal itself (Birkby et al., 2017; Sanchez-Lopez et al., 2019). We test this by injecting a model planetary signal into the spectra and recovering it, as described in Section 2.5, after different numbers of PCA iterations have been applied to detrend the spectra. We find that the planetary signal, injected at the expected planetary velocity, is degraded to some extent even after just one PCA iteration. In the case of this particular dataset, \(\sim\)15% of the planetary signal is lost after one PCA iteration. After 18 iterations less than 20% of the planetary signal remains for this dataset (Figure 1). We note however that this level of degradation could depend on the orbital properties of the planet. Therefore, there exists an optimum number of iterations which when applied to the spectra will retrieve the planetary signal with the greatest S/N. This optimum can vary between different spectral orders. Typically, it can be found by temporarily injecting a Doppler-shifted model planetary signal and maximising the retrieved detection significance at the desired location in velocity space (Birkby et al., 2017; Nugroho et al., 2017; Sanchez-Lopez et al., 2019). The robustness of different such injection-based detrending optimisations is explored in more detail in Section 3.
### High-Resolution Model Spectra
In order to cross-correlate with the data, we compute model templates for the transmission spectra of the hot Jupiters considered in this study. We model the transmission spectra using a variant of the AURA atmospheric modelling and retrieval code (Pinhas et al., 2018). The model computes line-by-line radiative transfer in transmission geometry assuming a plane-parallel atmosphere in hydrostatic equilibrium. The atmospheric structure is computed over a pressure range of \(10^{-7}\) - 100 bar. The chemical composition and temperature structure are free parameters in the model. We generate the spectra considering one molecule at a time, assuming no clouds/hazes and an isothermal temperature profile. The spectra are computed at
\begin{table}
\begin{tabular}{l l l} \hline \hline Parameter & Value & Reference \\ \hline P & \(2.21857567\pm 0.00000015\) d & Agol et al. (2010) \\ T\({}_{0}\) & \(2454279.436714\pm 0.000015\) BJD & Agol et al. (2010) \\ R\({}_{\rm star}\) & \(0.756\pm 0.018\) R\({}_{\odot}\) & Torres et al. (2008) \\ R\({}_{\rm p}\) & \(1.138\pm 0.027\) R\({}_{\rm Jup}\) & Torres et al. (2008) \\ Y\({}_{\rm sys}\) & \(-2.361\pm 0.003\) km s\({}^{-1}\) & Bouchy et al. (2005) \\ a & \(0.03120\pm 0.00027\) au & Triaud et al. (2009) \\ i & \(85.71\pm 0.024\)\({}^{\circ}\) & Agol et al. (2010) \\ T\({}_{14}\) & \(1.80\pm 0.04\) hr & Addison et al. (2019) \\ \hline \hline \end{tabular}
\end{table}
Table 1: System properties of HD 189733 b, following values used in Alonso-Floriano et al. (2019).
high resolution (\(R\gtrsim 10^{5}\)) over the CARMENES NIR spectral range, with opacity contributions due to prominent molecules expected in H\({}_{2}\)-rich atmospheres over this temperature range (H\({}_{2}\)O, CH\({}_{4}\), NH\({}_{3}\), HCN and OH) and assuming a nominal mixing ratio of \(10^{-4}\) for each molecule. We consider nominal isothermal temperature profiles at 1000 K, 1000 K and 2000 K for HD 189733 b, HD 209458 b, and WASP-76 b, respectively. The molecular cross-sections were obtained following the methods of Gandhi & Madhusudhan (2017) using absorption line lists from the following sources: H\({}_{2}\)O (Barber et al., 2006; Rothman et al., 2010), CH\({}_{4}\) (Yurchenko & Tennyson, 2014), NH\({}_{3}\) (Yurchenko et al., 2011), HCN (Harris et al., 2006; Barber et al., 2014), and OH (Bernath & Colin, 2009; Gordon et al., 2022). We also include collision-induced absorption from H\({}_{2}\)-H\({}_{2}\) and H\({}_{2}\)-He (Borysow et al., 1988; Orton et al., 2007; Abel et al., 2011; Richard et al., 2012).
The model templates are separated into orders before undergoing the same normalisation as the observed spectra, as described in section 2.2. We note that it is the relative depths and positions of the template absorption lines which matter in cross-correlation, and not their absolute depths (Sanchez-Lopez et al., 2019). The model templates are then convolved with the point spread function of the instrument before cross-correlating with the detrended data.
### Signal Extraction
Cross-correlating the detrended residuals with a model template allows us to combine the information from each line, such that we are able to extract a significant detection from the residuals (Snellen et al., 2010; Birkby et al., 2013; Brogi et al., 2016; Birkby, 2018). A strong cross-correlation with a certain model indicates the presence of that molecule in the exoplanet's atmosphere.
For each spectral order independently, we start with the residual spectra at different phases. For each spectrum we first calculate the cross-correlation function (CCF) over a pre-determined velocity grid (-400 km s\({}^{-1}\) to 400 km s\({}^{-1}\) in steps of 1 km s\({}^{-1}\)). To do so, we Doppler shift the model spectrum by each velocity in this grid and then cross-correlate it with the observed spectrum. This gives a cross-correlation value for each point in the velocity grid. Linear interpolation is used to project the Doppler-shifted model template onto the data wavelength grid (Sanchez-Lopez et al., 2019). By repeating this over all phases we have for each order a CCF matrix in velocity and phase. The peak traces out the radial velocity of the planet with time. The mean is subtracted from each row, i.e. along the velocity axis at each phase, to remove broad variations between each spectrum (Alonso-Floriano et al., 2019; Sanchez-Lopez et al., 2019). All the order-wise CCF matrices are then summed to give a single CCF matrix for the entire spectral range. We subsequently shift the co-added CCF matrix into the planet-frame, for each point in a grid over planetary velocity space. Assuming a circular orbit, the planetary radial velocity \(V_{\rm p}\) is given by
\[V_{\rm p}=K_{\rm p}\sin(2\pi\phi)+V_{\rm sys}-V_{\rm bary}+V_{\rm wind} \tag{1}\]
where \(K_{\rm p}\) is the semi-amplitude of the planet's orbital motion, \(V_{\rm sys}\) is the systemic velocity of the planetary system, \(V_{\rm bary}\) is the barycentric velocity correction and \(V_{\rm wind}\) accounts for any planetary atmospheric winds. The values of \(V_{\rm sys}\) and \(V_{\rm bary}\) are accurately known from the literature for each of the planets considered in this work.
We explore a grid in \(K_{\rm p}\) - \(V_{\rm wind}\) space and obtain the total CCF at each point as follows. The planetary radial velocity at each point in this space, as calculated from equation (1), is a function of phase. The rows of the CCF matrix, one for each phase, are shifted by the corresponding radial velocity. The total CCF is calculated by summing the cross-correlation values over time to give a one-dimensional distribution against planetary velocity. Whilst we use all spectra in detrending, only in-transit spectra are included in this addition. The separation of spectra into the in- and out-of-transit regimes can be done using the known orbital parameters of the system. To then obtain a detection significance at each point in \(K_{\rm p}\) - \(V_{\rm wind}\) space, the signal is taken as the value of the total CCF at zero velocity, whilst the noise is estimated by the standard deviation of the total CCF distribution away from this point. In estimating the noise we exclude velocities within \(\pm 15\) km s\({}^{-1}\) to ensure that the measured signal does not influence the noise estimate.
For the correct point in \(K_{\rm p}\) - \(V_{\rm wind}\) space, the cross-correlation matrix will be shifted into the planet's true rest frame. In this case, a peak in the CCF is obtained at zero velocity, maximising the detection significance. We therefore expect a high S/N peak at the planet's location in \(K_{\rm p}\) - \(V_{\rm wind}\) space if the atmosphere contains the chemical species present in the model template and our analysis is sufficiently robust. For HD 189733 b, the expected value for \(K_{\rm p}\) is \(152.5^{+1.3}_{-1.8}\) km s\({}^{-1}\)(Brogi et al., 2016). Any offset from the expected systematic velocity may be attributed to atmospheric winds at the planetary terminator contributing an additional Doppler shift. Atmospheric winds have been constrained for a number of exoplanets in this way (Snellen et al., 2010; Brogi et al., 2016, 2018; Alonso-Floriano et al., 2019; Sanchez-Lopez et al., 2019). The continued retrieval of a signal in only the out-of-transit spectra suggests that the signal may be spurious.
The Welch t-test (Welch, 1947) is an alternative metric to quantify the detection significance (Birkby et al., 2017; Nugroho et al., 2017; Hawker et al., 2018; Alonso-Floriano et al., 2019; Cabot et al., 2019; Sanchez-Lopez et al., 2019). The shifted CCF matrix is split into two distributions: the 'in-trail' distribution, covering the planet signal, and the 'out-of-trail' distribution, which contains the cross-correlation noise. The Welch t-test is used to compare the two distributions and quantify the significance of the in-trail distribution's increased mean. In contrast to the S/N metric, this test's consideration of the standard deviation of each distribution means it may be less vulnerable to noisy pixels in the CCF falsely boosting the detection significance. However, Cabot et al. (2019) suggest that the Welch t-test may overestimate the confidence of detections due to oversam
Figure 1: The erosion of the total ACCF value of a planetary signal with each PCA iteration, for H\({}_{2}\)O, NH\({}_{3}\), CH\({}_{4}\) and HCN models. In each case, \(\lesssim 20\%\) of the planetary signal, injected at the expected planetary velocity of HD 189733 b, remains after 18 iterations of PCA. Calculation of the total \(\Delta\)CCF value is explained in Sections 2.5 and 3.2.
pling, since correlations within the two distributions are typically not accounted for; also see Collier Cameron et al. (2010). Despite this issue potentially being solved if the aforementioned correlations are accounted for, the simpler S/N metric does not have this same problem, and is therefore more commonly used. We henceforth use the S/N metric in this work. We do however note that other factors can influence this metric. The effect of the explored velocity range on the noise estimate is investigated in Section 4.1.
## 3 Robustness of Detrending Optimisation Methods
Across the literature there are different methods for optimising the number of PCA iterations to apply in detrending. In this section we explore the robustness of some commonly used methods which involve the optimisation of detrending using an injected signal. We initially consider global detrending, where an equal number of PCA iterations is applied to each and every spectral order for a single night of observations. In the latter part of this section, we additionally consider optimisation of order-wise detrending.
### Direct CCF Optimisation
In this method we optimise the detrending parameters based on the recovery of a synthetic signal injected into the data. The number of PCA iterations is selected to maximise the recovery of the signal which has been injected into the normalised in-transit spectra (Birkby et al., 2017; Nugroho et al., 2017; Alonso-Floriano et al., 2019; Sanchez-Lopez et al., 2019). For each PCA iteration, the detrended residuals are cross-correlated with the Doppler-shifted model template to derive the signal-injected CCF, which we refer to as CCF\({}_{\rm inj}\). The iteration which returns the maximum S/N from CCF\({}_{\rm inj}\) at the injected planetary velocity is then selected for the detrending of the observed spectra. This optimisation method can also be done directly without first injecting a model signal (Alonso-Floriano et al., 2019; Landman et al., 2021). In this case, the iteration which optimises the S/N from the direct, or 'observed', CCF, referred to as CCF\({}_{\rm obs}\), at the known planetary velocity is selected.
However, it has previously been suggested that such detrending optimisation methods are vulnerable to the overfitting of noise at the very point in planetary velocity space where we expect the signal, thereby falsely amplifying the detection significance (Cabot et al., 2019).
### Differential CCF Optimisation
When optimising the number of PCA iterations, we aim to maximise the recovery of the planetary signal itself, rather than select an optimum iteration based on its amplification of noise into a more significant but biased detection. A less commonly used approach involves selecting the number of PCA iterations by optimising the S/N from a noise-subtracted CCF (Spring et al., 2022; Holmberg and Madhusudan, 2022). Both CCF\({}_{\rm obs}\) and CCF\({}_{\rm inj}\) are found individually for each PCA iteration, as discussed in Section 3.1. A differential CCF, CCF (Brogi et al., 2016; Hoeijmakers et al., 2018), can then be calculated for each iteration as:
\[\Delta\rm CCF=CCF_{\rm inj}-CCF_{\rm obs}. \tag{2}\]
When calculating the S/N from \(\Delta\rm CCF\), the signal is obtained from the \(\Delta\rm CCF\) matrix whereas the noise is estimated using CCF\({}_{\rm obs}\). Detrending parameters can then be selected to optimise this S/N at the expected planetary velocity. This approach allows us to optimise the detrending parameters on a model planetary signal with minimal noise at its location in planetary velocity space. Although information about residual noise around the injected planetary velocity is lost, we avoid the amplification of any noise which can falsely increase the detection significance. The extent to which this method may be more robust is investigated throughout this section.
### Comparison of Methods
To compare the robustness of the above optimisation methods, we use the previously reported detection of H\({}_{2}\)O in observations of HD 189733 b (Alonso-Floriano et al., 2019) as a test case. Using the methods presented in Section 2, we recover this detection of H\({}_{2}\)O for a wide range of PCA iterations.
At this point, we do not consider the optimisation of the S/N from CCF\({}_{\rm inj}\): Detrending parameters found by optimising the S/N from CCF\({}_{\rm inj}\) are dependent on the strength and structure of the injected model. There is no agreed upon injection strength in the literature, and different works use independently generated models, so results given by this method are inconsistent and difficult to reproduce. Conversely, we find that detrending parameters derived by optimising the S/N from \(\Delta\rm CCF\) are relatively independent of injected model strength. We later show that optimised detrending parameters found in this way show little variation across atmospheric models with the different chemical species that we consider. Since CCF\({}_{\rm inj}\) is similar to CCF\({}_{\rm obs}\) in the case of a weak injection and comparable to \(\Delta\rm CCF\) for a strong injection, we consider CCF\({}_{\rm obs}\) and \(\Delta\rm CCF\) as two extremes, from which conclusions about the performance and robustness of optimising the S/N from CCF\({}_{\rm inj}\) can be drawn. We therefore only compare the optimisations of the S/N from CCF\({}_{\rm obs}\) and \(\Delta\rm CCF\) in the remainder of this section.
We begin by examining how the S/N at the expected planetary velocity, from each of CCF\({}_{\rm obs}\) and \(\Delta\rm CCF\), varies with the number of applied PCA iterations between 1 and 18 (Figure 2). Whilst the shape of the \(\Delta\rm CCF\) S/N variation is reasonably invariant to the injection strength of the model, the absolute S/N values are approximately proportional to this strength. Since the absolute values are therefore arbitrary, we rescale the S/N from \(\Delta\rm CCF\) to have the same median as that from CCF\({}_{\rm obs}\) for \(\geq\)2 PCA iterations, such that the injected signal mimics the real signal. This rescaled S/N from \(\Delta\rm CCF\) demonstrates a fairly smooth variation with the number of applied PCA iterations, and may provide an estimate of the significance of the planetary signal. We find that the optimum PCA iteration is relatively independent of the strength of our injected model, an advantage over optimising the S/N from CCF\({}_{\rm inj}\) alone.
We observe that the variation in S/N from CCF\({}_{\rm obs}\) with the number of PCA iterations is noisier than that from \(\Delta\rm CCF\). If we assume that the rescaled S/N from \(\Delta\rm CCF\) is representative of the true planetary signal then noise in CCF\({}_{\rm obs}\) will give a greater than expected observed S/N for some iterations, and a lower than expected observed S/N for others. If we optimise the S/N from CCF\({}_{\rm obs}\) in the detrending, then an iteration where noise components increase the detection significance will likely be selected. In other words, we could systematically inflate the detection significance by methodically selecting detrending parameters which amplify noise at the expected planetary velocity. This is further investigated throughout the remainder of this work.
### Optimisation Bias of Detrending Methods
The extent to which the detection S/N is systematically increased due to amplified noise is here referred to as the bias. A detrending method is robust if the detection significance is unbiased, such that the expected S/N is not systemically increased or decreased. For example, a simple detrending method is to consistently across datasets apply an arbitrary and fixed number of PCA iterations. Although this sometimes will give a greater than expected S/N, it will also sometimes return a lower than expected S/N, creating a distribution of S/N values about the expected significance level. Such a detrending method may therefore be expected to be unbiased. A robust detrending optimisation method maximises the expected detection significance without inducing a bias.
In Figure 2, the S/N from \(\mathrm{CCF_{obs}}\) is optimised by 3 PCA iterations, whereas the S/N from \(\mathrm{\Delta CCF}\) is optimised by 4 iterations. We detrend our spectra globally in each of these cases; the results obtained are shown in Figure 3, with retrieved S/N values at the expected planetary velocity of 6.4 and 6.1, respectively. Here we show a case where the optimisation of detrending using the \(\mathrm{\Delta CCF}\) metric provides a number of PCA iterations very close to that when using \(\mathrm{CCF_{obs}}\); greater by just one iteration. This leads to consistent values for the detection S/N. Generally however, the number of PCA iterations, and the subsequent detection S/N, can be significantly different when detrending is optimised using each of the two metrics, especially in the case of low S/N detections.
It may not always be possible to determine after the fact whether a detection has been falsely inflated by optimisation bias. Since the optimisation bias is intrinsic to the method, it is therefore necessary to examine the optimisation methods themselves, rather than the results produced, in order to evaluate the bias. We do this as follows.
### Measuring the Optimisation Bias
We now examine the bias induced in the detection S/N by each detrending optimisation. To do this, we optimise the detrending parameters and calculate an optimised S/N at each and every point in planetary velocity space. Considering each point across \(K_{\mathrm{p}}\) - \(V_{\mathrm{wind}}\) space individually, we find the number of PCA iterations to apply in detrending such that the derived S/N at that point, from each of \(\mathrm{CCF_{obs}}\) and \(\mathrm{\Delta CCF}\), is maximised. The observed S/N corresponding to that number of PCA iterations is then found at each point. Excluding from consideration the central band around the real planetary signal (\(|V_{\mathrm{wind}}|<10\) km s\({}^{-1}\)), the distribution of S/N values obtained covers regions of velocity space devoid of the majority of this signal. As a result, the statistical expectation is that a robust optimisation method should yield S/N values normally distributed about zero. On the other hand, an optimisation method which is systemically inflating detection significances should produce a shifted S/N distribution with median \(>0\).
Figure 4 shows the distribution of optimised S/N values across planetary velocity space, when optimising the S/N from each of \(\mathrm{CCF_{obs}}\) and \(\mathrm{\Delta CCF}\) in the detrending. PCA iterations from 2 to 18 inclusive are considered, except for H\({}_{2}\)O where a minimum of 3 iterations is enforced to aid the sufficient removal of telluric residuals. Across H\({}_{2}\)O, NH\({}_{3}\), HCN and CH\({}_{4}\) models, the median optimised S/N across velocity space is 0.9 and -0.2 for \(\mathrm{CCF_{obs}}\) and \(\mathrm{\Delta CCF}\), respectively. This suggests that optimising the detrending parameters using the S/N from \(\mathrm{CCF_{obs}}\) biases the detection significance due to
Figure 3: S/N maps showing the retrieval of H\({}_{2}\)O signals in the atmosphere of HD 189733 b via the detrending optimisation of the S/N from \(\mathrm{CCF_{obs}}\) and \(\mathrm{\Delta CCF}\), respectively. Panel (a): applying 3 PCA iterations to optimise the S/N from \(\mathrm{CCF_{obs}}\) retrieves a detection for H\({}_{2}\)O with a S/N of 6.4. The median \(\mathrm{S/N}\) across velocity space away from the expected planetary velocity (\(|V_{\mathrm{wind}}|>10\) km s\({}^{-1}\)) is 0.2. Panel (b): applying 4 PCA iterations to optimise the S/N from \(\mathrm{\Delta CCF}\) retrieves a detection for H\({}_{2}\)O with a S/N of 6.1. The median S/N away from the expected planetary velocity is 0.0.
Figure 2: The variation of retrieved S/N, from each of \(\mathrm{CCF_{obs}}\) (blue) and \(\mathrm{\Delta CCF}\) (red), at the expected planetary velocity against the number of applied PCA iterations. An H\({}_{2}\)O model is used, and iterations from 1 to 18 are considered. Faint red lines show the S/N from \(\mathrm{\Delta CCF}\) for different values of \(V_{\mathrm{wind}}\) in the interval \(\pm 50\) km s\({}^{-1}\), for constant \(K_{\mathrm{p}}\).
the method's vulnerability to noise. On the other hand, optimising the S/N from \(\Delta\)CCF produces a smaller, negative bias. Whilst this is not zero, its magnitude is consistent with the median S/N values found away from the expected planetary velocity in Figure 3, in which the optimisation of detrending was done only at the expected planetary velocity. Therefore, we do not observe an average increase in the S/N found at points away from the expected planetary velocity when the S/N from \(\Delta\)CCF is optimised at every point in this space during detrending i.e. no bias is observed to be introduced by this detrending optimisation.
The bias could alternatively be measured by building a distribution of optimised S/N values at the expected planetary velocity for a large sample of randomised model spectra. Each random model is a H\({}_{2}\)O model whose transit depth values have been randomly scrambled in wavelength space. There should therefore be no signal present in the CCF, and hence a detection S/N of zero is expected. For each random model, the detrending is optimised at the expected planetary velocity and a S/N is calculated. As before, the median of the S/N distribution returned can provide a measure of the optimisation bias induced at the expected planetary velocity.
A similar method of estimating optimisation bias at the expected planetary velocity is to randomise the time-ordering of spectra, rather than the wavelength-ordering of the model spectrum, prior to detrending (Zhang et al., 2020; Giacobbe et al., 2021). This results in the planetary signal no longer being sinusoidally Doppler-shifted with time, and therefore no peak in velocity space should be recovered. However, due to the short transit duration of hot Jupiters, and the therefore narrow range in planetary radial velocity during the observations, correlated signals may remain in the randomly ordered spectra (Giacobbe et al., 2021). Since significant peaks could hence be recovered even if the detrending is robust, we do not use this method to measure the optimisation bias.
We now provide an illustration of how optimising the S/N from CCF\({}_{\mathrm{obs}}\) in the detrending can amplify noise into a more significant detection. We inject a model signal into an arbitrary point in velocity space away from the real signal. This injected signal is now treated as a real signal in the data. To recover this signal, during detrending we optimise the S/N from CCF\({}_{\mathrm{obs}}\) (which contains the injected signal) at the injected velocity. Figure 5 compares the optimised total CCF\({}_{\mathrm{obs}}\) to the total \(\Delta\)CCF, which is the isolated contribution of the injected signal to CCF\({}_{\mathrm{obs}}\), after the same number of PCA iterations. The optimised CCF\({}_{\mathrm{obs}}\) signal is greater than the injected signal itself due to the amplification of noise resulting in a more significant detection. When optimising the S/N from CCF\({}_{\mathrm{obs}}\), such systematic amplification leads to the bias observed in Figure 4.
We conclude that bias is introduced when the S/N from CCF\({}_{\mathrm{obs}}\) is optimised at the expected planetary velocity in the detrending, whereas using \(\Delta\)CCF is more robust.
Figure 4: The distribution of optimised S/N values across planetary velocity space when optimising everywhere the S/N from each of CCF\({}_{\mathrm{obs}}\) and \(\Delta\)CCF. In each case, the central band around the expected planetary velocity (\(|V_{\mathrm{wind}}|<10\) km s\({}^{-1}\)) is excluded from consideration, and the black vertical line represents the median of the combined distribution of all 4 models. An equally-populated normal distribution about zero is shown for reference in each panel. Panel (a): when the S/N from CCF\({}_{\mathrm{obs}}\) is optimised everywhere in the detrending, a distribution of S/N values with a median of 0.9 is found across the 4 models, demonstrating the inflated detection significances achieved using this method. Panel (b): when the S/N from \(\Delta\)CCF is optimised everywhere in the detrending, a distribution of S/N values with a median of -0.2 is found across the 4 models, suggesting that this method is more robust.
Figure 5: A comparison between the optimised total CCF\({}_{\mathrm{obs}}\) (blue) and the total \(\Delta\)CCF (orange) for a model signal injected into an arbitrary point in planetary velocity space away from the expected planetary velocity. The S/N from CCF\({}_{\mathrm{obs}}\) is optimised in the detrending, with both signals shown after the same number of PCA iterations. The total ACCF is the isolated contribution of the injected signal to CCF\({}_{\mathrm{obs}}\). As can be seen, the peak of the optimised total CCF\({}_{\mathrm{obs}}\) is greater than the peak of the injection itself, directly demonstrating the amplification of noise into a more significant detection. The planetary velocity considered for each molecule is not necessarily the same, nor is the absolute scale of the CCF value axis.
### Detrending Performance
When judging the robustness of a detrending optimisation method, it is important to consider how a good detrending method should perform. An ideal detrending method would remove telluric, stellar and instrumental effects from our spectra, leaving only the planetary signal and white noise. Ideally, such removal would be independent of the velocity shift (assuming a sufficiently large change in the planet's radial velocity over the transit), strength and model of the signal for which we are optimising. It would also not matter if this signal is actually present in the data or not. The optimal detrending parameters derived would therefore show little variation across velocity space and between different models. In light of these expectations, we here investigate the detrending behaviour when optimising the S/N from each of CCFobs and \(\Delta\)CCF.
Figure 6 shows the optimal detrending parameters derived by optimising the S/N from each of CCFobs and \(\Delta\)CCF over velocity space and between different models. As in Section 3.5, we inject a model at each location in planetary velocity space and optimise its recovery using each of these metrics. Whereas Figure 4 shows the optimised S/N distributions, in Figure 6 we show the distributions of the corresponding detrending parameters used to optimise each of the metrics at each point in planetary velocity space. The relative consistency of optimising the S/N from \(\Delta\)CCF across velocity space and between different models is demonstrated. For this dataset, across H2O, NH3 and HCN models, the S/N from \(\Delta\)CCF is most commonly optimised by either 7-8 PCA iterations of 3-4 PCA iterations. This tightly constrained bimodal distribution is somewhat characteristic of the trend seen in Figure 2, in which two local peaks in S/N appear at \(\sim\)4 and \(\sim\)7 PCA iterations for an H2O model. These results therefore support the finding in Figure 2 that there is little change in the shape, and hence the optimum, of the \(\Delta\)CCF S/N variation as we move across planetary velocity space, and the findings of Figure 1 that the erosion of an injected planetary signal with each PCA iteration is somewhat consistent across models. No such consistency across velocity space is observed when optimising the S/N from CCFobs. The optimised detrending parameters in this case are highly dependent on planetary velocity, which is expected due to the noise in CCFobs being variable across planetary velocity space.
Since the detrending parameters derived by optimising the S/N from \(\Delta\)CCF appear consistent across velocity space and between the models we consider, as demonstrated in Figure 6, it is unlikely that significant bias will be introduced due to the specific choice of injection velocity or atmospheric model used in the detrending optimisation. For this dataset, we find that the set of likely optimised detrending parameters is relatively independent of such choices. We note, however, that we have not investigated more extreme models, e.g. CO2-dominated atmospheres, considering that HD 189733 b is a gas giant with an H2-rich atmosphere.
As discussed in Section 3.2, noise around the injected planetary velocity is subtracted and therefore not considered when calculating the S/N from \(\Delta\)CCF. The robustness of the derived detrending parameters against velocity, as demonstrated here, may however suggest that such loss of noise is perhaps not overly consequential. At each different planetary velocity, different regions of the CCF are not considered, but similar optimised detrending parameters are found.
### Extension to Order-wise Optimisation
Given the varying telluric contamination and planetary signal strength in each spectral order, it is reasonable to assume that a different amount of detrending is required for each order (Alonso-Floriano et al., 2019; Spring et al., 2022). However, order-wise optimisation is typically avoided due to the significantly greater number of free parameters in the analysis, which increases the risk of amplifying noise into false detections (Cabot et al., 2019; Spring et al., 2022). We investigate this by optimising separately the number of PCA iterations applied to each order. Again, we consider iterations between 3 and 18 for H2O, and between 2 and 18 for other species.
To assess the robustness of order-wise detrending, we again optimise the S/N for each and every point in planetary velocity space, this time allowing different numbers of PCA iterations to be applied to each order. We do this optimisation using the S/N from each of CCFobs and \(\Delta\)CCF. The distributions of S/N values retrieved away from the expected planetary velocity are shown in Figure 7. When the S/N from CCFobs is optimised order-wise in the detrending (Figure 7a), a median S/N of 2.6 is found across the 4 models, demonstrating that a large bias is present when using this method. This bias is considerably greater than in the case of global detrending. We find that 32% and 9% of points in velocity space return S/N \(\geq\)3 and \(\geq\)4, respectively. This is in agreement with Cabot et al. (2019), who showed that detection significances of more than 4\(\sigma\) can be obtained by optimising the detrending order-wise at incorrect locations in planetary velocity space. These findings suggest that optimising the S/N from CCFobs order-wise is vulnerable to the recovery of spurious signals with significant S/N values, or the inflation of weak signals into much stronger ones dominated by an amplified noise component.
We demonstrate the potential for these effects in Figure 8, in which we find significant signals for H2O (S/N = 9.7) and NH3 (S/N = 3.9) in the atmosphere of HD 189733 b using order-wise optimisation of the S/N from CCFobs during detrending. Using global detrending, or when optimising the S/N from \(\Delta\)CCF order-wise, we are not able to recover a significant NH3 signal in this dataset. The signal may therefore be spurious, and only introduced by the bias attributed with optimising order-wise the S/N from CCFobs. Likewise, the significance of the H2O signal is largely inflated compared to what was found robustly in Figure 3b (S/N = 6.1). There is no conclusive information in Figure 8, e.g. the median S/N across velocity space, which could indicate whether either detection S/N has been biased by contributions from residual noise. This motivates the above analysis of the optimisation methods and their intrinsic biases themselves, rather than just the results produced.
On the other hand, when the S/N from \(\Delta\)CCF is optimised order-wise at each and every point across velocity space, a median S/N of -0.5 is found across the 4 models (Figure 7b). Whilst this is again non-zero, there is no positive bias and the absolute bias is much less than that obtained using CCFobs. Figure 7c shows the optimised S/N values across a region of velocity space for H2O, with a clear and significant peak at the expected planetary velocity. We additionally find that, within each order, the distribution of optimised detrending parameters across planetary velocity space shows similar behaviour as in Figure 6. We conclude that optimising the S/N from \(\Delta\)CCF in each order is therefore more robust than using CCFobs.
Using this optimisation method, the retrieved S/N for H2O is 5.4 (Figure 9), which is consistent with that found via global detrending, albeit slightly lower. In other datasets and/or with different models, however, there may be an increase in S/N by allowing each order to be detrended separately. In this example, we note the persistent telluric contamination in the form of a spurious second peak at low \(K_{\rm p}\) and negative \(V_{\rm wind}\); this is discussed further in Section 6. Despite this, the planetary signal at the expected planetary velocity does not appear to have been over-optimised to an inflated S/N, like that in Figure 8a, suggesting that minimal bias is introduced at the expected planetary velocity by this detrending optimisation.
Figure 6: The distribution of the optimum PCA iteration across velocity space for each of H\({}_{2}\)O, NH\({}_{3}\) and HCN. Optimising the S/N from ACCF (blue) gives a tightly constrained distribution for this parameter across velocity space. There is also consistency between models. On the other hand, detrending parameters derived by optimising the S/N from CCF\({}_{\rm obs}\) show large variation across planetary velocity space and between models.
Figure 7: The S/N is optimised order-wise at every point across planetary velocity space. This is done by optimising order-wise the S/N from each of CCF\({}_{\rm obs}\) and \(\Delta\)CCF in the detrending. The distributions of optimised S/N values across planetary velocity space are shown. In panels (a) and (b), the central band around the expected planetary velocity (\(|V_{\rm wind}|<10\)km s\({}^{-1}\)) is excluded from consideration, and the black vertical lines represent the median of the combined distributions of all 4 models. An equally-populated normal distribution about zero is shown for reference in each of these panels. Panel (a): when the S/N from CCF\({}_{\rm obs}\) is optimised order-wise in the detrending, a median S/N of 2.6 is found across the 4 models, demonstrating the bias introduced by this method. Panel (b): when the S/N from ACCF is optimised order-wise in the detrending, a median S/N of -0.5 is found across the 4 models, suggesting that this method is more robust. Panel (c): for H\({}_{2}\)O, the optimised S/N values are now shown as a function of velocity space, when the S/N from \(\Delta\)CCF is optimised order-wise in the detrending. The distribution of these S/N values (for \(|V_{\rm wind}|>10\) km s\({}^{-1}\)) is shown in panel (b).
We therefore find that, as in the case of global detrending, significant bias is introduced when the S/N from CCF\({}_{\rm obs}\) is optimised order-wise at the expected planetary velocity in the detrending, whereas using \(\Delta\)CCF is more robust.
### A Robust Detrending Recipe
We find that the selection of detrending parameters by optimising the S/N from \(\Delta\)CCF will minimally bias the S/N at the expected planetary velocity, whether done globally or order-wise. We here summarise this method for obtaining a robust S/N measurement, starting with cleaned and normalised spectra:
1. Apply iterations of PCA to the spectra. After each iteration, cross-correlate the residual with a model template (as described in Section 2.5) to derive an observed cross-correlation function, CCF\({}_{\rm obs}\), for each iteration.
2. Inject a model signal into the spectra at or close to the expected planetary velocity of the real signal. Repeat step (i) on the signal-injected spectra to derive a signal-injected cross-correlation function, CCF\({}_{\rm inj}\), for each iteration. The planetary velocity of the injection can be somewhat approximate, as discussed in Section 3.6.
3. For each PCA iteration, derive the differential cross-correlation function, \(\Delta\)CCF = CCF\({}_{\rm inj}\) - CCF\({}_{\rm obs}\). Calculate the S/N from \(\Delta\)CCF at the injected planetary velocity, as described in Sections 2.5 and 3.2, and find the number of PCA iterations which maximises this S/N.
4. Apply this optimal number of PCA iterations to the observed spectra. Cross-correlate the residuals with a Doppler-shifted model template, as in Section 2.5, over planetary velocity space to derive the detection S/N as a function of planetary velocity.
The above procedure is for global detrending optimisation. We have shown in Section 3.7 that it may also be robust to optimise separately the number of PCA iterations applied to each spectral order. In this case, steps 1-3 above can be applied to a single order to find the optimum number of PCA iterations required to detrend that order. The detrended residuals are then cross-correlated with the model template as before.
The choice of the maximum number of PCA iterations to consider during optimisation is somewhat arbitrary in the literature. In Figure 2, the S/N from \(\Delta\)CCF is clearly decreasing after 18 PCA iterations, hence this is a reasonable point at which to stop. This will not necessarily always be the case as we consider different datasets however. For a consistent determination of the maximum iteration to consider, we simulate the erosion of a model planetary signal, as in Figure 1. A minimum acceptable fraction of the remaining signal can be nominally defined, such that only iterations up to and including this are allowed. We here use \(\sim\)20-30% as this minimum fraction, and take the same maximum PCA iteration for different species due to the demonstrated consistency between the erosion of different models. We hence obtain a maximum of 18 PCA iterations for the HD 189733 b dataset.
It should be noted that, even when using identical detrending methods on the same dataset with the same model, the final S/N value will not necessarily be the same. We find that the addition of small amounts of Gaussian noise into the normalised spectra before detrending can produce a wide spread of observed S/N values. Unsubstantial differences in the cleaning, normalisation, calibration and masking of spectra prior to detrending can therefore consider
Figure 8: S/N maps showing non-robust signals for H\({}_{2}\)O and NH\({}_{3}\) in the atmosphere of HD 189733 b, via the order-wise detrending optimisation of the S/N from CCF\({}_{\rm obs}\). Panel (a): a S/N of 9.7 is retrieved for H\({}_{2}\)O, with a median S/N across velocity space (for \(|V_{\rm wind}|>10\) km s\({}^{-1}\)) of 0.1. Panel (b): a S/N of 3.9 is retrieved for NH\({}_{3}\), with a median S/N across velocity space (for \(|V_{\rm wind}|>10\) km s\({}^{-1}\)) of -0.3.
Figure 9: S/N map showing the detection of H\({}_{2}\)O in the atmosphere of HD 189733 b with a S/N of 5.4, via the order-wise detrending optimisation of the S/N from \(\Delta\)CCF.
ably alter the reported detection significance, even when subsequent methods are identical.
## 4 Additional factors in determination of detection significance
In addition to the optimisation of detrending, there are other method and parameter choices within the data analysis which are inconsistent across the literature. It is important to understand the extent to which such choices can impact the detection S/N. In this section we present some examples.
### Velocity Range
We now demonstrate how the retrieved detection S/N can be dependent on the planet-frame velocity range over which we calculate the noise in the total CCF. This dependency has previously been noted by Spring et al. (2022). We use the globally detrended H\({}_{2}\)O signal in the atmosphere of HD 189733 b to demonstrate this. Until now, we have cross-correlated the detrended spectra with the model template for velocities ranging from -400 km s\({}^{-1}\) to 400 km s\({}^{-1}\) in intervals of 1 km s\({}^{-1}\). The CCF is then shifted into the planet-frame, and the noise is calculated by taking the standard deviation of the total CCF values between velocities of -300 km s\({}^{-1}\) to 300 km s\({}^{-1}\), excluding the \(\pm\)15 km s\({}^{-1}\) region as described in Section 2.5. We examine the dependence of the retrieved S/N on this velocity range over which we calculate the noise.
Figure 10 shows the variation of the retrieved S/N, for two different PCA iterations, against the maximum velocity we consider when calculating the standard deviation of the total CCF away from the central peak. Considerably amplified significances can be achieved when velocity ranges narrower than \(\pm\)150 km s\({}^{-1}\) are considered. Values found here are in line with the detection of H\({}_{2}\)O (S/N = 6.6) reported by Alonso-Floriano et al. (2019), where a velocity range of \(\pm\)65 km s\({}^{-1}\) was used. Our velocity range of \(\pm\)300 km s\({}^{-1}\) gives a more conservative detection significance.
When calculating the standard deviation of the total CCF, we therefore encourage the consideration of a wide velocity range to avoid domination by any relatively noiseless regions which can lead to amplified detection significances. Future work could develop a significance metric which is not so dependent on this parameter. For example, the signal and noise could perhaps be estimated using the mean and standard error of the CCF values within a pixel-wide in-trail distribution. In the meantime, potential amplification of the quoted detection S/N by this effect should be considered.
### Optimising Order Weighting and Selection
When optimising the detrending parameters order-wise, the number of PCA iterations which maximises the retrieved S/N in each order is selected. During global detrending, we can similarly calculate the S/N in each order but instead use it to weight each order's contribution to the total CCF. Previous works have done this to improve the S/N of a detection, by favouring the orders where the signal can be recovered to a higher significance (Giacobbe et al., 2021; Spring et al., 2022; van Sluijs et al., 2022). This may be the case in certain orders due to there being fewer telluric and stellar lines, or more planetary signal. However, it may also be due to there being a greater noise component in the spectrum in some orders. Favouring such orders, where amplified noise at the expected planetary velocity is falsely increasing the signal in that order, could bias the detection significance. We here investigate the robustness of such order weightings.
One method of weighting orders is to mask orders where there is little recoverable signal. For example, Giacobbe et al. (2021) mask orders in which the S/N recovered from CCF\({}_{\rm inj}\) at the expected planetary velocity is less than a threshold. As already discussed, this is model dependent and therefore difficult to reproduce. We therefore here examine the robustness of masking a fixed percentage of orders, using the order-wise S/N values from each of CCF\({}_{\rm obs}\) and \(\Delta\)CCF at the expected planetary velocity.
To do so, each spectral order is first detrended using the number of PCA iterations found by globally optimising the S/N from \(\Delta\)CCF. For this dataset, in the case of H\({}_{2}\)O, this means that 4 PCA iterations are applied, such that the unweighted case is equivalent to the detection shown in Figure 3b. We then apply the same robustness tests as in Figures 4 and 7. For each and every point in planetary velocity space individually, we calculate the S/N by only including in the CCF the 75% of orders which recover the greatest order-wise S/N, from each of CCF\({}_{\rm obs}\) and \(\Delta\)CCF, at that planetary velocity. The remaining 25% of orders are masked. We then observe the resulting distribution of S/N values. In the CCF\({}_{\rm obs}\) case, the distribution of retrieved S/N values has a median \(>\) 1, suggesting the introduction of a bias when orders are selected according to the order-wise S/N from CCF\({}_{\rm obs}\). On the other hand, selecting orders according to the order-wise S/N from \(\Delta\)CCF returns a distribution of S/N values with a median that is small in magnitude, suggesting that this method is considerably less biased. These findings remain true when the percentage of orders selected is varied.
We explore examples to demonstrate these findings. As when optimising order-wise the S/N from CCF\({}_{\rm obs}\) during detrending, we can again retrieve an NH\({}_{3}\) signal in the atmosphere of HD 189733 b, this time by selecting orders according to the order-wise S/N from CCF\({}_{\rm obs}\) at the expected planetary velocity. We first globally optimise the S/N from CCF\({}_{\rm obs}\) during detrending, corresponding to 17 PCA iterations, to find NH\({}_{3}\) with a S/N of 3.2. We subsequently only select the 15 orders out of 20 with the greatest order-wise S/N from CCF\({}_{\rm obs}\) at the expected planetary velocity. An updated S/N of 4.2 is returned (Figure 11), providing a further example of the recovery of a tentative signal via non-robust methods.
Figure 10: For H\({}_{2}\)O, the retrieved S/N at the expected planetary velocity varies with the maximum velocity we consider when calculating the standard deviation of the total CCF away from the peak. This is shown for the cases of global detrending using 5 (blue) and 6 (orange) PCA iterations.
We now instead apply robust order selection to our detection (S/N = 6.1) of H\({}_{2}\)O in the atmosphere of HD 189733 b (Figure 3b), achieved via global detrending using 4 PCA iterations. We do so by considering the order-wise S/N from \(\Delta\)CCF at the expected planetary velocity. There are 20 spectral orders remaining after a priori masking but only the 16 orders with the greatest S/N from \(\Delta\)CCF are included when calculating the final CCF. An updated S/N of 5.9 is achieved (Figure 12), which is consistent with the original detection. In other datasets and/or with different models, it is possible that such robust order masking may result in an increased S/N.
Alternatively, unequal weightings, rather than a binary mask, can be applied to each order when their CCFs are summed (Spring et al., 2022; van Sluijs et al., 2022). Thus far in this work the CCFs from each remaining order have been added with equal weighting. During this summation, we now instead weight the CCF from each order according to the order-wise S/N recovered at the expected planetary velocity. Spring et al. (2022) do so using the S/N from \(\Delta\)CCF in each order. The weight \(w\) for each \(i\)-th order is calculated as:
\[w_{i}=\frac{S/N_{i}-S/N_{\rm min}}{S/N_{\rm max}-S/N_{\rm min}} \tag{3}\]
where S/N\({}_{i}\) is the order-wise S/N at the expected planetary velocity for order \(i\), and S/N\({}_{\rm min}\) and S/N\({}_{\rm max}\) are the minimum and maximum of these S/N values across the orders, respectively.
We test this method for robustness as before. At each point in planetary velocity space, a final CCF is formed using order weightings calculated with equation 3, using the order-wise S/N values from each of CCF\({}_{\rm obs}\) and \(\Delta\)CCF. A S/N value is then derived at each planetary velocity. In the CCF\({}_{\rm obs}\) case, the distribution of S/N values has a median \(>\) 1, once again suggesting that a bias is introduced by this method. On the other hand, a median that is small in magnitude is obtained in the \(\Delta\)CCF case, implying once more that this method is more robust. This follows the observed trend; optimising or weighting according to the S/N from CCF\({}_{\rm obs}\) is vulnerable to residual noise and therefore more biased, whereas \(\Delta\)CCF is noise-subtracted and therefore its use is more robust. When these order weightings are applied to the globally detrended H\({}_{2}\)O detection (S/N = 6.1) in Figure 3b, updated S/N values of 5.7 and 6.8 are found for the \(\Delta\)CCF and CCF\({}_{\rm obs}\) cases, respectively.
To summarise, we find that a bias will likely be introduced if the contribution of each order to the final CCF is calculated according to the order-wise S/N values from CCF\({}_{\rm obs}\) at the expected planetary velocity. We conversely find that doing so using \(\Delta\)CCF is more robust.
## 5 Results: Case Studies
This work has so far used CARMENES observations of HD 189733 b as a test case to investigate the robustness of molecular detections using high-resolution transmission spectroscopy of exoplanets. We now systematically implement the robust methodologies described in Sections 2, 3.8, and 4.2 to archival CARMENES observations of two other hot Jupiters: HD 209458 b and WASP-76 b. Planet-specific models are independently generated for each dataset.
### HD 189733 b
We here summarise our findings for the CARMENES observations of HD 189733 b. Following the robust methods described in Sections 2 and 3.8, we find H\({}_{2}\)O in the atmosphere of HD 189733 b with a S/N of 6.1 (Figure 3b) when globally optimising the S/N from \(\Delta\)CCF during detrending. S/N values of 5.4 and 5.9 are alternatively found when optimising the S/N from \(\Delta\)CCF order-wise (Figure 9), or when using robust order selection as detailed in Section 4.2 (Figure 12), respectively. We do not robustly find any significant detections (S/N > 3.0) for NH\({}_{3}\) or HCN in the atmosphere of this exoplanet using this dataset.
We also outline our results when applying non-robust methods. By optimising the S/N from CCF\({}_{\rm obs}\) order-wise in the detrending, a S/N of 9.7 can be achieved for H\({}_{2}\)O using the same spectra and model as before (Figure 8a). This emphasises the extent to which detection significances can be inflated by non-robust detrending optimisations. We additionally obtain NH\({}_{3}\) signals using non-robust methods, such as when optimising the S/N from CCF\({}_{\rm obs}\) order-wise in the detrending (S/N = 3.9) (Figure 8b), and when selecting orders according to the order-wise S/N from CCF\({}_{\rm obs}\), after globally optimising the S/N from CCF\({}_{\rm obs}\) during detrending (S/N = 4.2) (Figure 11). Such detections of
Figure 11: S/N map showing a non-robust NH\({}_{3}\) signal (S/N = 4.2) in the atmosphere of HD 189733 b, achieved by masking orders according to the order-wise S/N from CCF\({}_{\rm obs}\). After global detrending using 17 PCA iterations, found by optimising the S/N from CCF\({}_{\rm obs}\) in the detrending, only the 15 out of 20 orders with the greatest S/N from CCF\({}_{\rm obs}\) at the expected planetary velocity are included in the final CCF.
Figure 12: The HD 189733 b H\({}_{2}\)O signal in Figure 3b, achieved via global detrending using 4 PCA iterations, is now robustly refined such that the final CCF only includes the 16 out of 20 orders with the greatest S/N from \(\Delta\)CCF\({}_{\rm obs}\) the expected planetary velocity. A new S/N of 5.9 is achieved, compared to 6.1 previously.
NH\({}_{3}\) in the atmosphere of HD 189733 b are not necessarily spurious, but we can only obtain them here using non-robust methods.
### HD 209458 b
We now re-analyse archival CARMENES data for a transit of the hot Jupiter HD 209458 b on the night of 5th September 2018. HD 209458 b (Charbonneau et al., 2000) is another extensively studied hot Jupiter which has been subject to multiple high-resolution studies in the NIR region. H\({}_{2}\)O has previously been detected using the same CARMENES observations as we use here (Sanchez-Lopez et al., 2019), whilst observations from other high-resolution spectrographs have also found H\({}_{2}\)O in both emission (Hawker et al., 2018) and transmission (Giacobbe et al., 2021). Hawketer et al. (2018) additionally detected CO and HCN in the atmosphere of HD 209458 b using two spectral bands of CRIRES (2.29-2.35 \(\mu\)m and 3.18-3.27 \(\mu\)m), whilst Giacobbe et al. (2021) found CO, HCN, CH\({}_{4}\), NH\({}_{3}\) and C\({}_{2}\)H\({}_{2}\) using GIANO over a spectral range of 0.95-2.45 \(\mu\)m. CO was also detected by Snellen et al. (2010) in emission spectra observed using CRIRES.
The data consists of 91 observations (spanning planetary orbital phases -0.0358 \(<\phi<\) 0.0368), of which 46 are in transit. Exposure times of 198 s were used throughout. The median S/N of the observations was 76, with the airmass increasing from a minimum of 1.05 to a maximum of 2.11 over the observing night. The system parameters for this planet are shown in Table 11, and a planetary velocity semi-amplitude (\(K_{\rm p}\)) of \(145\pm 1.5\) km s\({}^{-1}\) is expected (Giacobbe et al., 2021). We apply the robust methods presented in Sections 2 and 3.8 to this dataset. We remove a priori the same 6 orders (54-53, 45-42) and similar exposures (first 15 and last 7 spectra) as Sanchez-Lopez et al. (2019) due to low S/N. We find that the PCA erosion of the planetary signal is significantly slower for this dataset than that for HD 189733 b. As shown in Figure 13, the nominal \(\sim\)20-30% threshold discussed in Section 3.8 is reached only after roughly 34 iterations. We therefore consider PCA iterations up to and including 34 in our optimisations.
We find a tentative signal for H\({}_{2}\)O, with a S/N of 3.2 and a significantly greater than expected value for \(K_{\rm p}\) (Figure 14), when globally optimising the S/N from \(\Delta\)CCF during detrending. This corresponds to 21 PCA iterations. We are unable to robustly recover any signals for HCN, which may be due to there being no strong spectral features in this wavelength band. NH\({}_{3}\) and CH\({}_{4}\) signals are also not observed. It is possible that a more comprehensive exploration of model space could yield stronger signals using robust methods than observed here.
We now use this dataset to further demonstrate the effect of non-robustly optimising the order selection and weighting. We first detrend the spectra globally by optimising the S/N from \(\Delta\)CCF for each chemical species considered in this work. We then conduct order selection and weighting, as described in Section 4.2, using the S/N from each of \(\Delta\)CCF and CCF\({}_{\rm obs}\), and assess its effect on the detection significance for that species. When the orders are weighted or selected according to the S/N from \(\Delta\)CCF, the H\({}_{2}\)O detection in Figure 14 is preserved but we continue to retrieve no significant and robust signals for NH\({}_{3}\) or HCN. However, when orders are selected according to the S/N from CCF\({}_{\rm obs}\), the H\({}_{2}\)O detection S/N is enhanced from 3.2 to 5.1, and NH\({}_{3}\) and HCN signals are found at S/N of 3.5 and 3.3, respectively, as shown in Figure 15. Similar effects are seen when orders are optimally weighted rather than selected. This may demonstrate the lack of robustness in selecting or weighting orders according to the S/N from CCF\({}_{\rm obs}\), as discussed in Section 4.2.
As shown in Section 3.7, order-wise optimisation of the S/N from CCF\({}_{\rm obs}\) during detrending may be vulnerable to the retrieval of spurious yet significant signals, and the inflation of weak signals into much more significant ones. For HD 209458 b, we retrieve a signal for CH\({}_{4}\) with a S/N of 4.2 (Figure 16) when optimising order-wise the S/N from CCF\({}_{\rm obs}\), emphasising the bias induced by this detrending optimisation. This signal is completely removed when we mask the CCF within the Earth-frame velocity interval \(\pm\) 5km s\({}^{-1}\), suggesting that is likely the result of the optimisation of uncorrected telluric residuals.
### WASP-76 b
We additionally analyse CARMENES archival data for a transit of the ultra-hot Jupiter WASP-76 b (West et al., 2016) on the night of 4th October 2018. The data consists of 44 observations (spanning planetary orbital phases -0.0709 \(<\phi<\) 0.0815), of which 25 observations are in transit. Exposure times of 498 s were used throughout, with a median observation S/N of 59. Parameters for this planet can be seen in Table 2. We use \(196.5\pm 0.9\) km s\({}^{-1}\)(Ehrenreich et al., 2020) as the expected planetary velocity. This same CARMENES dataset
Figure 14: S/N map showing a tentative H\({}_{2}\)O signal in the atmosphere of HD 209458 b, with a S/N of 3.2 and a significantly greater than expected value for \(K_{\rm p}\). The S/N from \(\Delta\)CCF was optimised in the detrending such that 21 PCA iterations were applied.
Figure 13: As in Figure 1 but now for HD 209458 b, the total \(\Delta\)CCF values of model planetary signals are eroded with each PCA iteration. H\({}_{2}\)O, NH\({}_{3}\), CH\({}_{4}\) and HCN models are included here.
has been examined by a number of previous works. Sanchez-Lopez et al. (2022) found detections for H\({}_{2}\)O (S/N = 5.5) and HCN (S/N = 5.2), as well as finding an inconclusive detection of NH\({}_{3}\) (S/N = 4.2). Landman et al. (2021) meanwhile detected OH with a S/N of 6.1.
We remove orders 45-41 and 55-53 due to their low S/N. After observing the erosion of planetary signals synthetically injected into these spectra, as in Figures 1 and 13, we consider up to 32 PCA iterations when optimising the detrending. Since OH is the species responsible for sky emission lines in the Earth's atmosphere, like H\({}_{2}\)O we only consider PCA iterations of 3 or more for this species to aid the sufficient removal of emission line residuals. Optimising the S/N from \(\Delta\)CCF, first globally and then order-wise, we confirm the Landman et al. (2021) detection of OH with S/N values of 4.1 and 4.7, respectively (Figure 17). The position and structure of the cross-correlation signals in \(K_{\rm p}\) - \(V_{\rm sys}\) space could perhaps be explained by atmospheric dynamics and rotation (Wardenier et al., 2021), if not by noise. To test if the signals spuriously arise from uncorrected sky emission line residuals, we mask the final CCF within the Earth-frame velocity interval \(\pm\)5 km s\({}^{-1}\). The OH signals are maintained, indicating that the signal is indeed likely planetary.
Figure 16: S/N map showing a non-robust CH\({}_{4}\) signal (S/N = 4.2) recovered in the atmosphere of HD 209458 b when the S/N from CCF\({}_{\rm obs}\) is optimised order-wise in the detrending.
Figure 15: S/N maps showing non-robust signals retrieved in the atmosphere of HD 209458 b when the orders are weighted or selected according to the S/N from CCF\({}_{\rm obs}\). In each case, the number of PCA iterations applied is found by globally optimising the S/N from \(\Delta\)CCF for that species. There are initially 22 orders remaining after the a priori masking. Panel (a): H\({}_{2}\)O signal with a S/N of 4.2 obtained using order weighting. The contribution of each order to the final CCF is weighted according to its S/N from CCF\({}_{\rm obs}\), as described in Section 4.2. Panel (b): H\({}_{2}\)O signal with a S/N of 5.1 obtained using order selection. Only the 13 best orders according to the S/N from CCF\({}_{\rm obs}\) are included in the final CCF. Panel (c): NH\({}_{3}\) signal with a S/N of 3.5 obtained using order selection. Only the 14 best orders according to the S/N from CCF\({}_{\rm obs}\) are included in the final CCF. Panel (d): HCN signal with a S/N of 3.3 obtained using order selection. Only the 17 best orders according to the S/N from CCF\({}_{\rm obs}\) are included in the final CCF.
We are unable to detect any other species in this dataset using the robust methodologies established in Sections 3.8 and 4.2. However, as in Section 5.2, we are able to obtain examples of non-robust detections for a number of species. For example, Figure 18 shows the recovery of an H\({}_{2}\)O signal in the atmosphere of WASP-76 b with a S/N of 5.3, via the order-wise optimisation of the S/N from CCF\({}_{\rm obs}\) in the detrending. It is possible that some of the previous detections mentioned above could be reproduced robustly using a wider exploration of model space, although as discussed this is not the main goal of our work.
## 6 Summary and Discussion
The primary goal of this work is to investigate the robustness of methods for making chemical detections using high-resolution transmission spectra of exoplanets in the NIR. The purpose of this is twofold: to prevent false or biased detections, and to encourage consistency in such analyses across datasets. Using CARMENES observations of HD 189733 b as a case study, we examine the robustness of different PCA-based detrending optimisations, and confirm that selecting the detrending parameters to maximise the S/N of a cross-correlation signal in the presence of noise has the potential to bias detection significances at the planetary velocity of optimisation. To do this, we show that selecting detrending parameters by optimising the S/N from the direct, or 'observed', CCF, CCF\({}_{\rm obs}\), can lead to detection significances which are inflated by residual noise. On the other hand, we find that optimising the S/N from the differential CCF, ACCF, as defined in Section 3.2, allows more robust detections. This appears true for both global and order-wise detrending. The robustness of optimising the S/N from the signal-injected CCF, CCF\({}_{\rm inj}\), lies in between these two extremes, depending on the strength with which the model is injected into the data. As well as residual noise in CCF\({}_{\rm inj}\) being able to influence the selection of detrending parameters, and hence produce biased detection significances, this method is difficult to reproduce since it depends strongly on the specific models used for injection.
We also consider the robustness of weighting each spectral order's contribution to the final CCF, as is done in some previous works. Using CCF\({}_{\rm obs}\), we again demonstrate that selecting or weighting orders according to the order-wise S/N of a cross-correlation signal in the presence of noise can bias detection significances at the planetary velocity of optimisation. However, we find that such order weighting is more robust when done according to the order-wise S/N from ACCF at the planetary velocity of optimisation. We additionally explore how parameter choices in the analysis can influence the reported detection significance. We find that the velocity range over which we calculate the noise in the total CCF can affect the final S/N by a considerable amount (Figure 10).
We confirm a detection of H\({}_{2}\)O in the atmosphere of HD 189733 b with a S/N of 6.1 (Figure 3b). We then conduct case studies of two further exoplanetary atmospheres, of HD 209458 b and WASP-76 b, and retrieve a signal for OH in the atmosphere of WASP-76 b with a S/N of 4.7 (Figure 17b). It should be reiterated that our goal is not an exhaustive exploration of model space aimed at detecting molecular species. Instead we are focused on assessing the relative robustness of molecular detections using different detrending methods for the same model template. This therefore hinders our prospects of retrieving signals for all the species that have been previously detected in our targets, and may explain why we have robustly recovered just two planetary signals across this work. It also follows that detections achieved here only via non-robust optimisations are not necessarily spurious.
Figure 17: S/N maps showing OH signals robustly found in the atmosphere of WASP-76 b. Panel (a): OH signal with a S/N of 4.1, found by global optimisation of the S/N from ACCF during detrending. This corresponds to the application of 18 PCA iterations. Panel (b): OH signal with a S/N of 4.7, found by optimising order-wise the S/N from \(\Delta\)CCF during detrending.
Figure 18: S/N map showing a non-robust H\({}_{2}\)O signal recovered in the atmosphere of WASP-76 b with a S/N of 5.3. The S/N from CCF\({}_{\rm obs}\) is optimised order-wise in the detrending.
Considerations in this work can be carried forward into future high-resolution spectroscopic surveys of exoplanetary atmospheres. Firstly, we have shown relative consistency in the erosion by PCA of planetary signals of different models (Figures 1 and 13), and in the detrending parameters found by optimising the S/N from \(\Delta\)CCF for different models and injection velocities (Figure 6). Therefore, it is unlikely that significant bias will be introduced due to the specific choice of injection velocity or atmospheric model, with similar detrending parameters to optimise the S/N from \(\Delta\)CCF likely derived independent of such choices. Secondly, as we aim to characterise the atmospheres of smaller planets, more transits of a single target will need to be observed. The demonstrated robustness of optimising order-wise the S/N from \(\Delta\)CCF could become very useful when considering observations from multiple nights; it appears robust to globally optimise the detrending of spectra from each transit individually, before combining the resultant CCF matrices. This could be important given the considerable differences in observing conditions across multiple nights, producing variable levels of telluric contamination and potentially very different optimum detrending parameters.
Other than maximising the S/N from \(\Delta\)CCF, there could be alternative methods by which to robustly optimise the number of PCA iterations during detrending. For example, one could determine the number of PCA iterations after which the residual correlated noise in CCF\({}_{\text{obs}}\), e.g. from telluric, stellar and instrumental effects, has been sufficiently removed. This could be beneficial in cases where the number of PCA iterations which optimises the S/N from \(\Delta\)CCF is small and coincides with there being remaining correlated noise in CCF\({}_{\text{obs}}\); it is possible that this is the case in Figure 9, where further detrending could potentially remove the spurious peak. Such an approach would likely require the introduction of a metric to measure the correlated noise. Alternatively, whilst this work discusses the robustness of different detrending optimisations, it would also be worthwhile to consider the efficiency of the signal extraction by considering the fraction of the planetary signal retrieved. The S/N from \(\Delta\)CCF stabilises around the optimum number of PCA iterations (Figure 2). Applying as few PCA iterations as possible to reach this stabilised S/N peak, rather than always selecting the maximum as we do here, could retain significantly more planetary information at the expense of just a small decrease in S/N. For example, whilst the S/N from \(\Delta\)CCF does not change much between 3 and 13 iterations in Figure 2, the planetary signal drops from around 75% of its initial value after 3 iterations to around 30% after 13 iterations (Figure 1). Adjusting for this could be key if we aim use the recovered planetary signal to infer more about the exoplanetary atmosphere's properties e.g. in retrievals (Brogi and Line, 2019). For robustness, a generalised method to take this into account would need to be formalised and employed homogeneously. Equally, an alternative detrending method to PCA which does not remove the planetary signal could allow for the retention of more planetary information.
Our findings motivate robust approaches for atmospheric characterisation of exoplanets using high-resolution transmission spectroscopy in the infrared. Robust and consistent methodologies would be beneficial to undertake homogeneous surveys of exoplanetary atmospheres at high spectral resolution. Inconsistencies in approaches used across different works can make it difficult to compare and contrast findings for different planets. Homogeneous surveys will allow us to place important constraints on the compositional diversity of exoplanetary atmospheres.
## Acknowledgements
We thank the anonymous referee for the very helpful comments. This work is supported by research grants to N.M. from the MERAC Foundation, Switzerland, and the UK Science and Technology Facilities Council (STFC) Center for Doctoral Training (CPT) in Data Intensive Science at the University of Cambridge. N.M., C.C. and M.H. acknowledge support from these sources towards the doctoral studies of C.C. and M.H. We thank the CAHA Archive for making the data available. This research has made use of NASA's Astrophysics Data System Service.
## Data Availability
This work is based on publicly available CARMENES2 observations.
Footnote 2: [http://caha.sdc.edu.inta-csic.es/calto/jsp/searchform.jsp](http://caha.sdc.edu.inta-csic.es/calto/jsp/searchform.jsp)
| 地上波高分解能伝射分光法は、移動する外惑星大気中の化学物質を検出するための魅力的な技術として注目されています。複数の外惑星での化学推測と過去の堅牢性研究にもかかわらず、伝射スペクトルからテイルリックと恒星の特徴を除去する堅牢で一貫したデトリニング方法が確立されていない。この研究では、高分解能伝射スペクトルにおけるPCAに基づくデトリニングの効率性を最適化する際に使用される指標の堅牢性を調査しています。近赤外線で外惑星に適用した近赤外線伝射スペクトルを対象としています。Case Studyとして、CARMENES分光器を用いてHD189733bを捉えた観測結果を調査しています。CARMENES分光器は3.5m CAHA望遠鏡に搭載されています。この研究では、ノイズの多い背景下でクロス correlogram の |
2306.04330 | A note on non-empty cross-intersecting families | The families $\mathcal F_1\subseteq \binom{[n]}{k_1},\mathcal F_2\subseteq
\binom{[n]}{k_2},\dots,\mathcal F_r\subseteq \binom{[n]}{k_r}$ are said to be
cross-intersecting if $|F_i\cap F_j|\geq 1$ for any $1\leq i<j\leq r$ and
$F_i\in \mathcal F_i$, $F_j\in\mathcal F_j$. Cross-intersecting families
$\mathcal F_1,\mathcal F_2,\dots,\mathcal F_r$ are said to be non-empty if
$\mathcal F_i\neq\emptyset$ for any $1\leq i\leq r$. This paper shows that if
$\mathcal F_1\subseteq\binom{[n]}{k_1},\mathcal
F_2\subseteq\binom{[n]}{k_2},\dots,\mathcal F_r\subseteq\binom{[n]}{k_r}$ are
non-empty cross-intersecting families with $k_1\geq k_2\geq\cdots\geq k_r$ and
$n\geq k_1+k_2$, then $\sum_{i=1}^{r}|\mathcal
F_i|\leq\max\{\binom{n}{k_1}-\binom{n-k_r}{k_1}+\sum_{i=2}^{r}\binom{n-k_r}{k_i-k_r},\
\sum_{i=1}^{r}\binom{n-1}{k_i-1}\}$. This solves a problem posed by Shi, Frankl
and Qian recently. The extremal families attaining the upper bounds are also
characterized. | Menglong Zhang, Tao Feng | 2023-06-07T10:50:15 | http://arxiv.org/abs/2306.04330v1 | # A note on non-empty cross-intersecting families
###### Abstract
The families \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n]}{ k_{2}},\dots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) are said to be cross-intersecting if \(|F_{i}\cap F_{j}|\geqslant 1\) for any \(1\leqslant i<j\leqslant r\) and \(F_{i}\in\mathcal{F}_{i}\), \(F_{j}\in\mathcal{F}_{j}\). Cross-intersecting families \(\mathcal{F}_{1},\mathcal{F}_{2},\dots,\mathcal{F}_{r}\) are said to be _non-empty_ if \(\mathcal{F}_{i}\neq\emptyset\) for any \(1\leqslant i\leqslant r\). This paper shows that if \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n] }{k_{2}},\dots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) are non-empty cross-intersecting families with \(k_{1}\geqslant k_{2}\geqslant\dots\geqslant k_{r}\) and \(n\geqslant k_{1}+k_{2}\), then \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\{\binom{n}{k_{1}}-\binom{n-k_{r }}{k_{1}}+\sum_{i=2}^{r}\binom{n-k_{r}}{k_{i}-k_{r}},\,\sum_{i=1}^{r}\binom{n-1 }{k_{i}-1}\}\). This solves a problem posed by Shi, Frankl and Qian recently. The extremal families attaining the upper bounds are also characterized.
+
Footnote †: Supported by NSFC under Grant 12271023
**Keywords**: intersecting family; non-empty cross-intersecting family
## 1 Introduction
Let \(n\) and \(k\) be integers with \(1\leqslant k\leqslant n\). Write \([n]=\{1,2,\dots,n\}\). Denote by \(2^{[n]}\) and \(\binom{[n]}{k}\) the power set and the family of all \(k\)-subsets of \([n]\), respectively. For \(1\leqslant k\leqslant n\), a family \(\mathcal{F}\subseteq 2^{[n]}\) is said to be _\(k\)-uniform_ if every member of \(\mathcal{F}\) contains exactly \(k\) elements, i.e., \(\mathcal{F}\subseteq\binom{[n]}{k}\). Write \(\overline{\mathcal{F}}=\{[n]\setminus F:F\in\mathcal{F}\}\) for \(\mathcal{F}\subseteq 2^{[n]}\). Two families \(\mathcal{F},\mathcal{F}^{\prime}\subseteq 2^{[n]}\) are said to be _isomorphic_ if there exists a permutation \(\pi\) on \([n]\) such that \(\{\{\pi(x):x\in F\}:F\in\mathcal{F}\}=\mathcal{F}^{\prime}\).
A family \(\mathcal{F}\subseteq\binom{[n]}{k}\) is said to be _intersecting_ if \(|F_{1}\cap F_{2}|\geqslant 1\) for any \(F_{1},F_{2}\in\mathcal{F}\). The celebrated Erdos-Ko-Rado theorem [3] determines the size and the structure of the largest intersecting uniform families.
**Theorem 1.1**.: [3] _If \(n\geqslant 2k\) and \(\mathcal{F}\subseteq\binom{[n]}{k}\) is an intersecting family, then \(|\mathcal{F}|\leqslant\binom{n-1}{k-1}\). Moreover, for \(n>2k\), the equality holds if and only if \(\mathcal{F}\) is isomorphic to \(\{F\in\binom{[n]}{k}:1\in F\}\)._
Cross-intersecting families are a variation of intersecting families. Let \(r\geqslant 2\) and \(n,k_{1},k_{2},\dots,\)\(k_{r}\) be positive integers. The families \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n] }{k_{2}},\dots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) are said to be _cross-intersecting_ if \(|F_{i}\cap F_{j}|\geqslant 1\) for any \(1\leqslant i<j\leqslant r\) and \(F_{i}\in\mathcal{F}_{i}\), \(F_{j}\in\mathcal{F}_{j}\). If \(\mathcal{F}\) is intersecting, then the families \(\mathcal{F}_{i}=\mathcal{F}\)\((i=1,2\dots,r)\) are cross-intersecting. Another trivial example of cross-intersecting families is \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}}\) and \(\mathcal{F}_{i}=\emptyset\) for every \(2\leqslant i\leqslant r\). Cross-intersecting
families \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\ldots,\mathcal{F}_{r}\subseteq \binom{[n]}{k_{r}}\) are called _maximal_ if \(\mathcal{F}_{1},\ldots,(\mathcal{F}_{i}\cup\{A\}),\ldots,\mathcal{F}_{r}\) are not cross-intersecting for any \(1\leqslant i\leqslant r\) and any \(A\in\binom{[n]}{k_{i}}\setminus\mathcal{F}_{i}\).
There are two natural ways to measure the largeness of cross-intersecting families: either by the sum \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) or by the product \(\prod_{i=1}^{r}|\mathcal{F}_{i}|\) (cf. [2, 14]) of their sizes. This paper only focuses on the measure of the sum \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\). Hilton [8] settled the problem of what cross-intersecting families with the maximum sum are.
**Theorem 1.2**.: [8] _Let \(n\) and \(k\) be positive integers with \(n\geqslant 2k\). If \(\mathcal{F}_{1},\mathcal{F}_{2},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k}\) are cross-intersecting families, then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\left\{\begin{array}{rl}\binom{n}{k},&\text{if }r\leqslant\frac{n}{k};\\ r\binom{n-1}{k-1},&\text{if }r\geqslant\frac{n}{k}.\end{array}\right.\]
_If the equality holds, then_
1. _when_ \(r<\frac{n}{k}\)_,_ \(\mathcal{F}_{1}=\binom{[n]}{k}\) _and_ \(\mathcal{F}_{2}=\cdots=\mathcal{F}_{r}=\emptyset\)_;_
2. _when_ \(r>\frac{n}{k}\)_,_ \(\mathcal{F}_{i}=\mathcal{F}\) _for every_ \(i\in[r]\)_, where_ \(\mathcal{F}\subseteq\binom{[n]}{k}\) _is an intersecting family with_ \(|\mathcal{F}|=\binom{n-1}{k-1}\)_;_
3. _when_ \(r=\frac{n}{k}\)_, if_ \(r=2\)_, then_ \(\mathcal{F}_{1}=\binom{[n]}{k}\setminus\overline{\mathcal{F}_{2}}\) _and_ \(\mathcal{F}_{2}\subseteq\binom{[n]}{k}\) _with_ \(0\leqslant|\mathcal{F}_{2}|\leqslant\binom{n}{k}\)_; if_ \(r>2\)_, then_ \(\mathcal{F}_{1},\mathcal{F}_{2},\ldots,\mathcal{F}_{r}\) _are as in_ \((\ref{eq:1})\) _or_ \((\ref{eq:2})\)_._
Brog [1] gave a short proof for Theorem 1.2 and Frankl [4] recently provided another simple proof by using Katona's circle method. Theorem 1.2 shows that one of extremal cross-intersecting families is \(\{\binom{[n]}{k},\emptyset,\ldots,\emptyset\}\). Cross-intersecting families \(\mathcal{F}_{1},\mathcal{F}_{2},\ldots,\mathcal{F}_{r}\) are said to be _non-empty_ if \(\mathcal{F}_{i}\neq\emptyset\) for any \(1\leqslant i\leqslant r\). It is quite natural to ask what the structure of the largest non-empty cross-intersecting families is. Hilton and Milner gave the following result.
**Theorem 1.3**.: [9] _Let \(n\) and \(k\) be positive integers with \(n\geqslant 2k\). If \(\mathcal{F}_{1}\subseteq\binom{[n]}{k}\) and \(\mathcal{F}_{2}\subseteq\binom{[n]}{k}\) are non-empty cross-intersecting, then \(|\mathcal{F}_{1}|+|\mathcal{F}_{2}|\leqslant\binom{n}{k}-\binom{n-k}{k}+1\)._
Frankl and Tokushige established the following stronger result by using the Kruskal-Katona Theorem.
**Theorem 1.4**.: [5] _Let \(n,k\) and \(l\) be positive integers with \(k\geqslant l\) and \(n\geqslant k+l\). If \(\mathcal{F}_{1}\subseteq\binom{[n]}{k}\) and \(\mathcal{F}_{2}\subseteq\binom{[n]}{l}\) are non-empty cross-intersecting, then_
1. \(|\mathcal{F}_{1}|+|\mathcal{F}_{2}|\leqslant\binom{n}{k}-\binom{n-l}{k}+1\)_;_
2. _if_ \(|\mathcal{F}_{2}|\geqslant\binom{n-1}{l-1}\)_, then_ \(|\mathcal{F}_{1}|+|\mathcal{F}_{2}|\leqslant\left\{\begin{array}{rl}\binom{n} {k}-\binom{n-k}{k}+1,&\text{if }k=l\geqslant 2;\\ \binom{n-1}{k-1}+\binom{n-1}{l-1},&\text{otherwise}.\end{array}\right.\)__
Shi, Frankl and Qian [15] gave another generalization of Theorem 1.3 by extending two families to arbitrary number of families.
**Theorem 1.5**.: [15, Theorem 1.5] _Let \(r\geqslant 2\) and \(n,k\) be positive integers with \(n\geqslant 2k\). If \(\mathcal{F}_{1},\mathcal{F}_{2},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k}\) are non-empty cross-intersecting families, then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\left\{\binom{n}{k}-\binom{n-k}{k} +r-1,\ r\binom{n-1}{k-1}\right\}\]
_with equality if and only if_
1. _if_ \(n>2k\)_, then either there exists_ \(x\in[n]\) _such that_ \(\mathcal{F}_{i}=\{F\in\binom{[n]}{k_{i}}:x\in F\}\) _for every_ \(i\in[r]\)_, or there exist_ \(i^{*}\in[r]\) _and_ \(S\in\binom{[n]}{k}\) _such that_ \(\mathcal{F}_{i^{*}}=\{F\in\binom{[n]}{k}:F\cap S\neq\emptyset\}\) _and_ \(\mathcal{F}_{i}=\{S\}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\)_;_
2. _if_ \(n=2k\)_, then_ 1. _when_ \(r=2\)_,_ \(\mathcal{F}_{1}\subseteq\binom{[n]}{k}\) _with_ \(0<|\mathcal{F}_{1}|<\binom{n}{k}\) _and_ \(\mathcal{F}_{2}=\binom{[n]}{k}\setminus\overline{\mathcal{F}_{1}}\)_;_ 2. _when_ \(r\geqslant 3\)_,_ \(\mathcal{F}_{i}=\mathcal{F}\) _for every_ \(i\in[r]\)_, where_ \(\mathcal{F}\subseteq\binom{[n]}{k}\) _is an intersecting family with_ \(|\mathcal{F}|=\binom{n-1}{k-1}\)_._
In this paper, we examine the structure of the largest non-empty cross-intersecting families \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{i}},\mathcal{F}_{2}\subseteq\binom{[n] }{k_{2}},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\), where \(k_{1},k_{2},\ldots,k_{r}\) are positive integers. We are to prove the following theorem.
**Theorem 1.6**.: _Let \(r\geqslant 2\) and \(n,k_{1},k_{2},\ldots,k_{r}\) be positive integers. Let \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n] }{k_{2}},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) be non-empty cross-intersecting families with \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i}-1}\) for some \(i^{*}\in[r]\). Let \(\bar{k}=\min\{k_{i}:i\in[r]\setminus\{i^{*}\}\}\). If \(n\geqslant k_{i}+k_{i^{*}}\) for every \(i\in[r]\setminus\{i^{*}\}\), then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\left\{\binom{n}{k_{i^{*}}}- \binom{n-\bar{k}}{k_{i^{*}}}+\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-\bar{k} }{k_{i}-\bar{k}},\ \sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\right\}\]
_with equality if and only if_
1. _if_ \(k_{i}=\bar{k}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\) _and_ \(n=\bar{k}+k_{i^{*}}\)_, then_ 1. _when_ \(r=2\)_,_ \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}_{3-i^ {*}}}\) _and_ \(1\leqslant|\mathcal{F}_{3-i^{*}}|\leqslant\binom{n-1}{k_{-1}}\)_;_ 2. _when_ \(r>2\) _and_ \(n>2\bar{k}\)_, there exists_ \(x\in[n]\) _such that_ \(\mathcal{F}_{i^{*}}=\{F\in\binom{[n]}{k_{i^{*}}}:x\in F\}\) _and_ \(\mathcal{F}_{i}=\{F\in\binom{[n]}{k}:x\in F\}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\)_;_ 3. _when_ \(r>2\) _and_ \(n\leqslant 2\bar{k}\)_,_ \(\mathcal{F}_{i}=\mathcal{F}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\) _and_ \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}}\)_, where_ \(\mathcal{F}\subseteq\binom{[n]}{\bar{k}}\) _is an intersecting family with_ \(|\mathcal{F}|=\binom{n-1}{k-1}\)_;_
2. _if_ \(n>\bar{k}+k_{i^{*}}\)_, then there exists_ \(S\in\binom{[n]}{s}\) _with_ \(s=1\) _or_ \(\bar{k}\) _such that_ \(\mathcal{F}_{i^{*}}=\{F\in\binom{[n]}{k_{i^{*}}}:S\cap F\neq\emptyset\}\) _and_ \(\mathcal{F}_{i}=\{F\in\binom{[n]}{k_{i}}:S\subseteq F\}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\)_._
Applying Theorem 1.6, we can give a solution to Problems 4.3 in [15] as follows.
**Theorem 1.7**.: _Let \(r\geqslant 2\) and \(n,k_{1},k_{2},\ldots,k_{r}\) be positive integers. Let \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n] }{k_{2}},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) be non-empty cross-intersecting families with \(k_{1}\geqslant k_{2}\geqslant\cdots\geqslant k_{r}\) and \(n\geqslant k_{1}+k_{2}\). Then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\left\{\binom{n}{k_{1}}-\binom{n- k_{r}}{k_{1}}+\sum_{i=2}^{r}\binom{n-k_{r}}{k_{i}-k_{r}},\ \sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\right\}\]
_with equality if and only if_
1. _if_ \(n=k_{1}+k_{r}\)_, then_ 1. _when_ \(r=2\)_,_ \(\mathcal{F}_{1}=\binom{[n]}{k_{1}}\setminus\overline{\mathcal{F}_{2}}\) _and_ \(0<|\mathcal{F}_{2}|<\binom{n}{k_{2}}\)_;_ 2. _when_ \(r>2\) _and_ \(k_{1}>k_{2}=\cdots=k_{r}\)_, there exists_ \(x\in[n]\) _such that_ \(\mathcal{F}_{i}=\{F\in\binom{[n]}{k_{i}}:x\in F\}\) _for every_ \(i\in[r]\)_;_ 3. _when_ \(r>2\) _and_ \(k_{1}=k_{2}=\cdots=k_{r}\)_,_ \(\mathcal{F}_{i}=\mathcal{F}\) _for every_ \(i\in[r]\)_, where_ \(\mathcal{F}\subseteq\binom{[n]}{k_{1}}\) _is an intersecting family with_ \(|\mathcal{F}|=\binom{n-1}{k_{1}-1}\)_;_
_._
2. _if_ \(n>k_{1}+k_{r}\)_, then either there exists_ \(x\in[n]\) _such that_ \(\mathcal{F}_{i}=\{F\in{[n]\choose k_{i}}:x\in F\}\) _for every_ \(i\in[r]\)_, or there exists_ \(S\in{[n]\choose k_{r}}\) _such that_ \(\mathcal{F}_{j}=\{F\in{[n]\choose k_{j}}:F\cap S\neq\emptyset\}\) _for some_ \(j\in[r]\) _with_ \(k_{j}=k_{1}\) _and_ \(\mathcal{F}_{i}=\{F\in{[n]\choose k_{i}}:S\subseteq F\}\) _for every_ \(i\in[r]\setminus\{j\}\)_._
## 2 Preliminaries
Let \(\prec_{L}\), or \(\prec\) for short, be the lexicographic order on \({[n]\choose i}\) where \(i\in\{1,2,...,n\}\), that is, for any two sets \(A,B\in{[n]\choose i}\), \(A\prec B\) if and only if \(\min\{a:a\in A\setminus B\}<\min\{b:b\in B\setminus A\}\). For a family \(\mathcal{A}\in{[n]\choose k}\), let \(\mathcal{A}_{L}\) denote the family consisting of the first \(|\mathcal{A}|\)\(k\)-sets in order \(\prec\) in \({[n]\choose k}\), and call \(\mathcal{A}\)_\(L\)-initial_ if \(\mathcal{A}_{L}=\mathcal{A}\). A powerful tool in the study of cross-intersecting families is the Kruskal-Katona Theorem ([10, 11]), especially its reformulation due to Hilton [7].
**Theorem 2.1**.: [7] _If \(\mathcal{A}\subseteq{[n]\choose k}\) and \(\mathcal{B}\subseteq{[n]\choose l}\) are cross-intersecting, then \(\mathcal{A}_{L}\) and \(\mathcal{B}_{L}\) are cross-intersecting as well._
For positive integers \(k,l\) and a set \(S\subseteq[n]\), let
\[\mathcal{P}_{S}^{(l)}=\left\{P\in{[n]\choose l}:S\subseteq P\right\}\text{ and }\mathcal{R}_{S}^{(k)}=\left\{R\in{[n]\choose k}:|R\cap S|\geqslant 1 \right\}.\]
Then \(|\mathcal{P}_{S}^{(l)}|={n-|S|\choose l-|S|}\) and \(|\mathcal{R}_{S}^{(k)}|={n\choose k}-{n-|S|\choose k}\). Clearly \(\mathcal{P}_{S}^{(l)}\) and \(\mathcal{R}_{S}^{(k)}\) are cross-intersecting. For a positive integer \(s\), write \(\mathcal{P}_{s}^{(l)}=\mathcal{P}_{[s]}^{(l)}\) and \(\mathcal{R}_{s}^{(k)}=\mathcal{R}_{[s]}^{(k)}\). Clearly \(\mathcal{P}_{s}^{(l)}\) and \(\mathcal{R}_{s}^{(k)}\) are both \(L\)-initial. Note that \(\mathcal{P}_{s}^{(l)}\subseteq\mathcal{P}_{s-1}^{(l)}\) for any \(2\leqslant s\leqslant l\) and \(\mathcal{R}_{s}^{(k)}\subseteq\mathcal{R}_{s+1}^{(k)}\) for any \(s\geqslant 1\).
The following lemma is a slight generalization of [15, Lemma 2.1].
**Lemma 2.2**.: _Let \(n,k\) and \(l\) be positive integers with \(n\geqslant k+l\). For any \(S\subseteq[n]\) with \(1\leqslant|S|\leqslant l\), \(\mathcal{R}_{S}^{(k)}\)\((\)resp. \(\mathcal{P}_{S}^{(l)})\) is the largest family in \({[n]\choose k}\)\((\)resp. \({[n]\choose l})\) that is cross-intersecting with \(\mathcal{P}_{S}^{(l)}\)\((\)resp. \(\mathcal{R}_{S}^{(k)})\)._
**Proof.** To prove that \(\mathcal{R}_{S}^{(k)}\) is the largest family in \({[n]\choose k}\) that is cross-intersecting with \(\mathcal{P}_{S}^{(l)}\), take \(A\in{[n]\choose k}\setminus\mathcal{R}_{S}^{(k)}\). It suffices to show that there exists \(C\in\mathcal{P}_{S}^{(l)}\) such that \(A\cap C=\emptyset\). Since \(A\cap S=\emptyset\), we have \(|[n]\setminus(A\cup S)|=n-k-|S|\geqslant l-|S|\), and so there exists an \((l-|S|)\)-set \(B\subseteq[n]\setminus(A\cup S)\). Take \(C=S\cup B\). Then \(C\in\mathcal{P}_{S}^{(l)}\) and \(A\cap C=\emptyset\).
Similarly, to prove that \(\mathcal{P}_{S}^{(l)}\) is the largest family in \({[n]\choose l}\) that is cross-intersecting with \(\mathcal{R}_{S}^{(k)}\), take \(A\in{[n]\choose l}\setminus\mathcal{P}_{S}^{(l)}\). It suffices to show that there exists \(C\in\mathcal{R}_{S}^{(k)}\) such that \(A\cap C=\emptyset\). Write \(|A\cap S|=y\leqslant|S|-1\). If \(k\leqslant|S|-y=|S\setminus A|\), then take \(C\in{S\setminus A\choose k}\), and so \(C\in\mathcal{R}_{S}^{(k)}\) and \(A\cap C=\emptyset\). If \(k>|S|-y\), since \(|[n]\setminus(A\cup S)|=n-l-|S|+y\geqslant k-|S|+y>0\), there exists a \((k-|S|+y)\)-set \(B\subseteq[n]\setminus(A\cup S)\). Take \(C=(S\setminus A)\cup B\). Then \(C\in\mathcal{R}_{S}^{(k)}\) and \(A\cap C=\emptyset\).
**Lemma 2.3**.: _Let \(r\geqslant 2\) and \(k_{1},k_{2},\ldots,k_{r}\) be positive integers such that \(k_{r}=\min\{k_{i}:2\leqslant i\leqslant r\}\). Write_
\[g(s):={n\choose k_{1}}-{n-s\choose k_{1}}+\sum_{i=2}^{r}{n-s\choose k_{i}-s}.\]
_If \(n\geqslant k_{1}+k_{i}\) for every \(i\in[2,r-1]\) and \(n>k_{1}+k_{r}\), then \(\max\{g(s):1\leqslant s\leqslant k_{r}\}\) is either \(g(1)\) or \(g(k_{r})\)._
**Proof.** When \(k_{r}\in\{1,2\}\), the conclusion is straightforward. Assume that \(k_{r}\geqslant 3\). We claim that there does not exist \(s\in[2,k_{r}-1]\) such that \(g(s)\geqslant g(s+1)\) and \(g(s)\geqslant g(s-1)\). Otherwise, if there exists \(s\in[2,k_{r}-1]\) such that
\[\binom{n}{k_{1}}-\binom{n-s}{k_{1}}+\sum_{i=2}^{r}\binom{n-s}{k_{i}-s} \geqslant\binom{n}{k_{1}}-\binom{n-s-1}{k_{1}}+\sum_{i=2}^{r}\binom{n-s-1}{k_{ i}-s-1}\text{ and }\]
\[\binom{n}{k_{1}}-\binom{n-s}{k_{1}}+\sum_{i=2}^{r}\binom{n-s}{k_{i}-s} \geqslant\binom{n}{k_{1}}-\binom{n-s+1}{k_{1}}+\sum_{i=2}^{r}\binom{n-s+1}{k_ {i}-s+1},\]
which implies
\[\sum_{i=2}^{r}\binom{n-s-1}{k_{i}-s} \geqslant\binom{n-s-1}{k_{1}-1}\text{ and } \tag{2.1}\] \[\binom{n-s}{k_{1}-1} \geqslant\sum_{i=2}^{r}\binom{n-s}{k_{i}-s+1}. \tag{2.2}\]
\((\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eqeq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eqeq:eq:eq:eq:eq:eq:eq:eq:eqeq:eqeq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eqeq:eqeq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eqeq:eqeq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eqeq:eq:eqeq:eq:eq:eqeq:eq:eqeq:eqeq:eq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eqeq:eq:eqeq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eqeq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eq:eq:eqeq:eq:eqeq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eqeq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq
_with equality if and only if \(r=2\) and \(n=k_{1}+k_{2}\)._
**Proof.** If \(r=2\), then \(\bar{k}=k_{1}\) and \(\binom{n-1}{k_{1}-1}+\binom{n-1}{k_{2}-1}-\binom{n}{k_{2}}-\binom{n-k_{1}}{k_{2} }+1)=\binom{n-1}{k_{1}-1}-\binom{n-1}{k_{2}}+\binom{n-k_{1}}{k_{2}}-1\). Since \(n\geqslant k_{1}+k_{2}\) and \(k_{1}>k_{2}\), \(\binom{n-1}{k_{1}-1}\geqslant\binom{n-1}{k_{2}}\) with equality if and only if \(k_{1}=k_{2}+1\) or \(n=k_{1}+k_{2}\). Since \(n\geqslant k_{1}+k_{2}\) and \(k_{2}>0\), \(\binom{n-k_{1}}{k_{2}}-1\geqslant 0\) with equality if and only if \(n=k_{1}+k_{2}\). Then the desired conclusion is obtained.
Assume that \(r\geqslant 3\). We have
\[\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}-\left(\binom{n}{k_{2}}-\binom{n -\bar{k}}{k_{2}}+\binom{n-\bar{k}}{k_{1}-\bar{k}}+\sum_{i=3}^{r}\binom{n-\bar{ k}}{k_{i}-\bar{k}}\right)\] \[=\binom{n-1}{k_{1}-1}-\binom{n-\bar{k}}{k_{1}-\bar{k}}-\binom{n-1 }{k_{2}}+\binom{n-\bar{k}}{k_{2}}+\sum_{i=3}^{r}\left(\binom{n-1}{k_{i}-1}- \binom{n-\bar{k}}{k_{i}-\bar{k}}\right)\] \[=\binom{n-1}{k_{1}-1}-\binom{n-\bar{k}}{k_{1}-\bar{k}}-\binom{n-1 }{k_{2}}+\binom{n-\bar{k}}{k_{2}}+\sum_{i=3}^{r}\sum_{j=0}^{\bar{k}-2}\binom{n -j-2}{k_{i}-1-j},\]
where the last equality follows from Lemma 2.4 (2). If \(k_{2}\geqslant k_{1}-\bar{k}\), since \(n\geqslant k_{1}+k_{2}\) and \(k_{1}>k_{2}\), we have \(\binom{n-\bar{k}}{k_{2}}\geqslant\binom{n-\bar{k}}{k_{1}-\bar{k}}\) and \(\binom{n-1}{k_{1}-1}\geqslant\binom{n-1}{k_{2}}\), which yield the desired inequality. If \(k_{2}<k_{1}-\bar{k}\), it follows from Lemma 2.4 that
\[\binom{n-1}{k_{1}-1}-\binom{n-\bar{k}}{k_{1}-\bar{k}}-\binom{n-1}{k_{2}}+ \binom{n-\bar{k}}{k_{2}}=\sum_{i=1}^{\bar{k}-1}\left(\binom{n-i-1}{k_{1}-i}- \binom{n-i-1}{k_{2}-1}\right).\]
Since \(k_{2}<k_{1}-\bar{k}\), \(k_{2}-1<k_{1}-i\) for every \(1\leqslant i\leqslant\bar{k}-1\), and so \(\binom{n-i-1}{k_{1}-i}\geqslant\binom{n-i-1}{k_{2}-1}\). This completes the proof.
## 3 Proof of Theorem 1.6
For non-empty cross-intersecting families \(\mathcal{F}_{i}\subseteq\binom{[n]}{k_{i}}\), \(i\in[r]\), to determine the largest possible value of \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\), by Theorem 2.1, without loss of generality, one can assume that \(\mathcal{F}_{i}\subseteq\binom{[n]}{k_{i}}\), \(i\in[r]\), are all \(L\)-initial.
**Lemma 3.1**.: _Let \(r\geqslant 2\) and \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n ]}{k_{2}},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) be non-empty \(L\)-initial cross-intersecting families with \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i^{*}}-1}\) for some \(i^{*}\in[r]\). Let \(\bar{k}=\min\{k_{i}:i\in[r]\setminus\{i^{*}\}\}\). If \(n>\bar{k}+k_{i^{*}}\) and \(n\geqslant k_{i}+k_{i^{*}}\) for every \(i\in[r]\setminus\{i^{*}\}\), then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\left\{\binom{n}{k_{i^{*}}}- \binom{n-\bar{k}}{k_{i^{*}}}+\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-\bar{k}} {k_{i}-\bar{k}},\ \sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\right\} \tag{3.1}\]
_with equality if and only if \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for \(i\in[r]\setminus\{i^{*}\}\), where \(s=1\) or \(\bar{k}\)._
**Proof.** Without loss of generality, suppose that \(\mathcal{F}_{1},\mathcal{F}_{2},\ldots,\mathcal{F}_{r}\) are maximal cross-intersecting families.
If \(k_{i^{*}}=1\), write \(|\mathcal{F}_{i^{*}}|=s\geqslant 1\). Since \(\mathcal{F}_{i^{*}}\) is \(L\)-initial, \(\mathcal{F}_{i^{*}}=\{\{1\},\ldots,\{s\}\}=\mathcal{R}_{s}^{(1)}.\) For any \(i\in[r]\setminus\{i^{*}\}\), \(\mathcal{F}_{i}\) is cross-intersecting with \(\mathcal{F}_{i^{*}}\), so each element of \(\mathcal{F}_{i}\) contains \([s]\), i.e., \(\mathcal{F}_{i}\subseteq\mathcal{P}_{s}^{(k_{i})}\). It follows from the maximality of \(\mathcal{F}_{i}\) that \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for every \(i\neq i^{*}\). Note that \(s\leqslant k_{i}\) for
any \(i\neq i^{*}\) and so \(s\leqslant\bar{k}\). Therefore, there is \(1\leqslant s\leqslant\bar{k}\) such that \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s}^{(k_{i*})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for all \(i\in[r]\setminus\{i^{*}\}\). Apply Lemma 2.3 to obtain the upper bound (3.1) for \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) and the structure of extremal families.
Assume that \(k_{i^{*}}\geqslant 2\). Since \(\mathcal{F}_{i}\) is \(L\)-initial and nonempty for \(i\in[r]\setminus\{i^{*}\}\), we have \([k_{i}]\in\mathcal{F}_{i}\), and so each element of \(\mathcal{F}_{i^{*}}\) intersects with \([k_{i}]\) and \(\mathcal{F}_{i^{*}}\subseteq\mathcal{R}_{k_{i}}^{(k_{i^{*}})}\) for every \(i\neq i^{*}\). Since \(\bar{k}=\min\{k_{i}:i\in[r]\setminus\{i^{*}\}\}\), \(\mathcal{F}_{i^{*}}\subseteq\mathcal{R}_{\bar{k}}^{(k_{i^{*}})}\). By the assumption of \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i^{*}}-1}\), \(\mathcal{R}_{1}^{(k_{i^{*}})}\subseteq\mathcal{F}_{i^{*}}\). So \(\mathcal{R}_{1}^{(k_{i^{*}})}\subseteq\mathcal{F}_{i^{*}}\subseteq\mathcal{R} _{\bar{k}}^{(k_{i^{*}})}\).
If \(\bar{k}=1\), then \(\mathcal{F}_{i^{*}}=\mathcal{R}_{1}^{(k_{i^{*}})}.\) It follows from Lemma 2.2 that \(\mathcal{F}_{i}\subseteq\mathcal{P}_{1}^{(k_{i})}\) for \(i\neq i^{*}\). By the maximality of \(\mathcal{F}_{i}\), \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for all \(i\in[r]\setminus\{i^{*}\}\), and consequently, \(\sum_{i=1}^{r}|\mathcal{F}_{i}|=\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\).
Assume that \(\bar{k}\geqslant 2\). Define \(s\) to be the largest integer such that \(2\leqslant s+1\leqslant\bar{k}\) and
\[\mathcal{R}_{s}^{(k_{i^{*}})}\subseteq\mathcal{F}_{i^{*}}\subseteq\mathcal{R }_{s+1}^{(k_{i^{*}})}.\]
\(\mathcal{R}_{s}^{(k_{i^{*}})}\subseteq\mathcal{F}_{i^{*}}\) implies that \(\mathcal{F}_{i}\) for \(i\neq i^{*}\) is cross-intersecting with \(\mathcal{R}_{s}^{(k_{i^{*}})}\). It follows from Lemma 2.2 that \(\mathcal{F}_{i}\subseteq\mathcal{P}_{s}^{(k_{i})}\) for \(i\neq i^{*}\). \(\mathcal{F}_{i^{*}}\subseteq\mathcal{R}_{s+1}^{(k_{i^{*}})}\) implies that \(\mathcal{P}_{s+1}^{(k_{i})}\) is cross-intersecting with \(\mathcal{F}_{i^{*}}\) for any \(i\neq i^{*}\). Since \(\mathcal{P}_{s+1}^{(k_{i})}\) is cross-intersecting with \(\mathcal{F}_{i_{2}}\subseteq\mathcal{P}_{s}^{(k_{i_{2}})}\) for any \(i_{1}\neq i_{2}\) and \(i_{1},i_{2}\in[r]\setminus\{i^{*}\}\), \(\mathcal{P}_{s+1}^{(k_{i})}\subseteq\mathcal{F}_{i}\) for \(i\in[r]\setminus\{i^{*}\}\) by the maximality of \(\mathcal{F}_{i}\). Therefore, for any \(i\in[r]\setminus\{i^{*}\}\),
\[\mathcal{P}_{s+1}^{(k_{i})}\subseteq\mathcal{F}_{i}\subseteq\mathcal{P}_{s}^{ (k_{i})}.\]
Let \(\mathcal{G}_{i^{*}}=\mathcal{F}_{i^{*}}\setminus\mathcal{R}_{s}^{(k_{i^{*}})}\) and \(\mathcal{G}_{i}=\mathcal{F}_{i}\setminus\mathcal{P}_{s+1}^{(k_{i})}\) for \(i\in[r]\setminus\{i^{*}\}\). Then
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|=|\mathcal{R}_{s}^{(k_{i^{*}})}|+\sum_{i\in[r] \setminus\{i^{*}\}}|\mathcal{P}_{s+1}^{(k_{i})}|+\sum_{i=1}^{r}|\mathcal{G}_{ i}|.\]
Our goal is to maximize \(\sum_{i=1}^{r}|\mathcal{G}_{i}|\).
Since \(\mathcal{G}_{i^{*}}\subseteq\mathcal{R}_{s+1}^{(k_{i^{*}})}\setminus\mathcal{R }_{s}^{(k_{i^{*}})}=\{R\in\binom{[n]}{k_{i^{*}}}:R\cap[s+1]=\{s+1\}\}=\{T\cup\{s+ 1\}:T\in\binom{[s+2,n]}{k_{i^{*}}-1}\}\) and for \(i\in[r]\setminus\{i^{*}\}\), \(\mathcal{G}_{i}\subseteq\mathcal{P}_{s}^{(k_{i})}\setminus\mathcal{P}_{s+1}^{( k_{i})}=\{P\in\binom{[n]}{k_{i}}:P\cap[s+1]=[s]\}=\{T\cup[s]:T\in \binom{[s+2,n]}{k_{i^{*}}-8}\}\), we construct an auxiliary \(r\)-partite graph \(G=(\mathcal{X}_{1},\mathcal{X}_{2},\ldots,\mathcal{X}_{r},E(G))\) where \(\mathcal{X}_{i^{*}}=\binom{[s+2,n]}{k_{i^{*}}-1}\) and \(\mathcal{X}_{i}=\binom{[s+2,n]}{k_{i}-s}\) for \(i\in[r]\setminus\{i^{*}\}\) such that for \(X_{i^{*}}\in\mathcal{X}_{i^{*}}\) and \(X_{i}\in\mathcal{X}_{i}\) with \(i\neq i^{*}\), \(X_{i^{*}}X_{i}\) is an edge if and only if \(X_{i^{*}}\cap X_{i}=\emptyset\), and there is no edge between \(\mathcal{X}_{i_{1}}\) and \(\mathcal{X}_{i_{2}}\) for \(i_{1},i_{2}\in[r]\setminus\{i^{*}\}\). Clearly \(G\) can be also regarded as a bipartite graph with parts \(\mathcal{X}_{i^{*}}\) and \(VG(\mathcal{)}\setminus\mathcal{X}_{i^{*}}\). Note that the vertex set of \(G\) is a slight abuse of notation, since it is possible that \(\mathcal{X}_{i_{1}}=\mathcal{X}_{i_{2}}\) for some \(1\leqslant i_{1}<i_{2}\leqslant r\). Let \(\mathcal{I}_{i^{*}}=\{R\setminus\{s+1\}:R\in\mathcal{G}_{i^{*}}\}\) and \(\mathcal{I}_{i}=\{P\setminus[s]:P\in\mathcal{G}_{i}\}\) for \(i\in[r]\setminus\{i^{*}\}\). Then \(\mathcal{I}_{i}\subseteq\mathcal{X}_{i}\) for \(i\in[r]\) and \(\mathcal{I}=\bigcup_{i=1}^{r}\mathcal{I}_{i}\) is an independent set of \(G\). On the other hand, if \(\mathcal{I}^{\prime}\) is an independent set of \(G\) and \(\mathcal{I}^{\prime}_{i}=\mathcal{I}^{\prime}\cap\mathcal{X}_{i}\) for \(i\in[r]\), then \(\mathcal{G}^{\prime}_{1},\ldots,\mathcal{G}^{\prime}_{r}\) are cross-intersecting, where \(\mathcal{G}^{\prime}_{i^{*}}=\{\{s+1\}\cup X_{i^{*}}:X_{i^{*}}\in\mathcal{I}^{ \prime}_{i}\}\) and \(\mathcal{G}^{\prime}_{i}=\{[s]\cup X_{i}:X_{i}\in\mathcal{I}^{\prime}_{i}\}\) for \(i\in[r]\setminus\{i^{*}\}\). Therefore, to maximize \(\sum_{i=1}^{r}|\mathcal{G}_{i}|\), it suffices to examine the largest independent set of \(G\).
**Claim 1**.: _For any \(\mathcal{Q}\subseteq\mathcal{X}_{i^{*}}\) and any \(i\in[r]\setminus\{i^{*}\}\), \(|N_{\mathcal{X}_{i}}(\mathcal{Q})|\geqslant\frac{|\mathcal{X}_{i}||\mathcal{Q} |}{|\mathcal{X}_{i^{*}}|}\), where \(N_{\mathcal{X}_{i}}(\mathcal{Q})=\{X_{i}\in\mathcal{X}_{i}:\) there is \(X_{i^{*}}\in\mathcal{Q}\) such that \(X_{i^{*}}X_{i}\) is an edge\(\}\). Moreover, if \(n>k_{i}+k_{i^{*}}\), then the equality holds if and only if \(\mathcal{Q}=\emptyset\) or \(\mathcal{Q}=\mathcal{X}_{i^{*}}\)._
\(d_{i^{*}}|\mathcal{X}_{i^{*}}|=d_{i}|\mathcal{X}_{i}|\), where \(e(G[\mathcal{X}_{i^{*}},\mathcal{X}_{i}])\) is the number of edges in \(G[\mathcal{X}_{i^{*}},\mathcal{X}_{i}]\). On the other hand, \(d_{i^{*}}|\mathcal{Q}|\leqslant d_{i}|N_{\mathcal{X}_{i}}(\mathcal{Q})|\). So \(|N_{\mathcal{X}_{i}}(\mathcal{Q})|\geqslant\frac{|\mathcal{X}_{i}||\mathcal{ Q}|}{|\mathcal{X}_{i^{*}}|}\). When \(n>k_{i}+k_{i^{*}}\), the induced bipartite subgraph \(G[\mathcal{X}_{i^{*}},\mathcal{X}_{i}]\) is connected. Hence, the equality holds if and only if \(\mathcal{Q}=\emptyset\) or \(\mathcal{Q}=\mathcal{X}_{i^{*}}\).
Since \(\mathcal{I}_{i}\cap N_{\mathcal{X}_{i}}(\mathcal{I}_{i^{*}})=\emptyset\) for \(i\neq i^{*}\), applying Claim 1, we have
\[\frac{|\mathcal{I}_{i^{*}}|\binom{n-s-1}{k_{i-s}}}{\binom{n-s-1}{k_{i^{*}}-1}}+ |\mathcal{I}_{i}|\leqslant|N_{\mathcal{X}_{i}}(\mathcal{I}_{i^{*}})|+| \mathcal{I}_{i}|\leqslant|\mathcal{X}_{i}|=\binom{n-s-1}{k_{i}-s}.\]
Hence, for \(i\neq i^{*}\),
\[|\mathcal{I}_{i}|\leqslant\binom{n-s-1}{k_{i}-s}\left(1-\frac{|\mathcal{I}_{i^ {*}}|}{\binom{n-s-1}{k_{i^{*}}-1}}\right),\]
and when \(n>k_{i}+k_{i^{*}}\), the equality holds if and only if \(\mathcal{I}_{i^{*}}=\emptyset\) or \(\mathcal{X}_{i^{*}}\).
Let \(H_{1}=\{i\in[r]:n>k_{i}+k_{i^{*}},i\neq i^{*}\}\) and \(H_{2}=\{i\in[r]:n=k_{i}+k_{i^{*}},i\neq i^{*}\}\). Note that \(H_{1}\neq\emptyset\) because of \(n>\bar{k}+k_{i^{*}}\).
**Case 1**\(H_{2}=\emptyset\)**.** We claim that \(\mathcal{I}_{i^{*}}\) is either \(\emptyset\) or \(\mathcal{X}_{i^{*}}\) when \(\mathcal{I}\) is the largest independent set of \(G\). If \(\mathcal{I}_{i^{*}}=\emptyset\), then \(\mathcal{I}_{i}=\mathcal{X}_{i}\) for \(i\in[r]\setminus\{i^{*}\}\) and so
\[|\mathcal{I}|=\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-s-1}{k_{i}-s}.\]
If \(\mathcal{I}_{i^{*}}=\mathcal{X}_{i^{*}}\), then \(\mathcal{I}_{i}=\emptyset\) for \(i\in[r]\setminus\{i^{*}\}\) and so
\[|\mathcal{I}|=\binom{n-s-1}{k_{i^{*}}-1}.\]
If \(\mathcal{I}_{i^{*}}\neq\emptyset\) and \(\mathcal{I}_{i^{*}}\neq\mathcal{X}_{i^{*}}\), then
\[|\mathcal{I}|=\sum_{i=1}^{r}|\mathcal{I}_{i}|<|\mathcal{I}_{i^{*}}|+\sum_{i\in [r]\setminus\{i^{*}\}}\binom{n-s-1}{k_{i}-s}\left(1-\frac{|\mathcal{I}_{i^{*}} |}{\binom{n-s-1}{k_{i^{*}}-1}}\right).\]
If \(\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-s-1}{k_{i}-s}\leqslant\binom{n-s-1}{ k_{i^{*}}-1}\), then \(|\mathcal{I}|<\binom{n-s-1}{k_{i^{*}}-1}\). If \(\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-s-1}{k_{i}-s}>\binom{n-s-1}{k_{i^{*}} -1}\), then
\[|\mathcal{I}|<\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-s-1}{k_{i}-s}+| \mathcal{I}_{i^{*}}|\left(1-\frac{\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-s-1} {k_{i}-s}}{\binom{n-s-1}{k_{i}-s}}\right)<\sum_{i\in[r]\setminus\{i^{*}\}} \binom{n-s-1}{k_{i}-s}.\]
To sum up, if \(\mathcal{I}_{i^{*}}\neq\emptyset\) and \(\mathcal{I}_{i^{*}}\neq\mathcal{X}_{i^{*}}\), then
\[|\mathcal{I}|<\max\left\{\binom{n-s-1}{k_{i^{*}}-1},\sum_{i\in[r]\setminus\{i^ {*}\}}\binom{n-s-1}{k_{i}-s}}.\]
Therefore, if \(\mathcal{I}\) is the largest independent set of \(G\), then \(\mathcal{I}_{i^{*}}=\emptyset\) or \(\mathcal{X}_{i^{*}}\), and so \(\mathcal{I}=V(G)\setminus\mathcal{X}_{i^{*}}\) or \(\mathcal{X}_{i^{*}}\), which yields that either \(\mathcal{G}_{i^{*}}=\emptyset\) and \(\mathcal{G}_{i}=\{[s]\cup X_{i}:X_{i}\in\binom{[s+2,n]}{k_{i}-s}\}\) for all \(i\neq i^{*}\), or \(\mathcal{G}_{i^{*}}=\{\{s+1\}\cup X_{i^{*}}:X_{i^{*}}\in\binom{[s+2,n]}{k_{i}-1}\}\) and \(\mathcal{G}_{i}=\emptyset\) for all \(i\neq i^{*}\). That is, either \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for all \(i\in[r]\setminus\{i^{*}\}\), or \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s+1}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s+1}^{(k_{i})}\) for all \(i\in[r]\setminus\{i^{*}\}\). Hence, there is \(1\leqslant s\leqslant\bar{k}\) such that \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for all \(i\in[r]\setminus\{i^{*}\}\). By Lemma 2.3,
we obtain the upper bound (3.1) for \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) and the structure of extremal families.
**Case 2**\(H_{2}\neq\emptyset\)**.** For any \(i\in H_{2}\), the induced bipartite subgraph \(G[\mathcal{X}_{i},\mathcal{X}_{i^{*}}]\) is a perfect matching, and so \(|\mathcal{I}_{i}|+|\mathcal{I}_{i^{*}}|=|\mathcal{I}_{i}|+|N_{\mathcal{X}_{i}} (\mathcal{I}_{i^{*}})|\leqslant|\mathcal{X}_{i}|=\binom{n-s-1}{k_{i}-s}\). Thus
\[|\mathcal{I}|=\sum_{i=1}^{r}|\mathcal{I}_{i}| \leqslant\sum_{i\in H_{1}}\binom{n-s-1}{k_{i}-s}\left(1-\frac{| \mathcal{I}_{i^{*}}|}{\binom{n-s-1}{k_{i^{*}}-1}}\right)+\sum_{i\in H_{2}}| \mathcal{I}_{i}|+|\mathcal{I}_{i^{*}}|\] \[\leqslant\sum_{i\in H_{1}}\binom{n-s-1}{k_{i}-s}+\sum_{i\in H_{2} }\binom{n-s-1}{k_{i}-s}=\sum_{i\in|r|\setminus\{i^{*}\}}\binom{n-s-1}{k_{i}-s },\]
with equality if and only if \(\mathcal{I}_{i^{*}}=\emptyset\) and \(\mathcal{I}_{i}=\mathcal{X}_{i}\) for \(i\in[r]\setminus\{i^{*}\}\). Then \(\mathcal{G}_{i^{*}}=\emptyset\) and \(\mathcal{G}_{i}=\{[s]\cup X_{i}:X_{i}\in\binom{[s+2,n]}{k_{i}-s}\}\) for \(i\in[r]\setminus\{i^{*}\}\), and hence \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for \(i\in[r]\setminus\{i^{*}\}\). Therefore, there is \(1\leqslant s\leqslant\bar{k}\) such that \(\mathcal{F}_{i^{*}}=\mathcal{R}_{s}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{s}^{(k_{i})}\) for all \(i\in[r]\setminus\{i^{*}\}\). By Lemma 2.3, we obtain the upper bound (3.1) for \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) and the structure of extremal families.
Lemma 3.1 gives the structure of the largest non-empty \(L\)-initial cross-intersecting families. We will generalize Lemma 3.1 to general non-empty cross-intersecting families. To this end, we need a notation. For \(\mathcal{A}\subseteq\binom{[n]}{k}\), let
\[\mathcal{D}_{l}(\mathcal{A})=\{D\in\binom{[n]}{l}:\text{there exists }A\in \mathcal{A}\text{ such that }A\cap D=\emptyset\}.\]
Clearly \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{l}\) are cross-intersecting if and only if \(\mathcal{A}\subseteq\binom{[n]}{k}\setminus\mathcal{D}_{k}(\mathcal{B})\) or \(\mathcal{B}\subseteq\binom{[n]}{l}\setminus\mathcal{D}_{l}(\mathcal{A})\). The following lemma was implicitly given in [6, 13] and was stated explicitly in [15, Proposition 2.3].
**Lemma 3.2**.: [6, 13, 15] _If \(n>k+l\) and \(\mathcal{A}\subseteq\binom{[n]}{k}\) with \(|\mathcal{A}|=\binom{n-s}{k-s}\) for some \(1\leqslant s\leqslant k\), then \(|\mathcal{D}_{l}(\mathcal{A})|\geqslant\binom{n-s}{l}\) with equality if and only if \(\mathcal{A}=\mathcal{P}_{S}^{(k)}\) for some \(S\in\binom{[n]}{s}\)._
**Lemma 3.3**.: _Let \(r\geqslant 2\) and \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n] }{k_{2}},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) be non-empty cross-intersecting families with \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i^{*}}-1}\) for some \(i^{*}\in[r]\). Let \(\bar{k}=\min\{k_{i}:i\in[r]\setminus\{i^{*}\}\}\). If \(n>\bar{k}+k_{i^{*}}\) and \(n\geqslant k_{i}+k_{i^{*}}\) for every \(i\in[r]\setminus\{i^{*}\}\), then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\left\{\binom{n}{k_{i^{*}}}- \binom{n-\bar{k}}{k_{i^{*}}}+\sum_{i\in[r]\setminus\{i^{*}\}}\binom{n-\bar{k}} {k_{i}-\bar{k}},\ \sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\right\} \tag{3.2}\]
_with equality if and only if there is some \(S\in\binom{[n]}{s}\) with \(s=1\) or \(\bar{k}\) such that \(\mathcal{F}_{i^{*}}=\mathcal{R}_{S}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{S}^{(k_{i})}\) for \(i\in[r]\setminus\{i^{*}\}\)._
**Proof.** The upper bound (3.2) for \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) follows from Theorem 2.1 and Lemma 3.1. Assume that the equality in (3.2) holds. Then by Theorem 2.1 and Lemma 3.1, \(|\mathcal{F}_{i^{*}}|=\binom{n}{k_{i^{*}}}-\binom{n-s}{k_{i^{*}}}\) and \(|\mathcal{F}_{i}|=\binom{n-s}{k_{i}-s}\) for \(i\in[r]\setminus\{i^{*}\}\), where \(s=1\) or \(\bar{k}\). Let \(H_{1}=\{i\in[r]:n>k_{i}+k_{i^{*}},i\neq i^{*}\}\) and \(H_{2}=\{i\in[r]:n=k_{i}+k_{i^{*}},i\neq i^{*}\}\). Since \(n>\bar{k}+k_{i^{*}}\), \(H_{1}\neq\emptyset\). For any \(i\in H_{1}\), \(\binom{n}{k_{i^{*}}}-|\mathcal{D}_{k_{i^{*}}}(\mathcal{F}_{i})|\geqslant| \mathcal{F}_{i^{*}}|\), i.e., \(|\mathcal{D}_{k_{i^{*}}}(\mathcal{F}_{i})|\leqslant\binom{n-s}{k_{i^{*}}}\). By Lemma 3.2, \(|\mathcal{D}_{k_{i^{*}}}(\mathcal{F}_{i})|=\binom{n-s}{k_{i^{*}}}\) and there exists \(S_{i}\in\binom{[n]}{s}\) for \(i\in H_{1}\) such that \(\mathcal{F}_{i}=\mathcal{P}_{S_{i}}^{(k_{i^{*}})}\).
We claim that \(S_{i_{1}}=S_{i_{2}}\), written as \(S\), for any \(i_{1},i_{2}\in H_{1}\), and \(\mathcal{F}_{i^{*}}=\mathcal{R}_{S}^{(k_{i^{*}})}\). Indeed, given any \(h\in H_{1}\), let \(S=S_{h}\). Since \(\mathcal{F}_{i^{*}}\) and \(\mathcal{F}_{h}=\mathcal{P}_{S_{h}}^{(k_{h})}\) are cross-intersecting and \(|\mathcal{F}_{i^{*}}|=\binom{n}{k_{i^{*}}}-\binom{n-s}{k_{i^{*}}}\)
it follows from Lemma 2.2 that \(\mathcal{F}_{i^{*}}=\mathcal{R}_{S}^{(k_{i^{*}})}\). For any \(i\in H_{1}\setminus\{h\}\), since \(\mathcal{F}_{i^{*}}\) and \(\mathcal{F}_{i}\) are cross-intersecting and \(|\mathcal{F}_{i}|=\binom{n-s}{k_{i-s}}\), by Lemma 2.2, we have \(\mathcal{F}_{i}=\mathcal{P}_{S}^{(k_{i})}\).
For every \(i\in H_{2}\), since \(\mathcal{F}_{i^{*}}\) and \(\mathcal{F}_{i}\) are cross-intersecting, \(\mathcal{F}_{i}\subseteq\binom{[n]}{k_{i}}\setminus\overline{\mathcal{F}_{i^ {*}}}\). Since \(|\mathcal{F}_{i}|=\binom{n-s}{k_{i-s}}=|\binom{[n]}{k_{i}}\setminus\overline{ \mathcal{F}_{i^{*}}}|\), we have \(\mathcal{F}_{i}=\binom{[n]}{k_{i}}\setminus\overline{\mathcal{F}_{i^{*}}}= \mathcal{P}_{S}^{(k_{i})}\).
To complete the proof of Theorem 1.6, we still need to examine the case of \(n=\bar{k}+k_{i^{*}}\) and \(n\geqslant k_{i}+k_{i^{*}}\) for every \(i\in[r]\setminus\{i^{*}\}\), where \(\bar{k}=\min\{k_{i}:i\in[r]\setminus\{i^{*}\}\}\). In this case \(k_{i}=\bar{k}\) for every \(i\in[r]\setminus\{i^{*}\}\). The following theorem will be used.
**Theorem 3.4**.: [15, Theorem 1.4] _Let \(n,k,l\) and \(\tau\) be positive integers with \(n\geqslant k+l\) and \(l\geqslant\tau\). Let \(c\) be a positive integer. If \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{l}\) are cross-intersecting and \(\binom{n-\tau}{l-\tau}\leqslant|\mathcal{B}|\leqslant\binom{n-1}{l-1}\), then_
\[|\mathcal{A}|+c|\mathcal{B}|\leqslant\max\left\{\binom{n}{k}-\binom{n-\tau}{ k}+c\binom{n-\tau}{l-\tau},\binom{n-1}{k-1}+c\binom{n-1}{l-1}\right\}\]
_with equality if and only if, up to isomorphism, one of the following holds:_
1. _when_ \(n>k+l\)_,_ \(\mathcal{A}=\mathcal{R}_{s}^{(k)}\) _and_ \(\mathcal{B}=\mathcal{P}_{s}^{(l)}\)_, where_ \[s=\left\{\begin{array}{ll}1,&\mbox{if }\binom{n}{k}-\binom{n-\tau}{k}+c \binom{n-\tau}{l-\tau}<\binom{n-1}{k-1}+c\binom{n-1}{l-1};\\ \tau,&\mbox{if }\binom{n}{k}-\binom{n-\tau}{k}+c\binom{n-\tau}{l-\tau}> \binom{n-1}{k-1}+c\binom{n-1}{l-1};\\ 1\mbox{ or }\tau,&\mbox{if }\binom{n}{k}-\binom{n-\tau}{k}+c\binom{n-\tau}{l- \tau}=\binom{n-1}{k-1}+c\binom{n-1}{l-1};\end{array}\right.\]
2. _when_ \(n=k+l\)_,_ \(\mathcal{B}\subseteq\binom{[n]}{l}\) _with_ \(|\mathcal{B}|=\binom{n-\tau}{l-\tau}\) _if_ \(c<1\) _or_ \(\binom{n-\tau}{l-\tau}\leqslant|\mathcal{B}|\leqslant\binom{n-1}{l-1}\) _if_ \(c=1\) _or_ \(|\mathcal{B}|=\binom{n-1}{l-1}\) _if_ \(c>1\)_, and_ \(\mathcal{A}=\binom{[n]}{k}\setminus\overline{\mathcal{B}}\)_._
**Lemma 3.5**.: _Let \(r\geqslant 2\) and \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n ]}{k_{2}},\ldots,\mathcal{F}_{r}\subseteq\binom{[n]}{k_{r}}\) be non-empty cross-intersecting families with \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i^{*}}-1}\) for some \(i^{*}\in[r]\) and \(k_{i}=h\) for every \(i\in[r]\setminus[i^{*}]\) where \(h\) is a positive integer. If \(n=h+k_{i^{*}}\), then_
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\binom{n-1}{k_{i^{*}}-1}+(r-1)\binom{ n-1}{h-1}\]
_with equality if and only if_
1. _when_ \(r=2\)_,_ \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}_{3-i^{*}}}\) _and_ \(1\leqslant|\mathcal{F}_{3-i^{*}}|\leqslant\binom{n-1}{h-1}\)_;_
2. _when_ \(r>2\) _and_ \(n>2h\)_, there exists_ \(x\in[n]\) _such that_ \(\mathcal{F}_{i^{*}}=\{F\in\binom{[n]}{k_{i^{*}}}:x\in F\}\) _and_ \(\mathcal{F}_{i}=\{F\in\binom{[n]}{h}:x\in F\}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\)_;_
3. _when_ \(r>2\) _and_ \(n\leqslant 2h\)_,_ \(\mathcal{F}_{i}=\mathcal{F}\) _for every_ \(i\in[r]\setminus\{i^{*}\}\) _and_ \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}}\)_, where_ \(\mathcal{F}\subseteq\binom{[n]}{h}\) _is an intersecting family with_ \(|\mathcal{F}|=\binom{n-1}{h-1}\)_._
Proof.: Since \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i^{*}}-1}\), by Theorem 2.1 and Lemma 2.2, \(|\mathcal{F}_{i}|\leqslant\binom{n-1}{k_{i}-1}=\binom{n-1}{h-1}\) for every \(i\in[r]\setminus\{i^{*}\}\). If \(r=2\), then apply Theorem 3.4 with \(k=k_{i^{*}}\), \(l=\tau=h\) and \(c=1\) to complete the proof. Assume that \(r>2\). Since \(n=h+k_{i^{*}}\),
\[\binom{n-1}{k_{i^{*}}-1}+(r-1)\binom{n-1}{h-1}=\binom{n-1}{k_{i^{*}}-1}+\binom{ n-1}{k_{i^{*}}}+(r-2)\binom{n-1}{h-1}\geqslant\binom{n}{k_{i^{*}}}+r-2. \tag{3.3}\]
It follows from Theorem 3.4 with \(k=k_{i^{*}}\), \(l=\tau=h\) and \(c=r-1>1\) that for any \(i\in[r]\setminus\{i^{*}\}\),
\[|\mathcal{F}_{i^{*}}|+(r-1)|\mathcal{F}_{i}| \leqslant\max\left\{\binom{n}{k_{i^{*}}}+r-2,\ \binom{n-1}{k_{i^{*}}-1}+(r-1)\binom{n-1}{h-1}\right\}\] \[=\binom{n-1}{k_{i^{*}}-1}+(r-1)\binom{n-1}{h-1}\]
with equality if and only if \(|\mathcal{F}_{i}|=\binom{n-1}{h-1}\) and \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}_{i}}\). Therefore,
\[\sum_{i=1}^{r}|\mathcal{F}_{i}|=\frac{1}{r-1}\sum_{i\in[r]\setminus\{i^{*}\}} (|\mathcal{F}_{i^{*}}|+(r-1)|\mathcal{F}_{i}|)\leqslant\binom{n-1}{k_{i^{*}}- 1}+(r-1)\binom{n-1}{h-1}\]
with equality if and only if \(\mathcal{F}_{i}=\mathcal{F}\) with \(|\mathcal{F}|=\binom{n-1}{h-1}\) for every \(i\in[r]\setminus\{i^{*}\}\) and \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}}\), where \(\mathcal{F}\subseteq\binom{[n]}{h}\) is an intersecting family because of the cross-intersecting property of \(\mathcal{F}_{i}\) for \(i\in[r]\setminus\{i^{*}\}\). When \(n>2h\), by Theorem 1.1, \(\mathcal{F}=\{F\in\binom{[n]}{h}:x\in F\}\) for some \(x\in[n]\), and so \(\mathcal{F}_{i^{*}}=\binom{[n]}{k_{i^{*}}}\setminus\overline{\mathcal{F}}=\{F \in\binom{[n]}{k_{i^{*}}}:x\in F\}\).
Now we are in a position to give a proof of Theorem 1.6.
**Proof of Theorem 1.6.** When \(k_{i}=\bar{k}\) for every \(i\in[r]\setminus\{i^{*}\}\) and \(n=\bar{k}+k_{i^{*}}\), it follows from (3.3) that \(\binom{n-1}{k_{i^{*}}-1}+(r-1)\binom{n-1}{k-1}\geqslant\binom{n}{k_{i^{*}}}+ r-2\). Combine Lemmas 3.3 and 3.5 to complete the proof.
## 4 Proof of Theorem 1.7
We are ready to prove Theorem 1.7 by employing Theorem 1.6.
**Proof of Theorem 1.7.** If \(|\mathcal{F}_{i}|<\binom{n-1}{k_{i}-1}\) for every \(i\in[r]\), then \(\sum_{i=1}^{r}|\mathcal{F}_{i}|<\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\leqslant \max\{\binom{n}{k_{1}}-\binom{n-k_{r}}{k_{1}}+\sum_{i=2}^{r}\binom{n-k_{r}}{ k_{i}-k_{r}},\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\}\).
Assume that there exists \(i^{*}\in[r]\) such that \(|\mathcal{F}_{i^{*}}|\geqslant\binom{n-1}{k_{i^{*}}-1}\). We shall compare the sizes of extremal families coming from Theorems 1.6.
**Case 1**\(r=2\) and \(n=k_{1}+k_{2}\). In this case, \(\binom{n}{k_{1}}=\binom{n}{k_{2}}=\binom{n-1}{k_{1}-1}+\binom{n-1}{k_{2}-1}\). It follows from \((i)\) of Theorem 1.6 (1) that \(|\mathcal{F}_{1}|+|\mathcal{F}_{2}|\leqslant\binom{n}{k_{1}}\) with equality if and only if
\[\mathcal{F}_{1}=\binom{[n]}{k_{1}}\setminus\overline{\mathcal{F}_{2}}\text{ with }1\leqslant|\mathcal{F}_{2}|\leqslant\binom{n-1}{k_{2}-1}\]
or
\[\mathcal{F}_{2}=\binom{[n]}{k_{2}}\setminus\overline{\mathcal{F}_{1}}\text{ with }1\leqslant|\mathcal{F}_{1}|\leqslant\binom{n-1}{k_{1}-1}.\]
The later case is equivalent to \(\mathcal{F}_{1}=\binom{[n]}{k_{1}}\setminus\overline{\mathcal{F}_{2}}\) with \(\binom{n-1}{k_{2}-1}\leqslant|\mathcal{F}_{2}|\leqslant\binom{n}{k_{2}}-1\). Therefore, the equality holds if and only if
\[\mathcal{F}_{1}=\binom{[n]}{k_{1}}\setminus\overline{\mathcal{F}_{2}}\text{ with }0<|\mathcal{F}_{2}|<\binom{n}{k_{2}}.\]
**Case 2**\(r>2\) and \(n=k_{1}+k_{r}\). In this case, \(\binom{n-1}{k_{1}-1}+(r-1)\binom{n-1}{k_{r}-1}\geqslant\binom{n}{k_{1}}+r-2\). Since \(n\geqslant k_{1}+k_{2}\) and \(k_{2}\geqslant\cdots\geqslant k_{r}\), we have \(k_{2}=\cdots=k_{r}\).
If \(k_{1}=k_{2}\), then by Theorem 1.5\((2)(ii)\), \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\sum_{i=1}^{r}\binom{n-1}{k_{i-1}}\) with equality if and only if \(\mathcal{F}_{i}=\mathcal{F}\) for \(i\in[r]\), where \(\mathcal{F}\) is an intersecting family with \(|\mathcal{F}|=\binom{n-1}{k_{1}-1}\).
Assume that \(k_{1}>k_{2}\). By Theorem 1.6\((1)(ii)\) and \((2)\), if \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) reaches the largest possible value, then up to isomorphism, either \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\), or \(\mathcal{F}_{i^{*}}=\mathcal{R}_{k_{r}}^{(k_{i^{*}})}\) with \(i^{*}\in[2,r]\) and \(\mathcal{F}_{i}=\mathcal{P}_{k_{r}}^{(k_{i})}\) for every \(i\in[r]\setminus\{i^{*}\}\). If \(k_{r}=1\), then the above two classes of extremal families are the same and so \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\sum_{i=1}^{r}\binom{n-1}{k_{i-1}}\) with equality if and only if \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\) up to isomorphism. If \(k_{r}>1\), then
\[\sum_{i=1}^{r}|\mathcal{P}_{1}^{(k_{i})}| =\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}=\binom{n-1}{k_{1}-1}+\binom{n -1}{k_{2}-1}+\sum_{i=3}^{r}\binom{n-1}{k_{i}-1}>\binom{n}{k_{i^{*}}}+r-2\] \[=\binom{n}{k_{i^{*}}}+r-2+\binom{n-k_{r}}{k_{1}-k_{r}}-\binom{n-k _{r}}{k_{i^{*}}}=|\mathcal{R}_{k_{r}}^{(k_{i^{*}})}|+\sum_{i\in[r]\setminus\{ i^{*}\}}|\mathcal{P}_{k_{r}}^{(k_{i})}|,\]
and hence \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\) with equality if and only if \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\) up to isomorphism.
**Case 3**\(n>k_{1}+k_{r}\). Since \(k_{1}\geqslant\cdots\geqslant k_{r}\), we have \(n>k_{i^{*}}+\overline{k_{i^{*}}}\) for every \(i^{*}\in[r]\), where \(\overline{k_{i^{*}}}=\min\{k_{i}:i\in[r]\setminus\{i^{*}\}\}\). By Theorem 1.6\((2)\), if the value of \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) reaches the largest, then up to isomorphism, one of the following three cases holds:
* \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\);
* if \(i^{*}\neq r\), then \(\mathcal{F}_{i^{*}}=\mathcal{R}_{k_{r}}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{k_{r}}^{(k_{i})}\) for every \(i\in[r]\setminus\{i^{*}\}\);
* if \(i^{*}=r\), then \(\mathcal{F}_{i^{*}}=\mathcal{R}_{k_{r-1}}^{(k_{i^{*}})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{k_{r-1}}^{(k_{i})}\) for every \(i\in[r-1]\).
If \(k_{r-1}=k_{r}=1\), then the above three cases are the same and so \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\) with equality if and only if \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\) up to isomorphism.
If \(k_{r-1}>k_{r}=1\), then the cross-intersecting families of type \((\alpha)\) are the same as the cross-intersecting families of type \((\beta)\). Applying Lemma 2.5 (note that \(n\geqslant k_{1}+k_{2}\)) to compare the values of \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) with types \((\alpha)\) and \((\gamma)\), we have \(\sum_{i=1}^{r}|\mathcal{P}_{1}^{(k_{i})}|>\sum_{i=1}^{r-1}|\mathcal{P}_{k_{r-1} }^{(k_{i})}|+|\mathcal{R}_{k_{r-1}}^{(k_{r})}|\). Thus \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\sum_{i=1}^{r}\binom{n-1}{k_{i}-1}\) with equality if and only if \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\) up to isomorphism.
If \(k_{1}=k_{r}\geqslant 2\), then \(k_{1}=k_{2}=\cdots=k_{r}\) and so the cross-intersecting families of type \((\beta)\) are isomorphic to the cross-intersecting families of type \((\gamma)\). Thus \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\{\binom{n}{k_{1}}-\binom{n-k_{r} }{k_{1}}+\sum_{i=2}^{r}\binom{n-k_{r}}{k_{i}-k_{r}},\sum_{i=1}^{r}\binom{n-1}{k_ {i}-1}\}\}\) with equality if and only if up to isomorphism, either \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\), or \(\mathcal{F}_{j}=\mathcal{R}_{k_{r}}^{(k_{j})}\) for some \(j\in[r]\) and \(\mathcal{F}_{i}=\mathcal{P}_{k_{r}}^{(k_{i})}\) for every \(i\in[r]\setminus\{j\}\).
Assume that \(k_{1}>k_{r}\geqslant 2\). Apply Lemma 2.5 to compare the values of \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) with types \((\alpha)\) and \((\gamma)\) to obtain \(\sum_{i=1}^{r}|\mathcal{P}_{1}^{(k_{i})}|>\sum_{i=1}^{r-1}|\mathcal{R}_{k_{r-1} }^{(k_{i})}|+|\mathcal{R}_{k_{r-1}}^{(k_{r})}|\). When \(k_{1}>k_{i^{*}}\), compare the values of \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) with types \((\alpha)\) and \((\beta)\) to obtain \(\sum_{i=1}^{r}|\mathcal{P}_{1}^{(k_{i})}|>\sum_{i\in[r]\setminus\{i^{*}\}}| \mathcal{P}_{k_{r}}^{(k_{i})}|+|\mathcal{R}_{k_{r}}^{(k_{i^{*}})}|\). Therefore, \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\leqslant\max\{\binom{n}{k_{1}}-\binom{n-k_{r} }{k_{1}}+\sum_{i=2}^{r}\binom{n-k_{r}}{k_{i-k_{r}}},\sum_{i=1}^{r}\binom{n-1}{k_ {i-1}}\}\) with equality if and only if up to isomorphism, either \(\mathcal{F}_{i}=\mathcal{P}_{1}^{(k_{i})}\) for every \(i\in[r]\), or \(\mathcal{F}_{1}=\mathcal{R}_{k_{r}}^{(k_{1})}\) and \(\mathcal{F}_{i}=\mathcal{P}_{k_{r}}^{(k_{i})}\) for every \(i\in[r]\setminus\{1\}\).
## 5 Concluding remarks
For non-empty cross-intersecting families \(\mathcal{F}_{i}\subseteq\binom{[n]}{k_{i}}\), \(i\in[r]\), this paper examines the largest value of \(\sum_{i=1}^{r}|\mathcal{F}_{i}|\) and determines the structure of \(\mathcal{F}_{i}\)'s that achieves the largest sum. Theorems
1.6 and 1.7 generalize Theorems 1.4 and 1.5 by extending two families to arbitrary number of families allowing different sizes.
We remark that Theorem 1.6 is not equivalent to Theorem 1.7. For example, take \(r=3\), \(n=10\), \(k_{1}=5\), \(k_{2}=3\) and \(k_{3}=2\). Suppose that \(\mathcal{F}_{1}\subseteq\binom{[n]}{k_{1}},\mathcal{F}_{2}\subseteq\binom{[n]} {k_{2}}\) and \(\mathcal{F}_{3}\subseteq\binom{[n]}{k_{3}}\) are non-empty cross-intersecting families such that \(|\mathcal{F}_{3}|\geqslant\binom{n-1}{k_{3}-1}=9\). Apply Theorem 1.6 with \(i^{*}=3\) to obtain \(\sum_{i=1}^{3}|\mathcal{F}_{i}|\leqslant\max\{46,171\}=171\) with equality if and only if there is \(x\in[n]\) such that \(\mathcal{F}_{i}=\{F\in\binom{[n]}{k_{i}}:x\in F\}\) for \(1\leqslant i\leqslant 3\). Apply Theorem 1.7 to obtain \(\sum_{i=1}^{3}|\mathcal{F}_{i}|\leqslant\max\{205,171\}=205\) with equality if and only if there is \(S\in\binom{[n]}{k_{3}}\) such that \(\mathcal{F}_{1}=\{F\in\binom{[n]}{k_{1}}:F\cap S\neq\emptyset\},\mathcal{F}_ {2}=\{F\in\binom{[n]}{k_{2}}:S\subseteq F\}\) and \(\mathcal{F}_{3}=\{S\}\). Since it is required that \(|\mathcal{F}_{3}|\geqslant 9\), by Theorem 1.7, we only know \(\sum_{i=1}^{3}|\mathcal{F}_{i}|<205\).
| ```
families $\mathcal F_1\subseteq \binom{[n]}{k_1},\mathcal F_2\subseteq\binom{[n]}{k_2},\dots,\mathcal F_r\subseteq \binom{[n]}{k_r}$ はクロス交差すると言われますが、$1\leq i<j\leq r$ に対し $F_i\cap F_j$ は 1 以上である。 $F_i\in \mathcal F_i$、$F_j\in\mathcal F_j$ を満たす $r$ つの $\mathcal F_1,\mathcal F_2,\dots,\mathcal F_r$ はクロス交差すると言われます。 クロス交差する $r$ つの $\mathcal F_1,\mathcal F_2,\dots,\mathcal F_r$ は非空であると言われますが、$1\leq i\leq r$ に対し $\mathcal F_ |
2307.04019 | GP-guided MPPI for Efficient Navigation in Complex Unknown Cluttered
Environments | Robotic navigation in unknown, cluttered environments with limited sensing
capabilities poses significant challenges in robotics. Local trajectory
optimization methods, such as Model Predictive Path Intergal (MPPI), are a
promising solution to this challenge. However, global guidance is required to
ensure effective navigation, especially when encountering challenging
environmental conditions or navigating beyond the planning horizon. This study
presents the GP-MPPI, an online learning-based control strategy that integrates
MPPI with a local perception model based on Sparse Gaussian Process (SGP). The
key idea is to leverage the learning capability of SGP to construct a variance
(uncertainty) surface, which enables the robot to learn about the navigable
space surrounding it, identify a set of suggested subgoals, and ultimately
recommend the optimal subgoal that minimizes a predefined cost function to the
local MPPI planner. Afterward, MPPI computes the optimal control sequence that
satisfies the robot and collision avoidance constraints. Such an approach
eliminates the necessity of a global map of the environment or an offline
training process. We validate the efficiency and robustness of our proposed
control strategy through both simulated and real-world experiments of 2D
autonomous navigation tasks in complex unknown environments, demonstrating its
superiority in guiding the robot safely towards its desired goal while avoiding
obstacles and escaping entrapment in local minima. The GPU implementation of
GP-MPPI, including the supplementary video, is available at
https://github.com/IhabMohamed/GP-MPPI. | Ihab S. Mohamed, Mahmoud Ali, Lantao Liu | 2023-07-08T17:33:20 | http://arxiv.org/abs/2307.04019v3 | # GP-guided MPPI for Efficient Navigation in Complex Unknown Cluttered Environments
###### Abstract
Robotic navigation in unknown, cluttered environments with limited sensing capabilities poses significant challenges in robotics. Local trajectory optimization methods, such as Model Predictive Path Integral (MPPI), are a promising solution to this challenge. However, global guidance is required to ensure effective navigation, especially when encountering challenging environmental conditions or navigating beyond the planning horizon. This study presents the GP-MPPI, an _online_ learning-based control strategy that integrates MPPI with a local perception model based on Sparse Gaussian Process (SGP). The key idea is to leverage the learning capability of SGP to construct a variance (uncertainty) surface, which enables the robot to learn about the navigable space surrounding it, identify a set of suggested subgoals, and ultimately recommend the optimal subgoal that minimizes a predefined cost function to the local MPPI planner. Afterward, MPPI computes the optimal control sequence that satisfies the robot and collision avoidance constraints. Such an approach eliminates the necessity of a global map of the environment or an offline training process. We validate the efficiency and robustness of our proposed control strategy through both simulated and real-world experiments of 2D autonomous navigation tasks in complex unknown environments, demonstrating its superiority in guiding the robot safely towards its desired goal while avoiding obstacles and escaping entrapment in local minima. The GPU implementation of GP-MPPI, including the supplementary video, is available at [https://github.com/IhabMohamed/GP-MPPI](https://github.com/IhabMohamed/GP-MPPI).
Autonomous vehicle navigation, MPPI, sparse Gaussian process (SGP), occupancy grid map path planning.
## I Introduction and Related Work
Autonomous navigation of mobile robots in unknown, cluttered, and unpredictable environments with limited sensor capabilities is a challenging task owing to the inherent uncertainty and complexity of such environments. To tackle this challenge, a _receding-horizon_ strategy such as Model Predictive Control (MPC) is commonly employed. The MPC control framework allows the robot to simultaneously plan a short trajectory (sequence of actions), following which the robot executes the immediate action while planning a subsequent trajectory. To successfully achieve receding-horizon planning, the robot must consider both safety and persistent feasibility, where _safety_ is achieved by avoiding collisions with any obstacles while executing a planned trajectory, and _persistent feasibility_ is maintained by always generating a safe trajectory that does not result in dead-ends or local minima while progressing towards the desired goal.
One of the significant challenges in robot motion planning is that the desired goal is often situated beyond the planning horizon, which requires the use of local subgoals or _cost-to-go_ heuristics for motion safety and persistent feasibility. A common strategy is to rely on single-query motion planning algorithms, such as A\({}^{*}\) and RRT\({}^{\text{X}}\), to identify feasible paths that direct the local planner towards its desired goal [1, 2]. For instance, the RRT\({}^{\text{X}}\) algorithm, introduced in [2], incorporates replanning techniques from Dynamic Rapidly-exploring Random Trees (DRRT) and Rapid-exploring Random Trees (RRT\({}^{*}\)) algorithms to adjust the path during exploration based on environmental changes. However, due to its high computational demands, implementing this algorithm in _real-time_ on a robot can be challenging.
One alternative method to achieve efficient solutions for motion planning problems is the integration of MPC with data-driven methods, also known as learning-based MPC [3]. To name a few, a subgoal planning policy using Deep Reinforcement Learning (DRL) is recently proposed to guide the local MPC planner to navigate in crowded surroundings [4, 5]. Similarly, RL was utilized to choose the next subgoal from a set of predefined possibilities [6], which guides the robot through challenging environments with dead-end corridors while also prevents the MPC planner from getting trapped in local minima. Another related work that combines learning with MPC is POLO which aims to enhance MPC performance by learning a global value function [7]. Most of these approaches typically rely on either offline training or having access to the global map of the environment. In addition, many recent studies have suggested combining Gaussian Process (GP) with MPC to learn system dynamics, leading to better control performance and robustness to uncertainty [8].
Another research avenue employed gap-based techniques
Fig. 1: Architecture of our proposed GP-MPPI control strategy, which comprises two main components: the GP-subgoal recommender and the local planner, the MPPI. First, the GP-subgoal recommender observes the surrounding environment and suggests the optimal subgoal position \(\mathbf{g}^{*}\) to the local motion planner, where four colored circles represent the GP-recommended subgoals. MPPI then computes the optimal control sequence, which minimizes the distance to \(\mathbf{g}^{*}\) while avoiding collision with obstacles and respecting system constraints, followed by executing the first optimal control \(\mathbf{u}_{0}\) to the robot. | ロボットによる未知、混雑した環境でのナビゲーションが、感知能力が限られた場合、ロボット分野において大きな課題となります。ローカルな軌道最適化手法、例えばモデル予測パスインターガル（MPPI）は、この課題への有望な解決策です。しかし、効果的なナビゲーションを確実にするには、グローバルなガイダンスが必須であり、特に環境の課題に直面する時や、計画の限界を超える場合に特に重要です。この研究では、GP-MPPIという、オンライン学習に基づく制御戦略を提示します。この戦略は、MPPIを、スパースガウス過程（SGP）に基づいたローカルな認識モデルと統合しています。キーアイディアは、SGPの学習能力を利用して、ナビゲート可能な空間の変動性（不確かさ）を構築することで、ロボットが周囲の環境を学習し、推奨される subgoals（目標）を特定し、最終 |
2305.12002 | XuanYuan 2.0: A Large Chinese Financial Chat Model with Hundreds of
Billions Parameters | In recent years, pre-trained language models have undergone rapid development
with the emergence of large-scale models. However, there is a lack of
open-sourced chat models specifically designed for the Chinese language,
especially in the field of Chinese finance, at the scale of hundreds of
billions. To address this gap, we introduce XuanYuan 2.0, the largest Chinese
chat model to date, built upon the BLOOM-176B architecture. Additionally, we
propose a novel training method called hybrid-tuning to mitigate catastrophic
forgetting. By combining general-domain with domain-specific knowledge and
integrating the stages of pre-training and fine-tuning, XuanYuan 2.0 is capable
of providing accurate and contextually appropriate responses in the Chinese
financial domain. | Xuanyu Zhang, Qing Yang, Dongliang Xu | 2023-05-19T21:01:20 | http://arxiv.org/abs/2305.12002v1 | # XuanYuan 2.0: A Large Chinese Financial Chat Model
###### Abstract
In recent years, pre-trained language models have undergone rapid development with the emergence of large-scale models. However, there is a lack of open-sourced chat models specifically designed for the Chinese language, especially in the field of Chinese finance, at the scale of hundreds of billions. To address this gap, we introduce **XuanYuan 2.0** (\(\ddagger\) 2.0), the largest Chinese chat model to date, built upon the BLOOM-176B architecture. Additionally, we propose a novel training method called hybrid-tuning to mitigate catastrophic forgetting. By combining general-domain with domain-specific knowledge and integrating the stages of pre-training and fine-tuning, XuanYuan 2.0 is capable of providing accurate and contextually appropriate responses in the Chinese financial domain.
## 1 Introduction
In recent years, pre-trained language models have witnessed rapid development. Broadly speaking, they can be categorized into three main architectures: the Encoder architecture represented by BERT (Devlin et al., 2018), the Decoder architecture represented by GPT (Radford et al., 2018), and the Encoder-Decoder architecture represented by T5 (Raffel et al., 2020). Each architecture has its unique characteristics and advantages, catering to different NLP requirements.
The GPT series, with GPT-4 (OpenAI, 2023) being the latest addition, has gained considerable attention due to its remarkable performance in natural language generation tasks, including dialogue generation. The ChatGPT (OpenAI, 2022) model, in particular, has impressed researchers and practitioners with its ability to generate coherent and contextually relevant responses in conversational settings. As a result, the GPT series has become a focal point of research and development in the NLP community.
Moreover, the emergence of large-scale pre-trained models has further fueled the advancements in language modeling. Models such as OPT (Zhang et al., 2022), BLOOM (Scao et al., 2022), and LLaMA (Touvron et al., 2023), with parameter sizes reaching billions, have recently been open-sourced, enabling researchers and developers to explore the potential of these massive models. These models have demonstrated superior performance on various tasks, pushing the boundaries of what is possible in NLP.
While the general-purpose large models mentioned above have garnered significant attention, the importance of domain-specific models cannot be overlooked. In many domains, the distribution of language and the specific linguistic nuances require models that are fine-tuned or specifically trained for that particular domain. Consequently, a range of domain-specific large models has been proposed to cater to the unique needs of various fields. For example, BioBERT (Lee et al., 2020) and PubMedBERT (Gu et al., 2021) are proposed for the biomedical field, and BloombergGPT (Wu et al., 2023) are proposed for financial scenarios. These models have shown promising results in their respective domains, leveraging the domain-specific knowledge learned during pre-training.
Within the Chinese financial domain, there has been considerable progress in the development of pre-trained language models. Researchers have introduced models such as FinBERT (Araci, 2019; Yang et al., 2020; Liu et al., 2021), Mengzi (Zhang et al., 2021), and FinT5 (Lu et al., 2023), which have been tailored for financial text analysis and understanding. These models, though valuable for certain applications, have parameter sizes below one billion, limiting their ability to handle the increasing demands of the Chinese financial NLP landscape. As the volume of financial data and the complexity of language usage continue to grow, there is a pressing need for more powerful models
that can effectively process and understand Chinese financial text.
Despite significant advancements in chat models, there is currently no open-sourced chat model at the scale of hundreds of billions specifically designed for the Chinese language, let alone in the field of Chinese finance. To address this gap, we propose **XuanYuan 2.0** (\(\frac{\text{FT-14}}{\text{SR}}\) 2.0), the largest Chinese chat model to date, based on BLOOM-176B. XuanYuan 2.0 not only surpasses its predecessor, **XuanYuan 1.0** (\(\frac{\text{FT-14}}{\text{SR}}\) 1.0), which achieved first place at the leaderboard of CLUE classification in 2021, but also addresses the need for a large-scale chat model specifically designed for the Chinese financial domain.
Furthermore, domain-specific language models and chat models impose higher requirements on data distribution and training approaches compared to general-domain models. Domain-specific models need to capture the unique linguistic characteristics, terminologies, and contexts of a particular field to achieve optimal performance. However, training these models solely on domain-specific data may lead to catastrophic forgetting, where the model loses previously learned knowledge from the general domain, impacting its overall performance. To mitigate this issue, we propose a novel training method, hybrid-tuning, that combines the stages of pre-training and fine-tuning. By integrating the two stages, our approach guarantees that fine-tuning the model with financial-specific instructions does not impede its general generation capabilities acquired during pre-training. As a result, XuanYuan 2.0 can effectively leverage both its general-domain knowledge and domain-specific financial knowledge to provide accurate and contextually appropriate responses in the Chinese financial domain.
## 2 Related Work
The advancements in pre-trained language models have led to remarkable progress in various NLP tasks, attracting extensive research efforts. Among the notable contributions, the BERT Devlin et al. (2018) series stands out as a groundbreaking development in the field of pre-trained models. Following the success of BERT, the GPT Radford et al. (2018) series emerged as a prominent line of research, focusing on the decoding aspect of language modeling. GPT models, in contrast to BERT's bidirectional approach, leveraged autoregressive language modeling. By training on large amounts of unlabeled text data, GPT models acquired a rich understanding of language and demonstrated impressive capabilities in generating coherent and contextually relevant text. Subsequent iterations of the GPT series, such as GPT-4 OpenAI (2023), showcased superior performance in various language generation tasks. And Chat-GPT (OpenAI, 2022), an extension of the GPT series, demonstrated the ability to engage in interactive and contextually coherent conversations. This breakthrough sparked considerable interest in developing conversational AI agents capable of simulating human-like dialogue.
In addition to the general-purpose BERT and GPT models, there has been a growing interest in domain-specific pre-training. Researchers have recognized that incorporating domain-specific knowledge during pre-training can lead to substantial performance gains in downstream tasks within those domains. Domain-specific pre-trained models aim to capture domain-specific nuances, enabling them to excel in tasks relevant to the target domain. For instance, in the biomedical domain, BioBERT Lee et al. (2020) and PubMedBERT Gu et al. (2021) are proposed to leverage large-scale biomedical
\begin{table}
\begin{tabular}{p{113.8pt} p{56.9pt} p{56.9pt} p{56.9pt}} \hline \hline
**Model** & **Type** & **Parameter** & **Corpus Content** \\ \hline FinBERT Araci (2019) & PLM & 110M & News filtered by financial keywords \\ FinBERT Yang et al. (2020) & PLM & 110M & Corporate Reports, Earnings Call Transcripts, Analyst Reports \\ Mengzi-BERT-base-fin Zhang et al. (2021) & PLM & 110M & News, Analyse reports, Company announcements \\ FinT5 Lu et al. (2023) & PLM & 220M, 1B & Corporate Reports, Analyst Reports, Social media and Financial News \\ \hline XuanYuan 2.0 & ChatLM & 176B & Corporate Reports, Analyst Reports, Social media and Financial News \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of different financial language models.
corpora during pre-training. Similarly, in the financial domain, models such as BloombergGPT Wu et al. (2023) were developed to address the unique challenges and intricacies of the financial text.
Despite the advancements in domain-specific pre-training, the availability of large-scale open-source chat models specifically tailored for the Chinese language and the Chinese financial domain has remained limited. This gap motivates our work in proposing XuanYuan 2.0, a model built upon BLOOM-176B Scao et al. (2022) with hundreds of billions parameters, to address the unique requirements of the Chinese financial domain and facilitate the development of sophisticated conversational AI systems.
## 3 XuanYuan 2.0
### Model Architecture
We adopted the original BLOOM Scao et al. (2022) architecture, which is a decoder-only architecture. The joint probability of tokens in a text can be represented as:
\[p(w)=p(w_{1},\dots,w_{T})=\prod_{t=1}^{T}p(w_{t}|w_{<t}) \tag{1}\]
where \(w\) represents a sequence of tokens, \(w_{t}\) is the \(t^{\rm th}\) token, and \(w_{<t}\) is the sequence of tokens preceding \(w_{t}\). This method is called autoregressive language modeling, where we predict the probability of the next token in an iterative manner. And following BLOOM, we utilize ALBi positional embeddings Press et al. (2021) and embedding LayerNorm Dettmers et al. (2022) in the traditional decoder structure of Transformer Vaswani et al. (2017).
### Hybrid-tuning
To alleviate the problem of catastrophic forgetting, we propose a novel domain-specific training framework, hybrid-tuning. In terms of the training stage, it integrates the pre-training stage and instruction fine-tuning stage that are previously split together. In terms of the field of data, it integrates data from both general and financial domains.
As shown in Figure 1, different from traditional two-stage domain-specific training, our proposed hybrid-tuning randomly shuffles pre-training data (general pre-training, financial pre-training) and instruction data (general instruction, financial instruction) into one training data. And all the training
Figure 1: Our proposed hybrid-tuning.
process is done in one stage. In this way, the model can accurately handle instructions in the financial domain, while retaining general conversational capabilities.
For unsupervised pre-training data, we crawl them from the Internet and clean and filter them. For Instruction-tuning data, we use human-written seed instructions to collect general data by Self-Instruct Wang et al. (2022) and utilize unstructured and structured data in the financial field to gather domain-specific instruction data by Self-QA Zhang and Yang (2023). Unstructured financial data comprises a wide range of textual information, such as financial news articles, market reports, analyst commentary, and social media discussions. And structured financial data includes company information and so on. These sources offer valuable insights into market trends, investment strategies, and economic situations.
### Training
To train our complex and computationally intensive model, we employ the powerful NVIDIA A100 80GB GPU and the DeepSpeed Rasley et al. (2020) distributed training framework. For parallel processing, we primarily rely on pipeline parallelism, which involves distributing the layers of our model across several GPUs. This approach ensures that each GPU only handles a portion of the model's layers, a technique also known as vertical parallelism. Additionally, we adopt the Zero Redundancy Optimizer Rajbhandari et al. (2020) to enable different processes to store only a portion of the data (parameters, gradients, and optimizer states). Specifically, we use ZeRO stage 1, which means that only the optimizer states are divided using this method. The specific hyperparameters are presented in Table 2.
## 4 Experiment
We conducted a comparison between our model and other open-source Chinese conversational models. Simultaneously, we constructed evaluation datasets encompassing various dimensions in both general and financial domains, which were subsequently subject to manual assessment. The results revealed XuanYuan's robust knowledge base and conversational capabilities in the financial domain. Further insights and additional findings will be presented in the next version of the paper after the release of the evaluation rankings.
## 5 Conclusion
In this paper, we propose the largest Chinese financial chat model, XuanYuan 2.0 (\(\frac{\text{Total}}{\text{Total}}\) 2.0), to fill the gap of open-source billion-scale chat models specifically designed for the Chinese financial domain. Besides, we propose a novel training method called hybrid-tuning to mitigate catastrophic forgetting. By combining the general domain with domain-specific knowledge and integrating the stages of pre-training and finetuning, XuanYuan 2.0 achieves the remarkable ability to deliver precise and contextually relevant responses within the Chinese financial domain. We will continue to gather larger-scale Chinese financial domain data in order to further optimize our model.
\begin{table}
\begin{tabular}{l|c|c} \hline \hline Hyperparameter & XuanYuan2-7B & XuanYuan2 \\ \hline \multicolumn{4}{c}{_Architecture hyperparameters_} \\ \hline Parameters & 7,069M & 176,247M \\ Layers & 30 & 70 \\ Hidden dim. & 4096 & 14336 \\ Attention heads & 32 & 112 \\ Vocab size & 250,680 & \\ Sequence length & 2048 & \\ Precision & float16 & \\ Activation & GELU & \\ Position emb. & Alibi & \\ Tied emb. & True & \\ \hline \hline \multicolumn{4}{c}{_Pretraining hyperparameters_} \\ \hline Global Batch Size & 512 & 2048 \\ Learning rate & 1.2e-4 & 6e-5 \\ Total tokens & 341B & 366B \\ Min. learning rate & 1e-5 & 6e-6 \\ Warmup tokens & 375M & \\ Decay tokens & 410B & \\ Decay style & cosine & \\ Adam \((\beta_{1},\beta_{2})\) & (0.9, 0.95) & \\ Weight decay & 1e-1 & \\ Gradient clipping & 1.0 & \\ \hline \hline \multicolumn{4}{c}{_Multitask finetuning hyperparameters_} \\ \hline Global Batch Size & 2048 & 2048 \\ Learning rate & 2.0e-5 & 2.0e-5 \\ Total tokens & & 13B \\ Warmup tokens & & 0 \\ Decay style & constant & \\ Weight decay & 1e-4 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Training hyperparameters of XuanYuan 2.0. | 近年、大規模言語モデルの急速な発展に伴い、 pre-trained language models は多くの進展を見せた。しかし、特に中国語で設計されたオープンソースのチャットモデルが不足しており、数百億規模の中国金融分野では、その存在がほとんどない。この空白を埋めるために、私たちは XuanYuan 2.0 を導入した。XuanYuan 2.0 は、BLOOM-176Bアーキテクチャに基づいて、最も大きな中国語チャットモデルである。さらに、Hybrid-Tuning という新しいトレーニング手法を提案し、Catastrophic Forgetting を軽減する。一般領域と特定の領域の知識を組み合わせ、事前学習と微調整の段階を統合することで、XuanYuan 2.0 は中国金融分野における正確で、状況に適した応答を提供できる。
Please let me know if you need further assistance or have any other questions. |
2306.02999 | Von Neumann Dimensions and Trace Formulas I: Limit Multiplicities | Given a connected semisimple Lie group $G$ and an arithmetic subgroup
$\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$
occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of
$L^2(\Gamma\backslash G)$ with at most a finite multiplicity $m_{\Gamma}(\pi)$.
While $m_{\Gamma}(\pi)$ is unknown in general, we are interested in its limit
as $\Gamma$ is taken to be in a tower of lattices $\Gamma_1\supset
\Gamma_2\supset\dots$. For a bounded measurable subset $X$ of the unitary dual
$\widehat{G}$, we let $m_{\Gamma_n}(X)$ be the sum of the multiplicity
$m_{\Gamma_n}(\pi)$ of a representation $\pi$ over all $\pi$ in $X$. Let $H_X$
be the direct integral of the irreducible representations in $X$, which is also
a module over the group von Neumann algebra $\mathcal{L}\Gamma_n$. We prove:
\begin{center} $\lim\limits_{n\to
\infty}\cfrac{m_{\Gamma_n}(X)}{\dim_{\mathcal{L}\Gamma_n}H_X}=1$, \end{center}
for any bounded subset $X$ of $\widehat{G}$, when i) $\Gamma_n$'s are
cocompact, or, ii) $G=\SL(n,\mathbb{R})$ and $\{\Gamma_n\}$ are principal
congruence subgroups. | Jun Yang | 2023-06-05T16:10:07 | http://arxiv.org/abs/2306.02999v1 | # Von Neumann dimensions and trace formulas I: limit multiplicities
###### Abstract.
Given a connected semisimple Lie group \(G\) and an arithmetic subgroup \(\Gamma\), it is well-known that each irreducible representation \(\pi\) of \(G\) occurs in the discrete spectrum \(L^{2}_{\mathrm{disc}}(\Gamma\backslash G)\) of \(L^{2}(\Gamma\backslash G)\) with at most a finite multiplicity \(m_{\Gamma}(\pi)\). While \(m_{\Gamma}(\pi)\) is unknown in general, we are interested in its limit as \(\Gamma\) is taken to be in a tower of lattices \(\Gamma_{1}\supset\Gamma_{2}\supset\dots\). For a bounded measurable subset \(X\) of the unitary dual \(\widehat{G}\), we let \(m_{\Gamma_{n}}(X)\) be the sum of the multiplicity \(m_{\Gamma_{n}}(\pi)\) over all \(\pi\) in \(X\). Let \(H_{X}\) be the direct integral of the irreducible representations in \(X\) with respect to the Plancherel measure of \(G\), which is also a module over the group von Neumann algebra \(\mathcal{L}\Gamma_{n}\). We prove:
\[\lim_{n\to\infty}\frac{m_{\Gamma_{n}}(X)}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1,\]
for any bounded subset \(X\) of \(\widehat{G}\), when i) \(\Gamma_{n}\)'s are cocompact, or, ii) \(G=\mathrm{SL}(n,\mathbb{R})\) and \(\{\Gamma_{n}\}\) are principal congruence subgroups.
AMS 2010 Mathematics Subject Classification: 46L10, 20G05, 20G35. \(\mathbb{E}\)Jun Yang [email protected] Harvard University, Cambridge, MA, USA This work was supported in part by the ARO Grant W911NF-19-1-0302 and the ARO MURI Grant W911NF-20-1-0082.
## 1. Introduction: an example on \(\operatorname{SL}(2,\mathbb{R})\)
In this section, we introduce a multiplicity problem of square-integrable irreducible representations of \(G=\operatorname{SL}(2,\mathbb{R})\) on \(L^{2}_{\operatorname{cusp}}(\Gamma\backslash G)\) for some arithmetic subgroups \(\Gamma\). It is one of the motivations of this article.
We first \(\Gamma=\operatorname{SL}(2,\mathbb{Z})\) and \(\Gamma(N)\) be the principal congruence subgroup of level \(n\) defined by
\[\Gamma(N)=\Big{\{}\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{SL}(2,\mathbb{Z}):a,d\equiv 1\pmod{N},b,c\equiv 0 \pmod{N}\Big{\}}.\]
Consider the right quasi-regular representation of \(G\) on \(L^{2}(\Gamma(N)\backslash G)\) given by \((R(g)\phi)(x)=\phi(xg)\) for \(\phi\in L^{2}(\Gamma(N)\backslash G)\), \(g\in G\). It is well known (see [28]) to be reducible and can be decomposed as
\[L^{2}(\Gamma(N)\backslash G)=L^{2}_{\operatorname{cusp}}(\Gamma(N)\backslash G )\oplus L^{2}_{\operatorname{cont}}(\Gamma(N)\backslash G)\oplus\mathbb{C}.\]
Here \(L^{2}_{\operatorname{cusp}}(\Gamma(N)\backslash G)\) is the cuspidal part, which is a direct sum of irreducible representations with finite multiplicities, i.e.,
\[L^{2}_{\operatorname{cusp}}(\Gamma(N)\backslash G)=\sum m_{\Gamma(N)}(\pi) \cdot\pi,\,m_{\Gamma(N)}(\pi)<\infty\text{ for each }\pi,\]
and \(L^{2}_{\operatorname{cont}}(\Gamma(N)\backslash G)\) is a direct integral of irreducible representations given by the Eisenstein series.
The multiplicities \(m_{\Gamma(N)}(\pi)\) are still unknown in general, except for some special families of irreducible representations including the discrete series of \(\operatorname{SL}(2,\mathbb{R})\) (see [26] for an introduction of discrete series). Let \(S_{k}(\Gamma)\) be the space of cusp forms of weight \(k\) for a Fuchsian group \(\Gamma\). We have the following result (see [22] Theorem 2.10).
**Lemma 1.1**.: _For the discrete series \(\pi_{k}\), we have \(m_{\Gamma(N)}(\pi_{k})=\dim S_{k}(\Gamma(N))\)._
By applying the dimension formulas of cusp forms (see [14] Chapter 3.9), we obtain
\[m_{\Gamma(N)}(\pi_{k})=(\frac{k-1}{24}-\frac{1}{4N})N^{3}\prod_{p|N}(1-\frac{1} {p^{2}}) \tag{1}\]
for all \(N>2\).
On the other hand, let \(H_{k}\) be the underlying Hilbert space of the discrete series \(\pi_{k}\). As \(H_{k}\) is a module over the group \(\Gamma(N)\), we can further prove that it is also a module over the _group von Neumann algebra_\(\mathcal{L}(\Gamma(N))\) (see Section 4.1 for the definition). Hence \(H_{k}\) has a _von Neumann dimension_\(\dim_{\mathcal{L}(\Gamma(N))}H_{k}\) over \(\mathcal{L}(\Gamma(N))\). Indeed, if a discrete group \(\Gamma\) is ICC (infinite conjugacy class, see also 4.1), this dimension totally determines the equivalence class of \(\mathcal{L}(\Gamma)\)-module, i.e., \(\dim_{\mathcal{L}(\Gamma)}H_{1}=\dim_{\mathcal{L}(\Gamma)}H_{2}\) if and only if \(H_{1},H_{2}\) are isomorphic as \(\mathcal{L}(\Gamma)\)-module.
We consider a lattice \(\Gamma\) in a Lie group \(G\). Suppose \((\pi,H)\) is a discrete series representation of \(G\) and let \(d(\pi)\) be the formal dimension of \(\pi\) (see [33] Chapter 16). We have
**Lemma 1.2** (Goodman-la Harpe-Jones[23]).: \(\dim_{\mathcal{L}(\Gamma)}H=\operatorname{vol}(\Gamma\backslash G)\cdot d(\pi)\)__
By Example 3.3.4 in [23], we know \(\dim_{\mathcal{L}(\operatorname{PSL}(2,\mathbb{Z}))}H_{k}=\frac{k-1}{12}\). As \(\operatorname{SL}(2,\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})\rtimes\operatorname{PSL }(2,\mathbb{Z})\), we have \(\dim_{\mathcal{L}(\operatorname{SL}(2,\mathbb{Z}))}H_{k}=\frac{k-1}{24}\). Since \([\Gamma\colon\Gamma(n)]=N^{3}\prod_{p|N}(1-\frac{1}{24})\), we have \(\dim_{\mathcal{L}(\operatorname{SL}(2,\mathbb{Z}))}H_{k}=\frac{k-1}{24}\).
\(\frac{1}{p^{2}}\)), we can conclude
\[\dim_{\mathcal{L}(\Gamma(N))}H_{k}=\frac{k-1}{24}N^{3}\prod_{p|N}(1-\frac{1}{p^{2 }}). \tag{2}\]
Thus we obtain:
**Corollary 1.3**.: _For a discrete series \((\pi_{k},H_{k})\) of \(\operatorname{SL}(2,\mathbb{R})\), we have_
\[\lim_{N\to\infty}\frac{m_{\Gamma(N)}(\pi_{k})}{\dim_{\mathcal{L}(\Gamma(N))}H _{k}}=1.\]
**Proof**: Comparing Equations 1 and 2, we obtain \(\frac{m_{\Gamma(N)}(\pi_{k})}{\dim_{\mathcal{L}(\Gamma(N))}H_{k}}=\frac{k-1-6/ N}{k-1}\) and then take the limit.
While the explicit multiplicities of most irreducible representations are still unknown, the limit multiplicities have been studied since 1970s. In the case of towers of uniform lattices, DeGeorge and Wallach got the first results for discrete series of Lie groups [11] and later for bounded sets of irreducible representations in rank one group [12]. Delorme [13] finally solved the problem for bounded sets of irreducible representations in all Lie groups. See also [1] for a recent approach.
For the non-uniform lattices (or most arithmetic subgroups), Savin [36] obtained the results on discrete series in his thesis at first, which is based on the work by Rohlfs and Speh [34]. Then Deitmar and Hoffmann proved the results on certain towers of arithmetic subgroups in rank one group. Recently, Finis and Lapid solved the case of congruence subgroups in \(\operatorname{SL}(n,\mathbb{R})\)[20, 17], which are based on their study of the spectral side of Arthur's trace formulas [18, 16].
The goal of this paper is to extend Corollary 1.3 to some general settings. In the rest of this paper, we generalize this result mainly in the following aspects:
1. from a single discrete series representation to any bounded subset of the unitary dual \(\widehat{G}\) of \(G\);
2. from \(\operatorname{SL}(2,\mathbb{R})\) to the towers of uniform lattices in an arbitrary semisimple Lie group,
3. from \(\operatorname{SL}(2,\mathbb{R})\) to \(\operatorname{SL}(n,\mathbb{R})\) with its the principal congruence subgroups.
Finally, we are able to prove:
**Theorem 1.4** (**The Main Theorem**).: _Let \(G\) be a semisimple simply-connected Lie group. Let \(X\) be a bounded subset of the unitary dual of \(G\) and \(H_{X}\) be the direct integral of the irreducible representations of \(G\) in \(X\). We have:_
\[\lim_{n\to\infty}\frac{m_{\Gamma_{n}}(X)}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1\]
_when i) \(\Gamma_{n}\)'s are cocompact, or ii) \(G=\operatorname{SL}(n,\mathbb{R})\) and \(\{\Gamma_{n}\}\) are principal congruence subgroups._
## 2. The trace formulas and dominant terms
We have a brief review of the Arthur-Selberg trace formulas and give the dominant terms in these formulas. We mainly follow [3, 4, 19].
Let \(\mathbf{G}\) be a reductive group over \(\mathbb{Q}\). The group \(G(\mathbb{A})\) acts naturally on \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))\) by
\[R(g)\phi(x)=\phi(xg)\]
for \(\phi\in L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))\) and \(g\in G(\mathbb{A})\)). Let \(C_{\mathrm{c}}^{\infty}(G(\mathbb{A}))\) be the complex algebra of smooth, compactly supported function on \(G(\mathbb{A})\)). Given \(f\in C_{\mathrm{c}}^{\infty}(G(\mathbb{A}))\), we may define
\[(R(f)\phi)(x)=\int_{G(\mathbb{A}))}f(g)R(g)\phi(x)dg=\int_{G(\mathbb{A}))}f(g) \phi(xg)dg.\]
If we define the _kernel_
\[K(x,y)=K_{f}(x,y)\colon=\sum\limits_{\gamma\in G(\mathbb{Q})}f(x^{-1}\gamma y),\]
we have \((R(f)\phi)(x)=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}K(x,y)\phi(y)dy\).
### The Selberg trace formula
We first assume \(\mathbf{G}\) is anisotropic and hence the quotient space \(G(\mathbb{Q})\backslash G(\mathbb{A})\) is compact. Let \(\mathcal{O}\) be the set of conjugacy classes in \(G(\mathbb{Q})\) and \(o\in\mathcal{O}\) be a conjugacy class. We may define
\[K_{o}(x,y)=\sum\limits_{\gamma\in o}f(x^{-1}\gamma y)\]
and obtain \(K(x,y)=\sum\limits_{o\in\mathcal{O}}K_{o}(x,y)\). On the other hand, the representation \(R\) decomposes into a direct sum of irreducible representations with finite multiplicities, i.e., \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))=\oplus_{\chi\in\mathcal{X}}L^{2} (G(\mathbb{Q})\backslash G(\mathbb{A}))_{\chi}\). Here \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))_{\chi}=m(\chi)\cdot\chi\), which is \(m(\chi)\) copies of the irreducible representation \(\chi\). Assume \(\mathcal{B}_{\chi}\) is a orthonormal basis of \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))_{\chi}\). Then
\[K_{\chi}(x,y)=K_{f,\chi}(x,y)\colon=\sum_{\phi\in\mathcal{B}_{\chi}}(R(f)) \phi(x)\cdot\overline{\phi(y)}\]
converges. Now we let
1. \(k_{\chi}(x,f)=K_{\chi}(x,x)\) and \(J_{\chi}(f)=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}k_{\chi}(x,f)dx\),
2. \(k_{o}(x,f)=K_{o}(x,x)\) and \(J_{o}(f)=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})}k_{o}(x,f)dx\).
If we let \(\gamma\) be a representatives of \(o\in\mathcal{O}\) and \(H_{\gamma}=\{h\in H|h\gamma h^{-1}=\gamma\}\) for a group \(H\) containing \(\gamma\), we get
\[J_{o}(f)=\operatorname{vol}(G(\mathbb{Q})_{\gamma}\backslash G(\mathbb{A})_{ \gamma})\int_{G(\mathbb{A})_{\gamma}\backslash G(\mathbb{A})}f(x^{-1}\gamma x )dx.\]
**Theorem 2.1**.: _Assuming \(G(\mathbb{Q})\backslash G(\mathbb{A})\) is compact, we have_
\[\operatorname{tr}R(f)=\sum_{o\in O}J_{o}(f)=\sum_{\chi\in\mathcal{X}}J_{\chi}(f) \tag{3}\]
_for any \(f\in C_{c}^{\infty}(G(\mathbb{A}))\)._
For the classical setting, we start with a real Lie group \(G\) and a lattice \(\Gamma\subset G\). Consider the representation \(R_{\Gamma}\) of \(G\) on \(L^{2}(\Gamma\backslash G)\) given by \((R_{\Gamma}(g)\phi)(x)=\phi(xg)\) for \(x,g\in G\). Let \(C_{c}^{\infty}(G)\) be the space of smooth function on \(G\) with compact support. For \(f\in C_{c}^{\infty}(G)\) and a representation \((\pi,H)\) of \(G\), we let
\[\pi(f)v=\int_{G}f(g)\pi(g)vdg.\]
If \(\pi\) is irreducible, \(\pi(f)\) is a trace class operator and we let \(\theta_{\pi}(f)=\operatorname{tr}\pi(f)\). Note for the representation \(R_{\Gamma}\), we have \((R_{\Gamma}(f)\phi)(x)=\int_{G}f(g)R_{\Gamma}(g)\phi(x)dg=\int_{G}f(g)\phi(xg)dg\).
It is known that \(\Gamma\backslash G\) is compact if and only if the reductive part of \(\mathbf{G}\) is anisotropic (see [32] Theorem 4.12). In this case, \(L^{2}(\Gamma\backslash G)\) can be decomposed into a direct sum of irreducible representations of \(G\) with each of finite multiplicity, i.e.,
\[L^{2}(\Gamma\backslash G)=\oplus m_{\Gamma}(\pi)\cdot\pi\]
with \(m_{\Gamma}(\pi)=\dim\operatorname{Hom}_{G}(\pi,L^{2}(\Gamma\backslash G))<\infty\) for each \(\pi\).
By taking the test function in Theorem 2.1 to be \(f\otimes 1_{K}\) for a maximal compact subgroup \(K\) of \(G(\mathbb{A}^{\operatorname{fin}})\) with \(f\in C_{c}^{\infty}(G)\) (see Section 3.1), we get the following result for the lattice \(\Gamma\) in the real Lie group \(G\).
**Corollary 2.2** (The Selberg trace formula).: _If \(\Gamma\backslash G\) is compact, \(R_{\Gamma}(f)\) is of trace class and_
\[trR_{\Gamma}(f)=\sum_{\pi\in\widehat{G}}m_{\Gamma}(\pi)\theta_{\pi}(f)=\sum_{ \gamma\in[\Gamma]}\operatorname{vol}(\Gamma_{\gamma}\backslash G_{\gamma}) \int_{\Gamma_{\gamma}\backslash G}f(x^{-1}\gamma x)dx \tag{4}\]
### The Arthur trace formula
We assume \(\mathbf{G}\) is not necessarily anisotropic and \(G(\mathbb{Q})\backslash G(\mathbb{A})\) may not be compact. Assume \(B\) is a Borel subgroup defined over \(\mathbb{Q}\), \(M_{0}\) is a Levi factor of \(B\) defined over \(\mathbb{Q}\), \(P\) is a standard parabolic subgroup defined over \(\mathbb{Q}\) (i.e., \(P_{0}=B\subset P\)), \(N_{P}=R_{u}(P)\) (the unipotent radical of \(P\)), \(M_{P}\) is the unique Levi component of \(P\) such that \(M_{0}\subset M_{P}\).
We also assume \(A_{P}\) is the split component of the center of \(M_{P}\) and \(Z=A_{G}\), \(\Delta_{0}=\Delta_{B}\) is a base for a root system. We will mostly use the notations of [3, 4, 6] and [21] as follows:
* \(\mathfrak{a}_{P}=\operatorname{Hom}(X(M_{P})_{\mathbb{Q}},\mathbb{R})\) where \(X(M_{P})_{\mathbb{Q}}\) is the \(\mathbb{Q}\)-characters of \(M_{P}\), \(\mathfrak{a}_{P}^{*}=X(M_{P})_{\mathbb{Q}}\otimes\mathbb{R}\) and \(\mathfrak{a}_{P}^{+}=\{H\in\mathfrak{a}_{P}|\alpha(H)>0,\forall\alpha\in \Delta_{P}\}\).
* \(\gamma=\gamma_{s}\gamma_{u}\), which is the decomposition such that \(\gamma_{s}\) is semisimple and \(\gamma_{u}\) is unipotent.
* \(\mathcal{O}\) is the set of \(G(\mathbb{Q})\)-semisimple conjugacy class of \(G(\mathbb{Q})\) (\(\gamma\cong\beta\) if \(\gamma_{s}\) and \(\beta_{s}\) are \(G(\mathbb{Q})\)-conjugate).
* \(o\in\mathcal{O}\) is a conjugacy class in \(G(\mathbb{Q})\).
* \(\mathcal{X}\) is the set of equivalence classes of pairs \((M,\rho)\), where \(M\) is a Levi subgroup of \(G\) and \(\rho\in\widehat{M(\mathbb{A})}^{1}\) (\((M,\rho)\sim(M^{\prime},\rho^{\prime})\) if there is an \(s\in\Omega(\mathfrak{a},\mathfrak{a}^{\prime})\) such that the representation \((s\rho)(m^{\prime})=\rho(w_{s}^{-1}mw_{s})\) is unitarily equivalent to \(\rho^{\prime}\)).
* For a pair of parabolic groups \(P_{1}\subset P_{2}\), \(\Delta_{P_{1}}^{P_{2}}\) is the set of simple roots of \((M_{P_{2}}\cap P_{1},A_{P_{1}})\) and \(\hat{\Delta}_{P_{1}}^{P_{2}}=\{\varpi_{\alpha}|\alpha\in\Delta_{P_{1}}^{P_{2}}\}\), i.e, the dual basis for \(\Delta_{P_{1}}^{P_{2}}\).
* \(\hat{\tau}_{P}\) is the characteristic function on \(\mathfrak{a}_{0}\) of \(\{H\in\mathfrak{a}_{0}|\varpi(H)>0,\varpi\in\hat{\Delta}_{P}^{G}\}\).
* For \(m=\prod_{v}m_{v}\in M(\mathbb{A})\), let \(H_{M}(m)\in\mathfrak{a}_{p}\) given by \[e^{\langle H_{M}(m),\chi\rangle}=|\chi(m)|=\prod_{v}|\chi(m_{v})|_{v},\,\forall \chi\in X(M)_{\mathbb{Q}}.\]
* \(x=nmak\in G(\mathbb{A})\) with \(n\in G(\mathbb{A}),m\in M(\mathbb{A})^{1},a\in A(\mathbb{R})^{0}\) and \(k\in K\).
* \(H(x)=H_{M}(ma)=H_{M}(a)\in\mathfrak{a}_{p}\).
Let \(T\in\mathfrak{a}_{0}^{+}\) be suitably regular, i.e., \(\alpha(T)\) is sufficiently large for all \(\alpha\in\Delta_{0}\). For a parabolic subgroup \(P\), there are kernels \(K_{P,o}=\sum\limits_{\gamma\in M(\mathbb{Q})\cap o}\int_{N(\mathbb{A})}f(x^{ -1}\gamma ny)dn\) and \(K_{P,\chi}\) (see [3] p.923 and p.935 for the precise definitions). Then Arthur is able to define the _truncated kernels_ and distributions \(J_{o}^{T},J_{\chi}^{T}\) as follows:
1. \(k_{o}^{T}(x,f)=\sum\limits_{P}(-1)\dim(A_{P}/Z)\sum\limits_{\delta\in P( \mathbb{Q})\backslash G(\mathbb{Q})}K_{P,o}(\delta x,\delta x)\cdot\hat{\tau}_{p }(H(\delta x)-T)\).
2. \(k_{\chi}^{T}(x,f)=\sum\limits_{P}(-1)\dim(A_{P}/Z)\sum\limits_{\delta\in P( \mathbb{Q})\backslash G(\mathbb{Q})}K_{P,\chi}(\delta x,\delta x)\cdot\hat{\tau}_{ p}(H(\delta x)-T)\).
3. \(J_{o}^{T}(f)=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})^{1}}k_{o}^{T}(x,f)dx\).
4. \(J_{\chi}^{T}(f)=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})^{1}}k_{\chi}^{T}(x,f)dx\).
Let \(\mathcal{X}(G)=\{(M,\rho)\in\mathcal{X}|M=G\}\). We reach a coarse trace formula, which is firstly given in [4] Chapter 5.
**Theorem 2.3** (The Arthur trace formula).: _For any \(f\in C_{c}^{\infty}(G(\mathbb{A})^{1})\) and any suitably regular \(T\in\mathfrak{a}_{0}^{+}\), we have_
\[\sum_{o\in\mathcal{O}}J_{o}^{T}(f)=\sum_{\chi\in\mathcal{X}}J_{\chi}^{T}(f) \tag{5}\]
_Moreover, the trace formula of \(R(f)\) is given by_
\[\operatorname{tr}R_{\text{cusp}}(f)=\sum_{o\in\mathcal{O}}J_{o}^{T}(f)-\sum_{ \chi\in\mathcal{X}\backslash\mathcal{X}(G)}J_{\chi}^{T}(f).\]
### The dominant term on the geometric side
Consider the adelic case at first. Let \(F\) be a number field and \(V,V_{\infty}\) and \(V_{f}\) be the set of places, Archimedean and non-Archimedean places of \(F\) respectively. Let \(\mathbb{A}\) be adele ring of \(F\) and \(A_{\text{fin}}\subset\mathbb{A}\) be restricted product over the finite places.
Suppose \(S\subset V\) is a finite set containing \(V_{\infty}\). Let \(F_{S}=\prod_{v\in S}F_{s}\) and \(\mathbb{A}^{S}=\prod_{v\in V\backslash S}^{\prime}F_{s}\) so that \(\mathbb{A}=F_{S}\times\mathbb{A}^{S}\). We define
1. \(G(F_{S})^{1}=\bigcap_{\chi\in\text{Hom}(G(F_{S}),F^{\times})}\{\ker|\chi| \colon G(F_{S})\to\mathbb{R}_{+}\}\),
2. \(G(\mathbb{A})^{1}=\bigcap_{\chi\in\text{Hom}(G(\mathbb{A}),F^{\times})}\{\ker| \chi|\colon G(\mathbb{A})\to\mathbb{R}_{+}\}\),
where \(|\cdot|\) is the product of valuations on \(F_{S}\) and \(\mathbb{A}\) respectively.
We will consider the representation of \(G(F_{S})\) on \(L^{2}(G(F)\backslash G(\mathbb{A})^{1}/K)\) for an open compact subgroup \(K\) of \(G(\mathbb{A}^{S})\). In particular, it will reduce to the representation of \(G(F_{\infty})\) on \(L^{2}(\Gamma_{K}\backslash F_{\infty})\) if we take \(S=\{\infty\}\) and \(\Gamma_{K}=G(F)\cap K\).
Let \(J(f)\) be the distribution defined by Equation 3 or 5 in Section 2 for \(f\in C_{c}^{\infty}(G(F_{S}))\), which also depends on \(G(F)\backslash G(\mathbb{A})^{1}\) is compact or not. The goal of this subsection is to prove
\[\lim_{n\to\infty}\frac{\operatorname{vol}(G(F)\backslash G(\mathbb{A})^{1})f( 1)}{J(f\otimes 1_{K_{n}})}=1\]
for certain towers of open compact subgroups \(\{K_{n}\}_{n\geq 1}\) of \(G(\mathbb{A}^{S})\).
Let us assume \(\Gamma\) is a uniform lattice in the semisimple Lie group \(G\). We add a subscript such as \(R_{\Gamma}\) and \(J_{\Gamma}\) for the representation of \(G\) on \(L^{2}(\Gamma\backslash G)\) and the corresponding trace formulas as an emphasis on the lattice \(\Gamma\). Since \(\Gamma\backslash G\) is compact, \(J_{\Gamma}(f)\) is the trace \(\operatorname{tr}R_{\Gamma}(f)\) and we obtain
\[J_{\Gamma}(f)=\operatorname{tr}R_{\Gamma}(f)=\sum_{\pi\in\bar{G}}J_{\pi, \Gamma}(f)=\sum_{o\in\mathcal{O}}J_{o,\Gamma}(f).\]
Let \(J_{\{1\},\Gamma}(f)=\operatorname{vol}(\Gamma\backslash G)f(1)\), the contribution of the identity to the geometric side of the trace formula.
We take a tower of uniform lattices \(\{\Gamma_{n}\}_{n\geq 1}\), such that \(\Gamma_{n}\trianglelefteq\Gamma_{1}\), \([\Gamma_{1}:\Gamma_{n}]<\infty\) and \(\cap_{n\geq 1}\Gamma_{n}=\{1\}\).
**Proposition 2.4**.: _With the assumption of uniform lattice \(\{\Gamma_{n}\}\) above, we have_
\[\lim_{n\to\infty}\frac{J_{\{1\},\Gamma_{n}}(f)}{J_{\Gamma_{n}}(f)}=1.\]
**Proof**: Following the Equation (2) in [10], we obtain
\[\operatorname{tr}R_{\Gamma_{n}}(\phi)=J_{\{1\},\Gamma_{n}}(\phi)+\sum_{\gamma \neq 1}s_{n}(\gamma)\operatorname{vol}(\Gamma_{j}\backslash G)\operatorname{vol }(\Gamma_{\gamma}\backslash G_{\gamma})\int_{\Gamma_{\gamma}\backslash G} \phi(x^{-1}\gamma x)dx,\]
where \(0\leq s_{n}(\gamma)\leq\operatorname{vol}(\Gamma\backslash G)^{-1}\). As \(\cap_{n\geq 1}\Gamma_{n}=\{1\}\), \(\lim_{n\to\infty}s_{n}(\gamma)\) for all \(\gamma\neq 1\). By [10] Theorem 2, we have \(\operatorname{vol}(\Gamma_{n}\backslash G)^{-1}\cdot\lim_{n\to\infty} \operatorname{tr}R_{\Gamma_{n}}(\phi)=\phi(1)\). Hence \(\lim_{n\to\infty}\frac{J_{\{1\},\Gamma_{n}}(\phi)}{J_{\Gamma_{n}}(\phi)}=\lim_ {n\to\infty}\frac{J_{\{1\},\Gamma_{n}}(\phi)}{\operatorname{tr}R_{\Gamma_{n}}( \phi)}=1\).
Now we let \(G\) be a reductive group over a number field \(F\). Let \(K=K_{\infty}K_{\rm fin}\) be a maximal compact subgroup of \(G(\mathbb{A})=G(\mathbb{A}_{F})\). By fixing a faithful \(F\)-rational representation \(\rho\colon G(F)\to{\rm GL}(m,F)\) for some \(m>0\), we let \(\Lambda\subset F^{m}\) be an \(\mathcal{O}_{F}\)-lattice such that the stablilizer of \(\widehat{\Lambda}=\mathcal{O}_{F}\otimes_{F}\Lambda\) in \(G(A_{\rm fin})\) is \(K_{\rm fin}\).
For a non-trivial ideal \(I\) of \(\mathcal{O}_{F}\), we let
\[K(I)=\{g\in G(A_{\rm fin})|\rho(g)v\equiv v\pmod{I\cdot\widehat{\Lambda}},v \in\widehat{\Lambda}\}\]
be the _principal congruence subgroup_ of level \(I\). We also denote the ideal norm of \(I\) by \(N(I)=[\mathcal{O}_{F}\colon I]\). Consider a descending tower of ideals \(I_{1}\supsetneq I_{2}\supsetneq I_{3}\supsetneq\cdots\) such that each \(I_{k}\) is prime to (the prime ideals in) \(S\). We obtain the corresponding tower of principal congruence subgroups:
\[K_{1}\supsetneq K_{2}\supsetneq K_{3}\supsetneq\cdots,\]
where \(K_{n}=K(I_{n})\). By factoring into prime ideals, the family \(\{I_{n}\}_{n\geq 1}\) satisfies either one of the following properties:
1. there exists a prime ideal \(\mathfrak{p}\) such that each \(\mathfrak{p}^{k}\) is eventually contained in the tower, i.e., for any \(k\geq 1\), there is \(N_{k}>0\) such that \(\mathfrak{p}^{k}\subset I_{n}\) for all \(n\geq n_{k}\), or,
2. there exists infinitely many prime ideals \(\{\mathfrak{p}_{k}\}_{k\geq 1}\) such that for each \(k\), there exist \(M_{k}>0\) such that \(\mathfrak{p}_{k}\subset I_{n}\) for all \(n\geq M_{k}\).
In either of these two cases, we have
**Lemma 2.5**.: \(\cap_{n\geq 1}I_{n}=\{0\}\) _and \(\cap_{n\geq 1}K_{n}=\{1\}\)._
Recall the equivalence class of unipotent elements in \(G(F)\), which is the element \(\gamma=\gamma_{s}\gamma_{u}\) with the semisimple component \(r_{s}=1\) (see [5] p.1240). Let
\[J_{\rm unip}^{T}(f),\,f\in C_{c}^{\infty}(G(\mathbb{A})^{1}).\]
be the contribution of this equivalence class on the geometric side of the trace formula 5. We will consider the function of the form \(f=h_{S}\otimes 1_{K_{n}}\) with \(h_{S}\in C_{c}^{\infty}(G(F_{S})^{1})\).
**Lemma 2.6**.: _For \(h_{S}\in C_{c}^{\infty}(G(F_{S})^{1})\), \(\lim_{n\to\infty}J(h_{S}\otimes 1_{K_{n}})=\lim_{n\to\infty}J_{unip}(h_{S} \otimes 1_{K_{n}})\)._
**Proof**: Let \(D_{h}={\rm supp}(h_{S})\subset G(F_{S})^{1}\) be the compact support of \(h_{S}\). Then \({\rm supp}(h_{S}\otimes 1_{K_{n}}))=D_{h}K_{n}\) is compact and hence it intersects finitely many semisimple-conjugate class \(o\in\mathcal{O}\).
Consider the trace formula and Equation 5, only the classes \(o\)'s (and its \(G(\mathbb{A})\)-conjugations) which intersect infinitely many \(D_{h}K_{n}\) contributes a non-trivial \(J_{o}(h_{S}\otimes 1_{K_{n}})\) to the limit \(\lim_{n\to\infty}J(h_{S}\otimes 1_{K_{n}})\).
Suppose the \(G(\mathbb{A})\) conjugacy classes of elements in \(o\) intersects \(D_{h}K_{n}\) for infinitely many \(n\), i.e., \(\{g\gamma g^{-1}|g\in G(\mathbb{A}),\gamma\in o\}\cap D_{h}K_{n}\neq\emptyset\) for infinitely many \(n\). Take some \(\gamma\in o\). By fixing a faithful \(F\)-representation \(\rho\colon G(F)\to{\rm GL}(m)\), we let \(p(x)\in F[x]\) be the characteristic polynomial of \(\rho(\gamma)-1\) (a \(m\)-by-\(m\) matrix over \(F\)). Suppose \(p(x)=x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}\) with all \(a_{i}\in F\). By Lemma 2.5, we know \(a_{i}\) belongs to infinitely many \(I_{n}\), or, equivalently \(a_{i}=0\). Hence \(p(x)=x^{m}\) and \(\gamma\) is unipotent.
The unipotent contribution \(J_{\rm unip}(h_{S}\otimes 1_{K_{n}})\) can be further reduced to the the one from the identity as follows. We let \(I_{S}\) be a product of prime ideals in at the places of \(S\) and \(K_{S-S_{\infty}}(I_{S})\) be the \(S-S_{\infty}\) component of the compact group \(K(I_{S})\).
We also let \(C_{\Omega}^{\infty}(G(F_{S})^{1})\) be the set of smooth functions with compact support contained in a compact subset \(\Omega\) of \(G(F_{S})^{1}\). For each \(k\geq 0\), we let \(\mathcal{B}_{k}\) be the \(k\)-th
component of the universal enveloping algebra \(\mathcal{U}(\mathfrak{g}_{\mathbb{C}})\), where \(\mathfrak{g}_{\mathbb{C}}\) is the complexified Lie algebra of the Lie group \(G(F_{\infty})\). We set \(\|h\|_{k}=\sum_{X\in\mathcal{B}_{k}}\|X\circ h\|_{L^{1}(G(\mathbb{A})^{1}}\) for \(h\in C_{\Omega}^{\infty}(G(F_{S})^{1})\).
The following result is a special case of Proposition 3.1 in [20], whose proof is mainly based on Theorem 3.1 and Theorem 4.2 in [5].
**Proposition 2.7** (Finis-Lapid-Muller).: _There exists an integer \(k\geq 0\) such that for any compact subset \(\Omega\) of \(G(F_{S})^{1}\), we have a constant \(C_{\Omega}>0\) and_
\[|J_{\text{unip}}(h_{S}\otimes 1_{K_{n}})-\operatorname{vol}(G(F)\backslash G( \mathbb{A})^{1})h_{S}(1)|\leq C_{\Omega}\tfrac{(1+\log N(I_{S}I))^{d_{0}}}{N(I )}\|h_{S}\|_{k}\]
_for any bi-\(K_{S-S_{\infty}}(I_{S})\)-invariant function \(h_{S}\in C_{\Omega}^{\infty}(G(F_{S})^{1})\)._
Then, combining Lemma 2.6 we can obtain:
**Corollary 2.8**.: _For \(h_{S}\in C_{c}^{\infty}(G(F_{S})^{1})\), we have_
\[\lim_{n\to\infty}\frac{\operatorname{vol}(G(F)\backslash G(\mathbb{A})^{1})h_ {S}(1)}{J(h_{S}\otimes 1_{K^{S}(n)})}=1.\]
## 3. The multiplicities problem
This section is devoted to the multiplicity of bounded subsets of the unitary dual, instead of a single irreducible representation.
### The multiplicities in \(L^{2}(\Gamma\backslash G)\)
Let \(G=\mathbf{G}(\mathbb{R})^{0}\), the connected component of the real group obtained from an almost simple group \(\mathbf{G}\) over \(\mathbb{Q}\). By fixing a faithful \(\mathbb{Q}\)-embedding \(\rho:\mathbf{G}\to GL_{n}\), we have an arithmetic group \(\Gamma\) commensurable with \(G\cap\operatorname{GL}_{n}(\mathbb{Z})\). Let \(\widehat{G}\) be the unitary dual of \(G\) and \(\widehat{G}_{\text{temp}}\subset\widehat{G}\) be the tempered dual. Let us consider the following two cases.
1. \(\Gamma\backslash G\)_is compact_. As introduced in Section 2.1, \(L^{2}(\Gamma\backslash G)\) can be decomposed into a direct sum of irreducible representations of \(G\) with each of finite multiplicity, i.e.,
\[L^{2}(\Gamma\backslash G)=\oplus m_{\Gamma}(\pi)\cdot\pi\]
with \(m_{\Gamma}(\pi)\colon=\dim\operatorname{Hom}_{G}(\pi,L^{2}(\Gamma\backslash G ))<\infty\) for each \(\pi\in\widehat{G}\).
2. \(\Gamma\backslash G\)_is not compact_. If \(G\) is semisimple, we have \(\Gamma\backslash G\) is of finite (Haar) measure (see [31] Theorem 4.13). The regular representation has both discrete and continuous spectra: \(L^{2}(\Gamma\backslash G)=L^{2}_{\text{disc}}(\Gamma\backslash G)\oplus L^{2}_ {\text{disc}}(\Gamma\backslash G)\). The discrete spectrum can be written as the direct sum of cuspidal and residue subspaces: \(L^{2}_{\text{disc}}(\Gamma\backslash G)=L^{2}_{\text{cusp}}(\Gamma\backslash G )\oplus L^{2}_{\text{res}}(\Gamma\backslash G)\). which can be decomposed further into a direct sum of irreducible representations with finite multiplicities, i.e.,
\[L^{2}_{\text{disc}}(\Gamma\backslash G)=\oplus m_{\Gamma}(\pi)\cdot\pi\]
with \(m_{\Gamma}(\pi)\colon=\dim\operatorname{Hom}_{G}(\pi,L^{2}_{\text{disc}}( \Gamma\backslash G))=\dim\operatorname{Hom}_{G}(\pi,L^{2}(\Gamma\backslash G))\) is finite for each \(\pi\in\widehat{G}\).
We say \(X\subset\widehat{G}\) is _bounded_ if it is relatively compact under the Fell topology.
**Definition 3.1** (The multiplicity for \(X\subset\widehat{G}\)).: For a bounded \(X\subset\widehat{G}\), we define the _multiplicity of \(X\)_ to be the sum of the multiplicities of the irreducible representations in \(X\), i.e.,
\[m_{\Gamma}(X)\colon=\sum_{\pi\in X}m_{\Gamma}(\pi).\]
Borel and Garland proved the finiteness of \(m_{\Gamma}(X)\) by considering the spectrum of a certain Laplacian (see [9] Theorem 3, Theorem 4.6 and also [24] Theorem 1.1.3).
**Theorem 3.1** (Borel-Garland).: _Let \(G={\bf G}(\mathbb{R})^{0}\) for a connected semisimple group \({\bf G}\) over \(\mathbb{Q}\) and \(X\subset\widehat{G}\) is bounded. We have \(m_{\Gamma}(X)<\infty\)._
For a subset \(X\subset\widehat{G(F_{S})}^{1}\), we call it _bounded_ if it is relatively compact under the Fell topology (see [35]).
**Definition 3.2** (The multiplicity for \(\widehat{G(F_{S})}^{1}\)).: Suppose \(K\) is a compact open subgroup of \(G(\mathbb{A}^{S})\). Let \(\sigma\) be an irreducible representation of \(G(F_{S})^{1}\) and \(X\subset\widehat{G(F_{S})}^{1}\) be a bounded subset.
1. The _multiplicity of \(\sigma\) with respect to \(K\)_ is defined as \[m_{K}(\sigma)\colon=\dim\operatorname{Hom}_{G(F_{S})^{1}}(\sigma,L^{2}(G( \mathbb{Q})\backslash G(\mathbb{A})^{1}/K)).\]
2. The _multiplicity of \(X\) with respect to \(K\)_ is defined as \[m_{K}(X)\colon=\sum_{\sigma\in X}m_{K}(\sigma).\]
For an irreducible representation \(\pi\) of \(G(\mathbb{A})^{1}\), we write \(\pi=\pi_{S}\otimes\pi^{S}\), where \(\pi_{S}\) and \(\pi^{S}\) denote the components of the representations of \(G(F_{S})^{1}\) and \(G(\mathbb{A}^{S})\) respectively. As shown in Theorem 3.1, \(m_{K}(X)\) is finite and hence well-defined. If we treat \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1}/K)\) as the subspace of \(K\)-right invariant functions in \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1}))\), we have
\[m_{K}(\sigma)=\sum_{\pi\in\widehat{G(\mathbb{A})}^{1},\pi_{S}=\sigma}\dim \operatorname{Hom}_{G(\mathbb{A})^{1}}(\pi,L^{2}(G(\mathbb{Q})\backslash G( \mathbb{A})^{1}))\dim(\pi^{S})^{K}.\]
If we take \(S=V_{\infty}\) and \({\bf G}\) is semisimple, simply connected, and without any \(F\)-simple factors \(H\) such that \(H(F_{\infty})\) is compact and \(K\) is an open compact subgroup of \(G(\mathbb{A}_{\text{fin}})\), we know \(\Gamma_{K}=G(\mathbb{F})\cap K\) is a lattice in the seimisimple Lie group \(G(F_{\infty})\).
**Lemma 3.2**.: _With the assumption above, we have \(m_{\Gamma_{K}}(\pi)=m_{K}(\pi)\) for any \(\pi\in\widehat{G(F_{\infty})}^{1}\) and \(m_{\Gamma_{K}}(X)=m_{K}(X)\) for any bounded \(X\subset\widehat{G(F_{\infty})}^{1}\)_
**Proof**: It follows the fact \(G(\mathbb{Q})\backslash G(\mathbb{A})/K\) can be identified with \(\Gamma_{K}\backslash G(F_{\infty})\), which leads to a \(G(F_{\infty})\)-isomorphism \(L^{2}(\Gamma_{K}\backslash G(F_{\infty}))\cong L^{2}(G(\mathbb{Q})\backslash G (\mathbb{A})^{1}/K)\) (see [27] Chapter 6 and [32] Chapter 7.4.).
For a finite set \(S\) and a function \(\phi\) on \(\widehat{G(F_{S})}^{1}\), we define
\[m_{K}(\phi)\colon=\int_{\widehat{G(F_{S})}^{1}}\phi(\pi)dm_{K}(\pi)\]
as its integral with respect to the measure given by multiplicities above. If \(1_{X}\) is the characteristic function of \(X\), i.e., \(1_{X}(\pi)=1\) if \(\pi\in X\) and \(0\) otherwise, \(m_{K}(1_{X})=m_{K}(X)\).
For \(f\in C_{\text{c}}^{\infty}(G(F_{S})^{1})\), we let \(\widehat{f}(\pi)=\operatorname{tr}\pi(f)\), the distribution character of \(\pi\). Let \(R_{\text{disc}}\) denote the action of \(G(\mathbb{A})\) on the discrete subspace \(L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1})\).
**Proposition 3.3**.: _For \(f\in C_{\text{c}}^{\infty}(G(F_{S})^{1})\), we have_
\[\operatorname{tr}R_{disc}(f\otimes\tfrac{1_{K}}{\operatorname{vol}(K)})=m_{K}( \hat{f}).\]
**Proof**: Observe for the component \(\pi^{S}\) of representation of \(G(\mathbb{A}^{S})\), we have
\[\operatorname{tr}\pi^{S}(1_{K}) =\int_{G(\mathbb{A}^{S})}1_{K}(x)\pi^{S}(x^{-1})d\mu^{S}(x)\] \[=\int_{K}\pi^{S}(x^{-1})d\mu^{S}(x)=\operatorname{vol}(K)\dim(\pi^ {S})^{K},\]
where we apply the fact that \(\int_{K}\sigma(x)d\mu^{S}(x)=0\) for any non-trivial irreducible representation \(\sigma\) of \(K\). Hence we obtain
\[\operatorname{tr}R_{\operatorname{disc}}(f\otimes\frac{1_{K}}{ \operatorname{vol}(K)}) =\frac{1}{\operatorname{vol}(K)}\sum_{\pi\in\widehat{G(\mathbb{A}) }^{1}}m(\pi)\operatorname{tr}\pi(f\otimes 1_{K})\] \[=\frac{1}{\operatorname{vol}(K)}\sum_{\pi\in\widehat{G(\mathbb{A} )}^{1}}m(\pi)\operatorname{tr}\pi_{S}(f)\operatorname{tr}\pi^{S}(1_{K})\] \[=\frac{1}{\operatorname{vol}(K)}\sum_{\pi\in\widehat{G(\mathbb{A} )}^{1}}m(\pi)\operatorname{tr}\pi_{S}(f)\operatorname{vol}(K)\dim(\pi^{S})^{K}\] \[=\sum_{\sigma\in\widehat{G(F_{S})}^{1}}m_{K}(\sigma) \operatorname{tr}\sigma(f)=m_{K}(\widehat{f}).\]
We also give the following result which connects the trace formulas for adelic groups and Lie groups.
**Corollary 3.4**.: _Let \(\Gamma_{K}=G(F)\cap K\) with an open compact subgroup \(K\) of \(G(\mathbb{A}_{\operatorname{fin}})\). We have_
\[\operatorname{tr}R_{\operatorname{disc}}(f\otimes\tfrac{1_{K}}{\operatorname {vol}(K)})=\operatorname{tr}R_{\Gamma_{K}}(f).\]
_for all \(f\in C_{c}^{\infty}(G(F_{\infty})^{1})\)._
**Proof**: It follows the fact \(m_{K}(\widehat{f})=m_{\Gamma_{K}}(\widehat{f})\) in Lemma 3.2, \(m_{\Gamma_{K}}(\widehat{f})=\operatorname{tr}R_{\Gamma_{K}}(f)\) and Proposition 3.3.
### Sauvageot's density theorems
We have a brief review of the results in [35]. See also [37] for an alternative approach and corrections.
For an open compact subgroup \(K\) of \(G(\mathbb{A}^{S})\), we define a measure on \(\widehat{G(F_{S})}^{1}\) by
\[\nu_{K}(X)\colon=\tfrac{\operatorname{vol}(K)}{\operatorname{vol}(G(\mathbb{Q} )\setminus\widehat{G(\mathbb{A})}^{1})}m_{K}(X)\]
for any bounded subset \(X\) of \(\widehat{G(F_{S})}^{1}\) and \(m_{K}\) is the multipilicity defined in Chapter 3.1.
Let \(K_{1}\supsetneq K_{2}\supsetneq\cdots\) be a sequence of open compact subgroups of \(G(\mathbb{A}^{S})\). Given a bounded subset \(X\) of \(\widehat{G(F_{S})}^{1}\) and \(C\geq 0\), we write
\[\lim_{n\to\infty}\nu_{K}(X)=C,\]
if for any \(\varepsilon>0\), there exists \(N=N(\varepsilon)>0\) such that \(|\nu_{K_{n}}(X)-C|<\varepsilon\) for all \(n\geq N\).
Let \(\mathcal{H}(G(F_{S})^{1})\) be the complex algebra of smooth, compactly-supported, bi-\(K_{S}\)-finite functions on \(G(F_{S})^{1}\).
**Lemma 3.5** ([35] Corollaire 6.2).: _For \(\varepsilon>0\) and any bounded \(X\subset\widehat{G(F_{S})}^{1}\setminus\widehat{G(F_{S})}^{1}{}_{temp}\), there is \(\Psi\in\mathcal{H}(G(F_{S})^{1})\) such that_
\[\widehat{\Psi}|_{\widehat{G(F_{S})}^{1}}\geq 0\text{, }\nu(\widehat{\Psi})<\varepsilon \text{ and }\widehat{\Psi}|_{X}\geq 1.\]
Given a function \(f\) defined on \(\widehat{G(F_{S})}^{1}{}_{temp}\), we also denote by \(f\) the function on \(\widehat{G(F_{S})}^{1}\), which is extended by \(0\) on the untempered part.
**Lemma 3.6** ([35] Theoreme 7.3(b)).: _For \(\varepsilon>0\) and any \(\nu\)-integrable function \(f\) on \(\widehat{G(F_{S})^{1}}_{\text{temp}}\), there exists \(\phi,\psi\in\mathcal{H}(G(F_{S})^{1})\) such that_
\[|f(\pi)-\widehat{\phi}(\pi)|\leq\widehat{\psi}(\pi)\text{ and }\nu(\widehat{\psi})<\varepsilon.\]
Here we obtain one of the main results in [35] and we also provide a proof for completeness.
**Theorem 3.7** (Sauvageot).: _Suppose \(\lim_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)\) for all \(\phi\in\mathcal{H}(G(F_{S})^{1})\). We have_
\[\lim_{n\to\infty}\nu_{K_{n}}(X)=\nu(X)\]
_for all bounded subset \(X\) of \(\widehat{G(F_{S})^{1}}\)._
**Proof**: First, we show the contribution from the untempered part is negligible in the limit. For a bounded subset \(X_{0}\) of \(\widehat{G(F_{S})^{1}}_{\text{temp}}\) and \(\varepsilon>0\), we let \(\Psi\in\mathcal{H}(G(F_{S})^{1})\) satisfies Lemma 3.5 with respect to \(X\). We have \(\nu_{K_{n}}(X)\leq\nu_{K_{n}}(\widehat{\Psi})\leq|\nu_{K_{n}}(\widehat{\Psi})- \psi(1)|+\psi(1)<2\varepsilon\) for all \(n\geq N_{1}\) with some \(N_{1}\geq 0\).
For the tempered part, we fix a bounded subset \(X_{1}\) of \(\widehat{G(F_{S})^{1}}_{\text{temp}}\) with the same \(\varepsilon\) above. Let \(\phi,\psi\in\mathcal{H}(G(F_{S})^{1})\) satisfy Lemma 3.6 with respect to the function \(f=1_{X_{1}}\) on \(\widehat{G(F_{S})^{1}}_{\text{temp}}\) and \(\varepsilon\). By assumption, we have \(|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|<\varepsilon\) and \(|\nu_{K_{n}}(\widehat{\psi})-\psi(1)|<\varepsilon\) for all \(n\geq N_{2}\) with some \(N_{2}\geq 0\). Hence, for \(n\geq N_{2}\), we obtain
\[|\nu_{K_{n}}(X_{1})-\nu(X_{1})| \leq|\nu_{K_{n}}(X_{1})-\nu_{K_{n}}(\widehat{\phi})|+|\nu_{K_{n} }(\widehat{\phi})-\phi(1)|+|\phi(1)-\nu(X)|\] \[\leq|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|+\nu_{K_{n}}(\widehat{ \psi})+\psi(1)\] \[\leq|\nu_{K_{n}}(\widehat{\phi})-\phi(1)|+|+|\nu_{K_{n}}(\widehat {\psi})-\psi(1)|+2\psi(1)<4\varepsilon.\]
Hence, for the bounded set \(X\) of \(\widehat{G(F_{S})^{1}}\), let \(X=X_{0}\sqcup X_{1}\) be the decomposition into its untempered and tempered parts. We have
\[|\nu_{K_{n}}(X)-\nu(X)| =|\nu_{K_{n}}(X)-\nu(X_{1})|=|\nu_{K_{n}}(X_{1})-\nu(X)|+\nu_{K_{ n}}(X_{0})\] \[\leq 4\varepsilon+2\varepsilon=6\varepsilon\]
for all \(N\geq\max\{N_{1},N_{2}\}\).
## 4. The von Neumann dimensions of direct integrals
### The group von Neumann algebra and the trace
Let \(\Gamma\) be a countable group with the counting measure. Let \(\{\delta_{\gamma}\}_{\gamma\in\Gamma}\) be the usual orthonormal basis of \(l^{2}(\Gamma)\). We also let \(\lambda\) and \(\rho\) be the left and right regular representations of \(\Gamma\) on \(l^{2}(\Gamma)\) respectively. For all \(\gamma,\gamma^{\prime}\in\Gamma\), we have \(\lambda(\gamma^{\prime})\delta_{\gamma}=\delta_{\gamma^{\prime}\gamma}\) and \(\rho(\gamma^{\prime})\delta_{\gamma}=\delta_{\gamma\gamma^{\prime-1}}\). Let \(\mathcal{L}(\Gamma)\) be the strong operator closure of the complex linear span of \(\lambda(\gamma)\)'s (or equivalently, \(\rho(\gamma)\)'s). This is the _group von Neumann algebra of \(\Gamma\)_. There is a canonical faithful normal tracial state \(\tau_{\Gamma}\), or simply \(\tau\), on \(\mathcal{L}(\Gamma)\), which is given by
\[\tau(x)=\langle x\delta_{e},\delta_{e}\rangle_{l^{2}(\Gamma)},\,x\in\mathcal{L }(\Gamma).\]
Hence \(\mathcal{L}(\Gamma)\) is a finite von Neumann algebra (which must be of type I or II\({}_{1}\)).
More generally, for a tracial von Neumann algebra \(M\) with the trace \(\tau\), we consider the GNS representation of \(M\) on the Hilbert space constructed from the completion of \(M\) with respect to the inner product \(\langle x,y\rangle_{\tau}=\tau(xy^{*})\). The underlying space will be denoted by \(L^{2}(M,\tau)\), or simply \(L^{2}(M)\).
Consider a normal unital representation \(\pi\colon M\to B(H)\) with both \(M\) and \(H\) separable. There exists an isometry \(u\colon H\to L^{2}(M)\otimes l^{2}(\mathbb{N})\), which commutes with the actions of \(M\):
\[u\circ\pi(x)=(\lambda(x)\otimes\operatorname{id}_{l^{2}(\mathbb{N})})\circ u,\, \forall x\in M,\]
where \(\lambda\colon M\mapsto L^{2}(M)\) denotes the left action. Then \(p=uu^{*}\) is a projection in \(B(L^{2}(M)\otimes l^{2}(\mathbb{N}))\) such that \(H\cong p(L^{2}(M)\otimes l^{2}(\mathbb{N}))\). We have the following result (see [2] Proposition 8.2.3).
**Proposition 4.1**.: _The correspondence \(H\mapsto p\) above defines a bijection between the set of equivalence classes of left \(M\)-modules and the set of equivalence classes of projections in \((M^{\prime}\cap B(L^{2}(M)))\otimes B(l^{2}(\mathbb{N}))\)._
The _von Neumann dimension_ of the \(M\)-module \(H\) are defined to be \((\tau\otimes\operatorname{Tr})(p)\) and denoted by \(\dim_{M}(H)\), which takes its value in \([0,\infty]\). We have:
1. \(\dim_{M}(\oplus_{i}H_{i})=\sum_{i}\dim_{M}(H_{i})\).
2. \(\dim_{M}(L^{2}(M))=1\).
Note \(\dim_{M}(H)\) depends on the trace \(\tau\). If \(M\) is a finite factor, i.e., \(Z(M)\cong\mathbb{C}\), there is a unique normal tracial state (see [25, 29]) and we further have:
1. \(\dim_{M}(H)=\dim_{M}(H^{\prime})\) if and only if \(H\) and \(H^{\prime}\) are isomorphic as \(M\)-modules (provided \(M\) is a factor).
When \(M\) is not a factor, there is a \(Z(M)\)-valued trace which determines the isomorphism class of an \(M\)-module (see [8]).
In the following sections, we will consider the group von Neumann algebra \(\mathcal{L}(\Gamma)\) with the canonical trace \(tr(x)=\langle x\delta_{e},\delta_{e}\rangle\). Hence the von Neumann dimension of \(\mathcal{L}(\Gamma)\) is the one uniquely determined by this trace. Note a discrete group \(\Gamma\) is called an infinite conjugacy class (ICC) group if every nontrivial conjugacy class \(C_{\gamma}=\{g\gamma g^{-1}|g\in\Gamma\}\), \(\gamma\neq e\), is infinite. It is well-known that \(\mathcal{L}(\Gamma)\) is a II\({}_{1}\) factor if and only if \(\Gamma\) is a nontrivial ICC group.
Now we consider the case that \(\Gamma\) is a discrete subgroup of a locally compact unimodular type I group \(G\). Let \(\mu\) be a Haar measure of \(G\). A measurable set \(D\subset G\) is called a _fundamental domain_ for \(\Gamma\) if \(D\) satisfies \(\mu(G\setminus\cup_{\gamma\in\Gamma}\gamma D)=0\) and \(\mu(\gamma_{1}D\cap\gamma_{2}D)=0\) if \(\gamma_{1}\neq\gamma_{2}\) in \(\Gamma\). In this section, we always assume \(\Gamma\) is a lattice, i.e., \(\mu(D)<\infty\). The measure \(\mu(D)\) is called _covolume_ of \(\Gamma\) and will be denoted by \(\operatorname{covol}(\Gamma)\). Note the covolume depends on the Haar measure \(\mu\) (see Remark 4.3).
There is a natural isomorphism \(L^{2}(G)\cong l^{2}(\Gamma)\otimes L^{2}(D,\mu)\) given by
\[\phi\mapsto\sum_{\gamma\in\Gamma}\delta_{\gamma}\otimes\phi_{\gamma}\text{ with }\phi_{\gamma}(z)=\phi(\gamma\cdot z),\]
where \(z\in D\) and \(\gamma\in\Gamma\). The restriction representation \(\lambda_{G}|_{\Gamma}\) of \(\Gamma\) is the tensor product of \(\lambda_{\Gamma}\) on \(l^{2}(\Gamma)\) and the identity operator \(\operatorname{id}\) on \(L^{2}(D,\mu)\). Hence we obtain the von Neumann algebra \(\lambda_{G}(\Gamma)^{\prime\prime}\cong\mathcal{L}(\Gamma)\otimes\mathbb{C}= \mathcal{L}(\Gamma)\), which will be denoted by \(M\) throughout this section. Please note \(L^{2}(M)=l^{2}(\Gamma)\).
### A theorem of von Neumann dimension
Suppose \(X\) is a measurable subset of \(\widehat{G}\) with the Plancherel measure \(\nu(X)<\infty\). Define
\[H_{X}=\int_{X}^{\oplus}H_{\pi}d\nu(\pi),\]
which is the direct integral of the spaces \(H_{\pi}\) with \(\pi\in X\). It is a module over \(G\), its lattice \(\Gamma\), and also the group von Neumann algebra \(\mathcal{L}(\Gamma)\).
We state a result on the von Neumann dimension of direct integrals. One may refer to [38] Section 4 for the proof.
**Theorem 4.2**.: _Let \(G\) be a locally compact unimodular type I group with Haar measure \(\mu\). Let \(\nu\) be the Plancherel measure on the unitary dual \(\widehat{G}\) of \(G\). Suppose \(\Gamma\) is a lattice in \(G\) and \(\mathcal{L}(\Gamma)\) is the group von Neumann algebra of \(\Gamma\). Let \(X\subset\widehat{G}\) such that \(\nu(X)<\infty\) and \(H_{X}=\int_{X}^{\oplus}H_{\pi}d\nu(\pi)\). We have_
\[\dim_{\mathcal{L}(\Gamma)}(H_{X})=\operatorname{covol}(\Gamma)\cdot\nu(X).\]
_Remark 4.3_.:
1. If \(\mu^{\prime}=k\cdot\mu\) is another Haar measure on \(G\) for some \(k>0\), the covolumes are related by \(\operatorname{covol}^{\prime}(\Gamma)=\mu^{\prime}(G/\Gamma)=k^{\prime}\cdot \mu(G/\Gamma)=k\cdot\operatorname{covol}(\Gamma)\). But the induced Plancherel measure \(\nu^{\prime}=k^{-1}\cdot\nu\) and the dependencies cancel out in the formula above.
2. There is a relevant approach by H. Peterson and A. Valette [31]. They study the von Neumann dimension over locally compact groups. The group von Neumann algebra is equipped with a semifinite tracial weight instead of a tracial state for a discrete group. It is motivated by the study of \(L^{2}\)-Betti number of locally compact groups [30].
If \(\pi\) is an atom in \(\widehat{G}\), i.e., \(\nu(\{\pi\})>0\), the irreducible representation \(\pi\) is a discrete series and \(\nu(\{\pi\})\) is just the formal dimension of \(\pi\)[15, 33]. Under this assumption, if \(G\) is a real Lie group that has discrete series and \(\Gamma\) is an ICC group, the theorem reduces to the special case of a single representation (see [23] Theorem 3.3.2)
\[\dim_{\mathcal{L}(\Gamma)}(H_{\pi})=\operatorname{covol}(\Gamma)\cdot d_{\pi}.\]
This is motivated by the geometric construction of discrete series of Lie groups by M. Atiyah and W. Schmid [7].
### The proof of the main theorem
We will prove the main theorem. We first give the proof for a tower of uniform lattices.
**Theorem 4.4**.: _[a tower of uniform lattices] Let \(\Gamma_{1}\supsetneq\Gamma_{2}\supsetneq\cdots\) be a normal tower of cocompact lattice in a semisimple real Lie group \(G\) such that \(\cap_{n\geq 1}\Gamma_{n}=\{1\}\). For any bounded subset \(X\) of \(\widehat{G}\), we have_
\[\lim_{n\to\infty}\frac{m(X,\Gamma_{n})}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1.\]
**Proof**: Recall that \(m_{\Gamma_{K}}(X)=\operatorname{vol}(\Gamma_{K}\backslash G(F_{\infty}))\nu_{ K}(X)\) by definition and \(\dim_{\mathcal{L}\Gamma_{n}}H_{X}=\operatorname{vol}(\Gamma_{K}\backslash G(F_{ \infty}))\nu(X)\) by Theorem 4.2. We need to show \(\lim_{n\to\infty}\nu_{K_{n}}(X)=\nu(X)\), which reduces to \(\lim_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)\) for all \(\phi\in C_{c}^{\infty}(G(F_{\infty})^{1})\) by Theorem 3.7.
From Proposition 3.3, we know
\[\operatorname{tr}R_{\operatorname{disc}}(\phi\otimes\frac{1_{K}}{\operatorname {vol}(K)})=m_{K}(\widehat{\phi})=\operatorname{vol}(G(\mathbb{Q})\backslash G (\mathbb{A})^{1})/K)\cdot\nu_{K}(\widehat{\phi}),\]
which is to say \(\operatorname{tr}R_{\operatorname{disc}}(\phi\otimes 1_{K})=\operatorname{vol}(G( \mathbb{Q})\backslash G(\mathbb{A})^{1}))\cdot\nu_{K}(\widehat{\phi})\). By Proposition 2.4, we have \(\lim_{n\to\infty}\operatorname{tr}R_{\operatorname{disc}}(\phi\otimes 1_{K_{n}})= \operatorname{vol}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1}))\cdot\phi(1)\). Hence \(\lim_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)\).
For the non-uniform case, the distribution \(J(f)\) in Equation 5 will no longer be the trace of \(R_{\operatorname{disc}}(f)\), which leads to the main task for most arithmetic subgroups. Fortunately, Finis-Lapid-Muller proved the following result on the limit of the spectral side of Equation 5 (see [20] Corollary 7.8).
**Theorem 4.5**.: _[Finis-Lapid-Muller] Suppose \(G=\operatorname{SL}(n)\). Let \(\{I_{n}\}\) be a family of descending integral ideals in \(\mathcal{O}_{F}\) prime to \(S\) and \(K_{n}=K(I_{n})\) be the compact subgroups of \(G(\mathbb{A}^{S})\) given by \(I_{n}\). We have_
\[\lim_{n\to\infty}J(h_{S}\otimes 1_{K_{n}})=\lim_{n\to\infty}\operatorname{tr}R _{\text{disc}}(h_{S}\otimes 1_{K_{n}})\]
_for any \(h_{S}\in C_{c}^{\infty}(G(F_{S})^{1})\)._
Then we are able to prove:
**Corollary 4.6**.: _[principal congruence subgroups in \(\operatorname{SL}(n,\mathbb{R})\)] Let \(\Gamma_{1}\supsetneq\Gamma_{2}\supsetneq\cdots\) be a tower of principal congruence subgroups in \(G=\operatorname{SL}(n,\mathbb{R})\). For any bounded subset \(X\) of \(\widehat{G}\), we have_
\[\lim_{n\to\infty}\frac{m(X,\Gamma_{n})}{\dim_{\mathcal{L}\Gamma_{n}}H_{X}}=1.\]
**Proof**: As shown in Theorem 4.4, it suffices to prove \(\lim_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)\) for all \(\phi\in C_{c}^{\infty}(G(F_{\infty})^{1})\). By Proposition 3.3 and Theorem 4.5, we know
\[\lim_{n\to\infty}\operatorname{vol}(G(\mathbb{Q})\backslash G(\mathbb{A})^{1 }))\cdot\nu_{K_{n}}(\widehat{\phi})=\lim_{n\to\infty}\operatorname{tr}R_{ \text{disc}}(\phi\otimes 1_{K_{n}})=\lim_{n\to\infty}J(\phi\otimes 1_{K_{n}}).\]
As \(\lim_{n\to\infty}\frac{\operatorname{vol}(G(F)\backslash G(\mathbb{A})^{1}) \phi(1)}{J(\phi\otimes 1_{K_{n}})}=1\) in Corollary 2.8, we obtain \(\lim_{n\to\infty}\nu_{K_{n}}(\widehat{\phi})=\phi(1)\).
| Given a connected semisimple Lie group G and an arithmetic subgroup Γ, it is well-known that each irreducible representation π of G occurs in the discrete spectrum L²(disc)(Γbackslash G) of L²(Γbackslash G) with at most a finite multiplicity mΓ(π).While mΓ(π) is unknown in general, we are interested in its limit as Γ is taken to be in a tower of lattices Γ₁supset Γ₂supset ... . For a bounded measurable subset X of the unitary dual widehatG, we let mΓₙ(X) be the sum of the multiplicity mΓₙ(π) of a representation π over all π in X. Let Hₓ be the direct integral of the irreducible representations in X, which is also a module over the group von Neumann algebra ℒΓₙ. We prove: limₙ→∞ (mΓₙ(X) / dimℒΓₙ Hₓ) = 1, for any bounded subset |
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